Factor Analysis. Factor Analysis


 Malcolm Hall
 1 years ago
 Views:
Transcription
1 Factor Analysis Principal Components Analysis, e.g. of stock price movements, sometimes suggests that several variables may be responding to a small number of underlying forces. In the factor model, we assume that such latent variables, or factors, exist. NC STATE UNIVERSITY 1 / 38
2 The Orthogonal Factor Model equation: X 1 µ 1 = l 1,1 F 1 + l 1,2 F l 1,m F m + ɛ 1, X 2 µ 2 = l 2,1 F 1 + l 2,2 F l 2,m F m + ɛ 2,.. X p µ p = l p,1 F 1 + l p,2 F l p,m F m + ɛ p, where: F1, F 2,..., F m are the common factors (latent variables); li,j is the loading of variable i, X i, on factor j, F j ; ɛi is a specific factor, affecting only X i. NC STATE UNIVERSITY 2 / 38
3 In matrix form: X µ = L F + ɛ. p 1 p 1 p m 1 p 1 To make this identifiable, we further assume, with no loss of generality: E(F) = 0 m 1 Cov(F) = I m m E(ɛ) = 0 p 1 Cov(ɛ, F) = 0 p m NC STATE UNIVERSITY 3 / 38
4 and with serious loss of generality: Cov(ɛ) = Ψ = diag (ψ 1, ψ 2,..., ψ p ). In terms of the observable variables X, these assumptions mean that E(X) = µ, Cov(X) = Σ = L L p m p + Ψ p p. Usually X is standardized, so Σ = R. The observable X and the unobservable F are related by Cov(X, F) = L. NC STATE UNIVERSITY 4 / 38
5 Some terminology: the (i, i) entry of the matrix equation Σ = LL + Ψ is or where σ i,i }{{} Var(X i ) is the i th communality. = li,1 2 + li, li,m 2 } {{ } Communality σ i,i = h 2 i + ψ i h 2 i = l 2 i,1 + l 2 i,2 + + l 2 i,m + ψ }{{} i, Specific variance Note that if T is (m m) orthogonal, then (LT)(LT) = LL, so loadings LT generate the same Σ as L: loadings are not unique. NC STATE UNIVERSITY 5 / 38
6 Existence of Factor Representation For any p, every (p p) Σ can be factorized as Σ = LL for (p p) L, which is a factor representation with m = p and Ψ = 0; however, m = p is not much use we usually want m p. For p = 3, every (3 3) Σ can be represented as Σ = LL + Ψ for (3 1) L, which is a factor representation with m = 1, but Ψ may have negative elements. NC STATE UNIVERSITY 6 / 38
7 In general, we can only approximate Σ by LL + Ψ. Principal components method: the spectral decomposition of Σ is with m = p. Σ = EΛE = ( EΛ 1/2) ( EΛ 1/2) = LL If λ 1 + λ λ m λ m λ p, and L (m) is the first m columns of L, then Σ L (m) L (m) gives such an approximation with Ψ = 0. NC STATE UNIVERSITY 7 / 38
8 The remainder term Σ L (m) L (m) is nonnegative definite, so its diagonal entries are nonnegative we can get a closer approximation as Σ L (m) L (m) + Ψ (m), ( where Ψ (m) = diag Σ L (m) L (m) ). SAS proc factor program and output: proc factor data = all method = prin; var cvx  xom; title Method = Principal Components ; proc factor data = all method = prin nfact = 2 plot; var cvx  xom; title Method = Principal Components, 2 factors ; NC STATE UNIVERSITY 8 / 38
9 Principal Factor Solution Recall the Orthogonal Factor Model which implies X = LF + ɛ Σ = LL + Ψ. The mfactor Principal Component solution is to approximate Σ (or, if we standardize the variables, R) by a rankm matrix using the spectral decomposition Σ = λ 1 e 1 e λ m e m e m + λ m+1 e m+1 e m λ p e p e p. The first m terms give the best rankm approximation to Σ. NC STATE UNIVERSITY 9 / 38
10 We can sometimes achieve higher communalities (= diag (LL )) by either: specifying an initial estimate of the communalities iterating the solution or both. Suppose we are working with R. Given initial communalities hi 2, form the reduced correlation matrix h1 2 r 1,2... r 1,p r 2,1 h r 2,p R r = r p,1 r.. p,2 h 2 p NC STATE UNIVERSITY 10 / 38
11 Now use the spectral decomposition of R r to find its best rankm approximation R r L r L r. New communalities are h 2 i = m j=1 Find Ψ by equating the diagonal terms: l 2 i,j. ψ i = 1 h 2 i, or Ψ = I diag ( L r L r ). NC STATE UNIVERSITY 11 / 38
