Bayes Estimators & Ridge Regression
|
|
- Kristina Banks
- 6 years ago
- Views:
Transcription
1 Bayes Estimators & Ridge Regression Readings ISLR 6 STA 521 Duke University Merlise Clyde October 27, 2017
2 Model Assume that we have centered (as before) and rescaled X o (original X) so that X j = X o j X o j i (Xo ij X o j )2 Equivalent to using r scale(x) divided by n 1 Model: Y = 1β 0 + Xβ + ϵ X T X = Cor(X) (correlation matrix of X) Eigenvalue Decomposition X T X = UΛU T if smallest eigen value is 0, X has columns that are linearly dependent
3 How Good are Various Estimators Quadratic loss for estimating β using estimator a L(β, a) = (β a) T (β a) Consider our expected loss (before we see the data) of taking an action a Under OLS or the Reference prior the Expected Mean Square Error E Y [(β ˆβ) T (β ˆβ) = σ 2 tr[(x T X) 1 ] = p σ 2 j=1 λ 1 j If smallest λ j 0 then MSE
4 Problems Estimates: ˆβ = (X T X) 1 X T Y or with g-prior ˆβ = g 1 + g (XT X) 1 X T Y may be unstable without variable selection. Solutions: remove redundant variables (model selection) (AIC, BIC, other approches) 2 p models combinatorial hard problem even with MCMC add constant to X T X: β = (X T X + ki) 1 X T Y to stabilise eigenvalues - alternative shrinkage estimator/prior
5 Independent Prior Reference prior p(β 0, ϕ) ϕ 1 Prior Distribution on β ϕ, β 0, k N(0 p, 1 ϕk I p) log likelihood (integrated) for β plus prior Posterior mean ϕ 2 ( Y 1Ȳ Xβ 2 + k β 2) b n = (X T X + ki) 1 X T Xˆβ importance of standardizing Choice of k in practice? k = 0 OLS k = estimates are 0 (intercept only)
6 Alternative Motivation If ˆβ is unconstrained expect high variance with nearly singular X Control how large coefficients may grow min β (Y 1Ȳ Xβ)T (Y 1Ȳ Xβ) subject to β 2 j t Equivalent Quadratic Programming Problem min β Yc X c β 2 + k β 2 penalized likelihood Ridge Regression
7 Geometry 1 1 onlinecourses.science.pse.edu
8 Longley Data: library(mass); data(longley) GNP.deflator GNP Unemployed Armed.Force Population Year Employed
9 OLS > longley.lm = lm(employed ~., data=longley) > summary(longley.lm) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e e ** GNP.deflator 1.506e e GNP e e Unemployed e e ** Armed.Forces e e *** Population e e Year 1.829e e ** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 9 degrees of freedom Multiple R-squared: ,^^IAdjusted R-squared: F-statistic: on 6 and 9 DF, p-value: 4.984e-10
10 Ridge Regression # from library MASS longley.ridge = lm.ridge(employed ~., data=longley, lambda=seq(0, 0.1, )) # lambda = k in notes summary(longley.ridge) ## Length Class Mode ## coef none- numeric ## scales 6 -none- numeric ## Inter 1 -none- numeric ## lambda none- numeric ## ym 1 -none- numeric ## xm 6 -none- numeric ## GCV none- numeric ## khkb 1 -none- numeric ## klw 1 -none- numeric
11 Ridge Trace Plot t(x$coef) x$lambda
12 Choice of k k = seq(0, 0.1, ) n.k = length(k); n = nrow(longley) cv.lambda = matrix(na, n, n.k) rmse.ridge = function(data, i, j, k) { m.ridge = lm.ridge(employed ~., data = data, lambda=k[j] subset = -i) yhat = scale(data[i,1:6, drop=f],center = m.ridge$xm, scale = m.ridge$scales) %*% m.ridge$coef + m.ridge$ym (yhat - data$employed[i])^2 } for (i in 1:n) { for (j in 1:n.k) { cv.lambda[i,j] = rmse.ridge(longley, i, j, k) }
13 Cross Validation Error cv.error = apply(cv.lambda, 2, mean) plot(k, cv.error, type="l") cv.error k Best k =
14 Generalized Cross-validation select(lm.ridge(employed ~., data=longley, lambda=seq(0, 0.1, ))) ## modified HKB estimator is ## modified L-W estimator is ## smallest value of GCV at best.k = longley.ridge$lambda[which.min(longley.ridge$gcv)] longley.rreg = lm.ridge(employed ~., data=longley, lambda=best.k) coef(longley.rreg) ## GNP.deflator GNP Unemployed Arme ## e e e e ## Population Year ## e e+00
15 Priors on k X is centered and standardized Y = 1β 0 + Xβ + ϵ Hierarchical prior p(β 0, ϕ β, κ) ϕ 1 β ϕ, κ N(0, I(ϕκ) 1 ) prior on κ? Take κ ϕ Gamma(1/2, 1/2) What is induced prior on β ϕ?
