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+# Project 2 
+BY
+
+A20539949-Usha Devaraju
+
+A20548244-Roopashri Kommana
+
+A20550565-Sai Sandeep Neerukonda
+
+# Gradient Boosting for Regression
+
+This repository contains a custom implementation of a Gradient Boosting model for regression tasks, using decision trees as base learners. The model is designed to be versatile and easily adjustable to fit various regression problems.
+
+# 1.What does the model you have implemented do and when should it be used?
+## Model Description
+
+The Gradient Boosting model implemented here constructs an ensemble of decision trees in a sequential manner, where each tree is built to correct the errors made by the previous ones. The model is particularly useful for datasets where relationships between features and the target variable are complex and non-linear.
+
+### When to Use This Model
+
+This model should be used when:
+- Dealing with regression tasks requiring robust predictive power.
+- Handling datasets with complex and non-linear relationships.
+- Situations where other simpler models (like linear regression) are insufficient.
+
+# 2.How did you test your model to determine if it is working reasonably correctly?
+## Testing the Model
+
+The model has been rigorously tested using the California Housing dataset, which is a standard dataset for evaluating regression models. The testing involves:
+- Splitting the data into training and testing sets.
+- Scaling the feature matrix to standardize the input data.
+- Training the Gradient Boosting model on the training data.
+- Evaluating its performance using Mean Squared Error (MSE) on the test set.
+
+# 3.What parameters have you exposed to users of your implementation in order to tune performance? (Also perhaps provide some basic usage examples.
+## Exposed Parameters
+
+Users can tune the following parameters to optimize the model's performance:
+- `n_estimators`: The number of trees to build (default is 100).
+- `learning_rate`: The step size at each iteration to control overfitting (default is 0.1).
+- `max_depth`: The maximum depth of each decision tree (default is 3).
+
+### Prerequisites
+
+Ensure you have Python installed along with the following libraries:
+- `numpy`
+- `scikit-learn`
+
+To install missing dependencies, use:
+```bash
+pip install numpy scikit-learn
+```
+
+### Basic Usage Example
+
+```python
+from gradient_boosting import GradientBoosting
+from sklearn.model_selection import train_test_split
+from sklearn.datasets import fetch_california_housing
+from sklearn.preprocessing import StandardScaler
+
+# Load data
+data = fetch_california_housing()
+X, y = data.data, data.target
+
+# Split data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
+
+# Scale features
+scaler = StandardScaler()
+X_train_scaled = scaler.fit_transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+
+# Initialize and train the gradient boosting model
+model = GradientBoosting(n_estimators=50, learning_rate=0.1, max_depth=3)
+model.fit(X_train_scaled, y_train)
+
+# Predict and evaluate
+predictions = model.predict(X_test_scaled)
+print(predictions)
+```
+
+## Running Tests
+To test the model on the California Housing dataset, run:
+```python
+python testing.py
+```
+The script will:
+- **Load the dataset.**
+- **Train and test the Gradient Boosting model.**
+- **Output the Mean Squared Error (MSE) of the predictions.**
+
+# 4.Are there specific inputs that your implementation has trouble with? Given more time, could you work around these or is it fundamental?
+## Potential Issues and Workarounds
+
+The model may encounter difficulties with specific types of inputs such as:
+
+- **Extremely Noisy Data**: High levels of noise can lead to overfitting, where the model learns the noise as patterns, degrading prediction accuracy on new data.
+- **Outliers**: Outliers can disproportionately influence the decision boundaries established by the decision trees, leading to suboptimal models.
+
+### Workarounds
+
+To enhance model robustness and performance:
+- **Preprocessing Steps**: Implement robust preprocessing steps to handle outliers and noise, such as outlier detection algorithms or robust scaling methods.
+- **Advanced Techniques**: Explore integrating outlier detection algorithms and advanced noise filtering techniques before fitting the model to improve its generalization capabilities.
+
diff --git a/gradient_boosting.py b/gradient_boosting.py
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+import numpy as np
+
+# Define the function to calculate the gradient of the squared loss
+def squared_loss_gradient(y, f):
+    """
+    Compute the gradient for the squared loss function.
+
+    Parameters:
+    - y (np.array): The target values.
+    - f (np.array): The predicted values.
+
+    Returns:
+    - np.array: The gradient of the squared loss.
+    """
+    return y - f
+
+# Define the Node class to represent each node in the decision tree
+class Node:
+    """
+    A node in the decision tree.
+
+    Attributes:
+    - value (float): The value at the node, used for leaf nodes.
+    - left (Node): Left child node.
+    - right (Node): Right child node.
+    - threshold (float): The threshold for splitting.
+    - feature (int): The index of the feature used for splitting.
+    """
+    def __init__(self, value=None, left=None, right=None, threshold=None, feature=None):
+        self.value = value
+        self.left = left
+        self.right = right
+        self.threshold = threshold
+        self.feature = feature
+
+# Define the DecisionTree class for building the regression tree
+class DecisionTree:
+    """
+    A simple decision tree for regression.
+    
+    Attributes:
+    - max_depth (int): The maximum depth of the tree.
+    - root (Node): The root node of the tree.
+    """
+    def __init__(self, max_depth=3):
+        self.max_depth = max_depth
+        self.root = None
+
+    def fit(self, X, residuals):
+        """
+        Fit the decision tree to the residuals.
+
+        Parameters:
+        - X (np.array): Feature matrix.
+        - residuals (np.array): Residuals to fit.
+        """
+        self.root = self._build_tree(X, residuals, depth=0)
+
+    def _build_tree(self, X, residuals, depth):
+        """
+        Recursively build the decision tree.
