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2 - Principal Component Analysis

from I - Classical Methods

Published online by Cambridge University Press:  05 June 2014

Inge Koch
Affiliation:
University of Adelaide
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Summary

Mathematics, rightly viewed, possesses not only truth, but supreme beauty (Bertrand Russell, Philosophical Essays No. 4, 1910).

Introduction

One of the aims in multivariate data analysis is to summarise the data in fewer than the original number of dimensions without losing essential information. More than a century ago, Pearson (1901) considered this problem, and Hotelling (1933) proposed a solution to it: instead of treating each variable separately, he considered combinations of the variables. Clearly, the average of all variables is such a combination, but many others exist. Two fundamental questions arise:

  1. How should one choose these combinations?

  2. How many such combinations should one choose?

There is no single strategy that always gives the right answer. This book will describe many ways of tackling at least the first problem.

Hotelling's proposal consisted in finding those linear combinations of the variables which best explain the variability of the data. Linear combinations are relatively easy to compute and interpret. Also, linear combinations have nice mathematical properties. Later methods, such as Multidimensional Scaling, broaden the types of combinations, but this is done at a cost: The mathematical treatment becomes more difficult, and the practical calculations will be more complex. The complexity increases with the size of the data, and it is one of the major reasons why Multidimensional Scaling has taken rather longer to regain popularity.

The second question is of a different nature, and its answer depends on the solution to the first.

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Publisher: Cambridge University Press
Print publication year: 2013

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  • Principal Component Analysis
  • Inge Koch, University of Adelaide
  • Book: Analysis of Multivariate and High-Dimensional Data
  • Online publication: 05 June 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139025805.003
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  • Principal Component Analysis
  • Inge Koch, University of Adelaide
  • Book: Analysis of Multivariate and High-Dimensional Data
  • Online publication: 05 June 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139025805.003
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Principal Component Analysis
  • Inge Koch, University of Adelaide
  • Book: Analysis of Multivariate and High-Dimensional Data
  • Online publication: 05 June 2014
  • Chapter DOI: https://doi.org/10.1017/CBO9781139025805.003
Available formats
×