Kernel Principal Component Analysis (KPCA) is a popular generalization of linear PCA that allows non-linear feature extraction. In KPCA, data in the input space is mapped to higher (usually) dimensional feature space where the data can be linearly modeled. The feature space is typically induced implicitly by a kernel function, and linear PCA in the feature space is performed via the kernel tric...

This tutorial is designed to give the reader a short overview of Principal Component Analysis (PCA) using R. PCA is a useful statistical method that has found application in a variety of fields and is a common technique for finding patterns in data of high dimension. Consider we are confronted with the following situation: The data, we want to work with, are in form of a matrix (xij)i=1...N,j=1...

1. Preliminaries Theorem A-1. (Theorem 3.1, (Chang, 2012)) Let A ∈ Rm×n be of full column rank with QR factorization A = QR, ∆A be a perturbation in A, and A + ∆A = (Q + ∆Q)(R + ∆R) be the QR-factorization of A + ∆A. Let PA and PA⊥ be the orthogonal projectors onto the range of A and the orthogonal complement of the range of A, respectively. LetQ⊥ be an orthonormal matrix such that matrix [Q,Q⊥...

Functional principal component analysis (FPCA) is a popular approach in functional data analysis to explore major sources of variation in a sample of random curves. These major sources of variation are represented by functional principal components (FPCs). Most existing FPCA approaches use a set of flexible basis functions such as B-spline basis to represent the FPCs, and control the smoothness...

PETER D. WENTZELL, DARREN T. ANDREWS, DAVID C. HAMILTON, KLAAS FABER AND BRUCE R. KOWALSKI 1 Trace Analysis Research Centre, Department of Chemistry, Dalhousie University, Halifax, Nova Scotia B3H 4J3, Canada 2 Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada 3 Center for Process Analytical Chemistry, University of Washingto...

High-dimensional Principal Component Analysis by Arash Ali Amini Doctor of Philosophy in Electrical Engineering University of California, Berkeley Associate Professor Martin Wainwright, Chair Advances in data acquisition and emergence of new sources of data, in recent years, have led to generation of massive datasets in many fields of science and engineering. These datasets are usually characte...

We present an extension of sparse PCA, or sparse dictionary learning, where the sparsity patterns of all dictionary elements are structured and constrained to belong to a prespecified set of shapes. This structured sparse PCA is based on a structured regularization recently introduced by [1]. While classical sparse priors only deal with cardinality, the regularization we use encodes higher-orde...

multivariate statistical techniques such as cluster analysis, multidimensional scaling and principal component analysis were applied to evaluate the temporal and spatial variations in water quality data set generated for two years (2008-2010) from six monitoring stations of veli-akkulam lake and compared with a regional reference lake vellayani of south india. seasonal variations of 14 differen...