12 This is the Principal Factor solution. The Principal Component solution is the special case where the initial communalities are all 1. In proc factor, use method = prin as for the Principal Component solution, but also specify the initial communalities: the priors =... option on the proc factor statement specifies a method, such as squared multiple correlations (priors = SMC); the priors statement provides explicit numerical values. NC STATE UNIVERSITY 12 / 38
13 SAS program and output: proc factor data = all method = prin priors = smc; title Method = Principal Factors ; var cvx  xom; In this case, the communalities are smaller than for the Principal Component solution. NC STATE UNIVERSITY 13 / 38
14 Other choices for the priors option include: MAX maximum absolute correlation with any other variable; ASMC Adjusted SMC (adjusted to make their sum equal to the sum of the maximum absolute correlations); ONE 1; RANDOM uniform on (0, 1). NC STATE UNIVERSITY 14 / 38
15 Iterated Principal Factors One issue with both Principal Components and Principal Factors: if S or R is exactly in the form LL + Ψ (or, more likely, approximately in that form), neither method produces L and Ψ (unless you specify the true communalities). Solution: iterate! Use the new communalities as initial communalities to get another set of Principal Factors. Repeat until nothing much changes. NC STATE UNIVERSITY 15 / 38
16 In proc factor, use method = prinit; may also specify the initial communalities (default = ONE). SAS program and output: proc factor data = all method = prinit; title Method = Iterated Principal Factors ; var cvx  xom; The communalities are still smaller than for the Principal Component solution, but larger than for Principal Factors. NC STATE UNIVERSITY 16 / 38
17 Likelihood Methods If we assume that X N p (µ, Σ) with Σ = LL + Ψ, we can fit by maximum likelihood: ˆµ = x; L is not identified without a constraint (uniqueness condition) such as L Ψ 1 L = diagonal; still no closed form equation for ˆL; numerical optimization required. NC STATE UNIVERSITY 17 / 38
18 We can also test hypotheses about m with the likelihood ratio test (Bartlett s correction improves the χ 2 approximation): H0 : m = m 0 ; H A : m > m 0 ; ] 2 log likelihood ratio χ 2 with 1 2 [(p m 0 ) 2 p m 0 degrees of freedom. ( Degrees of freedom > 0 m0 < 1 ) 2 2p + 1 8p + 1. E.g. for p = 5, m 0 < m 0 2: p m 0 degrees of freedom NC STATE UNIVERSITY 18 / 38
19 In proc factor, use method = ml; may also specify the initial communalities (default = SMC); SAS program and output: proc factor data = all method = ml; var cvx  xom; title Method = Maximum Likelihood ; proc factor data = all method = ml heywood plot; var cvx  xom; title Method = Maximum Likelihood with Heywood fixup ; proc factor data = all method = ml ultraheywood plot; var cvx  xom; title Method = Maximum Likelihood with UltraHeywood fixup ; NC STATE UNIVERSITY 19 / 38
20 Note that the iteration can produce communalities > 1! Two fixes: use the Heywood option on the proc factor statement; caps the communalities at 1; use the UltraHeywood option on the proc factor statement; allows the iteration to continue with communalities > 1. NC STATE UNIVERSITY 20 / 38
21 Scaling and the Likelihood If the maximum likelihood estimates for a data matrix X are ˆL and ˆΨ, and Y = X D n p n p p is a scaled data matrix, with the columns of X scaled by the entries of the diagonal matrix D, then the maximum likelihood estimates for Y are DˆL and D 2 ˆΨ. That is, the mle s are invariant to scaling: ˆΣ Y = D ˆΣ X D. NC STATE UNIVERSITY 21 / 38
22 Proof: L Y (µ, Σ) = L X (D 1 µ, D 1 ΣD 1 ). No distinction between covariance and correlation matrices. NC STATE UNIVERSITY 22 / 38
23 Weighting and the Likelihood Recall the uniqueness condition Write L Ψ 1 L =, diagonal. Σ = Ψ 1 2 ΣΨ 1 2 = Ψ 1 2 (LL + Ψ)Ψ 1 2 ) ( ) = (Ψ 1 2 L Ψ 1 2 L + Ip = L L + I p. Σ is the weighted covariance matrix. NC STATE UNIVERSITY 23 / 38
