# Multivariate Normal Distribution

Size: px
Start display at page:

Transcription

1 Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41

2 Last Time Matrices and vectors Eigenvalues Last Time Today s Lecture Parameters Likelihood Functions in Common Methods Eigenvectors Determinants Basic descriptive statistics using matrices: Mean vectors Covariance Matrices Correlation Matrices Lecture #4-7/21/2011 Slide 2 of 41

3 Today s Lecture Putting our new knowledge to use with a useful statistical distribution: the Multivariate Normal Distribution Last Time Today s Lecture This roughly maps onto Chapter 4 of Johnson and Wichern Parameters Likelihood Functions in Common Methods Lecture #4-7/21/2011 Slide 3 of 41

4 Multivariate Normal Distribution Univariate Review Contours Parameters Likelihood Functions in Common Methods The generalization of the univariate normal distribution to multiple variables is called the multivariate normal distribution () Many multivariate techniques rely on this distribution in some manner Although real data may never come from a true, the provides a robust approximation, and has many nice mathematical properties Furthermore, because of the central limit theorem, many multivariate statistics converge to the distribution as the sample size increases Lecture #4-7/21/2011 Slide 4 of 41

5 Univariate Normal Distribution The univariate normal distribution function is: Univariate Review Contours The mean is µ f(x) = 1 2πσ 2 e [(x µ)/σ]2 /2 Parameters Likelihood Functions The variance is σ 2 in Common Methods The standard deviation is σ Standard notation for normal distributions is N(µ, σ 2 ), which will be extended for the distribution Lecture #4-7/21/2011 Slide 5 of 41

6 Univariate Normal Distribution N(0, 1) Univariate Normal Distribution Univariate Review Contours Parameters Likelihood Functions in Common Methods f(x) x Lecture #4-7/21/2011 Slide 6 of 41

7 Univariate Normal Distribution N(0, 2) Univariate Normal Distribution Univariate Review Contours Parameters Likelihood Functions in Common Methods f(x) x Lecture #4-7/21/2011 Slide 7 of 41

8 Univariate Normal Distribution N(1.75, 1) Univariate Normal Distribution Univariate Review Contours Parameters Likelihood Functions in Common Methods f(x) x Lecture #4-7/21/2011 Slide 8 of 41

9 UVN - Notes The area under the curve for the univariate normal distribution is a function of the variance/standard deviation Univariate Review Contours Parameters In particular: P(µ σ X µ + σ) = P(µ 2σ X µ + 2σ) = Likelihood Functions in Common Methods Also note the term in the exponent: ( ) 2 (x µ) = (x µ)(σ 2 ) 1 (x µ) σ This is the square of the distance from x to µ in standard deviation units, and will be generalized for the Lecture #4-7/21/2011 Slide 9 of 41

10 The multivariate normal distribution function is: Univariate Review Contours Parameters Likelihood Functions f(x) = The mean vector is µ The covariance matrix is Σ 1 Σ 1 (x µ)/2 (2π) p/2 Σ 1/2e (x µ) in Common Methods Standard notation for multivariate normal distributions is N p (µ,σ) Visualizing the is difficult for more than two dimensions, so I will demonstrate some plots with two variables - the bivariate normal distribution Lecture #4-7/21/2011 Slide 10 of 41

11 Bivariate Normal Plot #1 µ = [ 0 0 ],Σ = [ ] Univariate Review Contours Parameters Likelihood Functions in Common Methods Lecture #4-7/21/2011 Slide 11 of 41

12 Bivariate Normal Plot #1a µ = [ 0 0 ],Σ = [ ] Univariate Review Contours Parameters Likelihood Functions in Common Methods Lecture #4-7/21/2011 Slide 12 of 41

13 Bivariate Normal Plot #2 µ = [ 0 0 ],Σ = [ ] Univariate Review Contours 0.2 Parameters Likelihood Functions 0.15 in Common Methods Lecture #4-7/21/2011 Slide 13 of 41