16 Posterior Distributions Joint Distribution β 0, β, ϕ κ, Y Normal-Gamma family given Y and κ κ Y not tractable Obtain marginal for β via MCMC Pick initial values β (0) 0, β(0), ϕ (0), Set t = 1 1. Sample κ (t) p(κ β (t 1) 0, β (t 1), ϕ (t 1), Y) 2. Sample β (t) 0, β(t), ϕ (t) κ(t), Y 3. Set t = t + 1 and repeat until t > T Use Samples β (t) 0, β(t), ϕ (t), κ (t) for t = B,..., T for inference
17 JAGS JAGS = Just Another Gibbs Sampler scripting language to express sampling models and priors derives full conditional distributions integrates with R typically faster than interpreted R code accounts for uncertainty about k How would you compare Bayes predictions with Ridge with Cross-validation?
Linear Model Selection and Regularization. especially usefull in high dimensions p>>100.
Linear Model Selection and Regularization especially usefull in high dimensions p>>100. 1 Why Linear Model Regularization? Linear models are simple, BUT consider p>>n, we have more features than data records
More informationLecture 13: Model selection and regularization
Lecture 13: Model selection and regularization Reading: Sections 6.1-6.2.1 STATS 202: Data mining and analysis October 23, 2017 1 / 17 What do we know so far In linear regression, adding predictors always
More informationLasso. November 14, 2017
Lasso November 14, 2017 Contents 1 Case Study: Least Absolute Shrinkage and Selection Operator (LASSO) 1 1.1 The Lasso Estimator.................................... 1 1.2 Computation of the Lasso Solution............................
More informationThis is called a linear basis expansion, and h m is the mth basis function For example if X is one-dimensional: f (X) = β 0 + β 1 X + β 2 X 2, or
STA 450/4000 S: February 2 2005 Flexible modelling using basis expansions (Chapter 5) Linear regression: y = Xβ + ɛ, ɛ (0, σ 2 ) Smooth regression: y = f (X) + ɛ: f (X) = E(Y X) to be specified Flexible
More informationLab #13 - Resampling Methods Econ 224 October 23rd, 2018
Lab #13 - Resampling Methods Econ 224 October 23rd, 2018 Introduction In this lab you will work through Section 5.3 of ISL and record your code and results in an RMarkdown document. I have added section
More informationLecture 16: High-dimensional regression, non-linear regression
Lecture 16: High-dimensional regression, non-linear regression Reading: Sections 6.4, 7.1 STATS 202: Data mining and analysis November 3, 2017 1 / 17 High-dimensional regression Most of the methods we
More informationGAMs semi-parametric GLMs. Simon Wood Mathematical Sciences, University of Bath, U.K.
GAMs semi-parametric GLMs Simon Wood Mathematical Sciences, University of Bath, U.K. Generalized linear models, GLM 1. A GLM models a univariate response, y i as g{e(y i )} = X i β where y i Exponential
More informationModel Selection and Inference
Model Selection and Inference Merlise Clyde January 29, 2017 Last Class Model for brain weight as a function of body weight In the model with both response and predictor log transformed, are dinosaurs
More informationCSSS 510: Lab 2. Introduction to Maximum Likelihood Estimation
CSSS 510: Lab 2 Introduction to Maximum Likelihood Estimation 2018-10-12 0. Agenda 1. Housekeeping: simcf, tile 2. Questions about Homework 1 or lecture 3. Simulating heteroskedastic normal data 4. Fitting
More informationVariable selection is intended to select the best subset of predictors. But why bother?