24 Here L = Ψ 1 2 L and L L = L Ψ 1 L =. Note: Σ L = L L L + L = L + L = L ( + I m ) so the columns of L are the (unnormalized) eigenvectors of Σ, the weighted covariance matrix. NC STATE UNIVERSITY 24 / 38
25 Also (Σ I p )L = L so the columns of L are also the eigenvectors of Σ I p = Ψ 1 2 (Σ Ψ)Ψ 1 2, the weighted reduced covariance matrix. Since the likelihood analysis is transparent to scaling, the weighted reduced correlation matrix gives essentially the same results as the weighted reduced covariance matrix. NC STATE UNIVERSITY 25 / 38
26 Factor Rotation In the orthogonal factor model X µ = LF + ɛ, factor loadings are not always easily interpreted. J&W (p 504): Ideally, we should like to see a pattern of loadings such that each variable loads highly on a single factor and has small to moderate loadings on the remaining factors. That is, each row of L should have a single large entry. NC STATE UNIVERSITY 26 / 38
27 Recall from the corresponding equation Σ = LL + Ψ that L and LT give the same Σ for any orthogonal T. We can choose T to make the rotated loadings LT more readily interpreted. Note that rotation changes neither Σ nor Ψ, and hence the communalities are also unchanged. NC STATE UNIVERSITY 27 / 38
28 The Varimax Criterion Kaiser proposed a criterion that measures interpretability: ˆL is some set of loadings with communalities ĥi 2, i = 1, 2,..., p; ˆL is a set of rotated loadings, ˆL = ˆLT; l i,j = ˆl i,j /ĥi are scaled loadings; criterion is ( V = 1 m p p ) 2 4 l i,j 1 2 l i,j. p p j=1 i=1 i=1 NC STATE UNIVERSITY 28 / 38
29 Note that the term in [ ]s is the variance of the l 2 i,j in column i. Making this variance large tends to produce two clusters of scaled loadings, one of small values and one of large values. So each column of the rotated loading matrix tends to contain: a group of large loadings, which identify the variables associated with the factor; the remaining loadings are small. NC STATE UNIVERSITY 29 / 38
30 Example: Weekly returns for the 30 Dow Industrials stocks from January, 2005 to March, 2007 (115 returns). R code to rotate Principal Components 2 10: dowprcomp = prcomp(dow, scale. = TRUE); dowvmax = varimax(dowprcomp$rotation[, 2:10], normalize = FALSE); loadings(dowvmax); Note: when R prints the loadings, entries with absolute value below a cutoff (default: 0.1) are printed as blanks, to draw attention to the larger values. NC STATE UNIVERSITY 30 / 38
31 Loadings: PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 AA AIG AXP BA CAT C DD DIS GE GM HD HON HPQ IBM INTC JNJ JPM KO NC STATE UNIVERSITY 31 / 38
32 MCD MMM MO MRK MSFT PFE PG T UTX VZ WMT XOM NC STATE UNIVERSITY 32 / 38
33 In proc factor, use rotate = varimax; may also request plots both before (preplot) and after (plot) rotation; SAS program and output: proc factor data = all method = prinit nfact = 2 rotate = varimax preplot plot out = stout; title Method = Iterated Principal Factors with Varimax Rotation ; var cvx  xom; NC STATE UNIVERSITY 33 / 38
34 Factor Scores Interpretation of a factor analysis is usually based on the factor loadings. Sometimes we need the (estimated) values of the unobserved factors for further analysis the factor scores. In Principal Components Analysis, typically the principal components are used, scaled to have variance 1. In other types of factor analysis, two methods are used. NC STATE UNIVERSITY 34 / 38
35 Bartlett s Weighted Least Squares Suppose that in the equation L is known. X µ = LF + ɛ, We can view the equation as a regression of X on L, with coefficients F and heteroscedastic errors ɛ with variance matrix Ψ. This suggests using to estimate F. ˆf = ( L Ψ 1 L ) 1 L Ψ 1 (x µ) NC STATE UNIVERSITY 35 / 38
36 With L, Ψ, and µ replaced by estimates, and for the j th observation x j, this gives as estimated values of the factors. ˆf j = (ˆL ˆΨ 1ˆL) 1 ˆL ˆΨ 1 (x j x) The sample mean of the scores is 0. If the factor loadings are ML estimates, ˆL ˆΨ 1ˆL is a diagonal matrix ˆ, and the sample covariance matrix of the scores is n ( I + ˆ 1). n 1 In particular, the sample correlations of the factor scores are zero. NC STATE UNIVERSITY 36 / 38
