On a Relation between Principal Components and Regression Analysis

By Jong, Jyh-Cherng; Kotz, Samuel | The American Statistician, November 1999 | Go to article overview

On a Relation between Principal Components and Regression Analysis


Jong, Jyh-Cherng, Kotz, Samuel, The American Statistician


A regression approach to principal component analysis is presented in this note. We provide an alternative interpretation of principal components that illustrates the relation between the extra sum of squares in regression analysis and the eigenvalues associated with the principal components.

KEY WORDS: Coefficient of determination; Eigenvalues; Eigenvectors; Extra sum of squares; Principal components regression; Residuals.

1. INTRODUCTION

Principal component analysis is one of the most well known and easily constructed techniques of classical multivariate analysis (see, e.g., Daultrey 1976; Dunteman 1989; Flury 1988; Jolliffe 1986; and Rao 1964). The objective here is to obtain a few uncorrelated "principal components" in terms of linear combinations of the original variables such that the variance of these components is maximized. This idea was originally implied in K. Pearson's work and developed by Hotelling (see, e.g., Dunteman 1989). It is usually described in numerous standard textbooks without a direct reference to the linear regression model. On the other hand, multivariate regression models are usually developed by using the criterion of minimizing residual sum of squares without employing the concepts of eigenvectors and eigenvalues. One may argue that the so-called principal components regression analysis (see, e.g., Coxe 1986 and Dunteman 1989) provides a relation between the principal components and the regression model. However, we must clarify that the principal components regression analysis deals with replacing the correlated predictors in the regression function by uncorrelated principal components. Roughly speaking, its purpose is not to obtain the best estimation of the linear transformation matrix, but rather to solve the problem of multicollinearities among the predictors.

In this article our goal is to derive principal components via a regression model. An approach which is similar to the one adopted in this article can be found in Jobson's textbook (1992) and Rao's classical paper (1964), where the authors use principal components to approximate the original variables, and then minimize the sum of squared deviations between the original variables and their approximations. However, their approach does not explicitly use the regression structure. Our literature search indicates that Meredith and Millsap (1985) were the first to derive the principal components directly by means of a linear regression model. Meredith and Millsap (1985) used expected loss function as their objective function in the regression model and applied the machinery developed by Darroch (1965) to minimize the expected loss. Although the expected loss function criterion is popular in psychological applications, it is less widespread in other fields of applied sciences and engineering.

The purpose of this article is to merge the concepts of principal components and multivariate regression, and to show the equivalence of the optimization criteria involved in each one of them. It turns out that the criterion of maximizing the variance of uncorrelated linear combinations is analogous to the criterion of minimizing error sum of squares in the regression model.

The authors were somewhat surprised that--to the best of their knowledge--this relation has not been discussed in statistical literature. Moreover, we also present the scale invariance property of principal components derived by regression approach. This property allows us to rescale principal components without changing their capabilities to interpret the original variables. A similar property can be found in Meredith and Millsap (1985).

2. A REGRESSION APPROACH TO PRINCIPAL COMPONENT ANALYSIS

Let X be a p-dimensional random vector; [X.sub.j](j = 1, [ldots], n) be its jth observation. For notational convenience, it is assumed, without loss of generality, that the variables [X.sub.j] are centered at their mean values so that [bar{X}] = 0.

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