From afa792feb81158c5f9f636c6e315dbc85f0aa14d Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 3 Oct 2024 09:49:20 -0500 Subject: [PATCH 01/36] Modified the initialization of ElasticNet.py --- elasticnet/models/ElasticNet.py | 11 ++++++++--- 1 file changed, 8 insertions(+), 3 deletions(-) diff --git a/elasticnet/models/ElasticNet.py b/elasticnet/models/ElasticNet.py index 017e925..2b52bcf 100644 --- a/elasticnet/models/ElasticNet.py +++ b/elasticnet/models/ElasticNet.py @@ -1,7 +1,12 @@ class ElasticNetModel(): - def __init__(self): + def __init__(self,lambdas,thresh = 0.5,max_iter=1000, tol=1e-4): + self.lambdas = lambdas + self.thresh = thresh# between [0,1],0 means only L1 regularization, 1 means only L2 regularization + self.max_iter = max_iter#Maximum number of iterations + self.tol = tol + self.w = None pass @@ -13,5 +18,5 @@ class ElasticNetModelResults(): def __init__(self): pass - def predict(self, x): - return 0.5 + def predict(self, X): + return X @ self.w From fa410fbd62959d32fc61c447b616bf57d32439ff Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 3 Oct 2024 10:30:25 -0500 Subject: [PATCH 02/36] Complete the iterative cycle, parameter update uses gradient descent (not completed) --- elasticnet/models/ElasticNet.py | 28 +++++++++++++++++++++++++--- 1 file changed, 25 insertions(+), 3 deletions(-) diff --git a/elasticnet/models/ElasticNet.py b/elasticnet/models/ElasticNet.py index 2b52bcf..d918d43 100644 --- a/elasticnet/models/ElasticNet.py +++ b/elasticnet/models/ElasticNet.py @@ -1,4 +1,4 @@ - +import numpy as np class ElasticNetModel(): def __init__(self,lambdas,thresh = 0.5,max_iter=1000, tol=1e-4): @@ -11,10 +11,32 @@ def __init__(self,lambdas,thresh = 0.5,max_iter=1000, tol=1e-4): def fit(self, X, y): - return ElasticNetModelResults() + def calculate_weights(X, residual): + pass + self.w = np.zeros(X.shape[1]) # Initialize weights including intercept + max_iter = self.max_iter + while max_iter > 0: + max_iter -= 1 + # Compute predictions and residuals + pred = X @ self.w + residual = pred - y + + # Calculate weight gradients + gradient = calculate_weights(X, residual) + # Update weights + w_old = self.w.copy() + self.w -= self.learning_rate * gradient + + # Check convergence + if np.linalg.norm(self.w - w_old) < self.tol: + break + + self.coef = self.w[1:] + return ElasticNetResults(self.coef) + -class ElasticNetModelResults(): +class ElasticNetResults(): def __init__(self): pass From dd49bb2b5b02990a93f31cea1d4d720e6f92ff83 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 3 Oct 2024 12:21:24 -0500 Subject: [PATCH 03/36] Complete gradient descent, increase parameter intercept --- elasticnet/models/ElasticNet.py | 31 +++++++++++++++++++++++-------- 1 file changed, 23 insertions(+), 8 deletions(-) diff --git a/elasticnet/models/ElasticNet.py b/elasticnet/models/ElasticNet.py index d918d43..d71065e 100644 --- a/elasticnet/models/ElasticNet.py +++ b/elasticnet/models/ElasticNet.py @@ -1,22 +1,34 @@ import numpy as np class ElasticNetModel(): - def __init__(self,lambdas,thresh = 0.5,max_iter=1000, tol=1e-4): + def __init__(self,lambdas,thresh = 0.5,max_iter=1000, tol=1e-4,learning_rate=0.01): self.lambdas = lambdas self.thresh = thresh# between [0,1],0 means only L1 regularization, 1 means only L2 regularization self.max_iter = max_iter#Maximum number of iterations self.tol = tol + self.learning_rate = learning_rate self.w = None + self.intercept_ = None pass def fit(self, X, y): def calculate_weights(X, residual): - pass + gradient = (2 / X.shape[0]) * (X.T @ residual) # MSE gradient + l1_grad = self.lambdas * self.thresh * np.sign(self.w) # L1 penalty gradient + l2_grad = 2 * self.lambdas * (1 - self.thresh) * self.w # L2 penalty gradient + + # Total gradient + gradient += l1_grad + l2_grad + return gradient + + # Add intercept term + X = np.c_[np.ones(X.shape[0]), X] self.w = np.zeros(X.shape[1]) # Initialize weights including intercept - max_iter = self.max_iter - while max_iter > 0: - max_iter -= 1 + Max_iter = self.max_iter + + while Max_iter > 0: + Max_iter -= 1 # Compute predictions and residuals pred = X @ self.w residual = pred - y @@ -30,14 +42,17 @@ def calculate_weights(X, residual): # Check convergence if np.linalg.norm(self.w - w_old) < self.tol: break - + # divide intercept and coef + self.intercept = self.w[0] self.coef = self.w[1:] - return ElasticNetResults(self.coef) + return ElasticNetResults(self.coef, self.intercept) class ElasticNetResults(): - def __init__(self): + def __init__(self, coef, intercept): + self.coef = coef + self.intercept = intercept pass def predict(self, X): From f9f996ed1924be2d7c1c17eea14d59db7a6c1fae Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 3 Oct 2024 12:26:21 -0500 Subject: [PATCH 04/36] modify some bugs after adding the parameter intercept --- elasticnet/models/ElasticNet.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/elasticnet/models/ElasticNet.py b/elasticnet/models/ElasticNet.py index d71065e..08dd0dd 100644 --- a/elasticnet/models/ElasticNet.py +++ b/elasticnet/models/ElasticNet.py @@ -56,4 +56,4 @@ def __init__(self, coef, intercept): pass def predict(self, X): - return X @ self.w + return X @ self.coef + self.intercept From 643f69d1ed2c0ca57136d31d7d646f7a2b3eb4e2 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 3 Oct 2024 18:34:50 -0500 Subject: [PATCH 05/36] Unfinished data generation function --- generate_positive_regression_data.py | 85 ++++++++++++++++++++++++++++ 1 file changed, 85 insertions(+) create mode 100644 generate_positive_regression_data.py diff --git a/generate_positive_regression_data.py b/generate_positive_regression_data.py new file mode 100644 index 0000000..6b83c78 --- /dev/null +++ b/generate_positive_regression_data.py @@ -0,0 +1,85 @@ +import numpy as np +import matplotlib.pyplot as plt + +def generate_rotated_positive_data(range_x, noise_scale, size, num_features, seed, rotation_angle=45, mode=0): + """ + Generate synthetic data with specified patterns: + - The first half of the features have a monotonic trend. + - The second half of the features have a wavy (slanted S-shaped) pattern adjusted by rotation. + + Parameters: + range_x - Range of feature values (min, max) + noise_scale - Standard deviation of the noise + size - Number of samples + num_features - Number of features + seed - Random seed for reproducibility + rotation_angle - Rotation angle to adjust the direction of the S-shape + mode - Determines scaling factors to use + + Returns: + X - Generated multi-dimensional feature dataset + y - Target values + """ + + def scale_random_rows(X, size, scale_factors, seed=None): + """ + Scale random rows of a given matrix X by specified scaling factors. + + Parameters: + X - Input matrix + size - Number of rows in the matrix, i.e., number of samples + scale_factors - A list of scaling factors, e.g., [0.5, 0.7, 0.3], indicating factors for scaling some rows + seed - Random seed for reproducibility + + Returns: + Modified matrix X + """ + rng = np.random.default_rng(seed=seed) + remaining_indices = np.arange(size) # Initialize with all row indices + + for scale in scale_factors: + # Randomly select 1/n of rows for scaling + selected_indices = rng.choice(remaining_indices, size // len(scale_factors), replace=False) + X[selected_indices, :] *= scale # Scale by the scaling factor + # Update the remaining indices + remaining_indices = np.setdiff1d(remaining_indices, selected_indices) + + return X + + rng = np.random.default_rng(seed=seed) + half_features = num_features // 2 # Calculate half the number of features + + # Generate X1 data + X1 = rng.uniform(low=range_x[0], high=range_x[1], size=(size, half_features)) + if mode == 0: + scale_factors = [0.5, 0.7, 0.3] # Scaling factors needed + else: + scale_factors = [0.8, 0.4] + X1 = scale_random_rows(X1, X1.shape[0], scale_factors, seed=42) + + # Generate S-shaped data for the second half + X2 = rng.uniform(low=range_x[0], high=range_x[1], size=(size, num_features - half_features)) + for i in range(X2.shape[1]): + # Create standard S-shaped curve + X2[:, i] = np.sin(X2[:, i] / 2) * 10 + 0.5 * X2[:, i] + X2 = scale_random_rows(X2, X2.shape[0], scale_factors, seed=52) + + # Create rotation matrix, convert angle to radians + theta = np.radians(rotation_angle) + rotation_matrix = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]]) + + # Rotate the first two dimensions of S-shaped data to make it slanted + rotated_X2 = X2[:, :2] @ rotation_matrix # Rotate only the first two columns since the matrix is 2x2 + X2[:, :2] = rotated_X2 # Assign the rotated part back + + # Combine the two parts + X = np.hstack((X1, X2)) + + # Generate target values y, adjusted by the contribution of each dimension + y = (2 * X1[:, 0] + # Contribution of the first dimension multiplied by 2 + 3 * X1[:, 1] + # Contribution of the second dimension multiplied by 3 + 4 * X1[:, 2] + # Contribution of the third dimension multiplied by 4 + np.sum(np.sin(X2), axis=1) + # Contribution of S-shaped data + rng.normal(loc=0, scale=noise_scale, size=size)) # Add noise + + return X, y From 83cc615895df0dba00177484cb14f42dcc5b6822 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Mon, 7 Oct 2024 13:02:31 -0500 Subject: [PATCH 06/36] An attempt --- README.md | 59 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 59 insertions(+) diff --git a/README.md b/README.md index c1e8359..2df5b70 100644 --- a/README.md +++ b/README.md @@ -6,3 +6,62 @@ Put your README here. Answer the following questions. * How did you test your model to determine if it is working reasonably correctly? * What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) * Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? +## 1. Overview + +This project implements an **Elastic Net regression model** (ElasticNetModel) that combines L1 (Lasso) and L2 (Ridge) regularization techniques. It is designed to handle high-dimensional datasets, feature selection, and multicollinearity problems. The model uses **gradient descent** for optimization, function consists of the MSE loss, L1 loss, L2 loss. + +## 2. Class and Function Descriptions + +### `ElasticNetModel` + +The `ElasticNetModel` class implements the Elastic Net regression model. Below are the key methods and their descriptions: + +#### `__init__(self, lambdas=0.1, thresh=0.5, max_iter=1000, tol=1e-4, learning_rate=0.01)` + +Initializes the Elastic Net model with the following parameters: + +- **`lambdas`** (*float*, default=0.1): Penalty coefficient for regularization. +- **`thresh`** (*float*, default=0.5): Mixing parameter between L1 and L2 regularization. Value ranges from 0 to 1, where 0 means L1 regularization only and 1 means L2 regularization only. +- **`max_iter`** (*int*, default=1000): Maximum number of iterations for gradient descent. +- **`tol`** (*float*, default=1e-4): Tolerance for the stopping condition. If changes in weights are smaller than `tol`, training stops. +- **`learning_rate`** (*float*, default=0.01): Step size for gradient descent updates. + +#### `fit(self, X, y)` + +Trains the Elastic Net regression model using gradient descent on the provided feature matrix `X` and target values `y`. + +- **Parameters**: + - **`X`** (*numpy array*): Feature matrix of the training dataset (without intercept). + - **`y`** (*numpy array*): Target values corresponding to the training dataset. + +- **Returns**: + - An instance of `ElasticNetResults`, containing the trained weight coefficients and intercept. + +### `ElasticNetResults` + +This class stores the results after training the Elastic Net model. + +#### `predict(self, X)` + +Predicts target values for the provided feature matrix `X`. + +- **Parameters**: + - **`X`** (*numpy array*): Feature matrix of the test dataset. + +- **Returns**: + - Predicted target values. + +### Example Usage + +```python +from elasticnet/models import ElasticNetModel + +# Initialize the model +model = ElasticNetModel(lambdas=0.1, thresh=0.5, max_iter=1000, tol=1e-4, learning_rate=0.01) + +# Fit the model on training data +model.fit(X_train, y_train) + +# Predict on test data +predictions = model.predict(X_test) +``` \ No newline at end of file From b3fa8a8dd279c35c70485ead70214fe209a7aacf Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Tue, 8 Oct 2024 16:40:14 -0500 Subject: [PATCH 07/36] Changed readme --- README.md | 43 ++++++++++++++++++++++++++----------------- 1 file changed, 26 insertions(+), 17 deletions(-) diff --git a/README.md b/README.md index 2df5b70..a7b419d 100644 --- a/README.md +++ b/README.md @@ -1,14 +1,15 @@ -# Project 1 +# Project 1 Put your README here. Answer the following questions. -* What does the model you have implemented do and when should it be used? -* How did you test your model to determine if it is working reasonably correctly? -* What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) -* Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? +- What does the model you have implemented do and when should it be used? +- How did you test your model to determine if it is working reasonably correctly? +- What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) +- Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? + ## 1. Overview -This project implements an **Elastic Net regression model** (ElasticNetModel) that combines L1 (Lasso) and L2 (Ridge) regularization techniques. It is designed to handle high-dimensional datasets, feature selection, and multicollinearity problems. The model uses **gradient descent** for optimization, function consists of the MSE loss, L1 loss, L2 loss. +This project implements an **Elastic Net regression model** (ElasticNetModel) that combines L1 (Lasso) and L2 (Ridge) regularization techniques. It is designed to handle high-dimensional datasets, feature selection, and multicollinearity problems. The model uses **gradient descent** for optimization, function consists of the MSE loss, L1 loss, L2 loss. ## 2. Class and Function Descriptions @@ -20,20 +21,19 @@ The `ElasticNetModel` class implements the Elastic Net regression model. Below a Initializes the Elastic Net model with the following parameters: -- **`lambdas`** (*float*, default=0.1): Penalty coefficient for regularization. -- **`thresh`** (*float*, default=0.5): Mixing parameter between L1 and L2 regularization. Value ranges from 0 to 1, where 0 means L1 regularization only and 1 means L2 regularization only. -- **`max_iter`** (*int*, default=1000): Maximum number of iterations for gradient descent. -- **`tol`** (*float*, default=1e-4): Tolerance for the stopping condition. If changes in weights are smaller than `tol`, training stops. -- **`learning_rate`** (*float*, default=0.01): Step size for gradient descent updates. +- **`lambdas`** (_float_, default=0.1): Penalty coefficient for regularization. +- **`thresh`** (_float_, default=0.5): Mixing parameter between L1 and L2 regularization. Value ranges from 0 to 1, where 0 means L1 regularization only and 1 means L2 regularization only. +- **`max_iter`** (_int_, default=1000): Maximum number of iterations for gradient descent. +- **`tol`** (_float_, default=1e-4): Tolerance for the stopping condition. If changes in weights are smaller than `tol`, training stops. +- **`learning_rate`** (_float_, default=0.01): Step size for gradient descent updates. #### `fit(self, X, y)` Trains the Elastic Net regression model using gradient descent on the provided feature matrix `X` and target values `y`. - **Parameters**: - - **`X`** (*numpy array*): Feature matrix of the training dataset (without intercept). - - **`y`** (*numpy array*): Target values corresponding to the training dataset. - + - **`X`** (_numpy array_): Feature matrix of the training dataset (without intercept). + - **`y`** (_numpy array_): Target values corresponding to the training dataset. - **Returns**: - An instance of `ElasticNetResults`, containing the trained weight coefficients and intercept. @@ -46,8 +46,7 @@ This class stores the results after training the Elastic Net model. Predicts target values for the provided feature matrix `X`. - **Parameters**: - - **`X`** (*numpy array*): Feature matrix of the test dataset. - + - **`X`** (_numpy array_): Feature matrix of the test dataset. - **Returns**: - Predicted target values. @@ -64,4 +63,14 @@ model.fit(X_train, y_train) # Predict on test data predictions = model.predict(X_test) -``` \ No newline at end of file +``` + +Team Members: +Aakash Shivanandappa Gowda +A20548984 +Dhyan Vasudeva Gowda +A20592874 +Hongi Jiang +A20506636 +Purnesh Shivarudrappa Vidhyadhara +A20552125 From 99b7b5db80e01badf0ef7ddca9850d4a7c87ab19 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Tue, 8 Oct 2024 21:47:47 -0500 Subject: [PATCH 08/36] Modidied README.md --- README.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/README.md b/README.md index a7b419d..c214b31 100644 --- a/README.md +++ b/README.md @@ -65,12 +65,12 @@ model.fit(X_train, y_train) predictions = model.predict(X_test) ``` -Team Members: -Aakash Shivanandappa Gowda -A20548984 -Dhyan Vasudeva Gowda -A20592874 -Hongi Jiang -A20506636 -Purnesh Shivarudrappa Vidhyadhara -A20552125 +## Team Members: +- Aakash Shivanandappa Gowda +- A20548984 +- Dhyan Vasudeva Gowda +- A20592874 +- Hongi Jiang +- A20506636 +- Purnesh Shivarudrappa Vidhyadhara +- A20552125 From a946763c8a3cccceff5fa8912d2699d71e7e8aba Mon Sep 17 00:00:00 2001 From: Purnesh S V Date: Tue, 8 Oct 2024 22:39:55 -0500 Subject: [PATCH 09/36] UPDATED --- README.md | 19 ++++++++++--------- 1 file changed, 10 insertions(+), 9 deletions(-) diff --git a/README.md b/README.md index a7b419d..d83932c 100644 --- a/README.md +++ b/README.md @@ -65,12 +65,13 @@ model.fit(X_train, y_train) predictions = model.predict(X_test) ``` -Team Members: -Aakash Shivanandappa Gowda -A20548984 -Dhyan Vasudeva Gowda -A20592874 -Hongi Jiang -A20506636 -Purnesh Shivarudrappa Vidhyadhara -A20552125 +## Team Members: +- **Aakash Shivanandappa Gowda** + A20548984 +- **Dhyan Vasudeva Gowda** + A20592874 +- **Hongi Jiang** + A20506636 +- **Purnesh Shivarudrappa Vidhyadhara** + A20552125 + From 490f1e703c1207097e4a1ffb7620c5e110df46da Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Tue, 8 Oct 2024 22:58:42 -0500 Subject: [PATCH 10/36] updated readme --- README.md | 189 ++++++++++++++++++++++++++++++++++++++++++++++++++---- 1 file changed, 175 insertions(+), 14 deletions(-) diff --git a/README.md b/README.md index a7b419d..1d31fe6 100644 --- a/README.md +++ b/README.md @@ -1,11 +1,15 @@ # Project 1 -Put your README here. Answer the following questions. +## Team Members: -- What does the model you have implemented do and when should it be used? -- How did you test your model to determine if it is working reasonably correctly? -- What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) -- Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? +- **Aakash Shivanandappa Gowda** + A20548984 +- **Dhyan Vasudeva Gowda** + A20592874 +- **Hongi Jiang** + A20506636 +- **Purnesh Shivarudrappa Vidhyadhara** + A20552125 ## 1. Overview @@ -65,12 +69,169 @@ model.fit(X_train, y_train) predictions = model.predict(X_test) ``` -Team Members: -Aakash Shivanandappa Gowda -A20548984 -Dhyan Vasudeva Gowda -A20592874 -Hongi Jiang -A20506636 -Purnesh Shivarudrappa Vidhyadhara -A20552125 +# 1. Linear regression with ElasticNet regularization (combination of L2 And L1 regularization) + +# Q1.What does the model you have implemented do and when should it be used? + +# ElasticNetModel Overview + +The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implement this model: + +## Loss Function + +The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows: + +$\text{Loss} = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2 + \lambda_1 \sum_{j=1}^m |w_j| + \lambda_2 \sum_{j=1}^m w_j^2$ + +## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. + +- **Gradient calculation results of MSE:**: + +$\nabla_{\text{MSE}} = -\frac{2}{n} X^T (\hat{y} - y)$ + +- **Gradient calculation results of L1:**: + +$\nabla_{\text{L1}} = \lambda_1 \cdot \text{sign}(w)$ + +- **Gradient calculation results of L2:**: + +$\nabla_{\text{L2}} = 2 \cdot \lambda_2 \cdot w$ + +- **Total Gradient**: + +$\nabla_{\text{Total}} = -\frac{2}{n} X^T (\hat{y} - y) + \lambda_1 \cdot \text{sign}(w) + 2 \cdot \lambda_2 \cdot w$ + +- **Weight Update**: + +$w = w - \eta \cdot \nabla_{\text{Total}}$ + +$\eta \text{ is the learning rate, } \lambda_1 \text{ and } \lambda_2 \text{ are the penalty coefficients}$ + +### ElasticNetModel Overview + +In summary, our model implements hybrid regularization by combining L1 and L2 penalties. This simplifies feature selection and improves model stability. Additionally, we uses gradient descent to efficiently update parameters and weights. + +### What the Model Does: + +- **Hybrid Regularization**: Elastic Net stands out because it uses a mix of L1 and L2 penalties. + + - **L1 penalty** helps thin out the less essential features, essentially cleaning up the model to focus on what really matters, making it simpler and easier to understand. This is especially useful when you’re dealing with loads of potential predictors and want to avoid a model that’s too complex to interpret or too tailored to the training data. + - **L2 penalty** helps when features are intertwined, which is quite common. It does this by distributing the influence of these features more evenly, which helps keep the model stable and reliable. + +- **Gradient Descent Optimization**: This technique is crucial for fine-tuning the model in big datasets where older methods might be too slow. It adjusts the model gradually and methodically to hone in on the best possible performance, based on the rate of learning and how precise we want the model to be. + +### When to Use This Model: + +- **Broad Application Spectrum**: This model shines in any scenario where you need precise and sturdy predictions. It’s fantastic for fields like economics or healthcare, where the data and relationships can be quite complex. + +- **Managing Complexity and Ensuring Accuracy**: It’s perfect for diving into complicated, noisy data and coming out with clear insights. Whether it’s studying genes or predicting trends in big industries, this model helps manage a lot of information without losing focus. + +The **ElasticNetModel** isn’t just about making predictions; it’s also about deeply understanding the data. It’s a powerful tool for anyone who needs to tackle complex data sets and derive meaningful conclusions, not just numbers. This model’s ability to simplify while handling complex relationships makes it invaluable for researchers and analysts who regularly face challenging modeling situations. + +# Q2.How did you test your model to determine if it is working reasonably correctly? + +## Model Testing and Validation Strategy + +To ensure our model is both robust and accurate, we’ve put a thorough testing and validation strategy into play. Let’s walk through each step of this strategy to see how it contributes to a full evaluation of the model’s performance: + +### 1. Synthetic Data Generation + +- **Method**: We created synthetic datasets using a method called `generate_rotated_s_wave_data` that clearly defines how features relate to target values. +- **Why It Matters**: This setup lets us dive deep into the model’s ability to recognize and learn the patterns we’ve set up. By working with synthetic data, where the outcomes are already known, we can rigorously test how well the model mimics these set relationships, giving us a clear picture of its learning precision. + +### 2. Training on Generated Data + +- **What We Did**: The model spent a good deal of time learning from these synthetic datasets. During this phase, we kept a close eye on its parameters, checking how well the learned coefficients and intercepts matched the expected patterns. +- **Why It Matters**: This step is crucial for confirming that the model understands the underlying data dynamics. It helps validate that the model’s foundation is solid and that it’s tuned to respond appropriately to the nuances it will face in real scenarios. + +### 3. Predictions on Unseen Data + +- **Method**: We tested the model’s predictions using a new set of data that it hadn’t seen during training. +- **Why It Matters**: This is key for assessing how well the model can apply what it’s learned to new, unknown datasets—a must for ensuring the model is genuinely useful in real-world settings where data variability is the norm. + +### 4. Model Evaluation Metrics + +- **Metrics Used**: We measured the model’s accuracy and reliability using several standard regression metrics like: + - Mean Squared Error (MSE) + - Root Mean Squared Error (RMSE) + - Mean Absolute Error (MAE) + - R-squared (R²) +- **Why They’re Important**: These metrics provide hard numbers that tell us how accurately the model predicts new data. They help us understand different aspects of performance, such as the average size of errors and how much of the variance in our target variable the model can explain. + +### 5. Visual Comparisons of Predicted vs. Actual Values + +- **How We Do It**: We use scatter plots where each point shows a predicted value against its actual value. The closer these points are to the diagonal line (where predicted equals actual), the better the model’s predictions. +- **The Benefits**: These visuals offer an instant, gut-level indication of the model’s accuracy. They’re particularly good for spotting whether the model tends to over or underestimate across the board (systematic bias) or if it’s inconsistent (high variance). They also let us see at a glance if some features yield more reliable predictions than others. + +### 6. Analysis and Iteration + +- **Ongoing Process**: Based on what we learn from the visual data and other metrics, we continuously refine the model. This might mean tweaking the model’s settings to better align predictions with actual results, especially where we notice gaps. +- **Why Keep At It**: These refinements help the model adapt to the subtleties of new data, ensuring it stays relevant and effective. This ongoing cycle of feedback and improvement is vital for the model to remain at the top of its game as it encounters new challenges. + +#### Insights from Graphs: + +Observing how “Feature 2 vs. Target” shows a tighter cluster around the diagonal than “Feature 6 vs. Target” suggests Feature 2 is being handled more precisely. This might lead us to look into whether Feature 6 needs more preprocessing or whether the model should employ advanced feature engineering strategies to better capture complex data patterns. + +By weaving together these detailed testing methods, we not only affirm that the model is theoretically sound but also ensure it’s equipped to perform reliably in a range of real-world applications. This rigorous validation and refinement process is crucial for building top-quality predictive models that can rise to new challenges and deliver dependable results. + +# Q3.What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) + +## Model Optimization Parameters + +To optimize the performance of our model, we’ve provided several adjustable parameters. Here’s a breakdown of each parameter, explaining their function and how adjustments can enhance model performance: + +### 1. Lambdas + +- **Function**: Lambdas are central to our model’s regularization strategies, utilized in both L1 (Lasso) and L2 (Ridge) techniques. This helps prevent overfitting, a common issue where the model learns noise as if it were true data signals. +- **Impact of Adjustment**: Increasing the lambdas value amplifies the penalty on the model’s coefficients, effectively pulling them towards zero. This adjustment is beneficial when dealing with complex datasets that could potentially lead to misleading noise. +- **When We Use It**: If we observe our model overfitting to training data, boosting the lambdas value can help mitigate this, promoting better generalization to new, unseen data. + +### 2. Thresh + +- **Function**: Thresh determines the cutoff for feature selection, which simplifies the model by focusing only on impactful features. +- **Impact of Adjustment**: A higher thresh value is likely to eliminate less significant features by setting their coefficients to zero, which can enhance both the interpretability and performance of the model on new data. +- **When We Use It**: If our dataset has an abundance of features, adjusting thresh helps us streamline the model, concentrating on the most influential factors for predictions. + +### 3. Max_iter + +- **Function**: This parameter sets the maximum number of iterations our algorithm will execute before stopping. +- **Impact of Adjustment**: Allowing more iterations gives the model additional opportunities to refine its weights to the data, potentially improving its accuracy. +- **When We Use It**: If our model hasn’t yet reached its optimal state, increasing `max_iter` may be necessary. However, this could lead to longer training times, requiring a balance between precision and efficiency. + +### 4. Tol + +- **Function**: Tol, or tolerance, defines how close the model needs to fit the training data trends before ceasing further adjustments. +- **Impact of Adjustment**: Setting a smaller `tol` value means the model won’t stop learning until it achieves a closer fit, which can enhance accuracy but also extend training duration. +- **When We Use It**: This is particularly valuable when high precision is crucial. However, setting the tolerance too tight might lead to diminishing returns, especially in terms of prolonged training without significant gains. + +### 5. Learning_rate + +- **Function**: In gradient descent optimization, this parameter controls the step size the model takes towards minimizing the loss function. +- **Impact of Adjustment**: Lowering the learning rate results in smaller, more cautious steps, potentially leading to more precise model optimization. +- **When We Use It**: If the model is overshooting the minimal loss point, reducing the learning rate can help achieve a more stable and accurate convergence. However, too low a rate may slow down the training process excessively. + +By fine-tuning these parameters, we can adjust our model to better fit the specific characteristics of the data and the demands of the task at hand, striking an optimal balance between accuracy, computational efficiency, and model simplicity. + +# Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? + +## Specific Inputs That May Challenge the ElasticNet Model + +### 1. Highly Nonlinear Data + +- **The Challenge**: Elastic Net is designed to see the world in straight lines, assuming a straightforward, linear relationship between features and the target variable. However, life often isn’t that simple, and data can show complex, curved relationships that the model struggles to grasp. +- **What Happens**: When faced with these complex patterns, Elastic Net might not hit the mark with its predictions since it can’t effectively capture the twists and turns in variable interactions. +- **What We Can Do**: + - **Feature Engineering**: To help the model better understand the underlying patterns, we can transform the input features. This might mean squaring or cubing features, or introducing interaction terms, and even applying mathematical transformations like logarithms or exponentials to straighten out those nonlinear relationships. + - **Kernel Methods**: Borrowing a trick from the support vector machine playbook, we can use kernel methods to project our data into a space where the once-complex relationships look linear, making it easier for Elastic Net to do its job. + - **Hybrid Models**: Sometimes, two heads (or models) are better than one. By teaming up Elastic Net with nonlinear models like decision trees or neural networks, we can