Chapter 10 Variable Selection Variable selection is intended to select the best subset of predictors. But why bother? 1. We want to explain the data in the simplest way redundant predictors should be removed.
More information22s:152 Applied Linear Regression
22s:152 Applied Linear Regression Chapter 22: Model Selection In model selection, the idea is to find the smallest set of variables which provides an adequate description of the data. We will consider
More information22s:152 Applied Linear Regression
22s:152 Applied Linear Regression Chapter 22: Model Selection In model selection, the idea is to find the smallest set of variables which provides an adequate description of the data. We will consider
More informationLinear Methods for Regression and Shrinkage Methods
Linear Methods for Regression and Shrinkage Methods Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer 1 Linear Regression Models Least Squares Input vectors
More informationPenalized regression Statistical Learning, 2011
Penalized regression Statistical Learning, 2011 Niels Richard Hansen September 19, 2011 Penalized regression is implemented in several different R packages. Ridge regression can, in principle, be carried
More informationPoisson Regression and Model Checking
Poisson Regression and Model Checking Readings GH Chapter 6-8 September 27, 2017 HIV & Risk Behaviour Study The variables couples and women_alone code the intervention: control - no counselling (both 0)
More informationStatistical Matching using Fractional Imputation
Statistical Matching using Fractional Imputation Jae-Kwang Kim 1 Iowa State University 1 Joint work with Emily Berg and Taesung Park 1 Introduction 2 Classical Approaches 3 Proposed method 4 Application:
More informationChapter 6: Linear Model Selection and Regularization
Chapter 6: Linear Model Selection and Regularization As p (the number of predictors) comes close to or exceeds n (the sample size) standard linear regression is faced with problems. The variance of the
More informationPredictive Checking. Readings GH Chapter 6-8. February 8, 2017
Predictive Checking Readings GH Chapter 6-8 February 8, 2017 Model Choice and Model Checking 2 Questions: 1. Is my Model good enough? (no alternative models in mind) 2. Which Model is best? (comparison
More informationPerformance Estimation and Regularization. Kasthuri Kannan, PhD. Machine Learning, Spring 2018
Performance Estimation and Regularization Kasthuri Kannan, PhD. Machine Learning, Spring 2018 Bias- Variance Tradeoff Fundamental to machine learning approaches Bias- Variance Tradeoff Error due to Bias:
More informationStat 8053, Fall 2013: Additive Models
Stat 853, Fall 213: Additive Models We will only use the package mgcv for fitting additive and later generalized additive models. The best reference is S. N. Wood (26), Generalized Additive Models, An
More information7. Collinearity and Model Selection
Sociology 740 John Fox Lecture Notes 7. Collinearity and Model Selection Copyright 2014 by John Fox Collinearity and Model Selection 1 1. Introduction I When there is a perfect linear relationship among
More informationGeneralized Additive Models
:p Texts in Statistical Science Generalized Additive Models An Introduction with R Simon N. Wood Contents Preface XV 1 Linear Models 1 1.1 A simple linear model 2 Simple least squares estimation 3 1.1.1
More informationExercise 2.23 Villanova MAT 8406 September 7, 2015
Exercise 2.23 Villanova MAT 8406 September 7, 2015 Step 1: Understand the Question Consider the simple linear regression model y = 50 + 10x + ε where ε is NID(0, 16). Suppose that n = 20 pairs of observations
More informationMultiple Linear Regression
Multiple Linear Regression Rebecca C. Steorts, Duke University STA 325, Chapter 3 ISL 1 / 49 Agenda How to extend beyond a SLR Multiple Linear Regression (MLR) Relationship Between the Response and Predictors
More informationDATA ANALYSIS USING HIERARCHICAL GENERALIZED LINEAR MODELS WITH R
