Parameter Estimation and Eigenvalue Analysis for Multivariate Regression Models Using MATLAB Toolbox

In multivariate regression analysis, MATLAB provides powerful toolboxes that enable users to efficiently perform parameter estimation and eigenvalue analysis. Multivariate regression models are commonly employed to study the effects of multiple independent variables on one or more dependent variables, with wide applications in finance, engineering, biostatistics, and other fields.

First, using MATLAB's Statistics and Machine Learning Toolbox, users can easily construct multivariate linear or nonlinear regression models. Through functions like `fitlm` (for linear models) or `mvregress` (for multivariate responses), users can input observed data to automatically compute regression coefficients and their significance levels. Parameter estimation typically employs least squares methods or maximum likelihood estimation, with MATLAB also supporting robust regression techniques to handle outliers through functions like `robustfit`.

For eigenvalue analysis, regression models may involve eigendecomposition of covariance matrices or design matrices. Using core functions like `eig` for eigenvalue decomposition or `svd` for singular value decomposition, users can extract eigenvalues and eigenvectors to assess multicollinearity issues in the model. This analysis facilitates dimensionality reduction techniques such as Principal Component Regression (PCR), implemented using the `pca` function combined with regression methods.

Additionally, MATLAB's visualization tools (including `plot`, `scatter`, and `plotResiduals`) provide intuitive displays of regression fitting results and eigenvalue distributions, helping users quickly validate model assumptions. Through these integrated functions, researchers can efficiently implement complete analytical workflows from data preprocessing to model optimization, with capabilities for cross-validation using `crossval` and model comparison via criteria like AIC or BIC.