combine their strengths, enhancing the model’s ability to predict accurately across a wider range of scenarios. + +### 2. Noisy Data + +- **The Challenge**: Noisy data is like trying to listen to a quiet radio station during a thunderstorm. The static (noise) can drown out the music (true patterns), making it hard for Elastic Net to hear what’s important. +- **What Happens**: If the data is too noisy, Elastic Net might start learning the static instead of the music, leading to models that perform well on their training data but falter in the real world. +- **What We Can Do**: + - **Increase Regularization**: By turning up the lambdas knob, we increase the penalties for complexity, encouraging the model to focus on the loud and clear signals and ignore the static. + - **Threshold Tuning**: Adjusting the thresh parameter helps us be stricter about which features we let into the model, cutting out those that are more noise than signal. + - **Noise Reduction Techniques**: Before we even feed data into our model, we can clean it up—filtering out oddball values or smoothing out jumpy signals—to make sure we’re working with the cleanest, clearest data possible. + +By understanding these limitations and employing smart strategies to mitigate them, we can enhance the capabilities of Elastic Net, making it a more versatile and reliable tool in our predictive modeling toolkit. From 6a34588b2bce069692ae4b6732eb2dab476319fe Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Tue, 8 Oct 2024 23:10:09 -0500 Subject: [PATCH 11/36] change the test file,it is still incompleted --- elasticnet/tests/test_ElasticNetModel.py | 99 ++++++++++++++++++++++-- 1 file changed, 94 insertions(+), 5 deletions(-) diff --git a/elasticnet/tests/test_ElasticNetModel.py b/elasticnet/tests/test_ElasticNetModel.py index 5022c3c..20f0571 100644 --- a/elasticnet/tests/test_ElasticNetModel.py +++ b/elasticnet/tests/test_ElasticNetModel.py @@ -1,10 +1,21 @@ +import sys import csv +import numpy as np +import os +# Add the project root directory to sys.path +sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), '..', '..'))) +import io -import numpy +sys.stdout = io.TextIOWrapper(sys.stdout.buffer, encoding='utf-8') +# Now you can import the elasticnet module from elasticnet.models.ElasticNet import ElasticNetModel +from generate_positive_regression_data import generate_rotated_positive_data +import matplotlib.pyplot as plt +from sklearn.metrics import mean_absolute_error, r2_score -def test_predict(): +# Original test code part +def test_predict_with_csv(): model = ElasticNetModel() data = [] with open("small_test.csv", "r") as file: @@ -12,8 +23,86 @@ def test_predict(): for row in reader: data.append(row) - X = numpy.array([[v for k,v in datum.items() if k.startswith('x')] for datum in data]) - y = numpy.array([[v for k,v in datum.items() if k=='y'] for datum in data]) - results = model.fit(X,y) + X = np.array([[v for k,v in datum.items() if k.startswith('x')] for datum in data], dtype=float) + y = np.array([[v for k,v in datum.items() if k=='y'] for datum in data], dtype=float) + results = model.fit(X, y) preds = results.predict(X) assert preds == 0.5 + +# Test code with your generated data +def test_with_generated_data(): + # Generate training data + X, y = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=200, num_features=6, seed=42, rotation_angle=45, mode=0) + + # Visualize the generated data + plt.figure(figsize=(15, 6)) + for i in range(X.shape[1]): + plt.subplot(1, X.shape[1], i + 1) + plt.scatter(X[:, i], y, label=f'Feature {i + 1} vs Target', alpha=0.6) + plt.xlabel(f'Feature {i + 1}') + plt.ylabel('Target') + plt.title(f'Feature {i + 1} vs Target') + plt.grid(True) + plt.tight_layout() + plt.show() + + # Generate test dataset + X_test, y_test = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=50, num_features=6, seed=100, rotation_angle=45, mode=1) + + # Visualize the test data + plt.figure(figsize=(15, 6)) + for i in range(X_test.shape[1]): + plt.subplot(1, X_test.shape[1], i + 1) + plt.scatter(X_test[:, i], y_test, label=f'Test Feature {i + 1} vs Target', alpha=0.6, color='orange') + plt.xlabel(f'Feature {i + 1}') + plt.ylabel('Target') + plt.title(f'Test Feature {i + 1} vs Target') + plt.grid(True) + plt.tight_layout() + plt.show() + + # Instantiate the ElasticNetModel and train + model = ElasticNetModel(lambdas=0.1, thresh=0.5, max_iter=1000, tol=1e-4, learning_rate=0.01) + model_results = model.fit(X, y) + + # Predict on the test data + y_pred = model_results.predict(X_test) + + # Visualize actual and predicted values + for i in range(X_test.shape[1]): + plt.subplot(2, 3, i + 1) + plt.scatter(X_test[:, i], y_test, color='orange', alpha=0.6, label='Actual Values') + plt.scatter(X_test[:, i], y_pred, color='blue', alpha=0.6, label='Predicted Values') + + # Fit line + z = np.polyfit(X_test[:, i], y_test, 1) + p = np.poly1d(z) + sorted_indices = np.argsort(X_test[:, i]) + X_sorted = X_test[:, i][sorted_indices] + y_fit = p(X_sorted) + plt.plot(X_sorted, y_fit, "r--", label='Fit Line') + + plt.xlabel(f'Feature {i + 1}') + plt.ylabel('Target') + plt.title(f'Feature {i + 1} vs. Actual and Predicted Target') + plt.grid(True) + plt.legend() + + plt.tight_layout() + plt.show() + + # Manually calculate MSE + mse = np.mean((y_test - y_pred) ** 2) + rmse = np.sqrt(mse) + mae = mean_absolute_error(y_test, y_pred) + r2 = r2_score(y_test, y_pred) + + print(f"Mean Squared Error (MSE): {mse:.4f}") + print(f"Root Mean Squared Error (RMSE): {rmse:.4f}") + print(f"Mean Absolute Error (MAE): {mae:.4f}") + print(f"R-squared (R²): {r2:.4f}") + +# Run the tests +if __name__ == "__main__": + # test_predict_with_csv() # Test the original CSV code + test_with_generated_data() # Test with generated data From 141126d71ea18c152636c351547e8ad3df75b9a2 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Tue, 8 Oct 2024 23:33:11 -0500 Subject: [PATCH 12/36] Updated test_ElasticNetModel.py and added new regression data generation script --- elasticnet/tests/test_ElasticNetModel.py | 4 +- generate_nagetive_regression_data.py | 52 ++++++++++++++++++++++++ generate_positive_regression_data.py | 1 - 3 files changed, 53 insertions(+), 4 deletions(-) create mode 100644 generate_nagetive_regression_data.py diff --git a/elasticnet/tests/test_ElasticNetModel.py b/elasticnet/tests/test_ElasticNetModel.py index 20f0571..c3f5847 100644 --- a/elasticnet/tests/test_ElasticNetModel.py +++ b/elasticnet/tests/test_ElasticNetModel.py @@ -2,13 +2,11 @@ import csv import numpy as np import os +import io # Add the project root directory to sys.path sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), '..', '..'))) -import io - sys.stdout = io.TextIOWrapper(sys.stdout.buffer, encoding='utf-8') -# Now you can import the elasticnet module from elasticnet.models.ElasticNet import ElasticNetModel from generate_positive_regression_data import generate_rotated_positive_data import matplotlib.pyplot as plt diff --git a/generate_nagetive_regression_data.py b/generate_nagetive_regression_data.py new file mode 100644 index 0000000..609b78b --- /dev/null +++ b/generate_nagetive_regression_data.py @@ -0,0 +1,52 @@ +import numpy as np + +def generate_negative_data(range_x, noise_scale, size, num_features, seed): + """ + Generate synthetic data with specified patterns: + - The first half of the features have a monotonic trend. + - The second half of the features have a linear decreasing (negative slope) pattern. + + Parameters: + range_x - Range of feature values (min, max) + noise_scale - Standard deviation of the noise + size - Number of samples + num_features - Number of features + seed - Random seed for reproducibility + + Returns: + X - Generated multi-dimensional feature dataset + y - Target values + """ + rng = np.random.default_rng(seed=seed) + half_features = num_features // 2 # Calculate half the number of features + + # Generate X1 data with a clear linear trend + X1 = np.zeros((size, half_features)) + for i in range(half_features): + # Generate linear data from low to high to ensure a clear positive slope + X1[:, i] = np.linspace(range_x[1], range_x[0], size) + rng.normal(loc=0, scale=noise_scale, size=size)### + + # Define X2 matrix as (size, num_features - half_features) + X2 = np.zeros((size, num_features - half_features)) + + # Generate data with a negative slope + for i in range(X2.shape[1]): + # Generate data from high to low with a negative slope + X2[:, i] = np.linspace(range_x[1], range_x[0], size) + # Add appropriate negative weights to ensure negative correlation between features and target values + X2[:, i] += rng.normal(loc=0, scale=noise_scale, size=size)##### + + # Flip to ensure X2 is arranged from large to small + X2 = np.flip(X2, axis=0) + + # Combine X1 and X2 + X = np.hstack((X1, X2)) + + # Generate target values y, enhancing the negative correlation of negative slope features + y = (2 * X1[:, 0] + # Contribution of the first dimension multiplied by 2 + 3 * X1[:, 1] + # Contribution of the second dimension multiplied by 3 + 4 * X1[:, 2] - # Contribution of the third dimension multiplied by 4 + 4 * np.sum(X2, axis=1) + # Strong negative slope contribution from decreasing data + rng.normal(loc=0, scale=noise_scale, size=size)) # Add noise + + return X, y \ No newline at end of file diff --git a/generate_positive_regression_data.py b/generate_positive_regression_data.py index 6b83c78..5812fe4 100644 --- a/generate_positive_regression_data.py +++ b/generate_positive_regression_data.py @@ -1,5 +1,4 @@ import numpy as np -import matplotlib.pyplot as plt def generate_rotated_positive_data(range_x, noise_scale, size, num_features, seed, rotation_angle=45, mode=0): """ From d0c383db8bcd6ae4404d29f9b9daad557d348ad7 Mon Sep 17 00:00:00 2001 From: Purnesh S V Date: Tue, 8 Oct 2024 23:48:40 -0500 Subject: [PATCH 13/36] updated group members --- README.md | 197 ++++++++++++++++++++++++++++++++++++++++++++++++++---- 1 file changed, 184 insertions(+), 13 deletions(-) diff --git a/README.md b/README.md index d83932c..98604cd 100644 --- a/README.md +++ b/README.md @@ -1,11 +1,15 @@ # Project 1 -Put your README here. Answer the following questions. +## Team Members: -- What does the model you have implemented do and when should it be used? -- How did you test your model to determine if it is working reasonably correctly? -- What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) -- Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? +- **Aakash Shivanandappa Gowda** + A20548984 +- **Dhyan Vasudeva Gowda** + A20592874 +- **Hongi Jiang** + A20506636 +- **Purnesh Shivarudrappa Vidhyadhara** + A20552125 ## 1. Overview @@ -65,13 +69,180 @@ model.fit(X_train, y_train) predictions = model.predict(X_test) ``` +# 1. Linear regression with ElasticNet regularization (combination of L2 And L1 regularization) + +# Q1.What does the model you have implemented do and when should it be used? + +# ElasticNetModel Overview + +The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implement this model: + +## Loss Function + +The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows: + +$\text{Loss} = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2 + \lambda_1 \sum_{j=1}^m |w_j| + \lambda_2 \sum_{j=1}^m w_j^2$ + +## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. + +- **Gradient calculation results of MSE:**: + +$\nabla_{\text{MSE}} = -\frac{2}{n} X^T (\hat{y} - y)$ + +- **Gradient calculation results of L1:**: + +$\nabla_{\text{L1}} = \lambda_1 \cdot \text{sign}(w)$ + +- **Gradient calculation results of L2:**: + +$\nabla_{\text{L2}} = 2 \cdot \lambda_2 \cdot w$ + +- **Total Gradient**: + +$\nabla_{\text{Total}} = -\frac{2}{n} X^T (\hat{y} - y) + \lambda_1 \cdot \text{sign}(w) + 2 \cdot \lambda_2 \cdot w$ + +- **Weight Update**: + +$w = w - \eta \cdot \nabla_{\text{Total}}$ + +$\eta \text{ is the learning rate, } \lambda_1 \text{ and } \lambda_2 \text{ are the penalty coefficients}$ + +### ElasticNetModel Overview + +In summary, our model implements hybrid regularization by combining L1 and L2 penalties. This simplifies feature selection and improves model stability. Additionally, we uses gradient descent to efficiently update parameters and weights. + +### What the Model Does: + +- **Hybrid Regularization**: Elastic Net stands out because it uses a mix of L1 and L2 penalties. + + - **L1 penalty** helps thin out the less essential features, essentially cleaning up the model to focus on what really matters, making it simpler and easier to understand. This is especially useful when you’re dealing with loads of potential predictors and want to avoid a model that’s too complex to interpret or too tailored to the training data. + - **L2 penalty** helps when features are intertwined, which is quite common. It does this by distributing the influence of these features more evenly, which helps keep the model stable and reliable. + +- **Gradient Descent Optimization**: This technique is crucial for fine-tuning the model in big datasets where older methods might be too slow. It adjusts the model gradually and methodically to hone in on the best possible performance, based on the rate of learning and how precise we want the model to be. + +### When to Use This Model: + +- **Broad Application Spectrum**: This model shines in any scenario where you need precise and sturdy predictions. It’s fantastic for fields like economics or healthcare, where the data and relationships can be quite complex. + +- **Managing Complexity and Ensuring Accuracy**: It’s perfect for diving into complicated, noisy data and coming out with clear insights. Whether it’s studying genes or predicting trends in big industries, this model helps manage a lot of information without losing focus. + +The **ElasticNetModel** isn’t just about making predictions; it’s also about deeply understanding the data. It’s a powerful tool for anyone who needs to tackle complex data sets and derive meaningful conclusions, not just numbers. This model’s ability to simplify while handling complex relationships makes it invaluable for researchers and analysts who regularly face challenging modeling situations. + +# Q2.How did you test your model to determine if it is working reasonably correctly? + +## Model Testing and Validation Strategy + +To ensure our model is both robust and accurate, we’ve put a thorough testing and validation strategy into play. Let’s walk through each step of this strategy to see how it contributes to a full evaluation of the model’s performance: + +### 1. Synthetic Data Generation + +- **Method**: We created synthetic datasets using a method called `generate_rotated_s_wave_data` that clearly defines how features relate to target values. +- **Why It Matters**: This setup lets us dive deep into the model’s ability to recognize and learn the patterns we’ve set up. By working with synthetic data, where the outcomes are already known, we can rigorously test how well the model mimics these set relationships, giving us a clear picture of its learning precision. + +### 2. Training on Generated Data + +- **What We Did**: The model spent a good deal of time learning from these synthetic datasets. During this phase, we kept a close eye on its parameters, checking how well the learned coefficients and intercepts matched the expected patterns. +- **Why It Matters**: This step is crucial for confirming that the model understands the underlying data dynamics. It helps validate that the model’s foundation is solid and that it’s tuned to respond appropriately to the nuances it will face in real scenarios. + +### 3. Predictions on Unseen Data + +- **Method**: We tested the model’s predictions using a new set of data that it hadn’t seen during training. +- **Why It Matters**: This is key for assessing how well the model can apply what it’s learned to new, unknown datasets—a must for ensuring the model is genuinely useful in real-world settings where data variability is the norm. + +### 4. Model Evaluation Metrics + +- **Metrics Used**: We measured the model’s accuracy and reliability using several standard regression metrics like: + - Mean Squared Error (MSE) + - Root Mean Squared Error (RMSE) + - Mean Absolute Error (MAE) + - R-squared (R²) +- **Why They’re Important**: These metrics provide hard numbers that tell us how accurately the model predicts new data. They help us understand different aspects of performance, such as the average size of errors and how much of the variance in our target variable the model can explain. + +### 5. Visual Comparisons of Predicted vs. Actual Values + +- **How We Do It**: We use scatter plots where each point shows a predicted value against its actual value. The closer these points are to the diagonal line (where predicted equals actual), the better the model’s predictions. +- **The Benefits**: These visuals offer an instant, gut-level indication of the model’s accuracy. They’re particularly good for spotting whether the model tends to over or underestimate across the board (systematic bias) or if it’s inconsistent (high variance). They also let us see at a glance if some features yield more reliable predictions than others. + +### 6. Analysis and Iteration + +- **Ongoing Process**: Based on what we learn from the visual data and other metrics, we continuously refine the model. This might mean tweaking the model’s settings to better align predictions with actual results, especially where we notice gaps. +- **Why Keep At It**: These refinements help the model adapt to the subtleties of new data, ensuring it stays relevant and effective. This ongoing cycle of feedback and improvement is vital for the model to remain at the top of its game as it encounters new challenges. + +#### Insights from Graphs: + +Observing how “Feature 2 vs. Target” shows a tighter cluster around the diagonal than “Feature 6 vs. Target” suggests Feature 2 is being handled more precisely. This might lead us to look into whether Feature 6 needs more preprocessing or whether the model should employ advanced feature engineering strategies to better capture complex data patterns. + +By weaving together these detailed testing methods, we not only affirm that the model is theoretically sound but also ensure it’s equipped to perform reliably in a range of real-world applications. This rigorous validation and refinement process is crucial for building top-quality predictive models that can rise to new challenges and deliver dependable results. + +# Q3.What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) + +## Model Optimization Parameters + +To optimize the performance of our model, we’ve provided several adjustable parameters. Here’s a breakdown of each parameter, explaining their function and how adjustments can enhance model performance: + +### 1. Lambdas + +- **Function**: Lambdas are central to our model’s regularization strategies, utilized in both L1 (Lasso) and L2 (Ridge) techniques. This helps prevent overfitting, a common issue where the model learns noise as if it were true data signals. +- **Impact of Adjustment**: Increasing the lambdas value amplifies the penalty on the model’s coefficients, effectively pulling them towards zero. This adjustment is beneficial when dealing with complex datasets that could potentially lead to misleading noise. +- **When We Use It**: If we observe our model overfitting to training data, boosting the lambdas value can help mitigate this, promoting better generalization to new, unseen data. + +### 2. Thresh + +- **Function**: Thresh determines the cutoff for feature selection, which simplifies the model by focusing only on impactful features. +- **Impact of Adjustment**: A higher thresh value is likely to eliminate less significant features by setting their coefficients to zero, which can enhance both the interpretability and performance of the model on new data. +- **When We Use It**: If our dataset has an abundance of features, adjusting thresh helps us streamline the model, concentrating on the most influential factors for predictions. + +### 3. Max_iter + +- **Function**: This parameter sets the maximum number of iterations our algorithm will execute before stopping. +- **Impact of Adjustment**: Allowing more iterations gives the model additional opportunities to refine its weights to the data, potentially improving its accuracy. +- **When We Use It**: If our model hasn’t yet reached its optimal state, increasing `max_iter` may be necessary. However, this could lead to longer training times, requiring a balance between precision and efficiency. + +### 4. Tol + +- **Function**: Tol, or tolerance, defines how close the model needs to fit the training data trends before ceasing further adjustments. +- **Impact of Adjustment**: Setting a smaller `tol` value means the model won’t stop learning until it achieves a closer fit, which can enhance accuracy but also extend training duration. +- **When We Use It**: This is particularly valuable when high precision is crucial. However, setting the tolerance too tight might lead to diminishing returns, especially in terms of prolonged training without significant gains. + +### 5. Learning_rate + +- **Function**: In gradient descent optimization, this parameter controls the step size the model takes towards minimizing the loss function. +- **Impact of Adjustment**: Lowering the learning rate results in smaller, more cautious steps, potentially leading to more precise model optimization. +- **When We Use It**: If the model is overshooting the minimal loss point, reducing the learning rate can help achieve a more stable and accurate convergence. However, too low a rate may slow down the training process excessively. + +By fine-tuning these parameters, we can adjust our model to better fit the specific characteristics of the data and the demands of the task at hand, striking an optimal balance between accuracy, computational efficiency, and model simplicity. + +# Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? + +## Specific Inputs That May Challenge the ElasticNet Model + +### 1. Highly Nonlinear Data + +- **The Challenge**: Elastic Net is designed to see the world in straight lines, assuming a straightforward, linear relationship between features and the target variable. However, life often isn’t that simple, and data can show complex, curved relationships that the model struggles to grasp. +- **What Happens**: When faced with these complex patterns, Elastic Net might not hit the mark with its predictions since it can’t effectively capture the twists and turns in variable interactions. +- **What We Can Do**: + - **Feature Engineering**: To help the model better understand the underlying patterns, we can transform the input features. This might mean squaring or cubing features, or introducing interaction terms, and even applying mathematical transformations like logarithms or exponentials to straighten out those nonlinear relationships. + - **Kernel Methods**: Borrowing a trick from the support vector machine playbook, we can use kernel methods to project our data into a space where the once-complex relationships look linear, making it easier for Elastic Net to do its job. + - **Hybrid Models**: Sometimes, two heads (or models) are better than one. By teaming up Elastic Net with nonlinear models like decision trees or neural networks, we can combine their strengths, enhancing the model’s ability to predict accurately across a wider range of scenarios. + +### 2. Noisy Data + +- **The Challenge**: Noisy data is like trying to listen to a quiet radio station during a thunderstorm. The static (noise) can drown out the music (true patterns), making it hard for Elastic Net to hear what’s important. +- **What Happens**: If the data is too noisy, Elastic Net might start learning the static instead of the music, leading to models that perform well on their training data but falter in the real world. +- **What We Can Do**: + - **Increase Regularization**: By turning up the lambdas knob, we increase the penalties for complexity, encouraging the model to focus on the loud and clear signals and ignore the static. + - **Threshold Tuning**: Adjusting the thresh parameter helps us be stricter about which features we let into the model, cutting out those that are more noise than signal. + - **Noise Reduction Techniques**: Before we even feed data into our model, we can clean it up—filtering out oddball values or smoothing out jumpy signals—to make sure we’re working with the cleanest, clearest data possible. + +By understanding these limitations and employing smart strategies to mitigate them, we can enhance the capabilities of Elastic Net, making it a more versatile and reliable tool in our predictive modeling toolkit. + ## Team Members: -- **Aakash Shivanandappa Gowda** - A20548984 -- **Dhyan Vasudeva Gowda** - A20592874 -- **Hongi Jiang** - A20506636 -- **Purnesh Shivarudrappa Vidhyadhara** - A20552125 +- Aakash Shivanandappa Gowda +- A20548984 +- Dhyan Vasudeva Gowda +- A20592874 +- Hongi Jiang +- A20506636 +- Purnesh Shivarudrappa Vidhyadhara +- A20552125 From 03d8c53e456d2f79eb39b0aa61beb1443d805f1f Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Wed, 9 Oct 2024 10:36:16 -0500 Subject: [PATCH 14/36] Removed unnecessary sentences --- README.md | 11 ----------- 1 file changed, 11 deletions(-) diff --git a/README.md b/README.md index 98604cd..1d31fe6 100644 --- a/README.md +++ b/README.md @@ -235,14 +235,3 @@ By fine-tuning these parameters, we can adjust our model to better fit the speci - **Noise Reduction Techniques**: Before we even feed data into our model, we can clean it up—filtering out oddball values or smoothing out jumpy signals—to make sure we’re working with the cleanest, clearest data possible. By understanding these limitations and employing smart strategies to mitigate them, we can enhance the capabilities of Elastic Net, making it a more versatile and reliable tool in our predictive modeling toolkit. - -## Team Members: - -- Aakash Shivanandappa Gowda -- A20548984 -- Dhyan Vasudeva Gowda -- A20592874 -- Hongi Jiang -- A20506636 -- Purnesh Shivarudrappa Vidhyadhara -- A20552125 From 92c5cb7d705f50ff82bdb519568a7b609c80bc97 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Wed, 9 Oct 2024 15:35:29 -0500 Subject: [PATCH 15/36] modify test model file --- elasticnet/tests/test_ElasticNetModel.py | 29 +++++++++++++++--------- 1 file changed, 18 insertions(+), 11 deletions(-) diff --git a/elasticnet/tests/test_ElasticNetModel.py b/elasticnet/tests/test_ElasticNetModel.py index c3f5847..906297a 100644 --- a/elasticnet/tests/test_ElasticNetModel.py +++ b/elasticnet/tests/test_ElasticNetModel.py @@ -9,6 +9,7 @@ from elasticnet.models.ElasticNet import ElasticNetModel from generate_positive_regression_data import generate_rotated_positive_data +from generate_nagetive_regression_data import generate_negative_data import matplotlib.pyplot as plt from sklearn.metrics import mean_absolute_error, r2_score @@ -28,10 +29,21 @@ def test_predict_with_csv(): assert preds == 0.5 # Test code with your generated data -def test_with_generated_data(): - # Generate training data - X, y = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=200, num_features=6, seed=42, rotation_angle=45, mode=0) +def test_with_generated_data(type = 0): + if(type == 1): + # Generate training data + X, y = generate_negative_data(range_x=[-40, 40], noise_scale=5, size=200, num_features=6, seed=42) + # Generate test dataset + X_test, y_test = generate_negative_data(range_x=[-40, 40], noise_scale=50, size=50, num_features=6, seed=526) + # Instantiate the ElasticNetModel and train + model = model = ElasticNetModel(lambdas=0.1, thresh=0.5, max_iter=1000, tol=1e-4, learning_rate=0.0001)#must turn down the learning rate + else: + X, y = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=200, num_features=6, seed=42, rotation_angle=45, mode=0) + X_test, y_test = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=50, num_features=6, seed=100, rotation_angle=45, mode=1) + + model = ElasticNetModel(lambdas=0.1, thresh=0.5, max_iter=1000, tol=1e-4, learning_rate=0.01) + # Visualize the generated data plt.figure(figsize=(15, 6)) for i in range(X.shape[1]): @@ -43,10 +55,6 @@ def test_with_generated_data(): plt.grid(True) plt.tight_layout() plt.show() - - # Generate test dataset - X_test, y_test = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=50, num_features=6, seed=100, rotation_angle=45, mode=1) - # Visualize the test data plt.figure(figsize=(15, 6)) for i in range(X_test.shape[1]): @@ -59,10 +67,8 @@ def test_with_generated_data(): plt.tight_layout() plt.show() - # Instantiate the ElasticNetModel and train - model = ElasticNetModel(lambdas=0.1, thresh=0.5, max_iter=1000, tol=1e-4, learning_rate=0.01) - model_results = model.fit(X, y) + model_results = model.fit(X, y) # Predict on the test data y_pred = model_results.predict(X_test) @@ -103,4 +109,5 @@ def test_with_generated_data(): # Run the tests if __name__ == "__main__": # test_predict_with_csv() # Test the original CSV code - test_with_generated_data() # Test with generated data + test_with_generated_data(data_type=0) # Test with generated positive regression data + test_with_generated_data(data_type=1) # Test with generated half positive and half nagetive regression data From 96d8c618a4c5eef09e63d540d7aa015a448753cf Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Wed, 9 Oct 2024 15:39:22 -0500 Subject: [PATCH 16/36] modify bugs of test file --- elasticnet/tests/test_ElasticNetModel.