DATA ANALYSIS USING HIERARCHICAL GENERALIZED LINEAR MODELS WITH R Lee, Rönnegård & Noh LRN@du.se Lee, Rönnegård & Noh HGLM book 1 / 25 Overview 1 Background to the book 2 A motivating example from my own
More informationPackage msgps. February 20, 2015
Type Package Package msgps February 20, 2015 Title Degrees of freedom of elastic net, adaptive lasso and generalized elastic net Version 1.3 Date 2012-5-17 Author Kei Hirose Maintainer Kei Hirose
More informationFMA901F: Machine Learning Lecture 3: Linear Models for Regression. Cristian Sminchisescu
FMA901F: Machine Learning Lecture 3: Linear Models for Regression Cristian Sminchisescu Machine Learning: Frequentist vs. Bayesian In the frequentist setting, we seek a fixed parameter (vector), with value(s)
More informationLecture 25: Review I
Lecture 25: Review I Reading: Up to chapter 5 in ISLR. STATS 202: Data mining and analysis Jonathan Taylor 1 / 18 Unsupervised learning In unsupervised learning, all the variables are on equal standing,
More informationModel selection. Peter Hoff. 560 Hierarchical modeling. Statistics, University of Washington 1/41
1/41 Model selection 560 Hierarchical modeling Peter Hoff Statistics, University of Washington /41 Modeling choices Model: A statistical model is a set of probability distributions for your data. In HLM,
More informationRegression Analysis and Linear Regression Models
Regression Analysis and Linear Regression Models University of Trento - FBK 2 March, 2015 (UNITN-FBK) Regression Analysis and Linear Regression Models 2 March, 2015 1 / 33 Relationship between numerical
More informationAlgorithms for LTS regression
Algorithms for LTS regression October 26, 2009 Outline Robust regression. LTS regression. Adding row algorithm. Branch and bound algorithm (BBA). Preordering BBA. Structured problems Generalized linear
More informationLecture 17: Smoothing splines, Local Regression, and GAMs
Lecture 17: Smoothing splines, Local Regression, and GAMs Reading: Sections 7.5-7 STATS 202: Data mining and analysis November 6, 2017 1 / 24 Cubic splines Define a set of knots ξ 1 < ξ 2 < < ξ K. We want
More informationSection 3.2: Multiple Linear Regression II. Jared S. Murray The University of Texas at Austin McCombs School of Business
Section 3.2: Multiple Linear Regression II Jared S. Murray The University of Texas at Austin McCombs School of Business 1 Multiple Linear Regression: Inference and Understanding We can answer new questions
More informationMultivariate Analysis Multivariate Calibration part 2
Multivariate Analysis Multivariate Calibration part 2 Prof. Dr. Anselmo E de Oliveira anselmo.quimica.ufg.br anselmo.disciplinas@gmail.com Linear Latent Variables An essential concept in multivariate data
More informationLast time... Coryn Bailer-Jones. check and if appropriate remove outliers, errors etc. linear regression
Machine learning, pattern recognition and statistical data modelling Lecture 3. Linear Methods (part 1) Coryn Bailer-Jones Last time... curse of dimensionality local methods quickly become nonlocal as
More informationDATA ANALYSIS USING HIERARCHICAL GENERALIZED LINEAR MODELS WITH R
DATA ANALYSIS USING HIERARCHICAL GENERALIZED LINEAR MODELS WITH R Lee, Rönnegård & Noh LRN@du.se Lee, Rönnegård & Noh HGLM book 1 / 24 Overview 1 Background to the book 2 Crack growth example 3 Contents
More informationModel selection and validation 1: Cross-validation
Model selection and validation 1: Cross-validation Ryan Tibshirani Data Mining: 36-462/36-662 March 26 2013 Optional reading: ISL 2.2, 5.1, ESL 7.4, 7.10 1 Reminder: modern regression techniques Over the
More informationLinear Modeling with Bayesian Statistics
Linear Modeling with Bayesian Statistics Bayesian Approach I I I I I Estimate probability of a parameter State degree of believe in specific parameter values Evaluate probability of hypothesis given the
More informationST512. Fall Quarter, Exam 1. Directions: Answer questions as directed. Please show work. For true/false questions, circle either true or false.
