Common Algorithms for Regression Analysis

Efficient Implementation of Regression Analysis in MATLAB

Regression analysis is a fundamental tool for data modeling and prediction. MATLAB offers various optimized algorithms designed to handle different types of regression problems. These algorithms undergo continuous improvement to efficiently process large-scale datasets while maintaining high computational accuracy.

**Linear Regression** For classical linear regression problems, MATLAB's `fitlm` and `regress` functions are the most common choices. The `fitlm` function implements ordinary least squares regression, automatically handles outliers, and provides comprehensive statistical outputs. For high-dimensional data, regularization techniques like LASSO and Ridge regression can be implemented using `lasso` or `ridge` functions to prevent overfitting.

**Nonlinear Regression** When dealing with nonlinear data relationships, the `fitnlm` function fits custom nonlinear models using the Levenberg-Marquardt optimization algorithm, known for its excellent convergence and stability. For more complex nonlinear relationships, `nlinfit` allows users to define custom model functions with iterative parameter estimation.

**Robust Regression** In datasets containing outliers, `robustfit` provides robust regression methods (including M-estimation) that minimize the influence of anomalous data points on model parameters.

**Polynomial Regression** The `polyfit` function enables rapid polynomial regression implementation by specifying the polynomial degree to fit curve relationships through least-squares approximation.

**Machine Learning Extensions** For advanced regression tasks, MATLAB's Statistics and Machine Learning Toolbox offers functions like `fitrensemble` (for ensemble methods including bagging and boosting) and `fitrgp` (for Gaussian process regression) to handle complex pattern recognition and uncertainty quantification.

Algorithm optimizations include efficient memory management, accelerated matrix operations, and parallel computing support, making these tools highly effective for practical data analysis in engineering and scientific research. Selecting appropriate algorithms based on data characteristics and modeling requirements significantly enhances the efficiency and accuracy of regression analysis.