37 Regression Method The second method depends on the normal distribution assumption. X and F have a joint multivariate normal distribution the conditional distribution of F given X is also multivariate normal. Best Linear Unbiased Predictor is the conditional mean. NC STATE UNIVERSITY 37 / 38
38 This leads to ˆfj = ˆL (ˆLˆL ˆΨ) 1 + (xj x) ( = I + ˆL ˆΨ 1ˆL) 1 ˆL ˆΨ 1 (x j x) The two methods are related by [ ) ] 1 ˆfLS j = I + (ˆL ˆΨ 1ˆL ˆfR j. In proc factor, use out = <data set name> on the proc factor statement; proc factor uses the regression method. NC STATE UNIVERSITY 38 / 38
Factor analysis. Angela Montanari
Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number
More informationExploratory Factor Analysis Brian Habing  University of South Carolina  October 15, 2003
Exploratory Factor Analysis Brian Habing  University of South Carolina  October 15, 2003 FA is not worth the time necessary to understand it and carry it out. Hills, 1977 Factor analysis should not
More informationSmith Barney Portfolio Manager Institute Conference
Smith Barney Portfolio Manager Institute Conference Richard E. Cripps, CFA Portfolio Strategy Group March 2006 The EquityCompass is an investment process focused on selecting stocks and managing portfolios
More informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,
More informationCommon factor analysis
Common factor analysis This is what people generally mean when they say "factor analysis" This family of techniques uses an estimate of common variance among the original variables to generate the factor
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationFactor Analysis. Principal components factor analysis. Use of extracted factors in multivariate dependency models
Factor Analysis Principal components factor analysis Use of extracted factors in multivariate dependency models 2 KEY CONCEPTS ***** Factor Analysis Interdependency technique Assumptions of factor analysis
More informationproblem arises when only a nonrandom sample is available differs from censored regression model in that x i is also unobserved
4 Data Issues 4.1 Truncated Regression population model y i = x i β + ε i, ε i N(0, σ 2 ) given a random sample, {y i, x i } N i=1, then OLS is consistent and efficient problem arises when only a nonrandom
More informationStatistics in Psychosocial Research Lecture 8 Factor Analysis I. Lecturer: Elizabeth GarrettMayer
This work is licensed under a Creative Commons AttributionNonCommercialShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationRachel J. Goldberg, Guideline Research/Atlanta, Inc., Duluth, GA
PROC FACTOR: How to Interpret the Output of a RealWorld Example Rachel J. Goldberg, Guideline Research/Atlanta, Inc., Duluth, GA ABSTRACT THE METHOD This paper summarizes a realworld example of a factor
More informationStatistics for Business Decision Making
Statistics for Business Decision Making Faculty of Economics University of Siena 1 / 62 You should be able to: ˆ Summarize and uncover any patterns in a set of multivariate data using the (FM) ˆ Apply
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationOverview of Factor Analysis
Overview of Factor Analysis Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 354870348 Phone: (205) 3484431 Fax: (205) 3488648 August 1,
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationSTATISTICA Formula Guide: Logistic Regression. Table of Contents
: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 SigmaRestricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary
More informationSections 2.11 and 5.8
Sections 211 and 58 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1/25 Gesell data Let X be the age in in months a child speaks his/her first word and
More informationRead chapter 7 and review lectures 8 and 9 from Econ 104 if you don t remember this stuff.