ST512 Fall Quarter, 2005 Exam 1 Name: Directions: Answer questions as directed. Please show work. For true/false questions, circle either true or false. 1. (42 points) A random sample of n = 30 NBA basketball
More informationBayesian model selection and diagnostics
Bayesian model selection and diagnostics A typical Bayesian analysis compares a handful of models. Example 1: Consider the spline model for the motorcycle data, how many basis functions? Example 2: Consider
More information1 StatLearn Practical exercise 5
1 StatLearn Practical exercise 5 Exercise 1.1. Download the LA ozone data set from the book homepage. We will be regressing the cube root of the ozone concentration on the other variables. Divide the data
More informationThe Data. Math 158, Spring 2016 Jo Hardin Shrinkage Methods R code Ridge Regression & LASSO
Math 158, Spring 2016 Jo Hardin Shrinkage Methods R code Ridge Regression & LASSO The Data The following dataset is from Hastie, Tibshirani and Friedman (2009), from a studyby Stamey et al. (1989) of prostate
More information14. League: A factor with levels A and N indicating player s league at the end of 1986
PENALIZED REGRESSION Ridge and The LASSO Note: The example contained herein was copied from the lab exercise in Chapter 6 of Introduction to Statistical Learning by. For this exercise, we ll use some baseball
More informationUsing the SemiPar Package
Using the SemiPar Package NICHOLAS J. SALKOWSKI Division of Biostatistics, School of Public Health, University of Minnesota, Minneapolis, MN 55455, USA salk0008@umn.edu May 15, 2008 1 Introduction The
More informationApplied Statistics and Econometrics Lecture 6
Applied Statistics and Econometrics Lecture 6 Giuseppe Ragusa Luiss University gragusa@luiss.it http://gragusa.org/ March 6, 2017 Luiss University Empirical application. Data Italian Labour Force Survey,
More informationThe linear mixed model: modeling hierarchical and longitudinal data
The linear mixed model: modeling hierarchical and longitudinal data Analysis of Experimental Data AED The linear mixed model: modeling hierarchical and longitudinal data 1 of 44 Contents 1 Modeling Hierarchical
More informationRepeated Measures Part 4: Blood Flow data
Repeated Measures Part 4: Blood Flow data /* bloodflow.sas */ options linesize=79 pagesize=100 noovp formdlim='_'; title 'Two within-subjecs factors: Blood flow data (NWK p. 1181)'; proc format; value
More informationStat 5303 (Oehlert): Response Surfaces 1
Stat 5303 (Oehlert): Response Surfaces 1 > data
More informationLecture 7: Linear Regression (continued)
Lecture 7: Linear Regression (continued) Reading: Chapter 3 STATS 2: Data mining and analysis Jonathan Taylor, 10/8 Slide credits: Sergio Bacallado 1 / 14 Potential issues in linear regression 1. Interactions
More informationRegularization Methods. Business Analytics Practice Winter Term 2015/16 Stefan Feuerriegel
Regularization Methods Business Analytics Practice Winter Term 2015/16 Stefan Feuerriegel Today s Lecture Objectives 1 Avoiding overfitting and improving model interpretability with the help of regularization
More informationPackage EBglmnet. January 30, 2016
Type Package Package EBglmnet January 30, 2016 Title Empirical Bayesian Lasso and Elastic Net Methods for Generalized Linear Models Version 4.1 Date 2016-01-15 Author Anhui Huang, Dianting Liu Maintainer
More informationPredictor Selection Algorithm for Bayesian Lasso
Predictor Selection Algorithm for Baesian Lasso Quan Zhang Ma 16, 2014 1 Introduction The Lasso [1] is a method in regression model for coefficients shrinkage and model selection. It is often used in the
More informationNONPARAMETRIC REGRESSION SPLINES FOR GENERALIZED LINEAR MODELS IN THE PRESENCE OF MEASUREMENT ERROR
NONPARAMETRIC REGRESSION SPLINES FOR GENERALIZED LINEAR MODELS IN THE PRESENCE OF MEASUREMENT ERROR J. D. Maca July 1, 1997 Abstract The purpose of this manual is to demonstrate the usage of software for
More informationMachine Learning. Topic 4: Linear Regression Models
Machine Learning Topic 4: Linear Regression Models (contains ideas and a few images from wikipedia and books by Alpaydin, Duda/Hart/ Stork, and Bishop. Updated Fall 205) Regression Learning Task There
More informationSplines and penalized regression
Splines and penalized regression November 23 Introduction We are discussing ways to estimate the regression function f, where E(y x) = f(x) One approach is of course to assume that f has a certain shape,
More informationSolution to Bonus Questions
Solution to Bonus Questions Q2: (a) The histogram of 1000 sample means and sample variances are plotted below. Both histogram are symmetrically centered around the true lambda value 20. But the sample
More informationthat is, Data Science Hello World.
