70 - Connecting Chemometrics to Statistics: Part 2 — The Statistics Side

Abstract

This chapter compares statistics with chemometrics by entering the multivariate domain and analyzes whether analysis of variance (ANOVA) can be calculated on multivariate data. The basic operation of any ANOVA is the partitioning of the sums of squares. A multivariate ANOVA, however, has some properties different than the univariate ANOVA. There are several critical facts that come out of the partitioning of sums of squares and its consequences. Standard deviation, correlation coefficient, and the whole process of decomposing a set of data into its component parts are very closely related to each other, because they all represent various outcomes obtained from the fundamental process of partitioning the sums of squares. This is where the convergence of statistics and chemometrics can be seen. The cross-product matrix, which appears so often in chemometric calculations and is so casually used in chemometrics, has a very close and fundamental connection to one of the most basic operations of statistics. The relationship is that the sums of squares and cross-products in the cross-product matrix equal the sum of squares of the original data. These relationships are not approximations, and not “within statistical variation,” but are mathematically (algebraically) exact quantities.