14 Bivariate Normal Plot #2 µ = [ 0 0 ],Σ = [ ] Univariate Review Contours Parameters Likelihood Functions in Common Methods Lecture #4-7/21/2011 Slide 14 of 41

15 Contours The lines of the contour plots denote places of equal probability mass for the distribution Univariate Review Contours Parameters Likelihood Functions in Common Methods The lines represent points of both variables that lead to the same height on the z-axis (the height of the surface) These contours can be constructed from the eigenvalues and eigenvectors of the covariance matrix The direction of the ellipse axes are in the direction of the eigenvalues The length of the ellipse axes are proportional to the constant times the eigenvector Specifically: (x µ) Σ 1 (x µ) = c 2 has ellipsoids centered at µ, and has axes ±c λ i e i Lecture #4-7/21/2011 Slide 15 of 41

16 Contours, Continued Contours are useful because they provide confidence regions for data points from the distribution Univariate Review Contours Parameters The multivariate analog of a confidence interval is given by an ellipsoid, where c is from the Chi-Squared distribution with p degrees of freedom Specifically: Likelihood Functions in Common Methods (x µ) Σ 1 (x µ) = χ 2 p(α) provides the confidence region containing 1 α of the probability mass of the distribution Lecture #4-7/21/2011 Slide 16 of 41

17 Contour Example Univariate Review Contours Parameters Likelihood Functions in Common Methods Imagine we had a bivariate normal distribution with: [ ] [ ] µ =,Σ = The covariance matrix has eigenvalues and eigenvectors: [ ] [ ] λ =, E = We want to find a contour where 95% of the probability will fall, corresponding to χ 2 2(0.05) = 5.99 Lecture #4-7/21/2011 Slide 17 of 41

18 Contour Example This contour will be centered at µ Axis 1: Univariate Review Contours µ ± [ ] = [ ], [ ] Parameters Likelihood Functions Axis 2: in Common Methods µ ± [ ] = [ ], [ ] Lecture #4-7/21/2011 Slide 18 of 41

19 The distribution has some convenient properties Parameters Likelihood Functions in Common Methods If X has a multivariate normal distribution, then: 1. Linear combinations of X are normally distributed 2. All subsets of the components of X have a distribution 3. Zero covariance implies that the corresponding components are independently distributed 4. The conditional distributions of the components are Lecture #4-7/21/2011 Slide 19 of 41

20 Linear Combinations If X N p (µ,σ), then any set of q linear combinations of variables A (q p) are also normally distributed as AX N q (Aµ, AΣA ) Parameters Likelihood Functions in Common Methods For example, let p = 3 and Y be the difference between X 1 and X 2. The combination matrix would be A = [ )1 1 0 ] For X µ = µ 1 µ 2 µ 3,Σ = σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 For Y = AX µ Y = [ ] µ 1 µ 2,Σ Y = [ σ 11 + σ 22 2σ 12 ] Lecture #4-7/21/2011 Slide 20 of 41

21 The distribution is characterized by two parameters: The mean vector µ The covariance matrix Σ Parameters UVN CLT Multi CLT Sufficient Stats Likelihood Functions in Common Methods The maximum likelihood estimates for these parameters are given by: The mean vector: x = 1 n n x i = 1 n X 1 i=1 The covariance matrix S = 1 n (x i x) 2 = 1 n n (X 1 x ) (X 1 x ) i=1 Lecture #4-7/21/2011 Slide 21 of 41

22 Distribution of x and S Recall back in Univariate statistics you discussed the Central Limit Theorem (CLT) Parameters UVN CLT Multi CLT Sufficient Stats Likelihood Functions in Common Methods It stated that, if the set of n observations x 1, x 2,...,x n were normal or not... The distribution of x would be normal with mean equal to µ and variance σ 2 /n We were also told that (n 1)s 2 /σ 2 had a Chi-Square distribution with n 1 degrees of freedom Note: We ended up using these pieces of information for hypothesis testing such as t-test and ANOVA. Lecture #4-7/21/2011 Slide 22 of 41

