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.
Contents
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 and visualization.
Eigenvectors and Eigenvalues
$ \vec{x}\in\mathbb{R}^{n} $
$ D(\lambda)=det\left(A-\lambda I\right) $
$ A=\left[\begin{matrix}-5 & 2\\ 2 & -2 \end{matrix}\right]. $
Technical Procedure of PCA
Examples
Questions and comments
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