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/elasticnet/tests/test_ElasticNetModel.py b/elasticnet/tests/test_ElasticNetModel.py index 906297a..4ef2b62 100644 --- a/elasticnet/tests/test_ElasticNetModel.py +++ b/elasticnet/tests/test_ElasticNetModel.py @@ -29,8 +29,8 @@ def test_predict_with_csv(): assert preds == 0.5 # Test code with your generated data -def test_with_generated_data(type = 0): - if(type == 1): +def test_with_generated_data(data_type = 0): + if(data_type == 1): # Generate training data X, y = generate_negative_data(range_x=[-40, 40], noise_scale=5, size=200, num_features=6, seed=42) # Generate test dataset @@ -109,5 +109,5 @@ def test_with_generated_data(type = 0): # Run the tests if __name__ == "__main__": # test_predict_with_csv() # Test the original CSV code - test_with_generated_data(data_type=0) # Test with generated positive regression data + #test_with_generated_data(data_type=0) # Test with generated positive regression data test_with_generated_data(data_type=1) # Test with generated half positive and half nagetive regression data From a2674f5e5f141ddc58daa214404f2a0a6299e931 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Wed, 9 Oct 2024 16:27:38 -0500 Subject: [PATCH 17/36] final version --- elasticnet/tests/test_ElasticNetModel.py | 50 ++++++++++++++++++------ 1 file changed, 37 insertions(+), 13 deletions(-) diff --git a/elasticnet/tests/test_ElasticNetModel.py b/elasticnet/tests/test_ElasticNetModel.py index 4ef2b62..3d312c2 100644 --- a/elasticnet/tests/test_ElasticNetModel.py +++ b/elasticnet/tests/test_ElasticNetModel.py @@ -2,37 +2,61 @@ import csv import numpy as np import os +''' import io # Add the project root directory to sys.path sys.path.append(os.path.abspath(os.path.join(os.path.dirname(__file__), '..', '..'))) sys.stdout = io.TextIOWrapper(sys.stdout.buffer, encoding='utf-8') - +''' from elasticnet.models.ElasticNet import ElasticNetModel from generate_positive_regression_data import generate_rotated_positive_data from generate_nagetive_regression_data import generate_negative_data import matplotlib.pyplot as plt from sklearn.metrics import mean_absolute_error, r2_score - +from sklearn.model_selection import train_test_split # Original test code part def test_predict_with_csv(): - model = ElasticNetModel() + # Initialize the ElasticNet model with the given parameters + model = ElasticNetModel(lambdas=0.1, thresh=0.6, max_iter=1000, tol=1e-4, learning_rate=0.01) + + # Read the CSV file and load data data = [] + #with open("elasticnet/tests/small_test.csv", "r") as file: with open("small_test.csv", "r") as file: reader = csv.DictReader(file) for row in reader: data.append(row) - X = np.array([[v for k,v in datum.items() if k.startswith('x')] for datum in data], dtype=float) - y = np.array([[v for k,v in datum.items() if k=='y'] for datum in data], dtype=float) - results = model.fit(X, y) - preds = results.predict(X) - assert preds == 0.5 + # Extract features (X) and target (y) from the data + X = np.array([[v for k, v in datum.items() if k.startswith('x')] for datum in data], dtype=float) + y = np.array([float(datum['y']) for datum in data], dtype=float) # Convert 'y' to a 1D array + + # Split the data into 80% training and 20% testing sets + X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42) + + # Fit the model on the training data + results = model.fit(X_train, y_train) + + # Make predictions on the test data + preds = results.predict(X_test) + + # Calculate various evaluation metrics + mse = np.mean((y_test - preds) ** 2) # Mean Squared Error + rmse = np.sqrt(mse) # Root Mean Squared Error + mae = mean_absolute_error(y_test, preds) # Mean Absolute Error + r2 = r2_score(y_test, preds) # R-squared + + # Print the results of the evaluation metrics + print(f"Mean Squared Error (MSE): {mse:.4f}") + print(f"Root Mean Squared Error (RMSE): {rmse:.4f}") + print(f"Mean Absolute Error (MAE): {mae:.4f}") + print(f"R-squared (R²): {r2:.4f}") + -# Test code with your generated data def test_with_generated_data(data_type = 0): if(data_type == 1): # Generate training data - X, y = generate_negative_data(range_x=[-40, 40], noise_scale=5, size=200, num_features=6, seed=42) + X, y = generate_negative_data(range_x=[-40, 40], noise_scale=5, size=300, num_features=6, seed=42) # Generate test dataset X_test, y_test = generate_negative_data(range_x=[-40, 40], noise_scale=50, size=50, num_features=6, seed=526) # Instantiate the ElasticNetModel and train @@ -40,7 +64,7 @@ def test_with_generated_data(data_type = 0): else: X, y = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=200, num_features=6, seed=42, rotation_angle=45, mode=0) - X_test, y_test = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=2, size=50, num_features=6, seed=100, rotation_angle=45, mode=1) + X_test, y_test = generate_rotated_positive_data(range_x=[-30, 30], noise_scale=50, size=50, num_features=6, seed=100, rotation_angle=45, mode=1) model = ElasticNetModel(lambdas=0.1, thresh=0.5, max_iter=1000, tol=1e-4, learning_rate=0.01) @@ -108,6 +132,6 @@ def test_with_generated_data(data_type = 0): # Run the tests if __name__ == "__main__": - # test_predict_with_csv() # Test the original CSV code - #test_with_generated_data(data_type=0) # Test with generated positive regression data + test_predict_with_csv() # Test the original CSV code + test_with_generated_data(data_type=0) # Test with generated positive regression data test_with_generated_data(data_type=1) # Test with generated half positive and half nagetive regression data From fbaa312a82cf7bb8e50e7795192c78aa9939a798 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Wed, 9 Oct 2024 16:28:57 -0500 Subject: [PATCH 18/36] fix bugs --- elasticnet/models/ElasticNet.py | 13 +++++++++++-- 1 file changed, 11 insertions(+), 2 deletions(-) diff --git a/elasticnet/models/ElasticNet.py b/elasticnet/models/ElasticNet.py index 08dd0dd..5cced38 100644 --- a/elasticnet/models/ElasticNet.py +++ b/elasticnet/models/ElasticNet.py @@ -13,12 +13,21 @@ def __init__(self,lambdas,thresh = 0.5,max_iter=1000, tol=1e-4,learning_rate=0.0 def fit(self, X, y): + def calculate_weights(X, residual): gradient = (2 / X.shape[0]) * (X.T @ residual) # MSE gradient l1_grad = self.lambdas * self.thresh * np.sign(self.w) # L1 penalty gradient - l2_grad = 2 * self.lambdas * (1 - self.thresh) * self.w # L2 penalty gradient + l2_grad = 2 * self.lambdas * (1 - self.thresh) * self.w # L2 penalty gradien + + # make sure l1_grad and l2_grad has same dimention with gradient + if l1_grad.ndim == 1: + l1_grad = l1_grad.reshape(-1) + if l2_grad.ndim == 1: + l2_grad = l2_grad.reshape(-1) + + if gradient.ndim > 1: + gradient = np.sum(gradient, axis=1) - # Total gradient gradient += l1_grad + l2_grad return gradient From 06041028a0092617faa3189d54419715547414ae Mon Sep 17 00:00:00 2001 From: Purnesh S V Date: Wed, 9 Oct 2024 21:13:14 -0500 Subject: [PATCH 19/36] 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zD0*9BhJW}x%^7RNodb{l=E72!@IKpUf%d7wVbk^*YxJ#7J|7{tdUX(6i4`8`?yfcE zAwzn%?AIJ`A9JHjU0nr_l7Q;ur=Xy~h58e%fc6_5{R-rvFXWfcsZ}|0?L#P%V5uDm zR-v7x59aE!mN;)38}P)d09Q+3DW`BK@~Lp#7SCVlw~ajTQs`sU>#twG9t& Date: Wed, 9 Oct 2024 21:38:42 -0500 Subject: [PATCH 20/36] introduce how to use the model --- README.md | 59 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 59 insertions(+) diff --git a/README.md b/README.md index 029bdd1..2cff29d 100644 --- a/README.md +++ b/README.md @@ -68,6 +68,65 @@ model.fit(X_train, y_train) # Predict on test data predictions = model.predict(X_test) ``` +### `generate_negative_data` + +The `generate_negative_data` function generates synthetic data with two distinct patterns: +1. The first half of the features has a **monotonic increasing trend**. +2. The second half of the features has a **linear decreasing trend** (i.e., negative slope). + +This function can be useful for testing model's ability in negative correlations data. + +#### Parameters: +- **`range_x`** (_tuple_): The range of feature values (min, max). +- **`noise_scale`** (_float_): Standard deviation of the Gaussian noise added to the data. +- **`size`** (_int_): Number of samples in the dataset. +- **`num_features`** (_int_): Total number of features (half with increasing trend, half with decreasing trend). +- **`seed`** (_int_): Random seed for reproducibility. + +#### Returns: +- **`X`** (_numpy array_): Generated feature matrix with both increasing and decreasing trends. +- **`y`** (_numpy array_): Target values with contributions from the features and added noise. + +#### Example Usage: + +```python +from generate_negative_regression_data import generate_negative_data + +# Generate negative slope data +X, y = generate_negative_data(range_x=(0, 10), noise_scale=1.0, size=100, num_features=6, seed=42) +``` + +--- + +### `generate_rotated_positive_data` + +The `generate_rotated_positive_data` function generates synthetic data with two patterns: +1. The first half of the features follows a **monotonic increasing trend**. +2. The second half exhibits a **wavy (S-shaped) pattern**, adjusted by a rotation matrix to create a slanted shape. + +This function generates more complex data to test model's ability. + +#### Parameters: +- **`range_x`** (_tuple_): Specifies the range of feature values (min, max). +- **`noise_scale`** (_float_): Standard deviation of the Gaussian noise added to the data. +- **`size`** (_int_): Number of samples in the dataset. +- **`num_features`** (_int_): Total number of features (half with increasing trend, half with a wavy pattern). +- **`seed`** (_int_): Random seed for reproducibility. +- **`rotation_angle`** (_float_, default=45): Angle (in degrees) to rotate the wavy pattern, introducing a slanted S-shape. +- **`mode`** (_int_, default=0): Determines the scaling factors for the feature values. + +#### Returns: +- **`X`** (_numpy array_): Generated feature matrix with increasing trends and rotated S-shaped patterns. +- **`y`** (_numpy array_): Target values influenced by the features and noise. + +#### Example Usage: + +```python +from generate_positive_regression_data import generate_rotated_positive_data + +# Generate positive data with rotation and S-shaped curves +X, y = generate_rotated_positive_data(range_x=(0, 10), noise_scale=1.0, size=100, num_features=6, seed=42, rotation_angle=45, mode=0) +``` ## 1. Linear regression with ElasticNet regularization (combination of L2 And L1 regularization) From 609f0bfd04feef6e45fadb2134d5463465854d90 Mon Sep 17 00:00:00 2001 From: Purnesh S V Date: Wed, 9 Oct 2024 21:39:43 -0500 Subject: [PATCH 21/36] FINAL UPDATE --- .DS_Store | Bin 6148 -> 6148 bytes 8.jpg | Bin 0 -> 28426 bytes README.md | 4 +++- 3 files changed, 3 insertions(+), 1 deletion(-) create mode 100644 8.jpg diff --git a/.DS_Store b/.DS_Store index 3960550ea4bfbfb8091297f06ae71826ba8113e1..a99aa7e08af2b21ab51f81dfc571653ebec8ce8d 100644 GIT binary patch delta 61 zcmZoMXfc@J&&a#}AO-~isnSu9s7Mnrib_=@(!0`COcVr^A_4*m3QARa?-DvT zM0yhvKv0?_mJm$gPJGMWcc1e--??X>@BTUq*2>6OvyCyw9Bs~Qj%^Yl{$nQQCJ+n; zL1y3&+MIz78HakfLXf2;qzFL}2Ly}ufDixzMIZ@qLlDbTmcKBp&{OzdF!XEd<>o4+ zchcQ2&@aH<@3OR}>OM&Ch`A+Vs|9fVhWGpi`G=(9#uK1Q#NtiPhbz=0*v(l!lDUx) z=A;$c#QeDN??Leq9WP(@X5oh*AKyTKE7K#=ws!W?>|Y=*h!x_5bRaE9r+~|cPo6x! z)#hKHzuW)&K^gfyc4$y#tF6D$|206s*(JaU5PuRBzvLX?8@?B#KE^ml`xZ6tt~4t8<_ 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The loss function is shown as follows : +- **The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows** : + $\text{Loss} = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2 + \lambda_1 \sum_{j=1}^m |w_j| + \lambda_2 \sum_{j=1}^m w_j^2$ + ## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. - **Gradient calculation results of MSE:** : From 486b3a6f4deaf5dee14ee107471e8c3badedb0d4 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Wed, 9 Oct 2024 22:08:01 -0500 Subject: [PATCH 22/36] modify the picture --- 1.png | Bin 143872 -> 140801 bytes 2.png | Bin 51716 -> 47766 bytes 3.png | Bin 108237 -> 135963 bytes 4.png | Bin 159916 -> 157011 bytes 5.png | Bin 54550 -> 51227 bytes 6.png | Bin 206995 -> 157893 bytes README.md | 32 +------------------------------- 7 files changed, 1 insertion(+), 31 deletions(-) diff --git a/1.png b/1.png index 51a5e24a94c6128e9f71fbb60999b845fe049a2a..0076104f011e666a6d639f91fa701178ab3bbec1 100644 GIT binary patch literal 140801 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The loss function is shown as follows : -$\text{Loss} = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2 + \lambda_1 \sum_{j=1}^m |w_j| + \lambda_2 \sum_{j=1}^m w_j^2$ +$\text{Loss}$ = $\frac{1}{n}$ $\sum_{i=1}^n$ $(y_i - \hat{y}_i)^2$ + $\lambda_1$ $\sum_{j=1}^m$ $|w_j|$ + $\lambda_2 \sum_{j=1}^m w_j^2$ ## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. @@ -546,36 +546,6 @@ This comprehensive analysis underscores the need for a broader approach in model - **Inverse Relationships:** Features 4, 5, and 6 exhibit inverse relationships, which are consistently evident from their respective plots. - **Model Performance:** The overall alignment of the fit lines across all features suggests that the regression model generally performs well, capturing the primary trends in the dataset effectively. - - -![alt text](7.png) - -## Scatter Plot Analysis of Features vs. Target - -### Feature 1 vs. Target -- **Distribution:** Shows a broad range of target values with no apparent trend or consistent pattern. -- **Interpretation:** Feature 1 exhibits high variability with the target, indicating no clear linear relationship, which suggests the influence of other factors or non-linear dynamics. - -### Feature 2 vs. Target -- **Distribution:** The data points are spread out across the target range, with some concentration in the middle. -- **Interpretation:** Slightly more structure than Feature 1, but still lacks a definitive pattern that would suggest a strong linear correlation. - -### Feature 3 vs. Target -- **Distribution:** Data points appear more clustered around the center of the target range, yet spread wide vertically. -- **Interpretation:** Indicates some degree of association, possibly non-linear, with the target, as points cluster around a central target value but do not form a distinct trend. - -### Feature 4 vs. Target -- **Distribution:** Similar to Feature 3 with a somewhat narrower spread across the target range. -- **Interpretation:** Might suggest a mild, potentially non-linear relationship with the target given the more concentrated clustering of data points. - -### Feature 5 vs. Target -- **Distribution:** Points are densely packed in a narrower range of the target, suggesting a more consistent relationship. -- **Interpretation:** This concentration and consistency could point to a stronger association with the target than the previous features, warranting further analysis. - -### Feature 6 vs. Target -- **Distribution:** Shows a wide distribution but with a clear boundary, not extending beyond certain target values. -- **Interpretation:** Although dispersed, the containment within specific target bounds suggests possible categorical or bounded influence of Feature 6 on the target. - ## General Observations - **Trends and Relationships:** These scatter plots generally do not show strong linear correlations, indicating the potential necessity for non-linear modeling approaches to capture the relationships between features and the target. - **Predictive Insights:** The variability and patterns observed suggest that some features might have more influence on the target, possibly in non-obvious or non-linear ways. From a65e91f8a8917c37abbdb231ebc59a6b2f00defe Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Wed, 9 Oct 2024 22:20:40 -0500 Subject: [PATCH 23/36] modify loss --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 1a19533..2578ff4 100644 --- a/README.md +++ b/README.md @@ -140,7 +140,7 @@ The model we've crafted, called **ElasticNetModel**, is an implementation of Ela The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows : -$\text{Loss}$ = $\frac{1}{n}$ $\sum_{i=1}^n$ $(y_i - \hat{y}_i)^2$ + $\lambda_1$ $\sum_{j=1}^m$ $|w_j|$ + $\lambda_2 \sum_{j=1}^m w_j^2$ +$\text{Loss}$(w) = $\frac{1}{n}$ $\sum_{i=1}^{n}$ $(y_i - w^T x_i)^2$ + $\lambda$ ($\rho$ $\sum_{j=1}^{d}$ $|w_j|$ + (1 - $\rho$) $\sum_{j=1}^{d}$ $w_j^2$) ## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. From 61bb46be2b965f1540f1e2ec4cd126eed3fda4d7 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 10 Oct 2024 11:02:31 -0500 Subject: [PATCH 24/36] change graph title --- elasticnet/tests/test_ElasticNetModel.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/elasticnet/tests/test_ElasticNetModel.py b/elasticnet/tests/test_ElasticNetModel.py index 3d312c2..62be01f 100644 --- a/elasticnet/tests/test_ElasticNetModel.py +++ b/elasticnet/tests/test_ElasticNetModel.py @@ -72,10 +72,10 @@ def test_with_generated_data(data_type = 0): plt.figure(figsize=(15, 6)) for i in range(X.shape[1]): plt.subplot(1, X.shape[1], i + 1) - plt.scatter(X[:, i], y, label=f'Feature {i + 1} vs Target', alpha=0.6) + plt.scatter(X[:, i], y, label=f'Feature {i + 1}', alpha=0.6) plt.xlabel(f'Feature {i + 1}') - plt.ylabel('Target') - plt.title(f'Feature {i + 1} vs Target') + plt.ylabel('Y_Train') + plt.title(f'Train Feature {i + 1}') plt.grid(True) plt.tight_layout() plt.show() @@ -83,10 +83,10 @@ def test_with_generated_data(data_type = 0): plt.figure(figsize=(15, 6)) for i in range(X_test.shape[1]): plt.subplot(1, X_test.shape[1], i + 1) - plt.scatter(X_test[:, i], y_test, label=f'Test Feature {i + 1} vs Target', alpha=0.6, color='orange') + plt.scatter(X_test[:, i], y_test, label=f'Test Feature {i + 1}', alpha=0.6, color='orange') plt.xlabel(f'Feature {i + 1}') - plt.ylabel('Target') - plt.title(f'Test Feature {i + 1} vs Target') + plt.ylabel('Y_test') + plt.title(f'Test Feature {i + 1}') plt.grid(True) plt.tight_layout() plt.show() From 724a9f9b44b851535c5315154f490ca0023d152c Mon Sep 17 00:00:00 2001 From: Purnesh S V Date: Thu, 10 Oct 2024 14:02:32 -0500 Subject: [PATCH 25/36] Updated README.md --- README.md | 249 +++++++++++++++--------------------------------------- 1 file changed, 69 insertions(+), 180 deletions(-) diff --git a/README.md b/README.md index c96bcac..d228be6 100644 --- a/README.md +++ b/README.md @@ -317,201 +317,104 @@ By acknowledging these limitations and using smart strategies to overcome them, - ## Visualization ![alt text](1.png) +### Scatter Plots of Train Features vs Target -## Visualization Overview +This image shows a series of scatter plots, each representing the relationship between a different feature (Feature 1 to Feature 6) and a target variable (Y_train) based on training data. Each plot displays the data points in two dimensions: -**Axes:** -The plots display the target variable on the Y-axis and the features on the X-axis for each plot. +- The horizontal axis (X-axis) shows values of a specific feature. +- The vertical axis (Y-axis) represents the values of Y_train. -**Purpose:** -These plots are commonly used to evaluate the relationships between each feature and the target variable. This evaluation helps in deciding which features to select, how to potentially transform them, and to understand the distribution and correlations within the data. +The purpose of these plots is typically to visually assess the relationship between the features and the target variable. Observations can be made regarding the distribution, trend, or any potential outliers in the data. From the plots: -## Analysis of Individual Plots +- **Feature 1, Feature 2, Feature 3:** These features show some vertical dispersion but no clear linear or nonlinear trend with Y_train. +- **Feature 4, Feature 5, Feature 6:** Similar to the first three, these features also display a scattered distribution with respect to Y_train, without a visible specific pattern. -### Feature 1 vs Target -- **Observation:** The scatter is loose and appears random, with no discernible trend towards any target line, implying little to no correlation with the target. +Overall, these scatter plots can help in determining which features might be relevant for modeling, although none of the features shown here have a distinct or strong linear relationship with the target variable. -### Feature 2 vs Target -- **Observation:** A few data points are noticeably scattered along the Y-axis, with no clear linear pattern, suggesting either a very low positive or possibly negative correlation. - -### Feature 3 vs Target -- **Observation:** Data points are vertically distributed, centered around zero on the X-axis. This plot suggests a very poor, if any, direct positive correlation between Feature 3 and the target. - -### Feature 4 vs Target -- **Observation:** While data points are widely dispersed, there is a slight indication of a weak relationship, as the spread is somewhat curved upwards. - -### Feature 5 vs Target -- **Observation:** Similar to Feature 4, with a slight upward curve in the spread, which may indicate a very weak positive correlation, though the slope is minimal. - -### Feature 6 vs Target -- **Observation:** Data points are concentrated around specific X-values, suggesting that Feature 6 might be categorical or discretized. The relationship to the target is not clearly linear. - -## General Observations - -### Lack of Strong Linear Relationships -- **Details:** No features show a strong linear relationship with the target, indicating that nonlinear models or transformations might be more suitable. - -### Potential for Non-linear Analysis -- **Details:** Exploring non-linear effects or interaction effects could be fruitful given the observed patterns. - -### Noise and Variability -- **Details:** The significant dispersion and overlap of data points suggest high variability within features affecting the target, or noise. ![alt text](2.png) -## Overview of Each Feature Plot Analysis - -### Test Feature 1 vs Target -- **Distribution:** The data points are randomly and evenly distributed across the range of Feature 1 without any obvious pattern. -- **Interpretation:** There appears to be no clear correlation between Feature 1 and the target, indicating high dispersion or heteroscedasticity and potentially non-linear relationships. - -### Test Feature 2 vs Target -- **Distribution:** Similar to Feature 1, points are spread across a wide range without a distinct pattern. -- **Interpretation:** The absence of clear linearity suggests that simple linear methods may be ineffective. - -### Test Feature 3 vs Target -- **Distribution:** Points are spread out, albeit slightly less than in previous cases. -- **Interpretation:** Despite a slight centralization, there is still very little indication of a linear relationship with the target. - -### Test Feature 4 vs Target -- **Distribution:** The scatter begins to resemble a downward sloping line, indicating that values may decrease as Feature 4 increases. -- **Interpretation:** This could suggest a negative dependency of Feature 4 on the target, although it might not be strong. Correlation coefficients or more complex regression analyses could further explore this relationship. - -### Test Feature 5 vs Target -- **Distribution:** Points display a steeper upward trend compared to other features. -- **Interpretation:** There is a limited yet visible positive correlation, indicating that the target may increase slightly as Feature 5 increases. This feature might offer some predictive value when integrated into a model. - -### General Observations -- **Lack of Strong Linear Relationships:** None of the features display strong linear relationships with the target. This suggests that relationships, if present, might be non-linear. -- **Potential for Non-linear Analysis:** Given the lack of clear linear correlations, non-linear modeling techniques such as decision trees or neural networks might better capture complex patterns between features and the target. -- **Importance of Feature Engineering:** Considering the subtle trends observed in Features 4 and 5, feature engineering might help in revealing more detailed relationships, possibly enhancing model performance. - -These plots are crucial for the preliminary analysis, guiding further statistical testing and modeling approaches. Exploring these relationships more deeply could yield insights critical to developing effective predictive models. - +### Overview of Each Feature Plot Analysis -## Test Feature 6 vs Target +The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against a target variable (labeled as "Y_Test"). Each subplot represents the relationship between a single feature and the target variable. -### Distribution -- The distribution is broad, similar to Features 1 and 2, with no discernible direction in the scatter of data points. +### Breakdown of Scatter Plots: -### Interpretation -- There is no visible significant linear relationship, indicating that Feature 6 may not effectively predict the target using linear methods alone. +- **Test Feature 1 through 6**: Each scatter plot represents the feature's values on the x-axis and the corresponding target variable values on the y-axis. +- **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. -## General Observations - -### No Strong Linear Relationships -- Across all features, there are no strong, clear linear relationships with the target. This suggests that the relationships, if present, might be non-linear, or the target is influenced by external factors or noise not directly captured by these features. - -### Potential for Non-linear Analysis -- Due to the absence of linear relationships, employing non-linear modeling techniques could be advantageous. Methods such as decision trees, random forests, or neural networks are capable of capturing more complex interactions between features and the target. - -### Feature Selection and Engineering -- Features 4 and 5 exhibit some patterns that may warrant further investigation. Engaging in feature engineering could help uncover more intricate relationships, like interactions between features or transformations that might better expose their relationships with the target. - -This comprehensive analysis underscores the need for a broader approach in modeling techniques to adequately predict the target variable. Such an approach should consider the non-linear dynamics of the dataset, along with a focused attempt at enhancing feature selection and engineering strategies to improve model accuracy and robustness. +### Visual Insights: +- For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. +- For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. +### Uses: +- **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). +- **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. ![alt text](3.png) -## Breakdown of Each Plot Analysis +### Breakdown of Each Plot Analysis -### Feature 1 vs. Actual and Predicted Target -- **Actual Values:** Represented by yellow dots. -- **Predicted Values:** Represented by blue dots. -- **Fit Line:** The red dashed line shows the regression line that represents the predicted values based on Feature 1. -- **Interpretation:** The fit line indicates moderate fitting, implying a certain degree of linear predictability from Feature 1 to the target. +The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against a target variable (labeled as "Y_Test"). Each subplot represents the relationship between a single feature and the target variable. -### Feature 2 vs. Actual and Predicted Target -- **Interpretation:** The spread of predicted values is wider, indicating more variance in the predictions relative to the actual values, suggesting less predictive accuracy from Feature 2. +### Breakdown of Scatter Plots: -### Feature 3 vs. Actual and Predicted Target -- **Interpretation:** The predictions closely follow the actual values with the fit line approximating a good linear fit, suggesting that Feature 3 might have a strong predictive power regarding the target. +- **Test Feature 1 through 6**: Each scatter plot represents the feature's values on the x-axis and the corresponding target variable values on the y-axis. +- **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. -### Feature 4 vs. Actual and Predicted Target -- **Interpretation:** The fit line indicates an inverse relationship, as the line slopes downward. This might suggest that as Feature 4 increases, the target decreases. - -### Feature 5 vs. Actual and Predicted Target -- **Interpretation:** Feature 5 shows a slightly poorer fitting compared to others, with the fit line sloping slightly, which might indicate only a weak predictive relationship. - -### Feature 6 vs. Actual and Predicted Target -- **Interpretation:** The predictions and actual values are somewhat aligned, but the spread is still notable. The fit line shows a moderate predictive relationship. - -## General Observations -- **Evaluation of Fit Lines:** The fit lines in these plots are crucial for understanding how well the model predicts the target based on each feature. A closer fit of the line to the actual values indicates a stronger predictive ability of the feature. -- **Comparison Across Features:** Some features (like Feature 3 and Feature 6) show a closer alignment between the predicted and actual values, indicating they might be more useful predictors of the target. -- **Predictive Power of Features:** The degree of scatter around the fit lines across different features illustrates the variability in the model's accuracy. Features with less scatter and closer fit lines are generally better at predicting the target. +### Visual Insights: +- For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. +- For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. +### Uses: +- **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). +- **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. ![alt text](4.png) -## Scatter Plot Analysis - -### Feature 1 vs. Target -- **Distribution:** Data points are spread throughout but show a slight vertical alignment, suggesting limited but noticeable variability with respect to the target. -- **Interpretation:** There is a wide spread in the data, which might indicate heteroscedasticity or the presence of outliers affecting the target. +### Scatter Plot Analysis -### Feature 2 vs. Target -- **Distribution:** Similar to Feature 1, with data points spread across a wide range but a less defined alignment. -- **Interpretation:** The scatter is less structured, suggesting even less of a linear relationship with the target. +The image shows scatter plots of six features (Feature 1 through Feature 6) plotted against the target variable labeled "Y_Train" on the y-axis. The x-axis in each subplot corresponds to the respective feature values. -### Feature 3 vs. Target -- **Distribution:** Data points extend vertically across a wide range of the target, with a tight clustering around the center, suggesting some degree of relationship. -- **Interpretation:** The clustering could indicate a potential relationship where Feature 3 moderately predicts the target variable. +### Key Observations: -### Feature 4 vs. Target -- **Distribution:** The scatter is less dispersed compared to other features, indicating a potential inverse relationship as data points seem to descend with higher values of Feature 4. -- **Interpretation:** This could suggest an inverse correlation where the target decreases as Feature 4 increases, warranting further investigation. +1. **Feature 1 to Feature 3**: + - All three features show a strong positive linear correlation with the target variable (Y_Train). + - The points align closely along a positively sloped line, indicating that as these feature values increase, the target variable increases in a linear fashion. -### Feature 5 vs. Target -- **Distribution:** Points are densely packed in a specific target range, which might indicate a stronger relationship than seen with other features. -- **Interpretation:** There seems to be a correlation, albeit not strong, where Feature 5 consistently associates with specific values of the target. - -### Feature 6 vs. Target -- **Distribution:** Shows a similar pattern to Feature 5 but with slightly more spread across the target range. -- **Interpretation:** Suggests a potential but weak relationship, possibly indicating a dependency of the target on Feature 6. - -### General Observations -- **Variability:** The target variable shows high variability across all features, suggesting complex underlying dynamics that may not be captured fully by linear models. -- **Potential for Non-linear Analysis:** Given the widespread and varied dispersion of points, non-linear analytical techniques might be more suitable to uncover deeper relationships. -- **Importance of Advanced Statistical Techniques:** Techniques such as polynomial regression or machine learning algorithms like decision trees might be necessary to model these relationships more effectively. +2. **Feature 4 to Feature 6**: + - These features show a strong negative linear correlation with the target variable. + - The points align closely along a negatively sloped line, suggesting that as these feature values increase, the target variable decreases. +### General Insights: +- All six features display clear linear relationships with the target, either positive (Feature 1 to 3) or negative (Feature 4 to 6). +- These patterns suggest that linear ![alt text](5.png) -## Scatter Plot Analysis for Test Features vs. Target - -### Feature 1 vs. Target -- **Distribution:** Data points are spread across a wide range of target values but show a vertical trend. -- **Interpretation:** Despite the broad spread, the vertical arrangement suggests minimal influence of Feature 1 on target variation. - -### Feature 2 vs. Target -- **Distribution:** Points exhibit a tighter vertical distribution compared to Feature 1. -- **Interpretation:** Feature 2 shows a slight hint of vertical clustering, which might imply a weak relationship with the target. +### Scatter Plot Analysis for Test Features vs. Target -### Feature 3 vs. Target -- **Distribution:** There is a clear vertical grouping around the center, suggesting that changes