23 Distribution of x and S We also have a Multivariate Central Limit Theorem (CLT) Parameters UVN CLT Multi CLT Sufficient Stats Likelihood Functions in Common Methods It states that, if the set of n observations x 1, x 2,...,x n are multivariate normal or not... The distribution of x would be normal with mean equal to µ and variance/covariance matrix Σ/n We are also told that (n 1)S will have a Wishart distribution, W p (n 1,Σ), with n 1 degrees of freedom This is the multivariate analogue to a Chi-Square distribution Note: We will end up using some of this information for multivariate hypothesis testing Lecture #4-7/21/2011 Slide 23 of 41

24 Distribution of x and S Therefore, let x 1, x 2,...,x n be independent observations from a population with mean µ and covariance Σ Parameters UVN CLT Multi CLT Sufficient Stats The following are true: n ( X µ ) is approximately Np (0,Σ) n (X µ) S 1 (X µ) is approximately χ 2 p Likelihood Functions in Common Methods Lecture #4-7/21/2011 Slide 24 of 41

25 Sufficient Statistics The sample estimates X and S) are sufficient statistics Parameters UVN CLT Multi CLT Sufficient Stats Likelihood Functions in Common Methods This means that all of the information contained in the data can be summarized by these two statistics alone This is only true if the data follow a multivariate normal distribution - if they do not, other terms are needed (i.e., skewness array, kurtosis array, etc...) Some statistical methods only use one or both of these matrices in their analysis procedures and not the actual data Lecture #4-7/21/2011 Slide 25 of 41

26 Density and Likelihood Functions The distribution is often the core statistical distribution for a uni- or multivariate statistical technique Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods Maximum likelihood estimates are preferable in statistics due to a set of desirable asymptotic properties, including: Consistency: the estimator converges in probability to the value being estimated Asymptotic Normality: the estimator has a normal distribution with a functionally known variance Efficiency: no asymptotically unbiased estimator has lower asymptotic mean squared error than the MLE The form of the ML function frequently appears in statistics, so we will briefly discuss MLE using normal distributions Lecture #4-7/21/2011 Slide 26 of 41

27 An Introduction to Maximum Likelihood Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods Maximum likelihood estimation seeks to find parameters of a statistical model (mapping onto the mean vector and/or covariance matrix) such that the statistical likelihood function is maximized The method assumes data follow a statistical distribution, in our case the More frequently, the log-likelihood function is used instead of the likelihood function The logged and un-logged version of the function have a maximum at the same point The logged version is easier mathematically Lecture #4-7/21/2011 Slide 27 of 41

28 Maximum Likelihood for Univariate Normal We will start with the univariate normal case and then generalize Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods Imagine we have a sample of data, X, which we will assume is normally distributed with an unknown mean but a known variance (say the variance is 1) We will build the maximum likelihood function for the mean Our function rests on two assumptions: 1. All data follow a normal distribution 2. All observations are independent Put into statistical terms: X is independent and identically distributed (iid) as N 1 (µ, 1) Lecture #4-7/21/2011 Slide 28 of 41

29 Building the Likelihood Function Each observation, then, follows a normal distribution with the same mean (unknown) and variance (1) Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods The distribution function begins with the density the function that provides the normal curve (with (1) in place of σ 2 ): f(x i µ) = 1 exp ( (X i µ) 2 ) 2π(1) 2(1) 2 The density provides the likelihood of observing an observation X i for a given value of µ (and a known value of σ 2 = 1 The likelihood is the height of the normal curve Lecture #4-7/21/2011 Slide 29 of 41

30 The One-Observation Likelihood Function The graph shows f(x i µ = 1) for a range of X Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods f(x mu) The vertical lines indicate: f(x i = 0 µ = 1) =.241 f(x i = 1 µ = 1) = X Lecture #4-7/21/2011 Slide 30 of 41

