Introduction PCA (Principal Component Analysis) Characteristics:

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Presentation transcript:

Introduction PCA (Principal Component Analysis) Characteristics: 2019/1/2 Introduction PCA (Principal Component Analysis) An effective statistical method for reducing dataset dimensions while keeping spatial characteristics as much as possible Applications Image compression Pattern recognition Characteristics: Designed for unlabeled data A linear transformation with solid mathematical foundation Easy to apply 2019/1/2

Problem Definition Input Output: 2019/1/2 Problem Definition Input A dataset of n d-dim points which are zero justified Output: A unity vector u such that the square sum of the dataset’s projection onto u is maximized. 2019/1/2

Math Formulation Formulation Dataset representation: 2019/1/2 Math Formulation Formulation Dataset representation: X is d by n, with n>d Projection of each column of X onto u: Square sum: Objective function with a constraint on u: 2019/1/2

Optimization of the Obj. Function 2019/1/2 Optimization of the Obj. Function Optimization Set the gradient to zero: Implication: u is the eigenvector while l is the eigenvalue When u is the eigenvector: If we arrange eigenvalues such that: Max of J(u) is l1, which occurs at u=u1 Min of J(u) is ld, which occurs at u=ud 2019/1/2

Steps for PCA Find the sample mean: Compute the covariance matrix: 2019/1/2 Steps for PCA Find the sample mean: Compute the covariance matrix: Find the eigenvalues of C and arrange them into descending order, with the corresponding eigenvectors The transformation is , with 2019/1/2

2019/1/2 Tidbits PCA is designed for unlabeled data. For classification problem, we usually resort to LDA (linear discriminant analysis) for dimension reduction. If d>>n, then we need to have a workaround for computing the eigenvectors 2019/1/2

2019/1/2 Example of PCA IRIS dataset projection 2019/1/2

Weakness of PCA Not designed for classification problem (with labeled 2019/1/2 Weakness of PCA Not designed for classification problem (with labeled training data. Ideal situation Adversary situation 2019/1/2

Linear Discriminant Analysis 2019/1/2 Linear Discriminant Analysis LDA projection onto directions that can best separate data of different classes. Adversary situation for PCA Ideal situation for LDA 2019/1/2