A tutorial on Principal Components Analysis

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A tutorial on Principal Components Analysis
Lindsay I Smith
February 26, 2002

Chapter 1

This tutorial is designed to give the reader an understanding of Principal Components
Analysis (PCA). PCA is a useful statistical technique that has found application in
fields such as face recognition and image compression, and is a common technique for
finding patterns in data of high dimension.
Before getting to a description of PCA, this tutorial first introduces mathematical
concepts that will be used in PCA. It covers standard deviation, covariance, eigenvectors and eigenvalues. This background knowledge is meant to make the PCA section
very straightforward, but can be skipped if the concepts are already familiar.
There are examples all the way through this tutorial that are meant to illustrate the
concepts being discussed. If further information is required, the mathematics textbook
“Elementary Linear Algebra 5e” by Howard Anton, Publisher John Wiley & Sons Inc,
ISBN 0-471-85223-6 is a good source of information regarding the mathematical background.


Chapter 2

Background Mathematics
This section will attempt to give some elementary background mathematical skills that
will be required to understand the process of Principal Components Analysis. The
topics are covered independently of each other, and examples given. It is less important
to remember the exact mechanics of a mathematical technique than it is to understand
the reason why such a technique may be used, and what the result of the operation tells
us about our data. Not all of these techniques are used in PCA, but the ones that are not
explicitly required do provide the grounding on which the most important techniques
are based.
I have included a section on Statistics which looks at distribution measurements,
or, how the data is spread out. The other section is on Matrix Algebra and looks at
eigenvectors and eigenvalues, important properties of matrices that are fundamental to



The entire subject of statistics is based around the idea that you have this big set of data,
and you want to analyse that set in terms of the relationships between the individual
points in that data set. I am going to look at a few of the measures you can do on a set
of data, and what they tell you about the data itself.


Standard Deviation

To understand standard deviation, we need a data set. Statisticians are usually concerned with taking a sample of a population. To use election...
A tutorial on Principal Components Analysis
Lindsay I Smith
February 26, 2002
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