Here is your teacher waiting for Steve Wynn to come on down so I could explain index options to him. He never showed so I guess that t he will have to download this lecture and figure it out like everyone
More informationIntroduction: Overview of Kernel Methods
Introduction: Overview of Kernel Methods Statistical Data Analysis with Positive Definite Kernels Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department of Statistical Science, Graduate University
More informationADVANCED FORECASTING MODELS USING SAS SOFTWARE
ADVANCED FORECASTING MODELS USING SAS SOFTWARE Girish Kumar Jha IARI, Pusa, New Delhi 110 012 gjha_eco@iari.res.in 1. Transfer Function Model Univariate ARIMA models are useful for analysis and forecasting
More informationPRINCIPAL COMPONENT ANALYSIS
1 Chapter 1 PRINCIPAL COMPONENT ANALYSIS Introduction: The Basics of Principal Component Analysis........................... 2 A Variable Reduction Procedure.......................................... 2
More informationExample: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.
Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation:  Feature vector X,  qualitative response Y, taking values in C
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationLesson 5 Save and Invest: Stocks Owning Part of a Company
Lesson 5 Save and Invest: Stocks Owning Part of a Company Lesson Description This lesson introduces students to information and basic concepts about the stock market. In a bingo game, students become aware
More information5.2 Customers Types for Grocery Shopping Scenario
 CHAPTER 5: RESULTS AND ANALYSIS 
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationMultivariate Analysis of Variance (MANOVA): I. Theory
Gregory Carey, 1998 MANOVA: I  1 Multivariate Analysis of Variance (MANOVA): I. Theory Introduction The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the
More informationTo do a factor analysis, we need to select an extraction method and a rotation method. Hit the Extraction button to specify your extraction method.
Factor Analysis in SPSS To conduct a Factor Analysis, start from the Analyze menu. This procedure is intended to reduce the complexity in a set of data, so we choose Data Reduction from the menu. And the
More information11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial Least Squares Regression
Frank C Porter and Ilya Narsky: Statistical Analysis Techniques in Particle Physics Chap. c11 2013/9/9 page 221 letex 221 11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial
More informationLinear Classification. Volker Tresp Summer 2015
Linear Classification Volker Tresp Summer 2015 1 Classification Classification is the central task of pattern recognition Sensors supply information about an object: to which class do the object belong
More informationStock Index Futures Spread Trading
S&P 500 vs. DJIA Stock Index Futures Spread Trading S&P MidCap 400 vs. S&P SmallCap 600 Second Quarter 2008 2 Contents Introduction S&P 500 vs. DJIA Introduction Index Methodology, Calculations and Weightings
More informationChapter 6: Multivariate Cointegration Analysis
Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration
More information9.2 User s Guide SAS/STAT. The FACTOR Procedure. (Book Excerpt) SAS Documentation
SAS/STAT 9.2 User s Guide The FACTOR Procedure (Book Excerpt) SAS Documentation This document is an individual chapter from SAS/STAT 9.2 User s Guide. The correct bibliographic citation for the complete
More informationVI. Introduction to Logistic Regression
VI. Introduction to Logistic Regression We turn our attention now to the topic of modeling a categorical outcome as a function of (possibly) several factors. The framework of generalized linear models
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
More informationDimensionality Reduction: Principal Components Analysis
Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely
More informationAnalysis of Financial Data Using NonNegative Matrix Factorization
International Mathematical Forum,, 008, no. 8, 8870 Analysis of Financial Data Using NonNegative Matrix Factorization Konstantinos Drakakis UCD CASL, University College Dublin Belfield, Dublin, Ireland
More informationMedical Information Management & Mining. You Chen Jan,15, 2013 You.chen@vanderbilt.edu
Medical Information Management & Mining You Chen Jan,15, 2013 You.chen@vanderbilt.edu 1 Trees Building Materials Trees cannot be used to build a house directly. How can we transform trees to building materials?