R 4 hackers Hello World that is, Data Science Hello World. We got some data... Sure, first we ALWAYS do some data exploration. data(longley) head(longley) GNP.deflator GNP Unemployed Armed.Forces Population
More informationDivide and Conquer Kernel Ridge Regression
Divide and Conquer Kernel Ridge Regression Yuchen Zhang John Duchi Martin Wainwright University of California, Berkeley COLT 2013 Yuchen Zhang (UC Berkeley) Divide and Conquer KRR COLT 2013 1 / 15 Problem
More informationSection 3.4: Diagnostics and Transformations. Jared S. Murray The University of Texas at Austin McCombs School of Business
Section 3.4: Diagnostics and Transformations Jared S. Murray The University of Texas at Austin McCombs School of Business 1 Regression Model Assumptions Y i = β 0 + β 1 X i + ɛ Recall the key assumptions
More informationDoubly Cyclic Smoothing Splines and Analysis of Seasonal Daily Pattern of CO2 Concentration in Antarctica
Boston-Keio Workshop 2016. Doubly Cyclic Smoothing Splines and Analysis of Seasonal Daily Pattern of CO2 Concentration in Antarctica... Mihoko Minami Keio University, Japan August 15, 2016 Joint work with
More informationTopics in Machine Learning-EE 5359 Model Assessment and Selection
Topics in Machine Learning-EE 5359 Model Assessment and Selection Ioannis D. Schizas Electrical Engineering Department University of Texas at Arlington 1 Training and Generalization Training stage: Utilizing
More informationWhat is machine learning?
Machine learning, pattern recognition and statistical data modelling Lecture 12. The last lecture Coryn Bailer-Jones 1 What is machine learning? Data description and interpretation finding simpler relationship
More informationStat 4510/7510 Homework 4
Stat 45/75 1/7. Stat 45/75 Homework 4 Instructions: Please list your name and student number clearly. In order to receive credit for a problem, your solution must show sufficient details so that the grader
More informationRegression III: Lab 4
Regression III: Lab 4 This lab will work through some model/variable selection problems, finite mixture models and missing data issues. You shouldn t feel obligated to work through this linearly, I would
More informationGeneralized Additive Models
Generalized Additive Models Statistics 135 Autumn 2005 Copyright c 2005 by Mark E. Irwin Generalized Additive Models GAMs are one approach to non-parametric regression in the multiple predictor setting.
