Principal component analysis
Mozafar Poshtam2002, Encyclopedia of Statistics in Behavioral Science
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Abstract
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This comprehensive update of the principal component analysis (PCA) book reflects significant advancements in the field since its first edition. While preserving foundational content, it includes new chapters focusing on recent methodologies and applications, particularly in atmospheric science, and enhances discussions on ordination methods, component retention, and outlier detection. New theoretical approaches to interpreting principal components are introduced, complementing PCA with alternative procedures. Overall, this edition broadens the scope of PCA's applicability, probing deeper into existing literature and innovative techniques.
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New Interpretation of Principal Components Analysis
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A new look on the principal component analysis has been presented. Firstly, a geometric interpretation of determination coefficient was shown. In turn, the ability to represent the analyzed data and their interdependencies in the form of easy-to-understand basic geometric structures was shown. As a result of the analysis of these structures it was proposed to enrich the classical PCA. In particular, it was proposed a new criterion for the selection of important principal components and a new algorithm for clustering primary variables by their level of similarity to the principal components. Virtual and real data spaces, as well as tensor operations on data, have also been identified.The anisotropy of the data was identified too.
Paper On PCA
Principal component analysis (PCA) is a mainstay of modern data analysis-a black box that is widely used but poorly understood. The goal of this paper is to dispel the magic behind this black box. This tutorial focuses on building a solid intuition for how and why principal component analysis works; furthermore, it crystallizes this knowledge by deriving from first principals , the mathematics behind PCA. This tutorial does not shy away from explaining the ideas informally , nor does it shy away from the mathematics. The hope is that by addressing both aspects, readers of all levels will be able to gain a better understanding of the power of PCA as well as the when, the how and the why of applying this technique.
An Overview of Principal Component Analysis
Journal of Signal and Information Processing, 2013
The principal component analysis (PCA) is a kind of algorithms in biometrics. It is a statistics technical and used orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables. PCA also is a tool to reduce multidimensional data to lower dimensions while retaining most of the information. It covers standard deviation, covariance, and eigenvectors. This background knowledge is meant to make the PCA section very straightforward, but can be skipped if the concepts are already familiar.
A Corrected Criterion for Selecting the Optimum Number of Principal Components
stat.tugraz.at
Abstract: Determining the optimum number of components to be retained is a key problem in principal component analysis (PCA). Besides the rule of thumb estimates there exist several sophisticated methods for automati-cally selecting the dimensionality of the data. Based ...
Multivariate Statistical Data Analysis-Principal Component Analysis (PCA
Principal component analysis (PCA) is a multivariate technique that analyzes a data table in which observations are described by several inter-correlated quantitative dependent variables. Its goal is to extract the important information from the statistical data to represent it as a set of new orthogonal variables called principal components, and to display the pattern of similarity between the observations and of the variables as points in spot maps. Mathematically, PCA depends upon the eigen-decomposition of positive semi-definite matrices and upon the singular value decomposition (SVD) of rectangular matrices. It is determined by eigenvectors and eigenvalues. Eigenvectors and eigenvalues are numbers and vectors associated to square matrices. Together they provide the eigen-decomposition of a matrix, which analyzes the structure of this matrix such as correlation, covariance, or cross-product matrices. Performing PCA is quite simple in practice. Organize a data set as an m × n matrix, where m is the number of measurement types and n is the number of trials. Subtract of the mean for each measurement type or row x i. Calculate the SVD or the eigenvectors of the co-variance. It was found that there were many interesting applications of PCA, out of which in day today life knowingly or unknowingly multivariate data analysis and image compression are being used alternatively.
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A general asymptotic framework is developed for studying consistency properties of principal component analysis (PCA). Our framework includes several previously studied domains of asymptotics as special cases and allows one to investigate interesting connections and transitions among the various domains. More importantly, it enables us to investigate asymptotic scenarios that have not been considered before, and gain new insights into the consistency, subspace consistency and strong inconsistency regions of PCA and the boundaries among them. We also establish the corresponding convergence rate within each region. Under general spike covariance models, the dimension (or the number of variables) discourages the consistency of PCA, while the sample size and spike information (the relative size of the population eigenvalues) encourages PCA consistency. Our framework nicely illustrates the relationship among these three types of information in terms of dimension, sample size and spike size, and rigorously characterizes how their relationships affect PCA consistency.