More informationA Brief Introduction to SPSS Factor Analysis
A Brief Introduction to SPSS Factor Analysis SPSS has a procedure that conducts exploratory factor analysis. Before launching into a step by step example of how to use this procedure, it is recommended
More informationChapter 4: Vector Autoregressive Models
Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...
More informationPrinciple Component Analysis and Partial Least Squares: Two Dimension Reduction Techniques for Regression
Principle Component Analysis and Partial Least Squares: Two Dimension Reduction Techniques for Regression Saikat Maitra and Jun Yan Abstract: Dimension reduction is one of the major tasks for multivariate
More informationSteven M. Ho!and. Department of Geology, University of Georgia, Athens, GA 306022501
PRINCIPAL COMPONENTS ANALYSIS (PCA) Steven M. Ho!and Department of Geology, University of Georgia, Athens, GA 306022501 May 2008 Introduction Suppose we had measured two variables, length and width, and
More information3. Regression & Exponential Smoothing
3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationIndices of Model Fit STRUCTURAL EQUATION MODELING 2013
Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit A recommended minimal set of fit indices that should be reported and interpreted when reporting the results of SEM analyses:
More informationMultivariate Normal Distribution
Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #47/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues
More informationGamma Distribution Fitting
Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 201112) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More informationWeek 5: Multiple Linear Regression
BUS41100 Applied Regression Analysis Week 5: Multiple Linear Regression Parameter estimation and inference, forecasting, diagnostics, dummy variables Robert B. Gramacy The University of Chicago Booth School
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationSales forecasting # 2
Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univrennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationPenalized regression: Introduction
Penalized regression: Introduction Patrick Breheny August 30 Patrick Breheny BST 764: Applied Statistical Modeling 1/19 Maximum likelihood Much of 20thcentury statistics dealt with maximum likelihood
More informationState Space Time Series Analysis
State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman http://staff.feweb.vu.nl/koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011 State
More informationSAS Software to Fit the Generalized Linear Model
SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
More informationOrganization & Analysis of Stock Option Market Data. A Professional Master's Project. Submitted to the Faculty of the WORCESTER
1 Organization & Analysis of Stock Option Market Data A Professional Master's Project Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationHLM software has been one of the leading statistical packages for hierarchical
Introductory Guide to HLM With HLM 7 Software 3 G. David Garson HLM software has been one of the leading statistical packages for hierarchical linear modeling due to the pioneering work of Stephen Raudenbush
More informationJoint models for classification and comparison of mortality in different countries.
Joint models for classification and comparison of mortality in different countries. Viani D. Biatat 1 and Iain D. Currie 1 1 Department of Actuarial Mathematics and Statistics, and the Maxwell Institute
More informationA Brief Introduction to Factor Analysis
1. Introduction A Brief Introduction to Factor Analysis Factor analysis attempts to represent a set of observed variables X 1, X 2. X n in terms of a number of 'common' factors plus a factor which is unique
More informationMachine Learning and Pattern Recognition Logistic Regression
Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,
More informationStatistics 104: Section 6!
Page 1 Statistics 104: Section 6! TF: Deirdre (say: Deardra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm3pm in SC 109, Thursday 5pm6pm in SC 705 Office Hours: Thursday 6pm7pm SC
More informationDegrees of Freedom and Model Search
Degrees of Freedom and Model Search Ryan J. Tibshirani Abstract Degrees of freedom is a fundamental concept in statistical modeling, as it provides a quantitative description of the amount of fitting performed
More informationLogistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression
Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max
More information1. Introduction to multivariate data
. Introduction to multivariate data. Books Chat eld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall Krzanowski, W.J. Principles of multivariate analysis. Oxford.000 Johnson,
More informationPartial Least Squares (PLS) Regression.
