Introduction to Principal Components Analysis

Understanding the inverse equivalent width - luminosity relationship (Baldwin Effect), the topic of this meeting, requires extracting information on continuum and emission line parameters from samples of AGN. We wish to discover whether, and how, different subsets of measured parameters may correlate with each other. This general problem is the domain of Principal Components Analysis (PCA). We discuss the purpose, principles, and the interpretation of PCA, using some examples from QSO spectroscopy. The hope is that identification of relationships among subsets of correlated variables may lead to new physical insight.Comment: Invited review to appear in ``Quasars and Cosmology'', A.S.P. Conference Series 1999. eds. G. J. Ferland, J. A. Baldwin, (San Francisco: ASP). 10 pages, 2 figure

Integrating Data Transformation in Principal Components Analysis

Principal component analysis (PCA) is a popular dimension-reduction method to reduce the complexity and obtain the informative aspects of high-dimensional datasets. When the data distribution is skewed, data transformation is commonly used prior to applying PCA. Such transformation is usually obtained from previous studies, prior knowledge, or trial-and-error. In this work, we develop a model-based method that integrates data transformation in PCA and finds an appropriate data transformation using the maximum profile likelihood. Extensions of the method to handle functional data and missing values are also developed. Several numerical algorithms are provided for efficient computation. The proposed method is illustrated using simulated and real-world data examples. Supplementary materials for this article are available online

Spectral components analysis of diffuse emission processes

We develop a novel method to separate the components of a diffuse emission process based on an association with the energy spectra. Most of the existing methods use some information about the spatial distribution of components, e.g., closeness to an external template, independence of components etc., in order to separate them. In this paper we propose a method where one puts conditions on the spectra only. The advantages of our method are: 1) it is internal: the maps of the components are constructed as combinations of data in different energy bins, 2) the components may be correlated among each other, 3) the method is semi-blind: in many cases, it is sufficient to assume a functional form of the spectra and determine the parameters from a maximization of a likelihood function. As an example, we derive the CMB map and the foreground maps for seven yeas of WMAP data. In an Appendix, we present a generalization of the method, where one can also add a number of external templates.Comment: 21 pages, 7 figure

Structural reliability analysis of laminated CMC components

For laminated ceramic matrix composite (CMC) materials to realize their full potential in aerospace applications, design methods and protocols are a necessity. The time independent failure response of these materials is focussed on and a reliability analysis is presented associated with the initiation of matrix cracking. A public domain computer algorithm is highlighted that was coupled with the laminate analysis of a finite element code and which serves as a design aid to analyze structural components made from laminated CMC materials. Issues relevant to the effect of the size of the component are discussed, and a parameter estimation procedure is presented. The estimation procedure allows three parameters to be calculated from a failure population that has an underlying Weibull distribution

Properties of Design-Based Functional Principal Components Analysis

This work aims at performing Functional Principal Components Analysis (FPCA) with Horvitz-Thompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and consistent. Under mild assumptions, asymptotic variances are derived for the FPCA' estimators and consistent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach.Comment: Revised version for J. of Statistical Planning and Inference (January 2009

Independent components in spectroscopic analysis of complex mixtures

We applied two methods of "blind" spectral decomposition (MILCA and SNICA) to quantitative and qualitative analysis of UV absorption spectra of several non-trivial mixture types. Both methods use the concept of statistical independence and aim at the reconstruction of minimally dependent components from a linear mixture. We examined mixtures of major ecotoxicants (aromatic and polyaromatic hydrocarbons), amino acids and complex mixtures of vitamins in a veterinary drug. Both MICLA and SNICA were able to recover concentrations and individual spectra with minimal errors comparable with instrumental noise. In most cases their performance was similar to or better than that of other chemometric methods such as MCR-ALS, SIMPLISMA, RADICAL, JADE and FastICA. These results suggest that the ICA methods used in this study are suitable for real life applications. Data used in this paper along with simple matlab codes to reproduce paper figures can be found at http://www.klab.caltech.edu/~kraskov/MILCA/spectraComment: 22 pages, 4 tables, 6 figure

Sparse logistic principal components analysis for binary data

We develop a new principal components analysis (PCA) type dimension reduction method for binary data. Different from the standard PCA which is defined on the observed data, the proposed PCA is defined on the logit transform of the success probabilities of the binary observations. Sparsity is introduced to the principal component (PC) loading vectors for enhanced interpretability and more stable extraction of the principal components. Our sparse PCA is formulated as solving an optimization problem with a criterion function motivated from a penalized Bernoulli likelihood. A Majorization--Minimization algorithm is developed to efficiently solve the optimization problem. The effectiveness of the proposed sparse logistic PCA method is illustrated by application to a single nucleotide polymorphism data set and a simulation study.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS327 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org