in Feature 3 might not significantly affect the target. -- **Interpretation:** This vertical alignment indicates low variability of the target with changes in Feature 3, pointing to a weak predictive power. +The image shows scatter plots of six features (Test Feature 1 through Test Feature 6) plotted against the target variable labeled "Y_Test." Each subplot represents the relationship between a specific feature (on the x-axis) and the target variable (on the y-axis). -### Feature 4 vs. Target -- **Distribution:** Scatter is more dispersed at higher target values, suggesting some variability influenced by Feature 4. -- **Interpretation:** The dispersion suggests a possible correlation where the target increases as Feature 4 increases, though the relationship may not be strong. +### Key Observations: -### Feature 5 vs. Target -- **Distribution:** Points are concentrated in a narrow range of the target, suggesting a potential moderate relationship. -- **Interpretation:** The concentration in a specific target range indicates that Feature 5 could have a consistent but limited impact on the target values. +1. **Feature 1 to Feature 6**: + - Each scatter plot shows a wide spread of points without a clear pattern, suggesting weak or no strong linear relationships between the features and the target variable. + - The lack of clustering along a line indicates that none of the features exhibit a strong direct correlation with the target. + +2. **Feature 5**: + - There appears to be a slight downward trend, indicating a weak negative correlation between Feature 5 and the target (Y_Test). -### Feature 6 vs. Target -- **Distribution:** Similar to Feature 5, but with slightly more spread across the target range. -- **Interpretation:** Indicates a potential relationship, with Feature 6 impacting the target across a broader range than Feature 5. +3. **Feature 6**: + - There is some clustering in the lower right quadrant, possibly indicating a negative influence on the target, but the scatter remains wide. -## General Observations -- **Trends:** None of the features show a strong linear relationship with the target, suggesting that relationships, if any, might be complex or non-linear. -- **Predictive Insights:** Features with tighter clusters or specific patterns (like Features 5 and 6) may warrant further analysis to understand their predictive capabilities better. -- **Implications for Modeling:** Given the lack of strong linear relationships, exploring non-linear models or machine learning techniques might be beneficial to capture the underlying patterns and improve predictive accuracy. +### General Insights: +- None of the features show a strong, easily identifiable linear correlation with the target variable, as the data points are dispersed across all plots. +- This suggests that more complex modeling techniques (e.g., non-linear methods) or additional feature engineering may be needed to capture meaningful patterns in the data. @@ -520,38 +423,24 @@ This comprehensive analysis underscores the need for a broader approach in model ![alt text](6.png) -## Regression Analysis for Features vs. Actual and Predicted Target Values - -### Feature 1 vs. Actual and Predicted Target -- **Actual Values:** Represented by yellow dots. -- **Predicted Values:** Represented by blue dots. -- **Fit Line:** Red dashed line indicating the regression model's predictions. -- **Interpretation:** The close alignment of the fit line with both actual and predicted values suggests that Feature 1 has a strong linear correlation with the target, making it a potentially strong predictor. - -### Feature 2 vs. Actual and Predicted Target -- **Interpretation:** This plot shows a tight correlation with a linear progression similar to Feature 1, indicating effective predictive capabilities of Feature 2 with a robust linear relationship. - -### Feature 3 vs. Actual and Predicted Target -- **Interpretation:** Despite the wide range of target values, the fit line closely follows the trend in actual and predicted values, suggesting that Feature 3 strongly influences the target, albeit with some potential outliers or high variance. +### Regression Analysis for Features vs. Actual and Predicted Target Values -### Feature 4 vs. Actual and Predicted Target -- **Interpretation:** The downward slope of the fit line suggests an inverse relationship between Feature 4 and the target, where increasing values of Feature 4 correlate with decreasing target values. This inverse relationship might require further analysis to understand causal factors. +The image shows six scatter plots comparing the actual target values and the predicted target values for six features (Feature 1 through Feature 6) against the target variable labeled "Y." Each subplot also includes a red dashed line representing the linear fit for each feature. -### Feature 5 vs. Actual and Predicted Target -- **Interpretation:** Feature 5 shows a downward trend similar to Feature 4, indicating a negative correlation. However, the scatter is somewhat more dispersed, suggesting less consistency in the predictive accuracy compared to other features. +### Key Observations: -### Feature 6 vs. Actual and Predicted Target -- **Interpretation:** The fit line demonstrates an inverse relationship, similar to Features 4 and 5, but with a tighter clustering of points around the line, enhancing its predictive validity. +1. **Feature 1 to Feature 3**: + - The fit lines have positive slopes, suggesting a weak positive relationship between these features and the target variable. + - The actual and predicted values are dispersed around the fit line with noticeable variance, indicating that the model's predictions are not tightly aligned with the actual values. The fit is somewhat loose, and the features do not have a strong linear relationship with the target. -## General Observations -- **Predictive Strength:** Features 1, 2, and 3 display strong linear correlations with the target, indicating high predictive strength. -- **Inverse Relationships:** Features 4, 5, and 6 exhibit inverse relationships, which are consistently evident from their respective plots. -- **Model Performance:** The overall alignment of the fit lines across all features suggests that the regression model generally performs well, capturing the primary trends in the dataset effectively. +2. **Feature 4 to Feature 6**: + - The fit lines have negative slopes, indicating a negative relationship between these features and the target variable. + - Similar to Features 1-3, there is substantial dispersion of the actual and predicted values around the fit line, suggesting weak predictive power and variability in the model's predictions for these features. -## General Observations -- **Trends and Relationships:** These scatter plots generally do not show strong linear correlations, indicating the potential necessity for non-linear modeling approaches to capture the relationships between features and the target. -- **Predictive Insights:** The variability and patterns observed suggest that some features might have more influence on the target, possibly in non-obvious or non-linear ways. -- **Implications for Modeling:** The lack of strong linear trends across all features reinforces the need for sophisticated analytical techniques, such as machine learning algorithms that can handle non-linearity and complex interactions more effectively. +### General Insights: +- All six features exhibit weak linear relationships with the target variable, as evidenced by the scattered points around the fit lines. The model does not capture the target values well, and the fit is not strong. +- **Features 4, 5, and 6** show negative relationships, meaning that as the feature values increase, the target decreases, though this is not strong. +- These scatter plots indicate that the model could benefit from more advanced techniques, as simple linear regression appears insufficient for accurately predicting the target values for these features. Non-linear modeling methods or additional feature engineering might improve the model's performance. From 1d2917c4b4d8aa08c37124e39c6aba5baf1adc2f Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Thu, 10 Oct 2024 15:27:01 -0500 Subject: [PATCH 26/36] Incoming changes --- README.md | 74 ++++++++++++++++++++++++++----------------------------- 1 file changed, 35 insertions(+), 39 deletions(-) diff --git a/README.md b/README.md index d228be6..31fe5b3 100644 --- a/README.md +++ b/README.md @@ -68,15 +68,18 @@ model.fit(X_train, y_train) # Predict on test data predictions = model.predict(X_test) ``` + ### `generate_negative_data` -The `generate_negative_data` function generates synthetic data with two distinct patterns: +The `generate_negative_data` function generates synthetic data with two distinct patterns: + 1. The first half of the features has a **monotonic increasing trend**. 2. The second half of the features has a **linear decreasing trend** (i.e., negative slope). This function can be useful for testing model's ability in negative correlations data. #### Parameters: + - **`range_x`** (_tuple_): The range of feature values (min, max). - **`noise_scale`** (_float_): Standard deviation of the Gaussian noise added to the data. - **`size`** (_int_): Number of samples in the dataset. @@ -84,6 +87,7 @@ This function can be useful for testing model's ability in negative correlations - **`seed`** (_int_): Random seed for reproducibility. #### Returns: + - **`X`** (_numpy array_): Generated feature matrix with both increasing and decreasing trends. - **`y`** (_numpy array_): Target values with contributions from the features and added noise. @@ -101,12 +105,14 @@ X, y = generate_negative_data(range_x=(0, 10), noise_scale=1.0, size=100, num_fe ### `generate_rotated_positive_data` The `generate_rotated_positive_data` function generates synthetic data with two patterns: + 1. The first half of the features follows a **monotonic increasing trend**. 2. The second half exhibits a **wavy (S-shaped) pattern**, adjusted by a rotation matrix to create a slanted shape. This function generates more complex data to test model's ability. #### Parameters: + - **`range_x`** (_tuple_): Specifies the range of feature values (min, max). - **`noise_scale`** (_float_): Standard deviation of the Gaussian noise added to the data. - **`size`** (_int_): Number of samples in the dataset. @@ -116,6 +122,7 @@ This function generates more complex data to test model's ability. - **`mode`** (_int_, default=0): Determines the scaling factors for the feature values. #### Returns: + - **`X`** (_numpy array_): Generated feature matrix with increasing trends and rotated S-shaped patterns. - **`y`** (_numpy array_): Target values influenced by the features and noise. @@ -134,16 +141,14 @@ X, y = generate_rotated_positive_data(range_x=(0, 10), noise_scale=1.0, size=100 ## ElasticNetModel Overview -The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implement this model: +The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implemented this model: ## Loss Function - **The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows** : - $\text{Loss}$(w) = $\frac{1}{n}$ $\sum_{i=1}^{n}$ $(y_i - w^T x_i)^2$ + $\lambda$ ($\rho$ $\sum_{j=1}^{d}$ $|w_j|$ + (1 - $\rho$) $\sum_{j=1}^{d}$ $w_j^2$) - ## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. - **Gradient calculation results of MSE:** : @@ -174,10 +179,10 @@ In summary, our model implements hybrid regularization by combining L1 and L2 pe ### What the Model Does: -- **Hybrid Regularization**: Out of all the classifiers Elastic Net is more advantageous because it uses a combination of L1 and L2 penalties. +- **Hybrid Regularization**: Out of all the classifiers Elastic Net is more advantageous because it uses a combination of L1 and L2 penalties. -- **L1 penalty** Reducing the effect of weak predictors makes the model purged of unnecessary items Skipping all the lesser significant variables makes the model slimmer and easily understandable and easy to interpret Especially when one has a large number of potential predictors often it is dreaded to end up with a bloated or overfit model. - **L2 penalty** This makes this approach more useful when the features are related in some form, which is very often the case, generally. This is because, when the influences are spread over various parameters of the model ; it ensures that the model can perform optimally in the long run. +- **L1 penalty** Reducing the effect of weak predictors makes the model purged of unnecessary items. Skipping all the lesser significant variables makes the model slimmer and easily understandable and easy to interpret, especially when one has a large number of potential predictors, often it is dreaded to end up with a bloated or overfit model. + **L2 penalty** This makes this approach more useful when the features are related in some form, which is very often the case.This is because, when the influences are spread over various parameters of the model it ensures that the model can perform optimally in the long run. - **Gradient Descent Optimization**: This technique is crucial when optimizing models in the cases where models are trained on large sets of data and older methods will slow down data processing. It does this in the sense that the different adjustments flow continuously and systematically so as to arrive at optimal value of the performance and hence its value is dependent on the learning rate and accuracy. @@ -210,18 +215,15 @@ For the purpose of the model’s strength and accuracy, we have developed an ela - **Method**: To check how well the model is doing, we fed it with data it did not learn with during the learning stage. - **Why It Matters**: This is essential for evaluating how good the model is in using the learning to solve other unseen dataset – which is very important in ascertaining the utility of the model in real world where dataset variation is the order of the day. - - ### 4. Model Evaluation Metrics - **Metrics Used**: To check the accuracy and reliability of the carried out predictions we used several standard regression parameters such as: - Mean Squared Error (MSE) - Linear – Root Mean Squared Error (RMSE) + Linear – Root Mean Squared Error (RMSE) - Mean Absolute Error (MAE) - R-squared (R²) - **Why They’re Important**: These measure give you actual numbers to explain the degree at which the model employed is capable of predicting new data. They assist us in getting an understanding of other aspects of performance including the average magnitude of the errors and variance in the target variable accounted for by the model. - ### 5. Probability versus Outcome Plots - **How We Do It**: We employ residual plots whereby each point represents a predicted value against its actual value. The closer that the observed points are to the 45-degree line, the better are the predictive capabilities of the model in question. @@ -254,20 +256,18 @@ raising the value of lambda leads to increase the amount the penalizes the coeff If a model is making sharp learning from data — infact picking up noise as the real signals, then making enhancements to the lambda values can be a good strategy. It does so while also tempering the learning of the model and its corresponding generalization to unseen data of any kind. - **Use Case:** -Thresh is most effective whenever a certain model is overly fitting the training data set. Finally, by making thresh larger, it improves the generalization capability of the model over new, never seen before data. + Thresh is most effective whenever a certain model is overly fitting the training data set. Finally, by making thresh larger, it improves the generalization capability of the model over new, never seen before data. ### 2. Thresh - **Function:** -Thresh decides the elimination point for feature selection after filtering in, shaving the model down to select characteristics that are necessary and excluding others that may distort. + Thresh decides the elimination point for feature selection after filtering in, shaving the model down to select characteristics that are necessary and excluding others that may distort. - **Adjustment Effect:** -Raising the thresh value may cause getting rid of secondary variables as their coefficients become zeroes. Such a reduction in the model leads to less accurate results but occasionally, the results are expected to be better for unseen data. + Raising the thresh value may cause getting rid of secondary variables as their coefficients become zeroes. Such a reduction in the model leads to less accurate results but occasionally, the results are expected to be better for unseen data. - **Where To Use It:** -Thresh is used to control the measure of variable importance, allowing identifying important features, and at the same time, excluding an excessive number of predictors that have no beneficial impact or, on the contrary, may negatively affect the model’s ability to make adequate predictions. This is more applicable especially where there are many features more than the dependent variables. - - + Thresh is used to control the measure of variable importance, allowing identifying important features, and at the same time, excluding an excessive number of predictors that have no beneficial impact or, on the contrary, may negatively affect the model’s ability to make adequate predictions. This is more applicable especially where there are many features more than the dependent variables. ### 3. Max_iter @@ -277,17 +277,17 @@ Thresh is used to control the measure of variable importance, allowing identifyi ### 4. Tol -- **Function**: This is the same as tolerance; it defines a small the error which must fit our training data before further adjustments are made to stop.r adjustments stop. +- **Function**: This is the same as tolerance; it defines a small the error which must fit our training data before further adjustments are made to stop.r adjustments stop. - **Impact of Adjustment**: Lowering the `tol` value ensures the model won’t stop learning until the fit is very tight, improving accuracy but extending training time. -- **When We Use It**: If rate of accuracy must be in the consideration. The most important one is that if you set the tolerance level too low then really long training can actually make no gain at all. +- **When We Use It**: If rate of accuracy must be in the consideration. The most important one is that if you set the tolerance level too low then really long training can actually make no gain at all. ### 5. Learning_rate -- **Function**: This hyperparameter controls by how much the model should change towards the optimization in the updating of the model. -- **Impact of Adjustment**: It reduces the steps taken towards implementation and is much slower but more precise, because it is a smaller learning rate and its implementations are more careful. -- **When We Use It**: However, if the model is over the point that represents the minimal loss, it means that the learning rate is oversized, thus should be decreased to improve the stability and accuracy of the convergence. Nonetheless, the rate of learning may be reduced to an extreme low in the presence of a low rate. +- **Function**: This hyperparameter controls by how much the model should change towards the optimization in the updating of the model. +- **Impact of Adjustment**: It reduces the steps taken towards implementation and is much slower but more precise, because it is a smaller learning rate and its implementations are more careful. +- **When We Use It**: However, if the model is over the point that represents the minimal loss, it means that the learning rate is oversized, thus should be decreased to improve the stability and accuracy of the convergence. Nonetheless, the rate of learning may be reduced to an extreme low in the presence of a low rate. -These two parameters, therefore, can be tuned to optimize accuracy for given data distribution and complexity of task, computational cost, and model complexity. +These two parameters, therefore, can be tuned to optimize accuracy for given data distribution and complexity of task, computational cost, and model complexity. ## Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? @@ -306,6 +306,7 @@ These two parameters, therefore, can be tuned to optimize accuracy for given dat - **The Challenge**:Think of your data as listening to one of those quiet FM radio stations during a thunderstorm, that is what this is like. This then becomes static (noise) that can drown the music (true patterns), and thus it makes it hard for the Elastic Net to hear what is important. - **What Happens**:If the data is too noisy, Elastic Net might fit to the noise instead of the music and learn something that works great on training data but will falter in practice. - **What We Can Do**: + - **Increase Regularization**:we have the option here to turn up the lambdas knob and increase penalties for complexity, allowing our model to focus in on every loud symptomatic signal while ignoring that static. - **Threshold Tuning**: Changing the thresh parameter to allow us to be more selective about which features we let into the model, and cut out those that tend to contain more noise than signal. @@ -313,10 +314,10 @@ These two parameters, therefore, can be tuned to optimize accuracy for given dat By acknowledging these limitations and using smart strategies to overcome them, we can increase the potential of Elastic Net because another graphical preferred in predictive modelling. - - ## Visualization ![alt text](1.png) + ### Scatter Plots of Train Features vs Target This image shows a series of scatter plots, each representing the relationship between a different feature (Feature 1 to Feature 6) and a target variable (Y_train) based on training data. Each plot displays the data points in two dimensions: @@ -331,10 +332,8 @@ The purpose of these plots is typically to visually assess the relationship betw Overall, these scatter plots can help in determining which features might be relevant for modeling, although none of the features shown here have a distinct or strong linear relationship with the target variable. - ![alt text](2.png) - ### Overview of Each Feature Plot Analysis The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against a target variable (labeled as "Y_Test"). Each subplot represents the relationship between a single feature and the target variable. @@ -345,10 +344,12 @@ The image shows scatter plots of six different features (labeled "Test Feature 1 - **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. ### Visual Insights: + - For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. - For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. ### Uses: + - **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). - **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. @@ -364,14 +365,15 @@ The image shows scatter plots of six different features (labeled "Test Feature 1 - **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. ### Visual Insights: + - For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. - For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. ### Uses: + - **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). - **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. - ![alt text](4.png) ### Scatter Plot Analysis @@ -381,6 +383,7 @@ The image shows scatter plots of six features (Feature 1 through Feature 6) plot ### Key Observations: 1. **Feature 1 to Feature 3**: + - All three features show a strong positive linear correlation with the target variable (Y_Train). - The points align closely along a positively sloped line, indicating that as these feature values increase, the target variable increases in a linear fashion. @@ -389,13 +392,12 @@ The image shows scatter plots of six features (Feature 1 through Feature 6) plot - The points align closely along a negatively sloped line, suggesting that as these feature values increase, the target variable decreases. ### General Insights: + - All six features display clear linear relationships with the target, either positive (Feature 1 to 3) or negative (Feature 4 to 6). - These patterns suggest that linear - ![alt text](5.png) - ### Scatter Plot Analysis for Test Features vs. Target The image shows scatter plots of six features (Test Feature 1 through Test Feature 6) plotted against the target variable labeled "Y_Test." Each subplot represents the relationship between a specific feature (on the x-axis) and the target variable (on the y-axis). @@ -405,24 +407,20 @@ The image shows scatter plots of six features (Test Feature 1 through Test Featu 1. **Feature 1 to Feature 6**: - Each scatter plot shows a wide spread of points without a clear pattern, suggesting weak or no strong linear relationships between the features and the target variable. - The lack of clustering along a line indicates that none of the features exhibit a strong direct correlation with the target. - 2. **Feature 5**: + - There appears to be a slight downward trend, indicating a weak negative correlation between Feature 5 and the target (Y_Test). 3. **Feature 6**: - There is some clustering in the lower right quadrant, possibly indicating a negative influence on the target, but the scatter remains wide. ### General Insights: + - None of the features show a strong, easily identifiable linear correlation with the target variable, as the data points are dispersed across all plots. - This suggests that more complex modeling techniques (e.g., non-linear methods) or additional feature engineering may be needed to capture meaningful patterns in the data. - - - - ![alt text](6.png) - ### Regression Analysis for Features vs. Actual and Predicted Target Values The image shows six scatter plots comparing the actual target values and the predicted target values for six features (Feature 1 through Feature 6) against the target variable labeled "Y." Each subplot also includes a red dashed line representing the linear fit for each feature. @@ -430,6 +428,7 @@ The image shows six scatter plots comparing the actual target values and the pre ### Key Observations: 1. **Feature 1 to Feature 3**: + - The fit lines have positive slopes, suggesting a weak positive relationship between these features and the target variable. - The actual and predicted values are dispersed around the fit line with noticeable variance, indicating that the model's predictions are not tightly aligned with the actual values. The fit is somewhat loose, and the features do not have a strong linear relationship with the target. @@ -438,10 +437,7 @@ The image shows six scatter plots comparing the actual target values and the pre - Similar to Features 1-3, there is substantial dispersion of the actual and predicted values around the fit line, suggesting weak predictive power and variability in the model's predictions for these features. ### General Insights: + - All six features exhibit weak linear relationships with the target variable, as evidenced by the scattered points around the fit lines. The model does not capture the target values well, and the fit is not strong. - **Features 4, 5, and 6** show negative relationships, meaning that as the feature values increase, the target decreases, though this is not strong. - These scatter plots indicate that the model could benefit from more advanced techniques, as simple linear regression appears insufficient for accurately predicting the target values for these features. Non-linear modeling methods or additional feature engineering might improve the model's performance. - - - - From 5fbeb206c3a63f4a9577da34c21589945e217d06 Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Thu, 10 Oct 2024 15:34:04 -0500 Subject: [PATCH 27/36] Changed 3rd question of readme --- README.md | 44 +++++++++++++++++++++----------------------- 1 file changed, 21 insertions(+), 23 deletions(-) diff --git a/README.md b/README.md index 1d31fe6..0e51e79 100644 --- a/README.md +++ b/README.md @@ -178,39 +178,37 @@ By weaving together these detailed testing methods, we not only affirm that the ## Model Optimization Parameters -To optimize the performance of our model, we’ve provided several adjustable parameters. Here’s a breakdown of each parameter, explaining their function and how adjustments can enhance model performance: +We gave some parameters to change in order to improve the performance of our model. This section is a breakdown of each parameter and what each does/means, as well as touches on how tuning them could lead to improved model performance: -### 1. Lambdas +1. **Lambdas** -- **Function**: Lambdas are central to our model’s regularization strategies, utilized in both L1 (Lasso) and L2 (Ridge) techniques. This helps prevent overfitting, a common issue where the model learns noise as if it were true data signals. -- **Impact of Adjustment**: Increasing the lambdas value amplifies the penalty on the model’s coefficients, effectively pulling them towards zero. This adjustment is beneficial when dealing with complex datasets that could potentially lead to misleading noise. -- **When We Use It**: If we observe our model overfitting to training data, boosting the lambdas value can help mitigate this, promoting better generalization to new, unseen data. + - **Function:** Lambdas are the key elements to the regularization strategies of our model, which we utilized in both L1 (Lasso) and L2 (Ridge) techniques. This alleviates a common overfitting problem in which the model learns noise as if it were real data signals. + - **Effect of Regularization:** When the value of lambdas is increased, it increases the penalty on coefficients so that they are constrained to be near zero. This adjustment is beneficial when dealing with complex datasets that could potentially lead to misleading noise. + - **When we use it:** When we observe our model is overfitting with training data, we then increase lambdas to rebuild our model in a much generalized way to new, unseen data. -### 2. Thresh +2. **Thresh** -- **Function**: Thresh determines the cutoff for feature selection, which simplifies the model by focusing only on impactful features. -- **Impact of Adjustment**: A higher thresh value is likely to eliminate less significant features by setting their coefficients to zero, which can enhance both the interpretability and performance of the model on new data. -- **When We Use It**: If our dataset has an abundance of features, adjusting thresh helps us streamline the model, concentrating on the most influential factors for predictions. + - **Operation:** Thresh For: Feature Selection Description: It creates a cutoff for feature selection. It simplifies the model by only considering a few significant features. A higher thresh value can remove less useful features by nulling their coefficients and it will improve the model's interpretability on new data. + - **Where We Use It:** Thresh is used when there were a lot of features in our dataset and we do not want to perform the data modeling on all those features (so using thresh makes sure that spearmake calculation applied only top x number of feature). -### 3. Max_iter +3. **Max_iter** -- **Function**: This parameter sets the maximum number of iterations our algorithm will execute before stopping. -- **Impact of Adjustment**: Allowing more iterations gives the model additional opportunities to refine its weights to the data, potentially improving its accuracy. -- **When We Use It**: If our model hasn’t yet reached its optimal state, increasing `max_iter` may be necessary. However, this could lead to longer training times, requiring a balance between precision and efficiency. + - **Function:** It decides the maximum loops (number of iterations) our algorithm performs before stopping. + - **Adjustment