Partial Least Squares (PLS) Regression. Hervé Abdi 1 The University of Texas at Dallas Introduction Pls regression is a recent technique that generalizes and combines features from principal component
More informationGLM, insurance pricing & big data: paying attention to convergence issues.
GLM, insurance pricing & big data: paying attention to convergence issues. Michaël NOACK  michael.noack@addactis.com Senior consultant & Manager of ADDACTIS Pricing Copyright 2014 ADDACTIS Worldwide.
More informationLinear Models and Conjoint Analysis with Nonlinear Spline Transformations
Linear Models and Conjoint Analysis with Nonlinear Spline Transformations Warren F. Kuhfeld Mark Garratt Abstract Many common data analysis models are based on the general linear univariate model, including
More informationDISCRIMINANT FUNCTION ANALYSIS (DA)
DISCRIMINANT FUNCTION ANALYSIS (DA) John Poulsen and Aaron French Key words: assumptions, further reading, computations, standardized coefficents, structure matrix, tests of signficance Introduction Discriminant
More informationANOVA. February 12, 2015
ANOVA February 12, 2015 1 ANOVA models Last time, we discussed the use of categorical variables in multivariate regression. Often, these are encoded as indicator columns in the design matrix. In [1]: %%R
More informationAPPRAISAL OF FINANCIAL AND ADMINISTRATIVE FUNCTIONING OF PUNJAB TECHNICAL UNIVERSITY
APPRAISAL OF FINANCIAL AND ADMINISTRATIVE FUNCTIONING OF PUNJAB TECHNICAL UNIVERSITY In the previous chapters the budgets of the university have been analyzed using various techniques to understand the
More information2. Linearity (in relationships among the variablesfactors are linear constructions of the set of variables) F 2 X 4 U 4
1 Neuendorf Factor Analysis Assumptions: 1. Metric (interval/ratio) data. Linearity (in relationships among the variablesfactors are linear constructions of the set of variables) 3. Univariate and multivariate
More informationA Multivariate Statistical Analysis of Crime Rate in US Cities
A Multivariate Statistical Analysis of Crime Rate in US Cities Kendall Williams Ralph Gedeon Howard University July 004 University of Florida Washington DC Gainesville, FL k_r_williams@howard.edu ralphael@ufl.edu
More informationTHE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok
THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan
More information1 Theory: The General Linear Model
QMIN GLM Theory  1.1 1 Theory: The General Linear Model 1.1 Introduction Before digital computers, statistics textbooks spoke of three procedures regression, the analysis of variance (ANOVA), and the
More informationBayesian logistic betting strategy against probability forecasting. Akimichi Takemura, Univ. Tokyo. November 12, 2012
Bayesian logistic betting strategy against probability forecasting Akimichi Takemura, Univ. Tokyo (joint with Masayuki Kumon, Jing Li and Kei Takeuchi) November 12, 2012 arxiv:1204.3496. To appear in Stochastic
More informationData analysis process
Data analysis process Data collection and preparation Collect data Prepare codebook Set up structure of data Enter data Screen data for errors Exploration of data Descriptive Statistics Graphs Analysis
More information7 Time series analysis
7 Time series analysis In Chapters 16, 17, 33 36 in Zuur, Ieno and Smith (2007), various time series techniques are discussed. Applying these methods in Brodgar is straightforward, and most choices are
More informationJava Modules for Time Series Analysis
Java Modules for Time Series Analysis Agenda Clustering Nonnormal distributions Multifactor modeling Implied ratings Time series prediction 1. Clustering + Cluster 1 Synthetic Clustering + Time series
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part X Factor analysis Whenwehavedatax (i) R n thatcomesfromamixtureofseveral Gaussians, the EM algorithm can be applied to fit a mixture model. In this setting, we usually
More informationThe Monte Carlo Framework, Examples from Finance and Generating Correlated Random Variables
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh The Monte Carlo Framework, Examples from Finance and Generating Correlated Random Variables 1 The Monte Carlo Framework Suppose we wish
More informationSTATISTICS AND DATA ANALYSIS IN GEOLOGY, 3rd ed. Clarificationof zonationprocedure described onpp. 238239
STATISTICS AND DATA ANALYSIS IN GEOLOGY, 3rd ed. by John C. Davis Clarificationof zonationprocedure described onpp. 3839 Because the notation used in this section (Eqs. 4.8 through 4.84) is inconsistent
More informationResponse variables assume only two values, say Y j = 1 or = 0, called success and failure (spam detection, credit scoring, contracting.
