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Basics and Examples of Principal Component Analysis (PCA)

A slecture by Sujin Jang

Partly based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.

Introduction

Principal Component Analysis (PCA) is one of famous techniqeus for dimension reduction, feature extraction, and data visualization. In general, PCA is defined by a transformation of a high dimensional vector space into a low dimensional space. Let's consider visualization of 10-dim data. It is barely possible to effectively show the shape of such high dimensional data distribution. PCA provides an efficient way to reduce the dimensionalty (i.e., from 10 to 2), so it is much easier to visualize the shape of data distribution. PCA is also useful in the modeling of robust classifier where considerably small number of high dimensional training data is provided. By reducing the dimensions of learning data sets, PCA provides an effective and efficient method for data description and classification.

This lecture is designed to provide a mathematical background of PCA and its applications. First, fundamentals of linear algebra is introduced that will be used in PCA. Technical procedure of PCA will be provided to aid understanding of practical implementation of PCA. Based on the procedure, several examples of PCA will be given in dimension reduction.

Eigenvectors, Eigenvalues, and Singular Vector Decompositoin

To understand mechanism of PCA, it is important to understand some important concepts in linear algebra. In this lecture, we will briefly discuss eigenvectors and eigenvalues of a matrix. Also singular vector decomposition (SVD) will be examined in the extraction of principal components.

Eigenvectors and Eigenvalues

Let define a n-by-n matrix A and a non-zero vector $\vec{x}\in\mathbb{R}^{n}$ . If there exists a scalar value $\lambda$ which satisfies the vector equation

A\vec{x}=\lambda\vec{x}

,

we define $\lambda$ as an eigenvalue of the matrix A, and the corresponding non-zero vector $\vec{x}$ is called an eigenvector of the matrix A. To determine eigenvalues and eigenvectors a characteristic equation

D(\lambda)=det\left(A-\lambda I\right)

is used. Here is an example of determining eigenvectors and eigenvalues where the matrix A is given by

A=\left[\begin{matrix}-5 & 2\\ 2 & -2 \end{matrix}\right].

Then the characteristic equation is given by

D(\lambda)=\left(-5-\lambda\right)\left(-2-\lambda\right)-4=\lambda^{2}+7\lambda+6=0.

By solving the quadratic equation for $\lambda$ , we will have two eigenvalues $\lambda_{1}=-1$ and $\lambda_{2}=-6$ . By substituting $\lambda's$ into the vector equation, we can obtain corresponding eigenvectors;

\lambda_{1}:\;\left(A-\lambda_{1}I\right)\vec{x}=0\Rightarrow\begin{cases} -4x_{1}+2x_{2} & =0\\ 2x_{1}-x_{2} & =0 \end{cases} \Rightarrow\vec{x}_{1}=\left[\begin{matrix}1\\ 2 \end{matrix}\right]

\lambda_{2}:\;\left(A-\lambda_{2}I\right)\vec{x}=0\Rightarrow\begin{cases} x_{1}+2x_{2} & =0\\ 2x_{1}+4x_{2} & =0 \end{cases} \Rightarrow\vec{x}_{2}=\left[\begin{matrix}2\\ -1 \end{matrix}\right]

Technical Procedure of PCA

Examples

Questions and comments

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