2.03 - Principal Component Analysis

Abstract

Principal Component Analysis (PCA) is a multivariate exploratory analysis method, useful to separate systematic variation from noise. It allows to define a space of reduced dimensions that preserves the relevant information of the original data and allows visualization of objects (scores) and variables (loadings). PCA requires multivariate data, meaning many variables measured on many objects. Data, vectors and matrices are defined and a short summary of necessary linear algebra is given. Purely mathematical almost identical definitions of PCA and Singular Value Decomposition (SVD) are shown, but in chemometrics, PCA always has a residual and a number of meaningful components, the rank. This leads to a discussion of numerical and visual diagnostics for finding the rank and checking the residual. The visualization of scores and loadings is introduced by means of two small examples. Data preprocessing is also given consideration.

Keywords

Data matrix; Eigenvalue; Eigenvector; Loading plot; Mean centering; Number of components; Objects; Preprocessing; Rank; Residual; Score plot; Scree plot; UV-scaling; Variable types; Vector

Published in

Title: Comprehensive Chemometrics (Second Edition) : Chemical and Biochemical Data Analysis
Publisher: Elsevier

SLU Authors

• Geladi, Paul

  • Department of Forest Biomaterials and Technology, Swedish University of Agricultural Sciences
    

UKÄ Subject classification

Mathematical Analysis
Probability Theory and Statistics

Publication identifier

• DOI: https://doi.org/10.1016/B978-0-12-409547-2.14892-9
• ISBN: 9780444641656

Permanent link to this page (URI)

https://res.slu.se/id/publ/129846