Impact:** The more iterations the model adjusts its weights with the data, the better it can learn and potentially yield higher accuracy. + - **Why We May Use It:** If our model has not yet been optimized, it may require an increase in max_iter. But, it may potentially take more training time which means a tradeoff between accuracy and speed. -### 4. Tol +4. **Tol** -- **Function**: Tol, or tolerance, defines how close the model needs to fit the training data trends before ceasing further adjustments. -- **Impact of Adjustment**: Setting a smaller `tol` value means the model won’t stop learning until it achieves a closer fit, which can enhance accuracy but also extend training duration. -- **When We Use It**: This is particularly valuable when high precision is crucial. However, setting the tolerance too tight might lead to diminishing returns, especially in terms of prolonged training without significant gains. + - **Function:** Tol — that is, tolerance — how closely does the model have to fit the training data trends before we stop adjusting further. + - **Impact of Tuning:** Lowering the value means the model won't stop learning until it achieves a closer fit, this can increase accuracy but would also prolong training time. + - **Where We Use This:** This is most useful when high precision matters a lot. However, making the tolerance too tight can result in diminishing returns, especially if more training leads to virtually no progress with prolonged training. -### 5. Learning_rate +5. **Learning_rate** + - **Function:** In gradient descent optimization, this parameter helps us to set the step size the model takes towards reducing the loss function. + - **Implication of Adjustment:** Lower learning rates give smaller and safer steps that might help in more finesse model optimization. + - **Where We Use It:** If the model reaches minimum loss and is going beyond it i.e., overshooting, by decreasing the learning rate we can make sure it converges more stably and accurately to its minima. However, if the rate is too low, it will slow the training process in an overly long way. -- **Function**: In gradient descent optimization, this parameter controls the step size the model takes towards minimizing the loss function. -- **Impact of Adjustment**: Lowering the learning rate results in smaller, more cautious steps, potentially leading to more precise model optimization. -- **When We Use It**: If the model is overshooting the minimal loss point, reducing the learning rate can help achieve a more stable and accurate convergence. However, too low a rate may slow down the training process excessively. - -By fine-tuning these parameters, we can adjust our model to better fit the specific characteristics of the data and the demands of the task at hand, striking an optimal balance between accuracy, computational efficiency, and model simplicity. +We can improve this model by tinkering with these parameters to help the model adapt well to the nuances of our data and the requirements of our task, ensuring that we have a good trade-off between accuracy, efficiency of computation, and model lean size. # Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? From 8ab33e7fcbf34e76822d44d96d7e63006db9db19 Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Thu, 10 Oct 2024 15:53:36 -0500 Subject: [PATCH 28/36] Changed 4th question --- README.md | 38 +++++++++++++++++++++++--------------- 1 file changed, 23 insertions(+), 15 deletions(-) diff --git a/README.md b/README.md index 0e51e79..f874ad3 100644 --- a/README.md +++ b/README.md @@ -212,24 +212,32 @@ We can improve this model by tinkering with these parameters to help the model a # Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? -## Specific Inputs That May Challenge the ElasticNet Model +True, there are certain inputs that the implementation finds difficult working with: -### 1. Highly Nonlinear Data +## 1. Data Dependencies (Highly Nonlinear Data): -- **The Challenge**: Elastic Net is designed to see the world in straight lines, assuming a straightforward, linear relationship between features and the target variable. However, life often isn’t that simple, and data can show complex, curved relationships that the model struggles to grasp. -- **What Happens**: When faced with these complex patterns, Elastic Net might not hit the mark with its predictions since it can’t effectively capture the twists and turns in variable interactions. -- **What We Can Do**: - - **Feature Engineering**: To help the model better understand the underlying patterns, we can transform the input features. This might mean squaring or cubing features, or introducing interaction terms, and even applying mathematical transformations like logarithms or exponentials to straighten out those nonlinear relationships. - - **Kernel Methods**: Borrowing a trick from the support vector machine playbook, we can use kernel methods to project our data into a space where the once-complex relationships look linear, making it easier for Elastic Net to do its job. - - **Hybrid Models**: Sometimes, two heads (or models) are better than one. By teaming up Elastic Net with nonlinear models like decision trees or neural networks, we can combine their strengths, enhancing the model’s ability to predict accurately across a wider range of scenarios. +- **The Challenge**: Elastic Net has an unspoiled ability to see a linear world, plots each point in the n-dimensional Euclidean space, and captures the straight line that passes near as many points as possible. The information might express intricate, curved relationships that the model finds difficult to learn. + +- **What Happens**: If the problem consists of complex patterns among variables, Elastic Net will fail to accurately predict target values due to its inability to model the variability and flexibility. + +- **Real Feature Engineering**: To capture the latent driving forces more effectively, we can manipulate input features to provide the model with a better perspective of the underlying patterns. This may involve creating new features by squaring or cubing the original variables, introducing interaction terms between variables, or using mathematical transformations such as logarithms or exponentials to linearize some nonlinear relationships. + +- **Kernel Methods**: Borrowing a trick from support vector machines, kernel methods can help by projecting data into a space where once-complex relationships look linear enough for Elastic Net to perform effectively. + +- **Hybrid Models**: Sometimes combining two models can be better than one. However, hybrid models have their own challenges, such as limited variance in terms of bias and variance, which Elastic Net aims to address. We can combine tree models or neural networks with Elastic Net to improve prediction across a variety of scenarios. -### 2. Noisy Data +## 2. Noisy Data: + +- **The Challenge**: Noisy data is like trying to listen to a quiet radio station during a thunderstorm. Noise can drown out the true patterns, making it hard for Elastic Net to capture what's important. + +- **Example**: If the data is too noisy, Elastic Net may learn the noise along with the true signal, leading to overfitting on the training set and poor performance in the real world. -- **The Challenge**: Noisy data is like trying to listen to a quiet radio station during a thunderstorm. The static (noise) can drown out the music (true patterns), making it hard for Elastic Net to hear what’s important. -- **What Happens**: If the data is too noisy, Elastic Net might start learning the static instead of the music, leading to models that perform well on their training data but falter in the real world. - **What We Can Do**: - - **Increase Regularization**: By turning up the lambdas knob, we increase the penalties for complexity, encouraging the model to focus on the loud and clear signals and ignore the static. - - **Threshold Tuning**: Adjusting the thresh parameter helps us be stricter about which features we let into the model, cutting out those that are more noise than signal. - - **Noise Reduction Techniques**: Before we even feed data into our model, we can clean it up—filtering out oddball values or smoothing out jumpy signals—to make sure we’re working with the cleanest, clearest data possible. -By understanding these limitations and employing smart strategies to mitigate them, we can enhance the capabilities of Elastic Net, making it a more versatile and reliable tool in our predictive modeling toolkit. + - **Increase Regularization**: Turning up the lambda parameter increases the penalties for model complexity, forcing Elastic Net to focus on the main signal and ignore noise. + + - **Filter**: Adjusting the threshold parameter can make the model more or less strict about what signals pass through, helping to filter out noise. + + - **Reduction of Noise**: We can filter out outliers or smooth signals before feeding the data into the model. + +This post aims to understand these limitations and the smart ways to get around them, solidifying Elastic Net as an even more powerful tool in predictive modeling. 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z!(B&c9d)UVlW?XNA>?v_NqDeXAYwjyZ92<2+F@#sguQMnV(U}Ki@c$KE|B-nb&US? zF;MIOMj6Qet7PSW>hJ$*#k~L2egB#3`g0VN{=+7 diff --git a/README.md b/README.md index d228be6..ed2c805 100644 --- a/README.md +++ b/README.md @@ -132,11 +132,11 @@ X, y = generate_rotated_positive_data(range_x=(0, 10), noise_scale=1.0, size=100 ## Q1.What does the model you have implemented do and when should it be used? -## ElasticNetModel Overview +### ElasticNetModel Overview The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implement this model: -## Loss Function +### Loss Function - **The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows** : @@ -144,7 +144,7 @@ The model we've crafted, called **ElasticNetModel**, is an implementation of Ela $\text{Loss}$(w) = $\frac{1}{n}$ $\sum_{i=1}^{n}$ $(y_i - w^T x_i)^2$ + $\lambda$ ($\rho$ $\sum_{j=1}^{d}$ $|w_j|$ + (1 - $\rho$) $\sum_{j=1}^{d}$ $w_j^2$) -## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. +### Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. - **Gradient calculation results of MSE:** : @@ -191,7 +191,7 @@ The **ElasticNetModel** It is an entire package where not only can you get to ma ## Q2.How did you test your model to determine if it is working reasonably correctly? -#### Model Testing and Validation Plan +### Model Testing and Validation Plan For the purpose of the model’s strength and accuracy, we have developed an elaborate testing and validation plan. Let’s walk through each step of this strategy to see how it contributes to a full evaluation of the model’s performance: @@ -242,76 +242,69 @@ By combining these specific types of testing, not only do we guarantee to oursel ## Model Optimization Parameters -## Lambdas in Regularization - -**Function:** -The lambdas are crucial in regularization methods and finding the optimal one is recommended for using in Lasso (L1) or Ridge (L2) regularization. These techniques are somewhat crucial for avoiding overfitting since a model can learn most of the noise in data set as a signal. +We gave some parameters to change in order to improve the performance of our model. This section is a breakdown of each parameter and what each does/means, as well as touches on how tuning them could lead to improved model performance: -**Effect of Regularization:** -raising the value of lambda leads to increase the amount the penalizes the coefficients of the model and thus reduces them towards zero but not to zero all the same. This mechanism is particularly beneficial for noisy large datasets where the distribution of data may vary thus leading to the generation of interpretive patterns that are misleading. +1.⁠ ⁠*Lambdas* -**When to Use:** -If a model is making sharp learning from data — infact picking up noise as the real signals, then making enhancements to the lambda values can be a good strategy. It does so while also tempering the learning of the model and its corresponding generalization to unseen data of any kind. + - *Function:* Lambdas are the key elements to the regularization strategies of our model, which we utilized in both L1 (Lasso) and L2 (Ridge) techniques. This alleviates a common overfitting problem in which the model learns noise as if it were real data signals. + - *Effect of Regularization:* When the value of lambdas is increased, it increases the penalty on coefficients so that they are constrained to be near zero. This adjustment is beneficial when dealing with complex datasets that could potentially lead to misleading noise. + - *When we use it:* When we observe our model is overfitting with training data, we then increase lambdas to rebuild our model in a much generalized way to new, unseen data. -- **Use Case:** -Thresh is most effective whenever a certain model is overly fitting the training data set. Finally, by making thresh larger, it improves the generalization capability of the model over new, never seen before data. +2.⁠ ⁠*Thresh* -### 2. Thresh + - *Operation:* Thresh For: Feature Selection Description: It creates a cutoff for feature selection. It simplifies the model by only considering a few significant features. A higher thresh value can remove less useful features by nulling their coefficients and it will improve the model's interpretability on new data. + - *Where We Use It:* Thresh is used when there were a lot of features in our dataset and we do not want to perform the data modeling on all those features (so using thresh makes sure that spearmake calculation applied only top x number of feature). -- **Function:** -Thresh decides the elimination point for feature selection after filtering in, shaving the model down to select characteristics that are necessary and excluding others that may distort. +3.⁠ ⁠*Max_iter* -- **Adjustment Effect:** -Raising the thresh value may cause getting rid of secondary variables as their coefficients become zeroes. Such a reduction in the model leads to less accurate results but occasionally, the results are expected to be better for unseen data. + - *Function:* It decides the maximum loops (number of iterations) our algorithm performs before stopping. + - *Adjustment Impact:* The more iterations the model adjusts its weights with the data, the better it can learn and potentially yield higher accuracy. + - *Why We May Use It:* If our model has not yet been optimized, it may require an increase in max_iter. But, it may potentially take more training time which means a tradeoff between accuracy and speed. -- **Where To Use It:** -Thresh is used to control the measure of variable importance, allowing identifying important features, and at the same time, excluding an excessive number of predictors that have no beneficial impact or, on the contrary, may negatively affect the model’s ability to make adequate predictions. This is more applicable especially where there are many features more than the dependent variables. +4.⁠ ⁠*Tol* + - *Function:* Tol — that is, tolerance — how closely does the model have to fit the training data trends before we stop adjusting further. + - *Impact of Tuning:* Lowering the value means the model won't stop learning until it achieves a closer fit, this can increase accuracy but would also prolong training time. + - *Where We Use This:* This is most useful when high precision matters a lot. However, making the tolerance too tight can result in diminishing returns, especially if more training leads to virtually no progress with prolonged training. +5.⁠ ⁠*Learning_rate* + - *Function:* In gradient descent optimization, this parameter helps us to set the step size the model takes towards reducing the loss function. + - *Implication of Adjustment:* Lower learning rates give smaller and safer steps that might help in more finesse model optimization. + - *Where We Use It:* If the model reaches minimum loss and is going beyond it i.e., overshooting, by decreasing the learning rate we can make sure it converges more stably and accurately to its minima. However, if the rate is too low, it will slow the training process in an overly long way. -### 3. Max_iter +We can improve this model by tinkering with these parameters to help the model adapt well to the nuances of our data and the requirements of our task, ensuring that we have a good trade-off between accuracy, efficiency of computation, and model lean size. -- **Function**: This parameter tells our algorithm to stop after running for the permitted number of iterations as stated below. -- **Impact of Adjustment**: Increasing the number of epochs may provide better chances for the model to adjust its weights with respect to data for better performance. -- **When We Use It**: If the model is not in its best condition yet, then it may be required to increase the `max_iter`. However, it is possible to have the model take more time for a training session, which calls for compromise between precision and time taken on the training session. +## Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? -### 4. Tol +True, there are certain inputs that the implementation finds difficult working with: -- **Function**: This is the same as tolerance; it defines a small the error which must fit our training data before further adjustments are made to stop.r adjustments stop. -- **Impact of Adjustment**: Lowering the `tol` value ensures the model won’t stop learning until the fit is very tight, improving accuracy but extending training time. -- **When We Use It**: If rate of accuracy must be in the consideration. The most important one is that if you set the tolerance level too low then really long training can actually make no gain at all. +## 1. Data Dependencies (Highly Nonlinear Data): -### 5. Learning_rate +- **The Challenge**: Elastic Net has an unspoiled ability to see a linear world, plots each point in the n-dimensional Euclidean space, and captures the straight line that passes near as many points as possible. The information might express intricate, curved relationships that the model finds difficult to learn. -- **Function**: This hyperparameter controls by how much the model should change towards the optimization in the updating of the model. -- **Impact of Adjustment**: It reduces the steps taken towards implementation and is much slower but more precise, because it is a smaller learning rate and its implementations are more careful. -- **When We Use It**: However, if the model is over the point that represents the minimal loss, it means that the learning rate is oversized, thus should be decreased to improve the stability and accuracy of the convergence. Nonetheless, the rate of learning may be reduced to an extreme low in the presence of a low rate. +- **What Happens**: If the problem consists of complex patterns among variables, Elastic Net will fail to accurately predict target values due to its inability to model the variability and flexibility. -These two parameters, therefore, can be tuned to optimize accuracy for given data distribution and complexity of task, computational cost, and model complexity. +- **Real Feature Engineering**: To capture the latent driving forces more effectively, we can manipulate input features to provide the model with a better perspective of the underlying patterns. This may involve creating new features by squaring or cubing the original variables, introducing interaction terms between variables, or using mathematical transformations such as logarithms or exponentials to linearize some nonlinear relationships. -## Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? +- **Kernel Methods**: Borrowing a trick from support vector machines, kernel methods can help by projecting data into a space where once-complex relationships look linear enough for Elastic Net to perform effectively. -## Specific Inputs That May Challenge the ElasticNet Model +- **Hybrid Models**: Sometimes combining two models can be better than one. However, hybrid models have their own challenges, such as limited variance in terms of bias and variance, which Elastic Net aims to address. We can combine tree models or neural networks with Elastic Net to improve prediction across a variety of scenarios. -### 1. Highly Nonlinear Data +## 2. Noisy Data: -- **The Challenge**: This is because Elastic Net has an Optimistic World View, it believes that the world can be described using straight lines and linearly separable features. But life is rarely that clean or bumpy, and data exhibits the kind of curved relationships that a model like biases against. -- **What Happens**: Elastic Net might miss with its predictions in these instances since they involve some of the sharp edges/turns and complex patterns that it cannot fully understand. - - **Feature Engineering**: One way to make the model learn the underlying patterns is by transforming those input features. This could involve squaring or cubing the features, maybe adding in interaction terms, possibly doing mathematical transformations (like logarithms or exponentials) to help straighten out those nonlinear relationships. - - **Kernel Methods**: Taking a cue from support vector machine, these can be kernelized as well. - - **Hybrid Models**:Two heads (or models) as well. We can combine the advantages of Elastic Net when teamwork with nonlinear models like decision trees or neural networks, which allows it to predict well in wider set of scenarios. +- **The Challenge**: Noisy data is like trying to listen to a quiet radio station during a thunderstorm. Noise can drown out the true patterns, making it hard for Elastic Net to capture what's important. -### 2. Noisy Data +- **Example**: If the data is too noisy, Elastic Net may learn the noise along with the true signal, leading to overfitting on the training set and poor performance in the real world. -- **The Challenge**:Think of your data as listening to one of those quiet FM radio stations during a thunderstorm, that is what this is like. This then becomes static (noise) that can drown the music (true patterns), and thus it makes it hard for the Elastic Net to hear what is important. -- **What Happens**:If the data is too noisy, Elastic Net might fit to the noise instead of the music and learn something that works great on training data but will falter in practice. - **What We Can Do**: - - **Increase Regularization**:we have the option here to turn up the lambdas knob and increase penalties for complexity, allowing our model to focus in on every loud symptomatic signal while ignoring that static. - - **Threshold Tuning**: Changing the thresh parameter to allow us to be more selective about which features we let into the model, and cut out those that tend to contain more noise than signal. - - **Noise Reduction Techniques**: Even before feeding data into our model, we can clean it up to filter out the oddball values or make some signal smoother so that what we pass on to our mod is as clean and clear as possible. + - **Increase Regularization**: Turning up the lambda parameter increases the penalties for model complexity, forcing Elastic Net to focus on the main signal and ignore noise. + + - **Filter**: Adjusting the threshold parameter can make the model more or less strict about what signals pass through, helping to filter out noise. + + - **Reduction of Noise**: We can filter out outliers or smooth signals before feeding the data into the model. -By acknowledging these limitations and using smart strategies to overcome them, we can increase the potential of Elastic Net because another graphical preferred in predictive modelling. +This post aims to understand these limitations and the smart ways to get around them, solidifying Elastic Net as an even more powerful tool in predictive modeling. - ## Visualization From 441aeb64be61005d2294b00d36bf21f9b919d2e3 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 10 Oct 2024 16:06:37 -0500 Subject: [PATCH 30/36] change half of the description of visualization --- README.md | 135 ++++++++++-------------------------------------------- 1 file changed, 24 insertions(+), 111 deletions(-) diff --git a/README.md b/README.md index 2578ff4..f32b33a 100644 --- a/README.md +++ b/README.md @@ -314,134 +314,47 @@ By acknowledging these limitations and using smart strategies to overcome them, - ## Visualization +### The first train dataset: ![alt text](1.png) +### Scatter Plots of Train Features -## Visualization Overview +This image shows a series of scatter plots, each representing the relationship between a different feature (Feature 1 to Feature 6) and Y_train based on training data. Each plot displays the data points in two dimensions: -**Axes:** -The plots display the target variable on the Y-axis and the features on the X-axis for each plot. +- The horizontal axis (X-axis) shows values of a specific feature. +- The vertical axis (Y-axis) represents the values of Y_train. -**Purpose:** -These plots are commonly used to evaluate the relationships between each feature and the target variable. This evaluation helps in deciding which features to select, how to potentially transform them, and to understand the distribution and correlations within the data. +The purpose of these plots is typically to visually assess the relationship between the features and the target variable. Observations can be made regarding the distribution, trend, or any potential outliers in the data. From the plots: -## Analysis of Individual Plots +- **Feature 1, Feature 2, Feature 3:** These features display a scattered distribution with respect to Y_train as well as relatively clear linear relationships. +- **Feature 4, Feature 5, Feature 6:** These features show some vertical dispersion and a less clear linear relationship. -### Feature 1 vs Target -- **Observation:** The scatter is loose and appears random, with no discernible trend towards any target line, implying little to no correlation with the target. - -### Feature 2 vs Target -- **Observation:** A few data points are noticeably scattered along the Y-axis, with no clear linear pattern, suggesting either a very low positive or possibly negative correlation. - -### Feature 3 vs Target -- **Observation:** Data points are vertically distributed, centered around zero on the X-axis. This plot suggests a very poor, if any, direct positive correlation between Feature 3 and the target. - -### Feature 4 vs Target -- **Observation:** While data points are widely dispersed, there is a slight indication of a weak relationship, as the spread is somewhat curved upwards. - -### Feature 5 vs Target -- **Observation:** Similar to Feature 4, with a slight upward curve in the spread, which may indicate a very weak positive correlation, though the slope is minimal. - -### Feature 6 vs Target -- **Observation:** Data points are concentrated around specific X-values, suggesting that Feature 6 might be categorical or discretized. The relationship to the target is not clearly linear. - -## General Observations - -### Lack of Strong Linear Relationships -- **Details:** No features show a strong linear relationship with the target, indicating that nonlinear models or transformations might be more suitable. - -### Potential for Non-linear Analysis -- **Details:** Exploring non-linear effects or interaction effects could be fruitful given the observed patterns. - -### Noise and Variability -- **Details:** The significant dispersion and overlap of data points suggest high variability within features affecting the target, or noise. +Overall, these scatter plots can help in determining which features might be relevant for modeling, although none of the features shown here have a distinct or strong linear relationship with the target variable. +### The first test dataset: ![alt text](2.png) +### Overview of Each Feature Plot Analysis -## Overview of Each Feature Plot Analysis - -### Test Feature 1 vs Target -- **Distribution:** The data points are randomly and evenly distributed across the range of Feature 1 without any obvious pattern. -- **Interpretation:** There appears to be no clear correlation between Feature 1 and the target, indicating high dispersion or heteroscedasticity and potentially non-linear relationships. - -### Test Feature 2 vs Target -- **Distribution:** Similar to Feature 1, points are spread across a wide range without a distinct pattern. -- **Interpretation:** The absence of clear linearity suggests that simple linear methods may be ineffective. +The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against "Y_Test". Each subplot represents the data of one specific demension of feature. -### Test Feature 3 vs Target -- **Distribution:** Points are spread out, albeit slightly less than in previous cases. -- **Interpretation:** Despite a slight centralization, there is still very little indication of a linear relationship with the target. - -### Test Feature 4 vs Target -- **Distribution:** The scatter begins to resemble a downward sloping line, indicating that values may decrease as Feature 4 increases. -- **Interpretation:** This could suggest a negative dependency of Feature 4 on the target, although it might not be strong. Correlation coefficients or more complex regression analyses could further explore this relationship. - -### Test Feature 5 vs Target -- **Distribution:** Points display a steeper upward trend compared to other features. -- **Interpretation:** There is a limited yet visible positive correlation, indicating that the target may increase slightly as Feature 5 increases. This feature might offer some predictive value when integrated into a model. - -### General Observations -- **Lack of Strong Linear Relationships:** None of the features display strong linear relationships with the target. This suggests that relationships, if present, might be non-linear. -- **Potential for Non-linear Analysis:** Given the lack of clear linear correlations, non-linear modeling techniques such as decision trees or neural networks might better capture complex patterns between features and the target. -- **Importance of Feature Engineering:** Considering the subtle trends observed in Features 4 and 5, feature engineering might help in revealing more detailed relationships, possibly enhancing model performance. - -These plots are crucial for the preliminary analysis, guiding further statistical testing and modeling approaches. Exploring these relationships more deeply could yield insights critical to developing effective predictive models. - - -## Test Feature 6 vs Target - -### Distribution -- The distribution is broad, similar to Features 1 and 2, with no discernible direction in the scatter of data points. - -### Interpretation -- There is no visible significant linear relationship, indicating that Feature 6 may not effectively predict the target using linear methods alone. - -## General Observations - -### No Strong Linear Relationships -- Across all features, there are no strong, clear linear relationships with the target. This suggests that the relationships, if present, might be non-linear, or the target is influenced by external factors or noise not directly captured by these features. - -### Potential for Non-linear Analysis -- Due to the absence of linear relationships, employing non-linear modeling techniques could be advantageous. Methods such as decision trees, random forests, or neural networks are capable of capturing more complex interactions between features and the target. - -### Feature Selection and Engineering -- Features 4 and 5 exhibit some patterns that may warrant further investigation. Engaging in feature engineering could help uncover more intricate relationships, like interactions between features or transformations that might better expose their relationships with