+        
+        Parameters:
+        - X (np.array): Feature matrix.
+        - residuals (np.array): Residuals to fit.
+        - depth (int): Current depth of the tree.
+        
+        Returns:
+        - Node: The constructed tree node.
+        """
+        num_samples = X.shape[0]
+        if depth >= self.max_depth or num_samples <= 1:
+            leaf_value = np.mean(residuals)
+            return Node(value=leaf_value)
+        
+        best_feature, best_threshold, best_var = None, None, np.inf
+        for feature in range(X.shape[1]):
+            thresholds = np.unique(X[:, feature])
+            for threshold in thresholds:
+                left_mask = X[:, feature] <= threshold
+                right_mask = X[:, feature] > threshold
+                if np.sum(left_mask) == 0 or np.sum(right_mask) == 0:
+                    continue
+                left_var = np.var(residuals[left_mask])
+                right_var = np.var(residuals[right_mask])
+                total_var = left_var + right_var
+                if total_var < best_var:
+                    best_feature, best_threshold, best_var = feature, threshold, total_var
+
+        left_mask = X[:, best_feature] <= best_threshold
+        right_mask = X[:, best_feature] > best_threshold
+        left_node = self._build_tree(X[left_mask], residuals[left_mask], depth + 1)
+        right_node = self._build_tree(X[right_mask], residuals[right_mask], depth + 1)
+        return Node(feature=best_feature, threshold=best_threshold, left=left_node, right=right_node)
+
+    def predict(self, X):
+        """
+        Make predictions using the decision tree.
+
+        Parameters:
+        - X (np.array): Feature matrix.
+
+        Returns:
+        - np.array: Predicted values.
+        """
+        return np.array([self._predict(x, self.root) for x in X])
+
+    def _predict(self, x, node):
+        """
+        Recursively predict by traversing the decision tree.
+
+        Parameters:
+        - x (np.array): Single feature vector.
+        - node (Node): Current node of the tree.
+
+        Returns:
+        - float: Predicted value.
+        """
+        if node.value is not None:
+            return node.value
+        if x[node.feature] <= node.threshold:
+            return self._predict(x, node.left)
+        else:
+            return self._predict(x, node.right)
+
+# Define the GradientBoosting class for boosting decision trees
+class GradientBoosting:
+    """
+    Gradient Boosting for regression.
+
+    Attributes:
+    - n_estimators (int): Number of boosting stages to perform.
+    - learning_rate (float): Learning rate shrinks the contribution of each tree.
+    - max_depth (int): Maximum depth of each decision tree.
+    - models (list): List of successive decision tree models.
+    - initial_prediction (float): Initial prediction to start the boosting.
+    """
+    def __init__(self, n_estimators=100, learning_rate=0.1, max_depth=3):
+        self.n_estimators = n_estimators
+        self.learning_rate = learning_rate
+        self.max_depth = max_depth
+        self.trees = []
+        self.initial_prediction = None
+
+    def fit(self, X, y):
+        """
+        Fit the gradient boosting model.
+
+        Parameters:
+        - X (np.array): Feature matrix.
+        - y (np.array): Target values.
+        """
+        # Initialize the first model to the mean of y
+        self.initial_prediction = np.mean(y)
+        f_m = np.full(y.shape, self.initial_prediction)
+
+        for _ in range(self.n_estimators):
+            residuals = y - f_m
+            tree = DecisionTree(max_depth=self.max_depth)
+            tree.fit(X, residuals)
+            predictions = tree.predict(X)
+            f_m += self.learning_rate * predictions
+            self.trees.append(tree)  # Store the tree instead of predictions
+
+    def predict(self, X):
+        """
+        Make predictions using the boosted model.
+
+        Parameters:
+        - X (np.array): Feature matrix.
+
+        Returns:
+        - np.array: Predicted values.
+        """
+        # Start with the initial mean prediction
+        f_m = np.full(X.shape[0], self.initial_prediction)
+
+        # Accumulate predictions from each tree
+        for tree in self.trees:
+            f_m += self.learning_rate * tree.predict(X)
+
+        return f_m
diff --git a/README.md b/question.md
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diff --git a/testing.py b/testing.py
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+# Import necessary libraries
+import numpy as np
+from sklearn.datasets import fetch_california_housing
+from sklearn.model_selection import train_test_split
+from sklearn.metrics import mean_squared_error
+from sklearn.preprocessing import StandardScaler
+
+# GradientBoosting class is in a file named gradient_boosting.py
+from gradient_boosting import GradientBoosting
+
+def main():
+    # Load the California housing dataset
+    data = fetch_california_housing()
+    X, y = data.data, data.target
+
+    # Split the dataset into training and testing sets
+    X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
+
+    # Initialize the StandardScaler
+    scaler = StandardScaler()
+
+    # Fit the scaler on the training data and transform it
+    X_train_scaled = scaler.fit_transform(X_train)
+
+    # Transform the testing data with the same scaler
+    X_test_scaled = scaler.transform(X_test)
+
+    # Initialize the GradientBoosting model
+    model = GradientBoosting(n_estimators=50, learning_rate=0.1, max_depth=3)
+
+    # Train the model on the scaled training data
+    model.fit(X_train_scaled, y_train)
+
+    # Predict the scaled test set
+    predictions = model.predict(X_test_scaled)
+
+    # Evaluate the model using mean squared error
+    mse = mean_squared_error(y_test, predictions)
+    print("Mean Squared Error on Test Set:", mse)
+
+if __name__ == "__main__":
+    main()