Multivariate Time Series Analysis and Applications

4Principle component analysis of multivariate time series

Principal component analysis (PCA) is a statistical technique used for explaining the variance–covariance matrix of a set of m‐dimensional variables through a few linear combinations of these variables. In this chapter, we will illustrate the method to show that a large m‐dimensional process can often be sufficiently explained by smaller k principal components and thus reduce a higher dimension problem to one with fewer dimensions.

4.1 Introduction

Because of the advances of computing technology, the dimension, m, used in data analysis has become larger and larger. PCA is a statistical method that converts a set of correlated variables into a set of uncorrelated variables through an orthogonal transformation. Hopefully, a small subset of the uncorrelated variables carries sufficient information of the original large set of correlated variables.

The PCA concept to achieve parsimony was first introduced by Karl Pearson (1901). However, it was Hotelling (1933) who developed the method of stochastic variables and officially introduced the term of principal components in 1933. The techniques can be used on general variables or standardized variables and hence either the covariance matrix or correlation matrix. To many people, these techniques seem related, but in reality, they could be quite different. The goal of the method is to represent a large m‐dimensional process with much smaller k principal components and hence ...