31 The Overall Likelihood Function Because we have a sample of N observations, our likelihood function is taken across all observations, not just one Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods The joint likelihood function uses the assumption that observations are independent to be expressed as a product of likelihood functions across all observations: L(x µ) = f(x 1 µ) f(x 2 µ)... f(x N µ) N ( ) N/2 ) N 1 i=1 L(x µ) = f(x i µ) = exp( (X i µ) 2 2π(1) 2(1) 2 i=1 The value of µ that maximizes f(x µ) is the MLE (in this case, it s the sample mean) In more complicated models, the MLE does not have a closed form and therefore must be found using numeric methods Lecture #4-7/21/2011 Slide 31 of 41

32 The Overall Log-Likelihood Function Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods For an unknown mean µ and variance σ 2, the likelihood function is: L(x µ, σ 2 ) = ( ) N/2 ) N 1 i=1 exp( 2πσ 2 (X i µ) 2 2σ 2 More commonly, the log-likelihood function is used: L(x µ, σ 2 ) = ( N 2 ) log ( ( N ) 2πσ 2) i=1 (X i µ) 2 2σ 2 Lecture #4-7/21/2011 Slide 32 of 41

33 The Multivariate Normal Likelihood Function For a set of N independent observations on p variables, X (N p), the multivariate normal likelihood function is formed by using a similar approach Parameters For an unknown mean vector µ and covariance Σ, the joint likelihood is: Likelihood Functions Intro to MLE Likelihood in Common Methods L(X µ,σ) = N i=1 1 (2π) p/2 Σ exp( (x 1/2 i µ) Σ 1 (x i µ)/2 ) = 1 1 (2π) np/2 Σ n/2 exp ( ) N (x i µ) Σ 1 (x i µ)/2 i=1 Lecture #4-7/21/2011 Slide 33 of 41

34 The Multivariate Normal Likelihood Function Occasionally, a more intricate form of the likelihood function shows up Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods Although mathematically identical to the function on the last page, this version typically appears without explanation: L(X µ,σ) = (2π) np/2 Σ n/2 ( [ ( N ) exp tr Σ 1 (x i x) (x i x) + n ( x µ) ( x µ) )]/2 i=1 Lecture #4-7/21/2011 Slide 34 of 41

35 The Log-Likelihood Function As with the univariate case, the likelihood function is typically converted into a log-likelihood function for simplicity Parameters Likelihood Functions Intro to MLE Likelihood in Common Methods The log-likelihood function is given by: l(x µ,σ) = np 2 log (2π) n log ( Σ ) 2 ( N ) 1 (x i µ) Σ 1 (x i µ) 2 i=1 Lecture #4-7/21/2011 Slide 35 of 41

36 But...Why? The distribution, likelihood, and log-likelihood functions show up frequently in statistical methods Parameters Likelihood Functions in Common Methods Motivation for in Mixed Models in SEM Commonly used methods rely on versions of the distribution, methods such as: Linear models (ANOVA, Regression) Mixed models (i.e., hierarchical linear models, random effects models, multilevel models) Path models/simultaneous equation models Structural equation models (and confirmatory factor models) Many versions of finite mixture models Understanding the form of the distribution will help to understand the commonalities between each of these models Lecture #4-7/21/2011 Slide 36 of 41

37 in Mixed Models From SAS manual for proc mixed: Parameters Likelihood Functions in Common Methods Motivation for in Mixed Models in SEM Lecture #4-7/21/2011 Slide 37 of 41

38 in Structural Equation Models From SAS manual for proc calis: Parameters Likelihood Functions This package uses only the covariance matrix, so the form of the likelihood function is phrased using only the Wishart Distribution: in Common Methods Motivation for in Mixed Models in SEM w (S Σ) = S (n p 2) exp [ tr [ SΣ 1] /2 ] 2 p(n 1)/2 π p(p 1)/4 Σ (n 1)/2 p i=1 Γ ( 1 2 (n i)) Lecture #4-7/21/2011 Slide 38 of 41

