COMMENTS ON THE RELATIONSHIP BETWEEN PRINCIPAL COMPONENTS-ANALYSIS AND WEIGHTED LINEAR-REGRESSION FOR BIVARIATE DATA SETS

Regression and principal components analysis (PCA) are two of the most widely used techniques in chemometrics. In this paper, these methods are compared by considering their application to linear, two-dimension al data sets with a zero intercept, The need for accommodating measure ment errors with these methods is addressed and various techniques to accomplish this are considered. Seven methods are examined: ordinary l east squares (OLS), weighted least squares (WLS), the effective varian ce method (EVM), multiply weighted regression (MWR), unweighted PCA (U PCA), and two forms of weighted PCA. Additionally, five error structur es in x and y are considered: homoscedastic equal, homoscedastic unequ al, proportional equal, proportional unequal, and random. It is shown that for certain error structures, several of the methods are mathemat ically equivalent. Furthermore, it is demonstrated that all of the met hods can be unified under the principle of maximum likelihood estimati on, embodied in the general case by MWR. Extensive simulations show th at MWR produces the most reliable parameter estimates in terms of bias and mean-squared error. Finally, implications for modeling in higher dimensions are considered.