the target. - -This comprehensive analysis underscores the need for a broader approach in modeling techniques to adequately predict the target variable. Such an approach should consider the non-linear dynamics of the dataset, along with a focused attempt at enhancing feature selection and engineering strategies to improve model accuracy and robustness. +### Visual Insights: +- For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. +- For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. +### Meaning of this graph: +These plots visualize the dataset to help understand our first dataset. +### Model fitting effect: ![alt text](3.png) -## Breakdown of Each Plot Analysis - -### Feature 1 vs. Actual and Predicted Target -- **Actual Values:** Represented by yellow dots. -- **Predicted Values:** Represented by blue dots. -- **Fit Line:** The red dashed line shows the regression line that represents the predicted values based on Feature 1. -- **Interpretation:** The fit line indicates moderate fitting, implying a certain degree of linear predictability from Feature 1 to the target. - -### Feature 2 vs. Actual and Predicted Target -- **Interpretation:** The spread of predicted values is wider, indicating more variance in the predictions relative to the actual values, suggesting less predictive accuracy from Feature 2. - -### Feature 3 vs. Actual and Predicted Target -- **Interpretation:** The predictions closely follow the actual values with the fit line approximating a good linear fit, suggesting that Feature 3 might have a strong predictive power regarding the target. - -### Feature 4 vs. Actual and Predicted Target -- **Interpretation:** The fit line indicates an inverse relationship, as the line slopes downward. This might suggest that as Feature 4 increases, the target decreases. - -### Feature 5 vs. Actual and Predicted Target -- **Interpretation:** Feature 5 shows a slightly poorer fitting compared to others, with the fit line sloping slightly, which might indicate only a weak predictive relationship. - -### Feature 6 vs. Actual and Predicted Target -- **Interpretation:** The predictions and actual values are somewhat aligned, but the spread is still notable. The fit line shows a moderate predictive relationship. - -## General Observations -- **Evaluation of Fit Lines:** The fit lines in these plots are crucial for understanding how well the model predicts the target based on each feature. A closer fit of the line to the actual values indicates a stronger predictive ability of the feature. -- **Comparison Across Features:** Some features (like Feature 3 and Feature 6) show a closer alignment between the predicted and actual values, indicating they might be more useful predictors of the target. -- **Predictive Power of Features:** The degree of scatter around the fit lines across different features illustrates the variability in the model's accuracy. Features with less scatter and closer fit lines are generally better at predicting the target. +### Image Description: +The figure above shows the effect of our model on the first test set based on the first training set, where the red line is the straight line fitted using the least squares method. The yellow points are true values and the blue points are predicted values. +It can be seen that the fitting effect of our model is very good at predicting the test level. +--- +### The second train dataset: ![alt text](4.png) ## Scatter Plot Analysis @@ -476,7 +389,7 @@ This comprehensive analysis underscores the need for a broader approach in model - **Importance of Advanced Statistical Techniques:** Techniques such as polynomial regression or machine learning algorithms like decision trees might be necessary to model these relationships more effectively. - +### The second test dataset: ![alt text](5.png) From 8b0adbc8244b68f9955d79f2c60a82e237dcde8b Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 10 Oct 2024 16:09:30 -0500 Subject: [PATCH 31/36] change half of the description of visualization --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index f32b33a..3cb9622 100644 --- a/README.md +++ b/README.md @@ -389,7 +389,7 @@ It can be seen that the fitting effect of our model is very good at predicting t - **Importance of Advanced Statistical Techniques:** Techniques such as polynomial regression or machine learning algorithms like decision trees might be necessary to model these relationships more effectively. -### The second test dataset: +### The test dataset: ![alt text](5.png) From 0f88cdd66d1892c2716b6181ef17620f46ae3203 Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Thu, 10 Oct 2024 16:19:51 -0500 Subject: [PATCH 32/36] fixed errors --- README.md | 342 ++++++++++++++++++++++++++++++++++++++++++------------ 1 file changed, 271 insertions(+), 71 deletions(-) diff --git a/README.md b/README.md index f874ad3..099f54c 100644 --- a/README.md +++ b/README.md @@ -69,23 +69,89 @@ model.fit(X_train, y_train) predictions = model.predict(X_test) ``` -# 1. Linear regression with ElasticNet regularization (combination of L2 And L1 regularization) +### `generate_negative_data` -# Q1.What does the model you have implemented do and when should it be used? +The `generate_negative_data` function generates synthetic data with two distinct patterns: -# ElasticNetModel Overview +1. The first half of the features has a **monotonic increasing trend**. +2. The second half of the features has a **linear decreasing trend** (i.e., negative slope). + +This function can be useful for testing model's ability in negative correlations data. + +#### Parameters: + +- **`range_x`** (_tuple_): The range of feature values (min, max). +- **`noise_scale`** (_float_): Standard deviation of the Gaussian noise added to the data. +- **`size`** (_int_): Number of samples in the dataset. +- **`num_features`** (_int_): Total number of features (half with increasing trend, half with decreasing trend). +- **`seed`** (_int_): Random seed for reproducibility. + +#### Returns: + +- **`X`** (_numpy array_): Generated feature matrix with both increasing and decreasing trends. +- **`y`** (_numpy array_): Target values with contributions from the features and added noise. + +#### Example Usage: + +```python +from generate_negative_regression_data import generate_negative_data + +# Generate negative slope data +X, y = generate_negative_data(range_x=(0, 10), noise_scale=1.0, size=100, num_features=6, seed=42) +``` + +--- + +### `generate_rotated_positive_data` + +The `generate_rotated_positive_data` function generates synthetic data with two patterns: + +1. The first half of the features follows a **monotonic increasing trend**. +2. The second half exhibits a **wavy (S-shaped) pattern**, adjusted by a rotation matrix to create a slanted shape. + +This function generates more complex data to test model's ability. + +#### Parameters: + +- **`range_x`** (_tuple_): Specifies the range of feature values (min, max). +- **`noise_scale`** (_float_): Standard deviation of the Gaussian noise added to the data. +- **`size`** (_int_): Number of samples in the dataset. +- **`num_features`** (_int_): Total number of features (half with increasing trend, half with a wavy pattern). +- **`seed`** (_int_): Random seed for reproducibility. +- **`rotation_angle`** (_float_, default=45): Angle (in degrees) to rotate the wavy pattern, introducing a slanted S-shape. +- **`mode`** (_int_, default=0): Determines the scaling factors for the feature values. + +#### Returns: + +- **`X`** (_numpy array_): Generated feature matrix with increasing trends and rotated S-shaped patterns. +- **`y`** (_numpy array_): Target values influenced by the features and noise. + +#### Example Usage: + +```python +from generate_positive_regression_data import generate_rotated_positive_data + +# Generate positive data with rotation and S-shaped curves +X, y = generate_rotated_positive_data(range_x=(0, 10), noise_scale=1.0, size=100, num_features=6, seed=42, rotation_angle=45, mode=0) +``` + +## 1. Linear regression with ElasticNet regularization (combination of L2 And L1 regularization) + +## Q1.What does the model you have implemented do and when should it be used? + +## ElasticNetModel Overview The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implement this model: ## Loss Function -The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows: +- **The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows** : -$\text{Loss} = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2 + \lambda_1 \sum_{j=1}^m |w_j| + \lambda_2 \sum_{j=1}^m w_j^2$ +$\text{Loss}$(w) = $\frac{1}{n}$ $\sum_{i=1}^{n}$ $(y_i - w^T x_i)^2$ + $\lambda$ ($\rho$ $\sum_{j=1}^{d}$ $|w_j|$ + (1 - $\rho$) $\sum_{j=1}^{d}$ $w_j^2$) ## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. -- **Gradient calculation results of MSE:**: +- **Gradient calculation results of MSE:** : $\nabla_{\text{MSE}} = -\frac{2}{n} X^T (\hat{y} - y)$ @@ -93,7 +159,7 @@ $\nabla_{\text{MSE}} = -\frac{2}{n} X^T (\hat{y} - y)$ $\nabla_{\text{L1}} = \lambda_1 \cdot \text{sign}(w)$ -- **Gradient calculation results of L2:**: +- **Gradient calculation results of L2:** : $\nabla_{\text{L2}} = 2 \cdot \lambda_2 \cdot w$ @@ -113,131 +179,265 @@ In summary, our model implements hybrid regularization by combining L1 and L2 pe ### What the Model Does: -- **Hybrid Regularization**: Elastic Net stands out because it uses a mix of L1 and L2 penalties. +- **Hybrid Regularization**: Out of all the classifiers Elastic Net is more advantageous because it uses a combination of L1 and L2 penalties. - - **L1 penalty** helps thin out the less essential features, essentially cleaning up the model to focus on what really matters, making it simpler and easier to understand. This is especially useful when you’re dealing with loads of potential predictors and want to avoid a model that’s too complex to interpret or too tailored to the training data. - - **L2 penalty** helps when features are intertwined, which is quite common. It does this by distributing the influence of these features more evenly, which helps keep the model stable and reliable. +- **L1 penalty** Reducing the effect of weak predictors makes the model purged of unnecessary items Skipping all the lesser significant variables makes the model slimmer and easily understandable and easy to interpret Especially when one has a large number of potential predictors often it is dreaded to end up with a bloated or overfit model. + **L2 penalty** This makes this approach more useful when the features are related in some form, which is very often the case, generally. This is because, when the influences are spread over various parameters of the model ; it ensures that the model can perform optimally in the long run. -- **Gradient Descent Optimization**: This technique is crucial for fine-tuning the model in big datasets where older methods might be too slow. It adjusts the model gradually and methodically to hone in on the best possible performance, based on the rate of learning and how precise we want the model to be. +- **Gradient Descent Optimization**: This technique is crucial when optimizing models in the cases where models are trained on large sets of data and older methods will slow down data processing. It does this in the sense that the different adjustments flow continuously and systematically so as to arrive at optimal value of the performance and hence its value is dependent on the learning rate and accuracy. ### When to Use This Model: -- **Broad Application Spectrum**: This model shines in any scenario where you need precise and sturdy predictions. It’s fantastic for fields like economics or healthcare, where the data and relationships can be quite complex. +- **Broad Application Spectrum**:It is on such aspects that this model actually shines when in areas where accurate and dependable forecasts are needed. It becomes particularly useful in such areas as economics or healthcare where the data may be very much complicated, and unraveling the relationships may well determine the difference between victory and defeat. -- **Managing Complexity and Ensuring Accuracy**: It’s perfect for diving into complicated, noisy data and coming out with clear insights. Whether it’s studying genes or predicting trends in big industries, this model helps manage a lot of information without losing focus. +- **Managing Complexity and Ensuring Accuracy**: It is great for getting to clean insights from messy, raucus data. Be it researching about human genetics or entire industry trends, this model enables to handle tonnes of information without diverting too much from the central theme. -The **ElasticNetModel** isn’t just about making predictions; it’s also about deeply understanding the data. It’s a powerful tool for anyone who needs to tackle complex data sets and derive meaningful conclusions, not just numbers. This model’s ability to simplify while handling complex relationships makes it invaluable for researchers and analysts who regularly face challenging modeling situations. +The **ElasticNetModel** It is an entire package where not only can you get to make predictions but also deep dive into the data. A weapon of productivity for anyone dealing with a large data set that needs to be analysed and read into, not just numbers. Its ability to generalize, while solving for intricate relationships among variables which are characteristic challenge settings of modellers and analysts; makes this model immensely useful as a statistical package. -# Q2.How did you test your model to determine if it is working reasonably correctly? +## Q2.How did you test your model to determine if it is working reasonably correctly? -## Model Testing and Validation Strategy +#### Model Testing and Validation Plan -To ensure our model is both robust and accurate, we’ve put a thorough testing and validation strategy into play. Let’s walk through each step of this strategy to see how it contributes to a full evaluation of the model’s performance: +For the purpose of the model’s strength and accuracy, we have developed an elaborate testing and validation plan. Let’s walk through each step of this strategy to see how it contributes to a full evaluation of the model’s performance: ### 1. Synthetic Data Generation -- **Method**: We created synthetic datasets using a method called `generate_rotated_s_wave_data` that clearly defines how features relate to target values. -- **Why It Matters**: This setup lets us dive deep into the model’s ability to recognize and learn the patterns we’ve set up. By working with synthetic data, where the outcomes are already known, we can rigorously test how well the model mimics these set relationships, giving us a clear picture of its learning precision. +- **Method**: To illustrate how features are related to target values, we generated so called synthetic data supported by the function `generate_rotated_s_wave_data`. +- **Why It Matters**: This setup allows reducing the amount of noise and a closer look at its capacity to read and learn the patterns we established. It means that when working with synthetic data which contains already known outcomes the degree of the model’s ability to reproduce these relationships can be assessed as an indicator of learning precision. ### 2. Training on Generated Data -- **What We Did**: The model spent a good deal of time learning from these synthetic datasets. During this phase, we kept a close eye on its parameters, checking how well the learned coefficients and intercepts matched the expected patterns. -- **Why It Matters**: This step is crucial for confirming that the model understands the underlying data dynamics. It helps validate that the model’s foundation is solid and that it’s tuned to respond appropriately to the nuances it will face in real scenarios. +- **What We Did**: The model used a lot of time studying these synthetic datasets. : The following figures show the parameters of the fitted coefficients and intercepts of the tested article in this phase, correlating them with the expected patterns. +- **Why It Matters**: This can be done by executing the step that ensures that the model has comprehension of the dynamics in the data set. It assists in affirming to a degree that what the model is based on is sound as well as ensuring that it is sensitive to how it will react to the subtleties it is like to encounter in actuality. ### 3. Predictions on Unseen Data -- **Method**: We tested the model’s predictions using a new set of data that it hadn’t seen during training. -- **Why It Matters**: This is key for assessing how well the model can apply what it’s learned to new, unknown datasets—a must for ensuring the model is genuinely useful in real-world settings where data variability is the norm. +- **Method**: To check how well the model is doing, we fed it with data it did not learn with during the learning stage. +- **Why It Matters**: This is essential for evaluating how good the model is in using the learning to solve other unseen dataset – which is very important in ascertaining the utility of the model in real world where dataset variation is the order of the day. ### 4. Model Evaluation Metrics -- **Metrics Used**: We measured the model’s accuracy and reliability using several standard regression metrics like: +- **Metrics Used**: To check the accuracy and reliability of the carried out predictions we used several standard regression parameters such as: - Mean Squared Error (MSE) - - Root Mean Squared Error (RMSE) + Linear – Root Mean Squared Error (RMSE) - Mean Absolute Error (MAE) - R-squared (R²) -- **Why They’re Important**: These metrics provide hard numbers that tell us how accurately the model predicts new data. They help us understand different aspects of performance, such as the average size of errors and how much of the variance in our target variable the model can explain. +- **Why They’re Important**: These measure give you actual numbers to explain the degree at which the model employed is capable of predicting new data. They assist us in getting an understanding of other aspects of performance including the average magnitude of the errors and variance in the target variable accounted for by the model. -### 5. Visual Comparisons of Predicted vs. Actual Values +### 5. Probability versus Outcome Plots -- **How We Do It**: We use scatter plots where each point shows a predicted value against its actual value. The closer these points are to the diagonal line (where predicted equals actual), the better the model’s predictions. -- **The Benefits**: These visuals offer an instant, gut-level indication of the model’s accuracy. They’re particularly good for spotting whether the model tends to over or underestimate across the board (systematic bias) or if it’s inconsistent (high variance). They also let us see at a glance if some features yield more reliable predictions than others. +- **How We Do It**: We employ residual plots whereby each point represents a predicted value against its actual value. The closer that the observed points are to the 45-degree line, the better are the predictive capabilities of the model in question. +- **The Benefits**: These visuals provide an immediate, tangiblerefreshing ofits gross approximation. They work especially well when used to discern whether the model bolts general or is inconsistent (high variation, systematic error). They also allow us to visualize readily whether some of the features provide better prediction accuracy than others. ### 6. Analysis and Iteration -- **Ongoing Process**: Based on what we learn from the visual data and other metrics, we continuously refine the model. This might mean tweaking the model’s settings to better align predictions with actual results, especially where we notice gaps. -- **Why Keep At It**: These refinements help the model adapt to the subtleties of new data, ensuring it stays relevant and effective. This ongoing cycle of feedback and improvement is vital for the model to remain at the top of its game as it encounters new challenges. +- **Ongoing Process**: Subsequently, the model is improved based on visual data and other collected items of measure. Such adjustments may entails the fine tuning of the parameters of the model so that predictions match actual outcomes particularly in the areas we identify discrepancies. +- **Why Keep At It**: These refinements assist in acquiring significant details in the new data patterns in order for the model to provide great results. This feedback and improvement cycle is essential to keep the model on the top of its form as new issues present themselves in the field. #### Insights from Graphs: -Observing how “Feature 2 vs. Target” shows a tighter cluster around the diagonal than “Feature 6 vs. Target” suggests Feature 2 is being handled more precisely. This might lead us to look into whether Feature 6 needs more preprocessing or whether the model should employ advanced feature engineering strategies to better capture complex data patterns. +Comparing the plot for “Feature 6 vs. Target” to the plot for “Feature 2 vs. Target” where despite the disperse nature of the former the latter seems to be more compact near the diagonal it is possible to conclude that Feature 2 is managed more accurately. This can put us in a position to ask ourselves whether Feature 6 should apply more data pre-processing or whether the model should apply more complex feature engineering methods to capture the fine relative data detail. -By weaving together these detailed testing methods, we not only affirm that the model is theoretically sound but also ensure it’s equipped to perform reliably in a range of real-world applications. This rigorous validation and refinement process is crucial for building top-quality predictive models that can rise to new challenges and deliver dependable results. +By combining these specific types of testing, not only do we guarantee to ourselves that the model is mathematically correct, but we also show that it can be used in practice in as many application fields as possible. This stringent check and balance system is well suited for creating accurate high quality predictive models that can climb even more for reliability. -# Q3.What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) +## Q3.What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) ## Model Optimization Parameters -We gave some parameters to change in order to improve the performance of our model. This section is a breakdown of each parameter and what each does/means, as well as touches on how tuning them could lead to improved model performance: - -1. **Lambdas** +## Lambdas in Regularization - - **Function:** Lambdas are the key elements to the regularization strategies of our model, which we utilized in both L1 (Lasso) and L2 (Ridge) techniques. This alleviates a common overfitting problem in which the model learns noise as if it were real data signals. - - **Effect of Regularization:** When the value of lambdas is increased, it increases the penalty on coefficients so that they are constrained to be near zero. This adjustment is beneficial when dealing with complex datasets that could potentially lead to misleading noise. - - **When we use it:** When we observe our model is overfitting with training data, we then increase lambdas to rebuild our model in a much generalized way to new, unseen data. +**Function:** +The lambdas are crucial in regularization methods and finding the optimal one is recommended for using in Lasso (L1) or Ridge (L2) regularization. These techniques are somewhat crucial for avoiding overfitting since a model can learn most of the noise in data set as a signal. -2. **Thresh** +**Effect of Regularization:** +raising the value of lambda leads to increase the amount the penalizes the coefficients of the model and thus reduces them towards zero but not to zero all the same. This mechanism is particularly beneficial for noisy large datasets where the distribution of data may vary thus leading to the generation of interpretive patterns that are misleading. - - **Operation:** Thresh For: Feature Selection Description: It creates a cutoff for feature selection. It simplifies the model by only considering a few significant features. A higher thresh value can remove less useful features by nulling their coefficients and it will improve the model's interpretability on new data. - - **Where We Use It:** Thresh is used when there were a lot of features in our dataset and we do not want to perform the data modeling on all those features (so using thresh makes sure that spearmake calculation applied only top x number of feature). +**When to Use:** +If a model is making sharp learning from data — infact picking up noise as the real signals, then making enhancements to the lambda values can be a good strategy. It does so while also tempering the learning of the model and its corresponding generalization to unseen data of any kind. -3. **Max_iter** +- **Use Case:** + Thresh is most effective whenever a certain model is overly fitting the training data set. Finally, by making thresh larger, it improves the generalization capability of the model over new, never seen before data. - - **Function:** It decides the maximum loops (number of iterations) our algorithm performs before stopping. - - **Adjustment Impact:** The more iterations the model adjusts its weights with the data, the better it can learn and potentially yield higher accuracy. - - **Why We May Use It:** If our model has not yet been optimized, it may require an increase in max_iter. But, it may potentially take more training time which means a tradeoff between accuracy and speed. +### 2. Thresh -4. **Tol** +- **Function:** + Thresh decides the elimination point for feature selection after filtering in, shaving the model down to select characteristics that are necessary and excluding others that may distort. - - **Function:** Tol — that is, tolerance — how closely does the model have to fit the training data trends before we stop adjusting further. - - **Impact of Tuning:** Lowering the value means the model won't stop learning until it achieves a closer fit, this can increase accuracy but would also prolong training time. - - **Where We Use This:** This is most useful when high precision matters a lot. However, making the tolerance too tight can result in diminishing returns, especially if more training leads to virtually no progress with prolonged training. +- **Adjustment Effect:** + Raising the thresh value may cause getting rid of secondary variables as their coefficients become zeroes. Such a reduction in the model leads to less accurate results but occasionally, the results are expected to be better for unseen data. -5. **Learning_rate** - - **Function:** In gradient descent optimization, this parameter helps us to set the step size the model takes towards reducing the loss function. - - **Implication of Adjustment:** Lower learning rates give smaller and safer steps that might help in more finesse model optimization. - - **Where We Use It:** If the model reaches minimum loss and is going beyond it i.e., overshooting, by decreasing the learning rate we can make sure it converges more stably and accurately to its minima. However, if the rate is too low, it will slow the training process in an overly long way. +- **Where To Use It:** + Thresh is used to control the measure of variable importance, allowing identifying important features, and at the same time, excluding an excessive number of predictors that have no beneficial impact or, on the contrary, may negatively affect the model’s ability to make adequate predictions. This is more applicable especially where there are many features more than the dependent variables. -We can improve this model by tinkering with these parameters to help the model adapt well to the nuances of our data and the requirements of our task, ensuring that we have a good trade-off between accuracy, efficiency of computation, and model lean size. +### 3. Max_iter -# Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? +- **Function**: This parameter tells our algorithm to stop after running for the permitted number of iterations as stated below. +- **Impact of Adjustment**: Increasing the number of epochs may provide better chances for the model to adjust its weights with respect to data for better performance. +- **When We Use It**: If the model is not in its best condition yet, then it may be required to increase the `max_iter`. However, it is possible to have the model take more time for a training session, which calls for compromise between precision and time taken on the training session. -True, there are certain inputs that the implementation finds difficult working with: +### 4. Tol -## 1. Data Dependencies (Highly Nonlinear Data): +- **Function**: This is the same as tolerance; it defines a small the error which must fit our training data before further adjustments are made to stop.r adjustments stop. +- **Impact of Adjustment**: Lowering the `tol` value ensures the model won’t stop learning until the fit is very tight, improving accuracy but extending training time. +- **When We Use It**: If rate of accuracy must be in the consideration. The most important one is that if you set the tolerance level too low then really long training can actually make no gain at all. -- **The Challenge**: Elastic Net has an unspoiled ability to see a linear world, plots each point in the n-dimensional Euclidean space, and captures the straight line that passes near as many points as possible. The information might express intricate, curved relationships that the model finds difficult to learn. +### 5. Learning_rate -- **What Happens**: If the problem consists of complex patterns among variables, Elastic Net will fail to accurately predict target values due to its inability to model the variability and flexibility. +- **Function**: This hyperparameter controls by how much the model should change towards the optimization in the updating of the model. +- **Impact of Adjustment**: It reduces the steps taken towards implementation and is much slower but more precise, because it is a smaller learning rate and its implementations are more careful. +- **When We Use It**: However, if the model is over the point that represents the minimal loss, it means that the learning rate is oversized, thus should be decreased to improve the stability and accuracy of the convergence. Nonetheless, the rate of learning may be reduced to an extreme low in the presence of a low rate. -- **Real Feature Engineering**: To capture the latent driving forces more effectively, we can manipulate input features to provide the model with a better perspective of the underlying patterns. This may involve creating new features by squaring or cubing the original variables, introducing interaction terms between variables, or using mathematical transformations such as logarithms or exponentials to linearize some nonlinear relationships. +These two parameters, therefore, can be tuned to optimize accuracy for given data distribution and complexity of task, computational cost, and model complexity. -- **Kernel Methods**: Borrowing a trick from support vector machines, kernel methods can help by projecting data into a space where once-complex relationships look linear enough for Elastic Net to perform effectively. +## Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? -- **Hybrid Models**: Sometimes combining two models can be better than one. However, hybrid models have their own challenges, such as limited variance in terms of bias and variance, which Elastic Net aims to address. We can combine tree models or neural networks with Elastic Net to improve prediction across a variety of scenarios. +## Specific Inputs That May Challenge the ElasticNet