Prof. Dr. J. Franke All of Statistics 1.52 Binary response variables  logistic regression Response variables assume only two values, say Y j = 1 or = 0, called success and failure (spam detection, credit
More informationCiti Volatility Balanced Beta (VIBE) Equity Eurozone Net Total Return Index Index Methodology. Citi Investment Strategies
Citi Volatility Balanced Beta (VIBE) Equity Eurozone Net Total Return Index Citi Investment Strategies 21 November 2011 Table of Contents Citi Investment Strategies Part A: Introduction 1 Part B: Key Information
More information1 Simple Linear Regression I Least Squares Estimation
Simple Linear Regression I Least Squares Estimation Textbook Sections: 8. 8.3 Previously, we have worked with a random variable x that comes from a population that is normally distributed with mean µ and
More informationPa8ern Recogni6on. and Machine Learning. Chapter 4: Linear Models for Classiﬁca6on
Pa8ern Recogni6on and Machine Learning Chapter 4: Linear Models for Classiﬁca6on Represen'ng the target values for classifica'on If there are only two classes, we typically use a single real valued output
More informationProbabilistic Linear Classification: Logistic Regression. Piyush Rai IIT Kanpur
Probabilistic Linear Classification: Logistic Regression Piyush Rai IIT Kanpur Probabilistic Machine Learning (CS772A) Jan 18, 2016 Probabilistic Machine Learning (CS772A) Probabilistic Linear Classification:
More informationMultivariate Statistical Inference and Applications
Multivariate Statistical Inference and Applications ALVIN C. RENCHER Department of Statistics Brigham Young University A WileyInterscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim
More informationCS 688 Pattern Recognition Lecture 4. Linear Models for Classification
CS 688 Pattern Recognition Lecture 4 Linear Models for Classification Probabilistic generative models Probabilistic discriminative models 1 Generative Approach ( x ) p C k p( C k ) Ck p ( ) ( x Ck ) p(
More informationANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEINUHLENBECK PROCESSES
ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEINUHLENBECK PROCESSES by Xiaofeng Qian Doctor of Philosophy, Boston University, 27 Bachelor of Science, Peking University, 2 a Project
More informationStephen du Toit Mathilda du Toit Gerhard Mels Yan Cheng. LISREL for Windows: SIMPLIS Syntax Files
Stephen du Toit Mathilda du Toit Gerhard Mels Yan Cheng LISREL for Windows: SIMPLIS Files Table of contents SIMPLIS SYNTAX FILES... 1 The structure of the SIMPLIS syntax file... 1 $CLUSTER command... 4
More informationExploratory data analysis for microarray data
Eploratory data analysis for microarray data Anja von Heydebreck Ma Planck Institute for Molecular Genetics, Dept. Computational Molecular Biology, Berlin, Germany heydebre@molgen.mpg.de Visualization
More information5. Multiple regression
5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful
More informationManifold Learning Examples PCA, LLE and ISOMAP
Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationMultiple Linear Regression in Data Mining
Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple
More informationThe Basic TwoLevel Regression Model
2 The Basic TwoLevel Regression Model The multilevel regression model has become known in the research literature under a variety of names, such as random coefficient model (de Leeuw & Kreft, 1986; Longford,
More information