39 Recall from earlier that IF the data have a Multivariate normal distribution then all of the previously discussed properties will hold Parameters Likelihood Functions in Common Methods Transformations to Near Normality There are a host of methods that have been developed to assess multivariate normality - just look in Johnson & Wichern Given the relative robustness of the distribution, I will skip this topic, acknowledging that extreme deviations from normality will result in poorly performing statistics More often than not, assessing MV normality is fraught with difficulty due to sample-estimated parameters of the distribution Lecture #4-7/21/2011 Slide 39 of 41

40 Transformations to Near Normality Historically, people have gone on an expedition to find a transformation to near-normality when learning their data may not be Parameters Likelihood Functions in Common Methods Transformations to Near Normality Modern statistical methods, however, make that a very bad idea More often than not, transformations end up changing the nature of the statistics you are interested in forming Furthermore, not all data need to be (think conditional distributions) Lecture #4-7/21/2011 Slide 40 of 41

41 Final Thoughts The multivariate normal distribution is an analog to the univariate normal distribution Parameters Likelihood Functions in Common Methods Final Thoughts The distribution will play a large role in the upcoming weeks We can finally put the background material to rest, and begin learning some statistics methods Tomorrow: lab with SAS - the fun of proc iml Up next week: Inferences about Mean Vectors and Multivariate ANOVA Lecture #4-7/21/2011 Slide 41 of 41

### Sections 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

### Stat 704 Data Analysis I Probability Review

1 / 30 Stat 704 Data Analysis I Probability Review Timothy Hanson Department of Statistics, University of South Carolina Course information 2 / 30 Logistics: Tuesday/Thursday 11:40am to 12:55pm in LeConte

### A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

### 15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

### Lecture 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,

### Introduction to Multivariate Analysis

Introduction to Multivariate Analysis Lecture 1 August 24, 2005 Multivariate Analysis Lecture #1-8/24/2005 Slide 1 of 30 Today s Lecture Today s Lecture Syllabus and course overview Chapter 1 (a brief

### Multivariate normal distribution and testing for means (see MKB Ch 3)

Multivariate normal distribution and testing for means (see MKB Ch 3) Where are we going? 2 One-sample t-test (univariate).................................................. 3 Two-sample t-test (univariate).................................................

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### Dimensionality 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

### Multivariate 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

### Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

### 2DI36 Statistics. 2DI36 Part II (Chapter 7 of MR)

2DI36 Statistics 2DI36 Part II (Chapter 7 of MR) What Have we Done so Far? Last time we introduced the concept of a dataset and seen how we can represent it in various ways But, how did this dataset came

### Service courses for graduate students in degree programs other than the MS or PhD programs in Biostatistics.

Course Catalog In order to be assured that all prerequisites are met, students must acquire a permission number from the education coordinator prior to enrolling in any Biostatistics course. Courses are

### Lecture 8: Signal Detection and Noise Assumption

ECE 83 Fall Statistical Signal Processing instructor: R. Nowak, scribe: Feng Ju Lecture 8: Signal Detection and Noise Assumption Signal Detection : X = W H : X = S + W where W N(, σ I n n and S = [s, s,...,

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

### Example: 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

### Multivariate Statistical Inference and Applications

Multivariate Statistical Inference and Applications ALVIN C. RENCHER Department of Statistics Brigham Young University A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim

Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models

### Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Structure of models in R Model Assessment (Part IA) Anova

: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

### BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

### Statistical 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

### How To Understand Multivariate Models

Neil H. Timm Applied Multivariate Analysis With 42 Figures Springer Contents Preface Acknowledgments List of Tables List of Figures vii ix xix xxiii 1 Introduction 1 1.1 Overview 1 1.2 Multivariate Models

### Least Squares Estimation

Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

### Section Format Day Begin End Building Rm# Instructor. 001 Lecture Tue 6:45 PM 8:40 PM Silver 401 Ballerini

NEW YORK UNIVERSITY ROBERT F. WAGNER GRADUATE SCHOOL OF PUBLIC SERVICE Course Syllabus Spring 2016 Statistical Methods for Public, Nonprofit, and Health Management Section Format Day Begin End Building

### MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

### Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts

### Indices 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:

### Basics of Statistical Machine Learning

CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar

### CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### Simple 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

### i=1 In practice, the natural logarithm of the likelihood function, called the log-likelihood function and denoted by

Statistics 580 Maximum Likelihood Estimation Introduction Let y (y 1, y 2,..., y n be a vector of iid, random variables from one of a family of distributions on R n and indexed by a p-dimensional parameter

### CS229 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

### On Mardia s Tests of Multinormality

On Mardia s Tests of Multinormality Kankainen, A., Taskinen, S., Oja, H. Abstract. Classical multivariate analysis is based on the assumption that the data come from a multivariate normal distribution.

### NCSS 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

### Canonical Correlation Analysis

Canonical Correlation Analysis Lecture 11 August 4, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #11-8/4/2011 Slide 1 of 39 Today s Lecture Canonical Correlation Analysis

Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.

### MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...

MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................

### 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

### 11 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 le-tex 221 11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### Linear 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

### From the help desk: Bootstrapped standard errors

The Stata Journal (2003) 3, Number 1, pp. 71 80 From the help desk: Bootstrapped standard errors Weihua Guan Stata Corporation Abstract. Bootstrapping is a nonparametric approach for evaluating the distribution

### Factor 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

### Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases:

Profile Analysis Introduction Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases: ) Comparing the same dependent variables

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### Comparison of Estimation Methods for Complex Survey Data Analysis

Comparison of Estimation Methods for Complex Survey Data Analysis Tihomir Asparouhov 1 Muthen & Muthen Bengt Muthen 2 UCLA 1 Tihomir Asparouhov, Muthen & Muthen, 3463 Stoner Ave. Los Angeles, CA 90066.

### Penalized regression: Introduction

Penalized regression: Introduction Patrick Breheny August 30 Patrick Breheny BST 764: Applied Statistical Modeling 1/19 Maximum likelihood Much of 20th-century statistics dealt with maximum likelihood

### Overview of Factor Analysis

Overview of Factor Analysis Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone: (205) 348-4431 Fax: (205) 348-8648 August 1,

### Goodness of fit assessment of item response theory models

Goodness of fit assessment of item response theory models Alberto Maydeu Olivares University of Barcelona Madrid November 1, 014 Outline Introduction Overall goodness of fit testing Two examples Assessing

### Chapter 6: Point Estimation. Fall 2011. - Probability & Statistics

STAT355 Chapter 6: Point Estimation Fall 2011 Chapter Fall 2011 6: Point1 Estimat / 18 Chap 6 - Point Estimation 1 6.1 Some general Concepts of Point Estimation Point Estimate Unbiasedness Principle of

### Master s Theory Exam Spring 2006

Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### SAS 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

### Factor Analysis. Chapter 420. Introduction

Chapter 420 Introduction (FA) is an exploratory technique applied to a set of observed variables that seeks to find underlying factors (subsets of variables) from which the observed variables were generated.

### [1] Diagonal factorization

8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### STAT 830 Convergence in Distribution

STAT 830 Convergence in Distribution Richard Lockhart Simon Fraser University STAT 830 Fall 2011 Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 Fall 2011 1 / 31

### Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software

STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used

### 1 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

### LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as

LOGISTIC REGRESSION Nitin R Patel Logistic regression extends the ideas of multiple linear regression to the situation where the dependent variable, y, is binary (for convenience we often code these values

### Spatial Statistics Chapter 3 Basics of areal data and areal data modeling

Spatial Statistics Chapter 3 Basics of areal data and areal data modeling Recall areal data also known as lattice data are data Y (s), s D where D is a discrete index set. This usually corresponds to data

### Multivariate Analysis of Ecological Data

Multivariate Analysis of Ecological Data MICHAEL GREENACRE Professor of Statistics at the Pompeu Fabra University in Barcelona, Spain RAUL PRIMICERIO Associate Professor of Ecology, Evolutionary Biology