Model -## 2. Noisy Data: +### 1. Highly Nonlinear Data -- **The Challenge**: Noisy data is like trying to listen to a quiet radio station during a thunderstorm. Noise can drown out the true patterns, making it hard for Elastic Net to capture what's important. +- **The Challenge**: This is because Elastic Net has an Optimistic World View, it believes that the world can be described using straight lines and linearly separable features. But life is rarely that clean or bumpy, and data exhibits the kind of curved relationships that a model like biases against. +- **What Happens**: Elastic Net might miss with its predictions in these instances since they involve some of the sharp edges/turns and complex patterns that it cannot fully understand. + - **Feature Engineering**: One way to make the model learn the underlying patterns is by transforming those input features. This could involve squaring or cubing the features, maybe adding in interaction terms, possibly doing mathematical transformations (like logarithms or exponentials) to help straighten out those nonlinear relationships. + - **Kernel Methods**: Taking a cue from support vector machine, these can be kernelized as well. + - **Hybrid Models**:Two heads (or models) as well. We can combine the advantages of Elastic Net when teamwork with nonlinear models like decision trees or neural networks, which allows it to predict well in wider set of scenarios. -- **Example**: If the data is too noisy, Elastic Net may learn the noise along with the true signal, leading to overfitting on the training set and poor performance in the real world. +### 2. Noisy Data +- **The Challenge**:Think of your data as listening to one of those quiet FM radio stations during a thunderstorm, that is what this is like. This then becomes static (noise) that can drown the music (true patterns), and thus it makes it hard for the Elastic Net to hear what is important. +- **What Happens**:If the data is too noisy, Elastic Net might fit to the noise instead of the music and learn something that works great on training data but will falter in practice. - **What We Can Do**: - - **Increase Regularization**: Turning up the lambda parameter increases the penalties for model complexity, forcing Elastic Net to focus on the main signal and ignore noise. + - **Increase Regularization**:we have the option here to turn up the lambdas knob and increase penalties for complexity, allowing our model to focus in on every loud symptomatic signal while ignoring that static. + - **Threshold Tuning**: Changing the thresh parameter to allow us to be more selective about which features we let into the model, and cut out those that tend to contain more noise than signal. + + - **Noise Reduction Techniques**: Even before feeding data into our model, we can clean it up to filter out the oddball values or make some signal smoother so that what we pass on to our mod is as clean and clear as possible. + +By acknowledging these limitations and using smart strategies to overcome them, we can increase the potential of Elastic Net because another graphical preferred in predictive modelling. + +- ## Visualization + +![alt text](1.png) + +### Scatter Plots of Train Features vs Target + +This image shows a series of scatter plots, each representing the relationship between a different feature (Feature 1 to Feature 6) and a target variable (Y_train) based on training data. Each plot displays the data points in two dimensions: + +- The horizontal axis (X-axis) shows values of a specific feature. +- The vertical axis (Y-axis) represents the values of Y_train. + +The purpose of these plots is typically to visually assess the relationship between the features and the target variable. Observations can be made regarding the distribution, trend, or any potential outliers in the data. From the plots: + +- **Feature 1, Feature 2, Feature 3:** These features show some vertical dispersion but no clear linear or nonlinear trend with Y_train. +- **Feature 4, Feature 5, Feature 6:** Similar to the first three, these features also display a scattered distribution with respect to Y_train, without a visible specific pattern. + +Overall, these scatter plots can help in determining which features might be relevant for modeling, although none of the features shown here have a distinct or strong linear relationship with the target variable. + +![alt text](2.png) + +### Overview of Each Feature Plot Analysis + +The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against a target variable (labeled as "Y_Test"). Each subplot represents the relationship between a single feature and the target variable. + +### Breakdown of Scatter Plots: + +- **Test Feature 1 through 6**: Each scatter plot represents the feature's values on the x-axis and the corresponding target variable values on the y-axis. +- **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. + +### Visual Insights: + +- For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. +- For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. + +### Uses: + +- **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). +- **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. + +![alt text](3.png) + +### Breakdown of Each Plot Analysis + +The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against a target variable (labeled as "Y_Test"). Each subplot represents the relationship between a single feature and the target variable. + +### Breakdown of Scatter Plots: + +- **Test Feature 1 through 6**: Each scatter plot represents the feature's values on the x-axis and the corresponding target variable values on the y-axis. +- **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. + +### Visual Insights: + +- For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. +- For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. + +### Uses: + +- **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). +- **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. + +![alt text](4.png) + +### Scatter Plot Analysis + +The image shows scatter plots of six features (Feature 1 through Feature 6) plotted against the target variable labeled "Y_Train" on the y-axis. The x-axis in each subplot corresponds to the respective feature values. + +### Key Observations: + +1. **Feature 1 to Feature 3**: + + - All three features show a strong positive linear correlation with the target variable (Y_Train). + - The points align closely along a positively sloped line, indicating that as these feature values increase, the target variable increases in a linear fashion. + +2. **Feature 4 to Feature 6**: + - These features show a strong negative linear correlation with the target variable. + - The points align closely along a negatively sloped line, suggesting that as these feature values increase, the target variable decreases. + +### General Insights: + +- All six features display clear linear relationships with the target, either positive (Feature 1 to 3) or negative (Feature 4 to 6). +- These patterns suggest that linear + +![alt text](5.png) + +### Scatter Plot Analysis for Test Features vs. Target + +The image shows scatter plots of six features (Test Feature 1 through Test Feature 6) plotted against the target variable labeled "Y_Test." Each subplot represents the relationship between a specific feature (on the x-axis) and the target variable (on the y-axis). + +### Key Observations: + +1. **Feature 1 to Feature 6**: + - Each scatter plot shows a wide spread of points without a clear pattern, suggesting weak or no strong linear relationships between the features and the target variable. + - The lack of clustering along a line indicates that none of the features exhibit a strong direct correlation with the target. +2. **Feature 5**: + + - There appears to be a slight downward trend, indicating a weak negative correlation between Feature 5 and the target (Y_Test). + +3. **Feature 6**: + - There is some clustering in the lower right quadrant, possibly indicating a negative influence on the target, but the scatter remains wide. + +### General Insights: + +- None of the features show a strong, easily identifiable linear correlation with the target variable, as the data points are dispersed across all plots. +- This suggests that more complex modeling techniques (e.g., non-linear methods) or additional feature engineering may be needed to capture meaningful patterns in the data. + +![alt text](6.png) + +### Regression Analysis for Features vs. Actual and Predicted Target Values + +The image shows six scatter plots comparing the actual target values and the predicted target values for six features (Feature 1 through Feature 6) against the target variable labeled "Y." Each subplot also includes a red dashed line representing the linear fit for each feature. + +### Key Observations: + +1. **Feature 1 to Feature 3**: + + - The fit lines have positive slopes, suggesting a weak positive relationship between these features and the target variable. + - The actual and predicted values are dispersed around the fit line with noticeable variance, indicating that the model's predictions are not tightly aligned with the actual values. The fit is somewhat loose, and the features do not have a strong linear relationship with the target. - - **Filter**: Adjusting the threshold parameter can make the model more or less strict about what signals pass through, helping to filter out noise. +2. **Feature 4 to Feature 6**: + - The fit lines have negative slopes, indicating a negative relationship between these features and the target variable. + - Similar to Features 1-3, there is substantial dispersion of the actual and predicted values around the fit line, suggesting weak predictive power and variability in the model's predictions for these features. - - **Reduction of Noise**: We can filter out outliers or smooth signals before feeding the data into the model. +### General Insights: -This post aims to understand these limitations and the smart ways to get around them, solidifying Elastic Net as an even more powerful tool in predictive modeling. +- All six features exhibit weak linear relationships with the target variable, as evidenced by the scattered points around the fit lines. The model does not capture the target values well, and the fit is not strong. +- **Features 4, 5, and 6** show negative relationships, meaning that as the feature values increase, the target decreases, though this is not strong. +- These scatter plots indicate that the model could benefit from more advanced techniques, as simple linear regression appears insufficient for accurately predicting the target values for these features. Non-linear modeling methods or additional feature engineering might improve the model's performance. From 8575fbc4fa79b25f40f0d82099a90a5827497063 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 10 Oct 2024 16:28:54 -0500 Subject: [PATCH 33/36] return old version --- README.md | 41 +++++++++++++---------------------------- 1 file changed, 13 insertions(+), 28 deletions(-) diff --git a/README.md b/README.md index 3cb9622..9ff771e 100644 --- a/README.md +++ b/README.md @@ -431,39 +431,24 @@ It can be seen that the fitting effect of our model is very good at predicting t ![alt text](6.png) -## Regression Analysis for Features vs. Actual and Predicted Target Values +### Regression Analysis for Features vs. Actual and Predicted Target Values -### Feature 1 vs. Actual and Predicted Target -- **Actual Values:** Represented by yellow dots. -- **Predicted Values:** Represented by blue dots. -- **Fit Line:** Red dashed line indicating the regression model's predictions. -- **Interpretation:** The close alignment of the fit line with both actual and predicted values suggests that Feature 1 has a strong linear correlation with the target, making it a potentially strong predictor. +The image shows six scatter plots comparing the actual target values and the predicted target values for six features (Feature 1 through Feature 6) against the target variable labeled "Y." Each subplot also includes a red dashed line representing the linear fit for each feature. -### Feature 2 vs. Actual and Predicted Target -- **Interpretation:** This plot shows a tight correlation with a linear progression similar to Feature 1, indicating effective predictive capabilities of Feature 2 with a robust linear relationship. +### Key Observations: -### Feature 3 vs. Actual and Predicted Target -- **Interpretation:** Despite the wide range of target values, the fit line closely follows the trend in actual and predicted values, suggesting that Feature 3 strongly influences the target, albeit with some potential outliers or high variance. +1. *Feature 1 to Feature 3*: + - The fit lines have positive slopes, suggesting a weak positive relationship between these features and the target variable. + - The actual and predicted values are dispersed around the fit line with noticeable variance, indicating that the model's predictions are not tightly aligned with the actual values. The fit is somewhat loose, and the features do not have a strong linear relationship with the target. -### Feature 4 vs. Actual and Predicted Target -- **Interpretation:** The downward slope of the fit line suggests an inverse relationship between Feature 4 and the target, where increasing values of Feature 4 correlate with decreasing target values. This inverse relationship might require further analysis to understand causal factors. - -### Feature 5 vs. Actual and Predicted Target -- **Interpretation:** Feature 5 shows a downward trend similar to Feature 4, indicating a negative correlation. However, the scatter is somewhat more dispersed, suggesting less consistency in the predictive accuracy compared to other features. - -### Feature 6 vs. Actual and Predicted Target -- **Interpretation:** The fit line demonstrates an inverse relationship, similar to Features 4 and 5, but with a tighter clustering of points around the line, enhancing its predictive validity. - -## General Observations -- **Predictive Strength:** Features 1, 2, and 3 display strong linear correlations with the target, indicating high predictive strength. -- **Inverse Relationships:** Features 4, 5, and 6 exhibit inverse relationships, which are consistently evident from their respective plots. -- **Model Performance:** The overall alignment of the fit lines across all features suggests that the regression model generally performs well, capturing the primary trends in the dataset effectively. - -## General Observations -- **Trends and Relationships:** These scatter plots generally do not show strong linear correlations, indicating the potential necessity for non-linear modeling approaches to capture the relationships between features and the target. -- **Predictive Insights:** The variability and patterns observed suggest that some features might have more influence on the target, possibly in non-obvious or non-linear ways. -- **Implications for Modeling:** The lack of strong linear trends across all features reinforces the need for sophisticated analytical techniques, such as machine learning algorithms that can handle non-linearity and complex interactions more effectively. +2. *Feature 4 to Feature 6*: + - The fit lines have negative slopes, indicating a negative relationship between these features and the target variable. + - Similar to Features 1-3, there is substantial dispersion of the actual and predicted values around the fit line, suggesting weak predictive power and variability in the model's predictions for these features. +### General Insights: +- All six features exhibit weak linear relationships with the target variable, as evidenced by the scattered points around the fit lines. The model does not capture the target values well, and the fit is not strong. +- *Features 4, 5, and 6* show negative relationships, meaning that as the feature values increase, the target decreases, though this is not strong. +- These scatter plots indicate that the model could benefit from more advanced techniques, as simple linear regression appears insufficient for accurately predicting the target values for these features. Non-linear modeling methods or additional feature engineering might improve the model's performance. From 5bac0c74d7b3243e7f42a7155242f49f73292e59 Mon Sep 17 00:00:00 2001 From: notjiang <128391769+Hj006@users.noreply.github.com> Date: Thu, 10 Oct 2024 17:05:43 -0500 Subject: [PATCH 34/36] change the description of visualazing --- README.md | 81 ++++++++++++++++++------------------------------------- 1 file changed, 26 insertions(+), 55 deletions(-) diff --git a/README.md b/README.md index 4251fff..1e5be96 100644 --- a/README.md +++ b/README.md @@ -316,7 +316,7 @@ By acknowledging these limitations and using smart strategies to overcome them, - ## Visualization -### The first train dataset: +### The First Train Dataset: ![alt text](1.png) ### Scatter Plots of Train Features @@ -332,7 +332,7 @@ The purpose of these plots is typically to visually assess the relationship betw Overall, these scatter plots can help in determining which features might be relevant for modeling, although none of the features shown here have a distinct or strong linear relationship with the target variable. -### The first test dataset: +### The First Test dDataset: ![alt text](2.png) ### Overview of Each Feature Plot Analysis @@ -340,87 +340,58 @@ Overall, these scatter plots can help in determining which features might be rel The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against "Y_Test". Each subplot represents the data of one specific demension of feature. -### Meaning of this graph: -These plots visualize the dataset to help understand our first dataset. +### Meaning of This Graph: +These plots visualize the dataset to help understand our first test dataset. +This test dataset is used to test the model's predictive ability on partially linear positively correlated data and data without obvious correlation. - -### Model fitting effect: +### Model Fitting Effect on first dataset: ![alt text](3.png) -### Image Description: -The figure above shows the effect of our model on the first test set based on the first training set, where the red line is the straight line fitted using the least squares method. The yellow points are true values and the blue points are predicted values. -It can be seen that the fitting effect of our model is very good at predicting the test level. - - +### Model Fitting Effect Image Description: +The figure above shows the effect of our model on the first test dataset based on the first training dataset, where the red line is the straight line fitted using the least squares method. The yellow points are true values and the blue points are predicted values. +It can be seen that the fitting effect of our model is very good on the first dataset. -### The second train dataset: +### The Second Train Dataset: ![alt text](4.png) ### Scatter Plot Analysis -The image shows scatter plots of six features (Feature 1 through Feature 6) plotted against the target variable labeled "Y_Train" on the y-axis. The x-axis in each subplot corresponds to the respective feature values. +The image shows scatter plots of six features (Feature 1 through Feature 6) plotted against "Y_Train" on the y-axis. The x-axis in each subplot corresponds to the respective feature values. ### Key Observations: 1. **Feature 1 to Feature 3**: - - All three features show a strong positive linear correlation with the target variable (Y_Train). + - All three features show a strong positive linear correlation. - The points align closely along a positively sloped line, indicating that as these feature values increase, the target variable increases in a linear fashion. 2. **Feature 4 to Feature 6**: - - These features show a strong negative linear correlation with the target variable. + - These features show a strong negative linear correlation. - The points align closely along a negatively sloped line, suggesting that as these feature values increase, the target variable decreases. -### General Insights: -- All six features display clear linear relationships with the target, either positive (Feature 1 to 3) or negative (Feature 4 to 6). -- These patterns suggest that linear -### The test dataset: +### The Second test dataset: ![alt text](5.png) -### Scatter Plot Analysis for Test Features vs. Target +### Scatter Plot Analysis for Test Features -The image shows scatter plots of six features (Test Feature 1 through Test Feature 6) plotted against the target variable labeled "Y_Test." Each subplot represents the relationship between a specific feature (on the x-axis) and the target variable (on the y-axis). - -### Key Observations: - -1. **Feature 1 to Feature 6**: - - Each scatter plot shows a wide spread of points without a clear pattern, suggesting weak or no strong linear relationships between the features and the target variable. - - The lack of clustering along a line indicates that none of the features exhibit a strong direct correlation with the target. - -2. **Feature 5**: - - There appears to be a slight downward trend, indicating a weak negative correlation between Feature 5 and the target (Y_Test). - -3. **Feature 6**: - - There is some clustering in the lower right quadrant, possibly indicating a negative influence on the target, but the scatter remains wide. +The image shows scatter plots of six features (Test Feature 1 through Test Feature 6) plotted against "Y_Test." Each subplot represents the data of one specific demension of feature. ### General Insights: -- None of the features show a strong, easily identifiable linear correlation with the target variable, as the data points are dispersed across all plots. -- This suggests that more complex modeling techniques (e.g., non-linear methods) or additional feature engineering may be needed to capture meaningful patterns in the data. - - +- The data of the first dimension are scattered in all the plots and do not show obvious characteristics. +- The data of the second and third dimensions show insignificant positive correlation. +- The data of the last three dimensions show insignificant negative correlation. +### Meaning of This Graph: +These plots visualize the dataset to help understand our second test dataset. +This test dataset is used to test the model's predictive ability on partially linear positively correlated data and partially linear negatively correlated data. +### Model Fitting Effect on second dataset: ![alt text](6.png) -### Regression Analysis for Features vs. Actual and Predicted Target Values - -The image shows six scatter plots comparing the actual target values and the predicted target values for six features (Feature 1 through Feature 6) against the target variable labeled "Y." Each subplot also includes a red dashed line representing the linear fit for each feature. - -### Key Observations: - -1. *Feature 1 to Feature 3*: - - The fit lines have positive slopes, suggesting a weak positive relationship between these features and the target variable. - - The actual and predicted values are dispersed around the fit line with noticeable variance, indicating that the model's predictions are not tightly aligned with the actual values. The fit is somewhat loose, and the features do not have a strong linear relationship with the target. - -2. *Feature 4 to Feature 6*: - - The fit lines have negative slopes, indicating a negative relationship between these features and the target variable. - - Similar to Features 1-3, there is substantial dispersion of the actual and predicted values around the fit line, suggesting weak predictive power and variability in the model's predictions for these features. - -### General Insights: -- All six features exhibit weak linear relationships with the target variable, as evidenced by the scattered points around the fit lines. The model does not capture the target values well, and the fit is not strong. -- *Features 4, 5, and 6* show negative relationships, meaning that as the feature values increase, the target decreases, though this is not strong. -- These scatter plots indicate that the model could benefit from more advanced techniques, as simple linear regression appears insufficient for accurately predicting the target values for these features. Non-linear modeling methods or additional feature engineering might improve the model's performance. \ No newline at end of file +### Model Fitting Effect Image Description: +The figure above shows the effect of our model on the second test dataset based on the second training dataset, where the red line is the straight line fitted using the least squares method. The yellow points are true values and the blue points are predicted values. +It can be seen that the fitting effect of our model is very good on linear positive correlation and linear negative correlation data sets. From f616c7011563a3142d2726743a3517b09f035830 Mon Sep 17 00:00:00 2001 From: Purnesh S V Date: Thu, 10 Oct 2024 18:19:03 -0500 Subject: [PATCH 35/36] Fixed case of README --- README.md => readme.md | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename README.md => readme.md (100%) diff --git a/README.md b/readme.md similarity index 100% rename from README.md rename to readme.md From cee9bfd66f6b0780d42e25471b97b84f2b980f9b Mon Sep 17 00:00:00 2001 From: DhyanVGowda Date: Thu, 10 Oct 2024 19:01:55 -0500 Subject: [PATCH 36/36] Final changes --- README.md | 254 ++++++++++++++++++++++-------------------------------- 1 file changed, 104 insertions(+), 150 deletions(-) diff --git a/README.md b/README.md index 099f54c..4e176d5 100644 --- a/README.md +++ b/README.md @@ -139,17 +139,17 @@ X, y = generate_rotated_positive_data(range_x=(0, 10), noise_scale=1.0, size=100 ## Q1.What does the model you have implemented do and when should it be used? -## ElasticNetModel Overview +### ElasticNetModel Overview -The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implement this model: +The model we've crafted, called **ElasticNetModel**, is an implementation of Elastic Net regression. What makes it special is that it not only uses MSE as the loss function, but also combines L1 (Lasso) and L2 (Ridge) regularization techniques. Here's a clearer view on how we implemented this model: -## Loss Function +### Loss Function - **The loss function consists of the MSE loss, L1 loss, and L2 loss. The loss function is shown as follows** : $\text{Loss}$(w) = $\frac{1}{n}$ $\sum_{i=1}^{n}$ $(y_i - w^T x_i)^2$ + $\lambda$ ($\rho$ $\sum_{j=1}^{d}$ $|w_j|$ + (1 - $\rho$) $\sum_{j=1}^{d}$ $w_j^2$) -## Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. +### Then use the gradient descent algorithm to calculate the gradient of the loss function and then update. - **Gradient calculation results of MSE:** : @@ -179,265 +179,219 @@ In summary, our model implements hybrid regularization by combining L1 and L2 pe ### What the Model Does: -- **Hybrid Regularization**: Out of all the classifiers Elastic Net is more advantageous because it uses a combination of L1 and L2 penalties. - -- **L1 penalty** Reducing the effect of weak predictors makes the model purged of unnecessary items Skipping all the lesser significant variables makes the model slimmer and easily understandable and easy to interpret Especially when one has a large number of potential predictors often it is dreaded to end up with a bloated or overfit model. - **L2 penalty** This makes this approach more useful when the features are related in some form, which is very often the case, generally. This is because, when the influences are spread over various parameters of the model ; it ensures that the model can perform optimally in the long run. +_Hybrid Regularization:_ Elastic Net is unique because it utilizes both L1 and L2 penalties. The L1 penalty naturally eliminates lower importance variables, which essentially refines the model to focus on more important aspects, thus simplifying feature selection and interpretation. This is especially helpful when you have tons of potential predictors and want to keep a model that is not overfitted or one that cannot be interpreted. The L2 penalty is also useful when variables are correlated with each other, which happens often. This will occur by spreading the influence of those features more broadly so that the model stays stable and reliable. -- **Gradient Descent Optimization**: This technique is crucial when optimizing models in the cases where models are trained on large sets of data and older methods will slow down data processing. It does this in the sense that the different adjustments flow continuously and systematically so as to arrive at optimal value of the performance and hence its value is dependent on the learning rate and accuracy. +_Gradient Descent Optimization:_ This is very useful in large datasets to optimize the model, where other methods would take too long. It slowly and systematically changes the model to find the perfect performance, based on how quick it is learning and how accurate we want our model. ### When to Use This Model: -- **Broad Application Spectrum**:It is on such aspects that this model actually shines when in areas where accurate and dependable forecasts are needed. It becomes particularly useful in such areas as economics or healthcare where the data may be very much complicated, and unraveling the relationships may well determine the difference between victory and defeat. +_Broad Application Spectrum:_ It works best in any scenario requiring accurate and tough predictions. Great for things like Economics or Healthcare where the data and relationships can be pretty complex. + +_Handling Complexity and Ensuring Accuracy:_ Great for understanding messy, complex data. From gene-centric analysis in the field of genomics to making sense of large-scale trends in powerful industries, this model is a way to juggle high-volume information without losing focus. -- **Managing Complexity and Ensuring Accuracy**: It is great for getting to clean insights from messy, raucus data. Be it researching about human genetics or entire industry trends, this model enables to handle tonnes of information without diverting too much from the central theme. +_Real-World Use Cases:_ In real estate, it can be used for predicting prices based on several factors (e.g., location, age) as well as through engineered features reflecting how the property compares to others. It may predict stock prices in finance by analyzing some economic metrics and company data. -The **ElasticNetModel** It is an entire package where not only can you get to make predictions but also deep dive into the data. A weapon of productivity for anyone dealing with a large data set that needs to be analysed and read into, not just numbers. Its ability to generalize, while solving for intricate relationships among variables which are characteristic challenge settings of modellers and analysts; makes this model immensely useful as a statistical package. +The ElasticNetModel is not only used for predictions; it is also indispensable for understanding the data. It is beneficial not only for numbers but also for anyone who deals with complex data puzzles and draws smart conclusions. Researchers and analysts who often encounter difficult modeling scenarios find the power to simplify this model while dealing with complex relationships invaluable. ## Q2.How did you test your model to determine if it is working reasonably correctly? -#### Model Testing and Validation Plan +We are testing and validating extensively to make our model not only accurate but also robust. So scrolling through, let us now see how it helps in providing a complete review of a model by each action described. -For the purpose of the model’s strength and accuracy, we have developed an elaborate testing