Applied Multivariate Statistical Analysis - UFPR

1 Applied MultivariateStatistical Analysis Wolfgang H ardleL eopold Simar Version: 29th April 2003 ContentsIDescriptive Techniques111 Comparison of Boxplots.. Histograms.. Kernel Densities.. Scatterplots.. Chernoff-Flury Faces.. Andrews Curves.. Parallel Coordinates Plots.. Boston Housing.. Exercises..52II Multivariate Random Variables552 A Short Excursion into Matrix Elementary Operations.. Spectral Decompositions.. Quadratic Forms.. Derivatives.. Partitioned Matrices.. Geometrical Aspects.. Exercises..793 Moving to Higher Covariance.. Correlation.. Summary Statistics.. Linear Model for Two Variables.. Simple Analysis of Variance.. Multiple Linear Model.. Boston Housing.. Exercises..1154 Multivariate Distribution and Density Function.. Moments and Characteristic Functions.

2 Transformations.. The Multinormal Distribution.. Sampling Distributions and Limit Theorems.. Bootstrap.. Exercises..1525 Theory of the Elementary Properties of the Multinormal.. The Wishart Distribution.. Hotelling Distribution.. Spherical and Elliptical Distributions.. Exercises..169 Contents36 Theory of The Likelihood Function.. The Cramer-Rao Lower Bound.. Exercises..1817 Hypothesis Likelihood Ratio Test.. Linear Hypothesis.. Boston Housing.. Exercises..212 III Multivariate Techniques2178 Decomposition of Data Matrices by The Geometric Point of View.. Fitting thep-dimensional Point Cloud.. Fitting then-dimensional Point Cloud.. Relations between Subspaces.. Practical Computation.. Exercises..2329 Principal Components Standardized Linear Combinations.

3 Principal Components in Practice.. Interpretation of the PCs.. Asymptotic Properties of the PCs.. Normalized Principal Components Analysis .. Principal Components as a Factorial Method.. Common Principal Components.. Boston Housing.. More Examples.. Exercises..27210 Factor The Orthogonal Factor Model.. Estimation of the Factor Model.. Factor Scores and Strategies.. Boston Housing.. Exercises..29811 cluster The Problem.. The Proximity between Objects.. cluster Algorithms.. Boston Housing.. Exercises..31812 Discriminant Allocation Rules for Known Distributions.. Discrimination Rules in Practice.. Boston Housing.. Exercises..33913 Correspondence Motivation.. Chi-square Decomposition.. Correspondence Analysis in Practice.. Exercises..358 Contents514 Canonical Correlation Most Interesting Linear Combination.

4 Canonical Correlation in Practice.. Exercises..37215 Multidimensional The Problem.. Metric Multidimensional Scaling.. The Classical Solution.. Nonmetric Multidimensional Scaling.. Shepard-Kruskal algorithm.. Exercises..39116 Conjoint Measurement Introduction.. Design of Data Generation.. Estimation of Preference Orderings.. Exercises..40517 Applications in Portfolio Choice.. Efficient Portfolio.. Efficient Portfolios in Practice.. The Capital Asset Pricing Model (CAPM).. Exercises..41818 Highly Interactive, Computationally Intensive Simplicial Depth.. Projection Pursuit.. Sliced Inverse Regression.. Boston Housing.. Exercises..440A Symbols and Notation443B Boston Housing Data.. Swiss Bank Notes.. Car Data.. Classic Blue Pullovers Data.. Companies Data.. French Food Data.. Car Marks.

5 French Baccalaur eat Frequencies.. Journaux Data.. Crime Data.. Plasma Data.. WAIS Data.. ANOVA Data.. Timebudget Data.. Geopol Data.. Health Data.. Vocabulary Data.. Athletic Records Data.. Unemployment Data.. Annual Population Data..478 Bibliography479 Index483 PrefaceMost of the observable phenomena in the empirical sciences are of a Multivariate financial studies, assets in stock markets are observed simultaneously and their jointdevelopment is analyzed to better understand general tendencies and to track indices. Inmedicine recorded observations of subjects in different locations are the basis of reliablediagnoses and medication. In quantitative marketing consumer preferences are collected inorder to construct models of consumer behavior. The underlying theoretical structure ofthese and many other quantitative studies of Applied sciences is Multivariate .

6 This bookon Applied Multivariate Statistical Analysis presents the tools and concepts of multivariatedata Analysis with a strong focus on aim of the book is to present Multivariate data Analysis in a way that is understandablefor non-mathematicians and practitioners who are confronted by Statistical data is achieved by focusing on the practical relevance and through the e-book character ofthis text. All practical examples may be recalculated and modified by the reader using astandard web browser and without reference or application of any specific book is divided into three main parts. The first part is devoted to graphical techniquesdescribing the distributions of the variables involved. The second part deals with multivariaterandom variables and presents from a theoretical point of view distributions, estimatorsand tests for various practical situations.

7 The last part is on Multivariate techniques andintroduces the reader to the wide selection of tools available for Multivariate data data sets are given in the appendix and are downloadable Thetext contains a wide variety of exercises the solutions of which are given in a separatetextbook. In addition a full set of transparencies provided making iteasier for an instructor to present the materials in this book. All transparencies contain hyperlinks to the Statistical web service so that students and instructors alike may recompute allexamples via a standard web first section on descriptive techniques is on the construction of the boxplot. Here thestandard data sets on genuine and counterfeit bank notes and on the Boston housing data areintroduced. Flury faces are shown in , followed by the presentation of Andrewscurves and parallel coordinate plots.

8 Histograms, kernel densities and scatterplots completethe first part of the book. The reader is introduced to the concept of skewness and correlationfrom a graphical point of the beginning of the second part of the book the reader goes on a short excursion intomatrix algebra. Covariances, correlation and the linear model are introduced. This sectionis followed by the presentation of the ANOVA technique and its application to the multiplelinear model. In Chapter4the Multivariate distributions are introduced and thereafterspecialized to the multinormal. The theory of estimation and testing ends the discussion onmultivariate random third and last part of this book starts with a geometric decomposition of data is influenced by the French school of analyse de donn ees. This geometric point of viewis linked to principal components Analysis in Chapter9.

9 An important discussion on factoranalysis follows with a variety of examples from psychology and economics. The section oncluster Analysis deals with the various cluster techniques and leads naturally to the problemof discrimination Analysis . The next chapter deals with the detection of correspondencebetween factors. The joint structure of data sets is presented in the chapter on canonicalcorrelation Analysis and a practical study on prices and safety features of automobiles isgiven. Next the important topic of multidimensional scaling is introduced, followed by thetool of conjoint measurement Analysis . The conjoint measurement Analysis is often usedin psychology and marketing in order to measure preference orderings for certain applications in finance (Chapter17) are numerous. We present here the CAPM modeland discuss efficient portfolio allocations.

10 The book closes with a presentation on highlyinteractive, computationally intensive book is designed for the advanced bachelor and first year graduate student as well asfor the inexperienced data analyst who would like a tour of the various Statistical tools ina Multivariate data Analysis workshop. The experienced reader with a bright knowledge ofalgebra will certainly skip some sections of the Multivariate random variables part but willhopefully enjoy the various mathematical roots of the Multivariate techniques. A graduatestudent might think that the first part on description techniques is well known to him from histraining in introductory statistics. The mathematical and the Applied parts of the book (II,III) will certainly introduce him into the rich realm of Multivariate Statistical data inexperienced computer user of this e-book is slowly introduced to an interdisciplinaryway of Statistical thinking and will certainly enjoy the various practical examples.