### Simple Predictive Analytics Curtis Seare

Using Excel to Solve Business Problems: Simple Predictive Analytics Curtis Seare Copyright: Vault Analytics July 2010 Contents Section I: Background Information Why use Predictive Analytics? How to use

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### lavaan: an R package for structural equation modeling

lavaan: an R package for structural equation modeling Yves Rosseel Department of Data Analysis Belgium Utrecht April 24, 2012 Yves Rosseel lavaan: an R package for structural equation modeling 1 / 20 Overview

### Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010

Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different

### Statistics in Retail Finance. Chapter 6: Behavioural models

Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics:- Behavioural

### CHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA

Examples: Multilevel Modeling With Complex Survey Data CHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA Complex survey data refers to data obtained by stratification, cluster sampling and/or

### The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.

Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium

### business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

### Module 3: Correlation and Covariance

Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

### Overview Classes. 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7)

Overview Classes 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7) 2-4 Loglinear models (8) 5-4 15-17 hrs; 5B02 Building and

### Introduction to Multilevel Modeling Using HLM 6. By ATS Statistical Consulting Group

Introduction to Multilevel Modeling Using HLM 6 By ATS Statistical Consulting Group Multilevel data structure Students nested within schools Children nested within families Respondents nested within interviewers

### Financial Time Series Analysis (FTSA) Lecture 1: Introduction

Financial Time Series Analysis (FTSA) Lecture 1: Introduction Brief History of Time Series Analysis Statistical analysis of time series data (Yule, 1927) v/s forecasting (even longer). Forecasting is often

### Chapter 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)...

### Elasticity Theory Basics

G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

### Statistics in Retail Finance. Chapter 2: Statistical models of default

Statistics in Retail Finance 1 Overview > We consider how to build statistical models of default, or delinquency, and how such models are traditionally used for credit application scoring and decision

### 171:290 Model Selection Lecture II: The Akaike Information Criterion

171:290 Model Selection Lecture II: The Akaike Information Criterion Department of Biostatistics Department of Statistics and Actuarial Science August 28, 2012 Introduction AIC, the Akaike Information

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Linear Models for Continuous Data

Chapter 2 Linear Models for Continuous Data The starting point in our exploration of statistical models in social research will be the classical linear model. Stops along the way include multiple linear

### Highly Efficient Incremental Estimation of Gaussian Mixture Models for Online Data Stream Clustering

Highly Efficient Incremental Estimation of Gaussian Mixture Models for Online Data Stream Clustering Mingzhou Song a,b and Hongbin Wang b a Department of Computer Science, Queens College of CUNY, Flushing,

### CONTENTS PREFACE 1 INTRODUCTION 1 2 DATA VISUALIZATION 19

PREFACE xi 1 INTRODUCTION 1 1.1 Overview 1 1.2 Definition 1 1.3 Preparation 2 1.3.1 Overview 2 1.3.2 Accessing Tabular Data 3 1.3.3 Accessing Unstructured Data 3 1.3.4 Understanding the Variables and Observations

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### 521493S Computer Graphics. Exercise 2 & course schedule change

521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 16-18 in TS128 Question 2.1 Given two nonparallel,

### Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round \$200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

### Section 14 Simple Linear Regression: Introduction to Least Squares Regression

Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

### STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct

### LOGNORMAL MODEL FOR STOCK PRICES

LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as

### A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS

A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS Eusebio GÓMEZ, Miguel A. GÓMEZ-VILLEGAS and J. Miguel MARÍN Abstract In this paper it is taken up a revision and characterization of the class of

### 11. Analysis of Case-control Studies Logistic Regression

Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

### Lecture 2: Descriptive Statistics and Exploratory Data Analysis

Lecture 2: Descriptive Statistics and Exploratory Data Analysis Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals

### Characteristics of Binomial Distributions

Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

### Introduction to Longitudinal Data Analysis

Introduction to Longitudinal Data Analysis Longitudinal Data Analysis Workshop Section 1 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section 1: Introduction