and validation plan. Let’s walk through each step of this strategy to see how it contributes to a full evaluation of the model’s performance: +### Synthetic Data Generation -### 1. Synthetic Data Generation +- **Method:** We used a method `generate_rotated_s_wave_data` for generating synthetic datasets which makes the relationship of features with target values explicit. -- **Method**: To illustrate how features are related to target values, we generated so called synthetic data supported by the function `generate_rotated_s_wave_data`. -- **Why It Matters**: This setup allows reducing the amount of noise and a closer look at its capacity to read and learn the patterns we established. It means that when working with synthetic data which contains already known outcomes the degree of the model’s ability to reproduce these relationships can be assessed as an indicator of learning precision. +- **Why It Matters:** This allows us to go deep into how the model is recognizing and learning the patterns we have defined. If we are using synthetic data where we obviously know the outcomes, it is possible to thoroughly test how much of this set relationship between elements that the model has actually learned. -### 2. Training on Generated Data +### Training on Generated Data -- **What We Did**: The model used a lot of time studying these synthetic datasets. : The following figures show the parameters of the fitted coefficients and intercepts of the tested article in this phase, correlating them with the expected patterns. -- **Why It Matters**: This can be done by executing the step that ensures that the model has comprehension of the dynamics in the data set. It assists in affirming to a degree that what the model is based on is sound as well as ensuring that it is sensitive to how it will react to the subtleties it is like to encounter in actuality. +- **What We Did:** The model spent a good deal of time learning from these synthetic datasets. During this phase, we kept a close eye on its parameters, checking how well the learned coefficients and intercepts matched the expected patterns. -### 3. Predictions on Unseen Data +- **Why It Matters:** It is important as it confers that the model captures the underlying phenomenon in data. It helps prove that the groundwork of the model is strong and it responds well to various nuances that come in real scenarios. -- **Method**: To check how well the model is doing, we fed it with data it did not learn with during the learning stage. -- **Why It Matters**: This is essential for evaluating how good the model is in using the learning to solve other unseen dataset – which is very important in ascertaining the utility of the model in real world where dataset variation is the order of the day. +### Predictions on Unseen Data -### 4. Model Evaluation Metrics +- **Trial:** We tried the model's predictions on new data that it has never seen in training. -- **Metrics Used**: To check the accuracy and reliability of the carried out predictions we used several standard regression parameters such as: - - Mean Squared Error (MSE) - Linear – Root Mean Squared Error (RMSE) - - Mean Absolute Error (MAE) - - R-squared (R²) -- **Why They’re Important**: These measure give you actual numbers to explain the degree at which the model employed is capable of predicting new data. They assist us in getting an understanding of other aspects of performance including the average magnitude of the errors and variance in the target variable accounted for by the model. +- **Why That Matters:** It's critical that the model be able to generalize its learning to new, unobserved data as this is essential in making sure the model is actually useful for the real world, where variability will always exist. -### 5. Probability versus Outcome Plots +### Model Evaluation Metrics -- **How We Do It**: We employ residual plots whereby each point represents a predicted value against its actual value. The closer that the observed points are to the 45-degree line, the better are the predictive capabilities of the model in question. -- **The Benefits**: These visuals provide an immediate, tangiblerefreshing ofits gross approximation. They work especially well when used to discern whether the model bolts general or is inconsistent (high variation, systematic error). They also allow us to visualize readily whether some of the features provide better prediction accuracy than others. +- **Metrics Used:** We measured the model’s accuracy and reliability using several standard regression metrics like Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and R-squared (R²). -### 6. Analysis and Iteration +- **Why They Matter:** These measures are objective figures that show us how well the model performs on unseen data. These metrics help us in differentiating so many of the things that we encounter, for example, what is the average size of errors, how much variance in our target variable has been explained by the model. -- **Ongoing Process**: Subsequently, the model is improved based on visual data and other collected items of measure. Such adjustments may entails the fine tuning of the parameters of the model so that predictions match actual outcomes particularly in the areas we identify discrepancies. -- **Why Keep At It**: These refinements assist in acquiring significant details in the new data patterns in order for the model to provide great results. This feedback and improvement cycle is essential to keep the model on the top of its form as new issues present themselves in the field. +### Predicted vs. Actual Value Visual Comparisons -#### Insights from Graphs: +- **How We Do It:** Scatter plots of actual vs. predicted values. The model would be making perfect predictions if every point is close to the diagonal line (x and y values are identical). -Comparing the plot for “Feature 6 vs. Target” to the plot for “Feature 2 vs. Target” where despite the disperse nature of the former the latter seems to be more compact near the diagonal it is possible to conclude that Feature 2 is managed more accurately. This can put us in a position to ask ourselves whether Feature 6 should apply more data pre-processing or whether the model should apply more complex feature engineering methods to capture the fine relative data detail. +- **The Benefits:** These visualizations provide an immediate, visceral understanding of the model's accuracy. These are especially useful for seeing whether the model tends to over or underestimate in general (systematic bias) or if it is all over the place (high variance). They also allow us to easily observe any preferable prediction ability of some characteristics over the others. -By combining these specific types of testing, not only do we guarantee to ourselves that the model is mathematically correct, but we also show that it can be used in practice in as many application fields as possible. This stringent check and balance system is well suited for creating accurate high quality predictive models that can climb even more for reliability. +### Analysis and Iteration -## Q3.What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) +- **Continuously Iterating Model:** We keep refining the model based on insights derived from the visual data and other metrics. It might entail adjusting the settings of the model to more closely reflect reality — particularly in areas where we see gaps. -## Model Optimization Parameters +- **Why Work On It:** It is to give the model adaptations so as to fit not very systematic characteristics the new data comes with, this guarantees that the model keeps effective and would still work properly. This endless circle of iteration is what keeps the model out front in confronting new obstacles. -## Lambdas in Regularization +### Graph Observations -**Function:** -The lambdas are crucial in regularization methods and finding the optimal one is recommended for using in Lasso (L1) or Ridge (L2) regularization. These techniques are somewhat crucial for avoiding overfitting since a model can learn most of the noise in data set as a signal. +The improved, tight clustering along the diagonal for Feature 2 vs. Target when compared to Feature 6 vs. Target indicates that we are handling Feature 2 better. In this case, we will wonder whether Feature 6 requires additional preprocessing, or whether the model should use more powerful feature engineering resources to learn better and deeper data patterns. By combining these detailed testing methodologies, we are able to not only validate the theoretical basis of the model but also certify that it can take on a wide range of real applications with reliable performance. This thorough validation and refinement front is paramount for developing robust predictive models that can scale to novel queries and provide you with reliable predictions. + +## Q3.What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.) + +## Model Optimization Parameters -**Effect of Regularization:** -raising the value of lambda leads to increase the amount the penalizes the coefficients of the model and thus reduces them towards zero but not to zero all the same. This mechanism is particularly beneficial for noisy large datasets where the distribution of data may vary thus leading to the generation of interpretive patterns that are misleading. +We gave some parameters to change in order to improve the performance of our model. This section is a breakdown of each parameter and what each does/means, as well as touches on how tuning them could lead to improved model performance: -**When to Use:** -If a model is making sharp learning from data — infact picking up noise as the real signals, then making enhancements to the lambda values can be a good strategy. It does so while also tempering the learning of the model and its corresponding generalization to unseen data of any kind. +### 1.⁠ ⁠*Lambdas* -- **Use Case:** - Thresh is most effective whenever a certain model is overly fitting the training data set. Finally, by making thresh larger, it improves the generalization capability of the model over new, never seen before data. +- _Function:_ Lambdas are the key elements to the regularization strategies of our model, which we utilized in both L1 (Lasso) and L2 (Ridge) techniques. This alleviates a common overfitting problem in which the model learns noise as if it were real data signals. +- _Effect of Regularization:_ When the value of lambdas is increased, it increases the penalty on coefficients so that they are constrained to be near zero. This adjustment is beneficial when dealing with complex datasets that could potentially lead to misleading noise. +- _When we use it:_ When we observe our model is overfitting with training data, we then increase lambdas to rebuild our model in a much generalized way to new, unseen data. -### 2. Thresh +### 2. _Thresh_ -- **Function:** +- _Function:_ Thresh decides the elimination point for feature selection after filtering in, shaving the model down to select characteristics that are necessary and excluding others that may distort. -- **Adjustment Effect:** +- _Adjustment Effect:_ Raising the thresh value may cause getting rid of secondary variables as their coefficients become zeroes. Such a reduction in the model leads to less accurate results but occasionally, the results are expected to be better for unseen data. -- **Where To Use It:** +- _Where To Use It:_ Thresh is used to control the measure of variable importance, allowing identifying important features, and at the same time, excluding an excessive number of predictors that have no beneficial impact or, on the contrary, may negatively affect the model’s ability to make adequate predictions. This is more applicable especially where there are many features more than the dependent variables. -### 3. Max_iter +### 3._Max_iter_ -- **Function**: This parameter tells our algorithm to stop after running for the permitted number of iterations as stated below. -- **Impact of Adjustment**: Increasing the number of epochs may provide better chances for the model to adjust its weights with respect to data for better performance. -- **When We Use It**: If the model is not in its best condition yet, then it may be required to increase the `max_iter`. However, it is possible to have the model take more time for a training session, which calls for compromise between precision and time taken on the training session. +- _Function_: This parameter tells our algorithm to stop after running for the permitted number of iterations as stated below. +- _Impact of Adjustment_: Increasing the number of epochs may provide better chances for the model to adjust its weights with respect to data for better performance. +- _When We Use It_: If the model is not in its best condition yet, then it may be required to increase the max_iter. However, it is possible to have the model take more time for a training session, which calls for compromise between precision and time taken on the training session. -### 4. Tol +### 4. _Tol_ -- **Function**: This is the same as tolerance; it defines a small the error which must fit our training data before further adjustments are made to stop.r adjustments stop. -- **Impact of Adjustment**: Lowering the `tol` value ensures the model won’t stop learning until the fit is very tight, improving accuracy but extending training time. -- **When We Use It**: If rate of accuracy must be in the consideration. The most important one is that if you set the tolerance level too low then really long training can actually make no gain at all. +- _Function_: This is the same as tolerance; it defines a small the error which must fit our training data before further adjustments are made to stop.r adjustments stop. +- _Impact of Adjustment_: Lowering the tol value ensures the model won’t stop learning until the fit is very tight, improving accuracy but extending training time. +- _When We Use It_: If rate of accuracy must be in the consideration. The most important one is that if you set the tolerance level too low then really long training can actually make no gain at all. -### 5. Learning_rate +### 5. _Learning_rate_ -- **Function**: This hyperparameter controls by how much the model should change towards the optimization in the updating of the model. -- **Impact of Adjustment**: It reduces the steps taken towards implementation and is much slower but more precise, because it is a smaller learning rate and its implementations are more careful. -- **When We Use It**: However, if the model is over the point that represents the minimal loss, it means that the learning rate is oversized, thus should be decreased to improve the stability and accuracy of the convergence. Nonetheless, the rate of learning may be reduced to an extreme low in the presence of a low rate. +- _Function_: This hyperparameter controls by how much the model should change towards the optimization in the updating of the model. +- _Impact of Adjustment_: It reduces the steps taken towards implementation and is much slower but more precise, because it is a smaller learning rate and its implementations are more careful. +- _When We Use It_: However, if the model is over the point that represents the minimal loss, it means that the learning rate is oversized, thus should be decreased to improve the stability and accuracy of the convergence. Nonetheless, the rate of learning may be reduced to an extreme low in the presence of a low rate. These two parameters, therefore, can be tuned to optimize accuracy for given data distribution and complexity of task, computational cost, and model complexity. +We can improve this model by tinkering with these parameters to help the model adapt well to the nuances of our data and the requirements of our task, ensuring that we have a good trade-off between accuracy, efficiency of computation, and model lean size. + ## Q4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental? ## Specific Inputs That May Challenge the ElasticNet Model ### 1. Highly Nonlinear Data -- **The Challenge**: This is because Elastic Net has an Optimistic World View, it believes that the world can be described using straight lines and linearly separable features. But life is rarely that clean or bumpy, and data exhibits the kind of curved relationships that a model like biases against. -- **What Happens**: Elastic Net might miss with its predictions in these instances since they involve some of the sharp edges/turns and complex patterns that it cannot fully understand. - - **Feature Engineering**: One way to make the model learn the underlying patterns is by transforming those input features. This could involve squaring or cubing the features, maybe adding in interaction terms, possibly doing mathematical transformations (like logarithms or exponentials) to help straighten out those nonlinear relationships. - - **Kernel Methods**: Taking a cue from support vector machine, these can be kernelized as well. - - **Hybrid Models**:Two heads (or models) as well. We can combine the advantages of Elastic Net when teamwork with nonlinear models like decision trees or neural networks, which allows it to predict well in wider set of scenarios. +- _The Challenge_: This is because Elastic Net has an Optimistic World View, it believes that the world can be described using straight lines and linearly separable features. But life is rarely that clean or bumpy, and data exhibits the kind of curved relationships that a model like biases against. +- _What Happens_: Elastic Net might miss with its predictions in these instances since they involve some of the sharp edges/turns and complex patterns that it cannot fully understand. + - _Feature Engineering_: One way to make the model learn the underlying patterns is by transforming those input features. This could involve squaring or cubing the features, maybe adding in interaction terms, possibly doing mathematical transformations (like logarithms or exponentials) to help straighten out those nonlinear relationships. + - _Kernel Methods_: Taking a cue from support vector machine, these can be kernelized as well. + - _Hybrid Models_:Two heads (or models) as well. We can combine the advantages of Elastic Net when teamwork with nonlinear models like decision trees or neural networks, which allows it to predict well in wider set of scenarios. ### 2. Noisy Data -- **The Challenge**:Think of your data as listening to one of those quiet FM radio stations during a thunderstorm, that is what this is like. This then becomes static (noise) that can drown the music (true patterns), and thus it makes it hard for the Elastic Net to hear what is important. -- **What Happens**:If the data is too noisy, Elastic Net might fit to the noise instead of the music and learn something that works great on training data but will falter in practice. -- **What We Can Do**: +- _The Challenge_:Think of your data as listening to one of those quiet FM radio stations during a thunderstorm, that is what this is like. This then becomes static (noise) that can drown the music (true patterns), and thus it makes it hard for the Elastic Net to hear what is important. +- _What Happens_:If the data is too noisy, Elastic Net might fit to the noise instead of the music and learn something that works great on training data but will falter in practice. +- _What We Can Do_: - - **Increase Regularization**:we have the option here to turn up the lambdas knob and increase penalties for complexity, allowing our model to focus in on every loud symptomatic signal while ignoring that static. - - **Threshold Tuning**: Changing the thresh parameter to allow us to be more selective about which features we let into the model, and cut out those that tend to contain more noise than signal. + - _Increase Regularization_:we have the option here to turn up the lambdas knob and increase penalties for complexity, allowing our model to focus in on every loud symptomatic signal while ignoring that static. + - _Threshold Tuning_: Changing the thresh parameter to allow us to be more selective about which features we let into the model, and cut out those that tend to contain more noise than signal. - - **Noise Reduction Techniques**: Even before feeding data into our model, we can clean it up to filter out the oddball values or make some signal smoother so that what we pass on to our mod is as clean and clear as possible. + - _Noise Reduction Techniques_: Even before feeding data into our model, we can clean it up to filter out the oddball values or make some signal smoother so that what we pass on to our mod is as clean and clear as possible. By acknowledging these limitations and using smart strategies to overcome them, we can increase the potential of Elastic Net because another graphical preferred in predictive modelling. - ## Visualization +### The First Train Dataset: + ![alt text](1.png) -### Scatter Plots of Train Features vs Target +### Scatter Plots of Train Features -This image shows a series of scatter plots, each representing the relationship between a different feature (Feature 1 to Feature 6) and a target variable (Y_train) based on training data. Each plot displays the data points in two dimensions: +This image shows a series of scatter plots, each representing the relationship between a different feature (Feature 1 to Feature 6) and Y_train based on training data. Each plot displays the data points in two dimensions: - The horizontal axis (X-axis) shows values of a specific feature. - The vertical axis (Y-axis) represents the values of Y_train. The purpose of these plots is typically to visually assess the relationship between the features and the target variable. Observations can be made regarding the distribution, trend, or any potential outliers in the data. From the plots: -- **Feature 1, Feature 2, Feature 3:** These features show some vertical dispersion but no clear linear or nonlinear trend with Y_train. -- **Feature 4, Feature 5, Feature 6:** Similar to the first three, these features also display a scattered distribution with respect to Y_train, without a visible specific pattern. +- **Feature 1, Feature 2, Feature 3:** These features display a scattered distribution with respect to Y_train as well as relatively clear linear relationships. +- **Feature 4, Feature 5, Feature 6:** These features show some vertical dispersion and a less clear linear relationship. Overall, these scatter plots can help in determining which features might be relevant for modeling, although none of the features shown here have a distinct or strong linear relationship with the target variable. +### The First Test dDataset: + ![alt text](2.png) ### Overview of Each Feature Plot Analysis -The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against a target variable (labeled as "Y_Test"). Each subplot represents the relationship between a single feature and the target variable. - -### Breakdown of Scatter Plots: - -- **Test Feature 1 through 6**: Each scatter plot represents the feature's values on the x-axis and the corresponding target variable values on the y-axis. -- **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. +The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against "Y_Test". Each subplot represents the data of one specific demension of feature. -### Visual Insights: +### Meaning of This Graph: -- For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. -- For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. +These plots visualize the dataset to help understand our first test dataset. +This test dataset is used to test the model's predictive ability on partially linear positively correlated data and data without obvious correlation. -### Uses: - -- **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). -- **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. +### Model Fitting Effect on first dataset: ![alt text](3.png) -### Breakdown of Each Plot Analysis - -The image shows scatter plots of six different features (labeled "Test Feature 1" through "Test Feature 6") plotted against a target variable (labeled as "Y_Test"). Each subplot represents the relationship between a single feature and the target variable. - -### Breakdown of Scatter Plots: - -- **Test Feature 1 through 6**: Each scatter plot represents the feature's values on the x-axis and the corresponding target variable values on the y-axis. -- **Y_Test**: This is the target variable (possibly the dependent variable or output you're trying to predict). It is plotted on the y-axis for each subplot. +### Model Fitting Effect Image Description: -### Visual Insights: +The figure above shows the effect of our model on the first test dataset based on the first training dataset, where the red line is the straight line fitted using the least squares method. The yellow points are true values and the blue points are predicted values. +It can be seen that the fitting effect of our model is very good on the first dataset. -- For **Feature 1, 2, 4, 5, and 6**, the scatter seems somewhat spread out without clear patterns. This may suggest that these features do not have a strong linear relationship with the target variable. -- For **Feature 3**, the scatter plot shows more of a linear relationship, indicating that this feature might have a stronger correlation with the target variable compared to others. - -### Uses: - -- **Feature Importance**: These plots can help you visually assess which features might be more important for predicting the target variable based on how well the feature correlates with the target (e.g., Feature 3 seems more relevant). -- **Correlation Analysis**: You might use these plots to further investigate whether linear or nonlinear regression models could capture the relationships seen in these features. +### The Second Train Dataset: ![alt text](4.png) ### Scatter Plot Analysis -The image shows scatter plots of six features (Feature 1 through Feature 6) plotted against the target variable labeled "Y_Train" on the y-axis. The x-axis in each subplot corresponds to the respective feature values. +The image shows scatter plots of six features (Feature 1 through Feature 6) plotted against "Y_Train" on the y-axis. The x-axis in each subplot corresponds to the respective feature values. ### Key Observations: 1. **Feature 1 to Feature 3**: - - All three features show a strong positive linear correlation with the target variable (Y_Train). + - All three features show a strong positive linear correlation. - The points align closely along a positively sloped line, indicating that as these feature values increase, the target variable increases in a linear fashion. 2. **Feature 4 to Feature 6**: - - These features show a strong negative linear correlation with the target variable. + - These features show a strong negative linear correlation. - The points align closely along a negatively sloped line, suggesting that as these feature values increase, the target variable decreases. -### General Insights: - -- All six features display clear linear relationships with the target, either positive (Feature 1 to 3) or negative (Feature 4 to 6). -- These patterns suggest that linear +### The Second test dataset: ![alt text](5.png) -### Scatter Plot Analysis for Test Features vs. Target - -The image shows scatter plots of six features (Test Feature 1 through Test Feature 6) plotted against the target variable labeled "Y_Test." Each subplot represents the relationship between a specific feature (on the x-axis) and the target variable (on the y-axis). - -### Key Observations: - -1. **Feature 1 to Feature 6**: - - Each scatter plot shows a wide spread of points without a clear pattern, suggesting weak or no strong linear relationships between the features and the target variable. - - The lack of clustering along a line indicates that none of the features exhibit a strong direct correlation with the target. -2. **Feature 5**: - - - There appears to be a slight downward trend, indicating a weak negative correlation between Feature 5 and the target (Y_Test). +### Scatter Plot Analysis for Test Features -3. **Feature 6**: - - There is some clustering in the lower right quadrant, possibly indicating a negative influence on the target, but the scatter remains wide. +The image shows scatter plots of six features (Test Feature 1 through Test Feature 6) plotted against "Y_Test." Each subplot represents the data of one specific demension of feature. ### General Insights: -- None of the features show a strong, easily identifiable linear correlation with the target variable, as the data points are dispersed across all plots. -- This suggests that more complex modeling techniques (e.g., non-linear methods) or additional feature engineering may be needed to capture meaningful patterns in the data. +- The data of the first dimension are scattered in all the plots and do not show obvious characteristics. +- The data of the second and third dimensions show insignificant positive correlation. +- The data of the last three dimensions show insignificant negative correlation. -![alt text](6.png) - -### Regression Analysis for Features vs. Actual and Predicted Target Values +### Meaning of This Graph: -The image shows six scatter plots comparing the actual target values and the predicted target values for six features (Feature 1 through Feature 6) against the target variable labeled "Y." Each subplot also includes a red dashed line representing the linear fit for each feature. +These plots visualize the dataset to help understand our second test dataset. +This test dataset is used to test the model's predictive ability on partially linear positively correlated data and partially linear negatively correlated data. -### Key Observations: - -1. **Feature 1 to Feature 3**: +### Model Fitting Effect on second dataset: - - The fit lines have positive slopes, suggesting a weak positive relationship between these features and the target variable. - - The actual and predicted values are dispersed around the fit line with noticeable variance, indicating that the model's predictions are not tightly aligned with the actual values. The fit is somewhat loose, and the features do not have a strong linear relationship with the target. - -2. **Feature 4 to Feature 6**: - - The fit lines have negative slopes, indicating a negative relationship between these features and the target variable. - - Similar to Features 1-3, there is substantial dispersion of the actual and predicted values around the fit line, suggesting weak predictive power and variability in the model's predictions for these features. +![alt text](6.png) -### General Insights: +### Model Fitting Effect Image Description: -- All six features exhibit weak linear relationships with the target variable, as evidenced by the scattered points around the fit lines. The model does not capture the target values well, and the fit is not strong. -- **Features 4, 5, and 6** show negative relationships, meaning that as the feature values increase, the target decreases, though this is not strong. -- These scatter plots indicate that the model could benefit from more advanced techniques, as simple linear regression appears insufficient for accurately predicting the target values for these features. Non-linear modeling methods or additional feature engineering might improve the model's performance. +The figure above shows the effect of our model on the second test dataset based on the second training dataset, where the red line is the straight line fitted using the least squares method. The yellow points are true values and the blue points are predicted values. +It can be seen that the fitting effect of our model is very good on linear positive correlation and linear negative correlation data sets.

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