Principal components analysis: From global to local
2010
Principal Components Analysis (PCA) is a widely used technique in the social and physical sciences. Originally developed by Pearson (1901), the details of extracting components for a data matrix were presented in an extensive two part paper by Hotelling (1933). In this paper we examine some problems concerning the extraction and interpretation of geographically weighted principal components (Fotheringham et al. 2002: p196-202).
Maximum likelihood principal component analysis
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The theoretical principles and practical implementation of a new method for multivariate data analysis, maximum likelihood principal component analysis (MLPCA), are described. MLCPA is an analog to principal component analysis (PCA) that incorporates information about measurement errors to develop PCA models that are optimal in a maximum likelihood sense. The theoretical foundations of MLPCA are initially established using a regression model and extended to the framework of PCA and singular value decomposition (SVD). An efficient and reliable algorithm based on an alternating regression method is described. Generalization of the algorithm allows its adaptation to cases of correlated errors provided that the error covariance matrix is known. Models with intercept terms can also be accommodated. Simulated data and near-infrared spectra, with a variety of error structures, are used to evaluate the performance of the new algorithm. Convergence times depend on the error structure but are typically around a few minutes. In all cases, models determined by MLPCA are found to be superior to those obtained by PCA when non-uniform error distributions are present, although the level of improvement depends on the error structure of the particular data set.
A Comparative Study of Principal Component Analysis Techniques
1998
Principal Component Analysis (PCA) is a useful technique for reducing the dimensionality of datasets for compression or recognition purposes. Many different methods have been proposed for performing PCA. This study aims to compare these methods by analysing the solutions which these methods find. We have estimated the correlation between these solutions and produced the errors using bootstrap resampling.
Generalizations of Principal Components Analysis
], they proposed the Centers and the Tops Methods to extend the known principal components analysis method, PCA, to a particular kind of symbolic objects characterized by multi-valued variables of interval-type. Nevertheless they utilize the classical circle of correlation to represent the variables. In this paper we derive duality relations for PCA of interval-type data and we propose a method to compute the symbolic correlation circle using the duality relations. Also, in this article we propose an algorithm for PCA when the variables are histogram type. This algorithm also works if the data table has variables of interval type and histogram type mixed. If all the variables are interval type it produces the same output as the one produced by the algorithm of the Centers Method.
History of Nonlinear Principal Component Analysis
2013
We discuss several forms of Nonlinear Principal Component Analysis (NLPCA) that have been proposed over the years: Linear PCA with optimal scaling, aspect analysis of correlations, Guttman’s MSA, Logit and Probit PCA of binary data, and Logistic Homogeneity Analysis. They are compared with Multiple Correspondence Analysis (MCA), which we also consider to be a form of NLPCA.
Principal Component Analysis -A Tutorial
Dimensionality reduction is one of the preprocessing steps in many machine learning applications and it is used to transform the features into a lower dimension space. Principal Component Analysis (PCA) technique is one of the most famous unsupervised dimensionality reduction techniques. The goal of the PCA is to find the space, which represents the direction of the maximum variance of the given data. This paper highlights the basic background needed to understand and implement the PCA technique. This paper starts with basic definitions of the PCA technique and the algorithms of two methods of calculating PCA, namely, the covariance matrix and Singular Value Decomposition (SVD) methods. Moreover, a number of numerical examples are illustrated to show how the PCA space is calculated in easy steps. Three experiments are conducted to show how to apply PCA in the real applications including biometrics, image compression, and visualization of high-dimensional datasets.
Singular value decomposition and principal component analysis
2004
This note is intended as a brief introduction to singular value decomposition (SVD) and principal component analysis (PCA). These are very useful techniques in data analysis and visualization. Further information can found for example in Numerical Recipes, section 2.6, available free online: http://www. nr. com/http://www. library. cornell. edu/nr/bookcpdf. html and in C. Bishop, Neural Networks for Pattern Recognition, Chapter 8.
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