Multivariate Linear Regression

Introduction to Multivariate Methods

Large, high-dimensional data sets are common in the modern era of computer-based instrumentation and electronic data storage. High-dimensional data present many challenges for statistical visualization, analysis, and modeling.

Data visualization, of course, is impossible beyond a few dimensions. As a result, pattern recognition, data preprocessing, and model selection must rely heavily on numerical methods.

A fundamental challenge in high-dimensional data analysis is the so-called curse of dimensionality. Observations in a high-dimensional space are necessarily sparser and less representative than those in a low-dimensional space. In higher dimensions, data over-represent the edges of a sampling distribution, because regions of higher-dimensional space contain the majority of their volume near the surface. (A d-dimensional spherical shell has a volume, relative to the total volume of the sphere, that approaches 1 as d approaches infinity.) In high dimensions, typical data points at the interior of a distribution are sampled less frequently.

Often, many of the dimensions in a data set—the measured features—are not useful in producing a model. Features may be irrelevant or redundant. Regression and classification algorithms may require large amounts of storage and computation time to process raw data, and even if the algorithms are successful the resulting models may contain an incomprehensible number of terms.

Because of these challenges, multivariate statistical methods often begin with some type of dimension reduction, in which data are approximated by points in a lower-dimensional space. Dimension reduction is the goal of the methods presented in this chapter. Dimension reduction often leads to simpler models and fewer measured variables, with consequent benefits when measurements are expensive and visualization is important.

Multivariate Linear Regression Model

The multivariate linear regression model expresses a d-dimensional continuous response vector as a linear combination of predictor terms plus a vector of error terms with a multivariate normal distribution. Let $$y_i={\left(y_{i1},\dots ,y_{id}\right)}^{\prime }$$ denote the response vector for observation i, i = 1,...,n. In the most general case, given the d-by-K design matrix $$X_i$$ and the K-by-1 vector of coefficients$$\beta$$, the multivariate linear regression model is

$$y_i=X_i\beta +{\epsilon }_i,$$

where the d-dimensional vector of error terms follows a multivariate normal distribution,

$${\epsilon }_i\sim MV{N}_d\left(0,\Sigma \right).$$

The model assumes independence between observations, meaning the error variance-covariance matrix for the n stacked d-dimensional response vectors is

$$I_n\otimes \Sigma =\left(\begin{array}{ccc}\Sigma & & 0\\ & \ddots & \\ 0& & \Sigma \end{array}\right).$$

If $$y$$ denotes the nd-by-1 vector of stacked d-dimensional responses, and $$X$$ denotes the nd-by-K matrix of stacked design matrices, then the distribution of the response vector is

$$y\sim MV{N}_{nd}(X\beta ,I_n\otimes \Sigma ).$$

Solving Multivariate Regression Problems

To fit multivariate linear regression models of the form

$$y_i=X_i\beta +{\epsilon }_i,{\epsilon }_i\sim MV{N}_d(0,\Sigma )$$

in Statistics and Machine Learning Toolbox™, use mvregress. This function fits multivariate regression models with a diagonal (heteroscedastic) or unstructured (heteroscedastic and correlated) error variance-covariance matrix, $$\Sigma ,$$ using least squares or maximum likelihood estimation.

Many variations of multivariate regression might not initially appear to be of the form supported by mvregress, such as:

• Multivariate general linear model

• Multivariate analysis of variance (MANOVA)

• Longitudinal analysis

• Panel data analysis

• Seemingly unrelated regression (SUR)

• Vector autoregressive (VAR) model

In many cases, you can frame these problems in the form used by mvregress (but mvregress does not support parameterized error variance-covariance matrices). For the special case of one-way MANOVA, you can alternatively use manova1. Econometrics Toolbox™ has functions for VAR estimation.

Note

The multivariate linear regression model is distinct from the multiple linear regression model, which models a univariate continuous response as a linear combination of exogenous terms plus an independent and identically distributed error term. To fit a multiple linear regression model, use fitlm.