More informationStatistics Lab #7 ANOVA Part 2 & ANCOVA
Statistics Lab #7 ANOVA Part 2 & ANCOVA PSYCH 710 7 Initialize R Initialize R by entering the following commands at the prompt. You must type the commands exactly as shown. options(contrasts=c("contr.sum","contr.poly")
More informationSolution to Series 7
Dr. Marcel Dettling Applied Statistical Regression AS 2015 Solution to Series 7 1. a) We begin the analysis by plotting histograms and barplots for all variables. > ## load data > load("customerwinback.rda")
More informationLast time... Bias-Variance decomposition. This week
Machine learning, pattern recognition and statistical data modelling Lecture 4. Going nonlinear: basis expansions and splines Last time... Coryn Bailer-Jones linear regression methods for high dimensional
More informationMissing Data and Imputation
Missing Data and Imputation Hoff Chapter 7, GH Chapter 25 April 21, 2017 Bednets and Malaria Y:presence or absence of parasites in a blood smear AGE: age of child BEDNET: bed net use (exposure) GREEN:greenness
More informationMCMC Methods for data modeling
MCMC Methods for data modeling Kenneth Scerri Department of Automatic Control and Systems Engineering Introduction 1. Symposium on Data Modelling 2. Outline: a. Definition and uses of MCMC b. MCMC algorithms
More informationPackage rgcvpack. February 20, Index 6. Fitting Thin Plate Smoothing Spline. Fit thin plate splines of any order with user specified knots
Version 0.1-4 Date 2013/10/25 Title R Interface for GCVPACK Fortran Package Author Xianhong Xie Package rgcvpack February 20, 2015 Maintainer Xianhong Xie
More informationBivariate (Simple) Regression Analysis
Revised July 2018 Bivariate (Simple) Regression Analysis This set of notes shows how to use Stata to estimate a simple (two-variable) regression equation. It assumes that you have set Stata up on your
More informationSplines. Patrick Breheny. November 20. Introduction Regression splines (parametric) Smoothing splines (nonparametric)
Splines Patrick Breheny November 20 Patrick Breheny STA 621: Nonparametric Statistics 1/46 Introduction Introduction Problems with polynomial bases We are discussing ways to estimate the regression function
More informationModel selection Outline for today
Model selection Outline for today The problem of model selection Choose among models by a criterion rather than significance testing Criteria: Mallow s C p and AIC Search strategies: All subsets; stepaic
More informationDS Machine Learning and Data Mining I. Alina Oprea Associate Professor, CCIS Northeastern University
DS 4400 Machine Learning and Data Mining I Alina Oprea Associate Professor, CCIS Northeastern University September 20 2018 Review Solution for multiple linear regression can be computed in closed form
More informationModel selection. Peter Hoff STAT 423. Applied Regression and Analysis of Variance. University of Washington /53
/53 Model selection Peter Hoff STAT 423 Applied Regression and Analysis of Variance University of Washington Diabetes example: y = diabetes progression x 1 = age x 2 = sex. dim(x) ## [1] 442 64 colnames(x)
More informationAnalysis of variance - ANOVA
Analysis of variance - ANOVA Based on a book by Julian J. Faraway University of Iceland (UI) Estimation 1 / 50 Anova In ANOVAs all predictors are categorical/qualitative. The original thinking was to try
More informationPS 6: Regularization. PART A: (Source: HTF page 95) The Ridge regression problem is:
Economics 1660: Big Data PS 6: Regularization Prof. Daniel Björkegren PART A: (Source: HTF page 95) The Ridge regression problem is: : β "#$%& = argmin (y # β 2 x #4 β 4 ) 6 6 + λ β 4 #89 Consider the
More informationDetecting and Circumventing Collinearity or Ill-Conditioning Problems
Chapter 8 Detecting and Circumventing Collinearity or Ill-Conditioning Problems Section 8.1 Introduction Multicollinearity/Collinearity/Ill-Conditioning The terms multicollinearity, collinearity, and ill-conditioning
More information1 Methods for Posterior Simulation
1 Methods for Posterior Simulation Let p(θ y) be the posterior. simulation. Koop presents four methods for (posterior) 1. Monte Carlo integration: draw from p(θ y). 2. Gibbs sampler: sequentially drawing
More informationCDAA No. 4 - Part Two - Multiple Regression - Initial Data Screening
CDAA No. 4 - Part Two - Multiple Regression - Initial Data Screening Variables Entered/Removed b Variables Entered GPA in other high school, test, Math test, GPA, High school math GPA a Variables Removed
