Polynomial_Regression_Analysis

Polynomial Regression with Confidence and Prediction Intervals

This project demonstrates cubic polynomial regression including residual analysis, confidence intervals, and prediction intervals. It shows how to fit a polynomial model to synthetic data and evaluate its quality.

The goal of this project is to generate synthetic data, fit a cubic polynomial regression model, and assess its performance. The project also visualizes confidence and prediction intervals to better understand model uncertainty.

Key steps performed:

1. Data Generation

   • Created a synthetic dataset with X uniformly distributed between 0 and 10.
   • Generated Y using a cubic polynomial relationship plus normal noise: [ Y = 1 + X - 0.3 X^2 + 0.05 X^3 + \epsilon ]

2. Polynomial Regression (Degree 3)

   • Estimated polynomial coefficients using Ordinary Least Squares (OLS).
   • Plotted original data points and the regression curve.

3. Residual Analysis

   • Calculated residuals (differences between observed and predicted values).
   • Plotted:
     • Residuals vs predicted values
     • Histogram of residuals

4. Confidence and Prediction Intervals

   • Computed 95% confidence intervals for the mean response.
   • Computed 95% prediction intervals for new observations.
   • Visualized both intervals along with the regression curve.

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Technologies Used

• Python 3
• NumPy (data generation and calculations)
• Matplotlib (plots and visualization)
• SciPy (t-distribution for interval estimation)