More informationCalibration and emulation of TIE-GCM
Calibration and emulation of TIE-GCM Serge Guillas School of Mathematics Georgia Institute of Technology Jonathan Rougier University of Bristol Big Thanks to Crystal Linkletter (SFU-SAMSI summer school)
More informationWeek 5: Multiple Linear Regression II
Week 5: Multiple Linear Regression II Marcelo Coca Perraillon University of Colorado Anschutz Medical Campus Health Services Research Methods I HSMP 7607 2017 c 2017 PERRAILLON ARR 1 Outline Adjusted R
More informationLecture on Modeling Tools for Clustering & Regression
Lecture on Modeling Tools for Clustering & Regression CS 590.21 Analysis and Modeling of Brain Networks Department of Computer Science University of Crete Data Clustering Overview Organizing data into
More informationGelman-Hill Chapter 3
Gelman-Hill Chapter 3 Linear Regression Basics In linear regression with a single independent variable, as we have seen, the fundamental equation is where ŷ bx 1 b0 b b b y 1 yx, 0 y 1 x x Bivariate Normal
More informationPackage SafeBayes. October 20, 2016
Type Package Package SafeBayes October 20, 2016 Title Generalized and Safe-Bayesian Ridge and Lasso Regression Version 1.1 Date 2016-10-17 Depends R (>= 3.1.2), stats Description Functions for Generalized
More informationOutline. Topic 16 - Other Remedies. Ridge Regression. Ridge Regression. Ridge Regression. Robust Regression. Regression Trees. Piecewise Linear Model
Topic 16 - Other Remedies Ridge Regression Robust Regression Regression Trees Outline - Fall 2013 Piecewise Linear Model Bootstrapping Topic 16 2 Ridge Regression Modification of least squares that addresses
More informationFast Ridge Regression with Randomized Principal Component Analysis and Gradient Descent
Fast Ridge Regression with Randomized Principal Component Analysis and Gradient Descent Yichao Lu and Dean P. Foster Department of Statistics Wharton, University of Pennsylvania Philadelphia, PA, 94-634
More informationP-spline ANOVA-type interaction models for spatio-temporal smoothing
P-spline ANOVA-type interaction models for spatio-temporal smoothing Dae-Jin Lee and María Durbán Universidad Carlos III de Madrid Department of Statistics IWSM Utrecht 2008 D.-J. Lee and M. Durban (UC3M)
More informationSimulation studies. Patrick Breheny. September 8. Monte Carlo simulation Example: Ridge vs. Lasso vs. Subset
Simulation studies Patrick Breheny September 8 Patrick Breheny BST 764: Applied Statistical Modeling 1/17 Introduction In statistics, we are often interested in properties of various estimation and model
More information9.1 Random coefficients models Constructed data Consumer preference mapping of carrots... 10
St@tmaster 02429/MIXED LINEAR MODELS PREPARED BY THE STATISTICS GROUPS AT IMM, DTU AND KU-LIFE Module 9: R 9.1 Random coefficients models...................... 1 9.1.1 Constructed data........................
More informationAmong those 14 potential explanatory variables,non-dummy variables are:
Among those 14 potential explanatory variables,non-dummy variables are: Size: 2nd column in the dataset Land: 14th column in the dataset Bed.Rooms: 5th column in the dataset Fireplace: 7th column in the
More informationA quick introduction to First Bayes
A quick introduction to First Bayes Lawrence Joseph October 1, 2003 1 Introduction This document very briefly reviews the main features of the First Bayes statistical teaching package. For full details,
More informationLecture 26: Missing data
Lecture 26: Missing data Reading: ESL 9.6 STATS 202: Data mining and analysis December 1, 2017 1 / 10 Missing data is everywhere Survey data: nonresponse. 2 / 10 Missing data is everywhere Survey data:
More informationPerforming Cluster Bootstrapped Regressions in R
Performing Cluster Bootstrapped Regressions in R Francis L. Huang / October 6, 2016 Supplementary material for: Using Cluster Bootstrapping to Analyze Nested Data with a Few Clusters in Educational and
More informationPractice in R. 1 Sivan s practice. 2 Hetroskadasticity. January 28, (pdf version)
Practice in R January 28, 2010 (pdf version) 1 Sivan s practice Her practice file should be (here), or check the web for a more useful pointer. 2 Hetroskadasticity ˆ Let s make some hetroskadastic data:
More informationRegression on SAT Scores of 374 High Schools and K-means on Clustering Schools
Regression on SAT Scores of 374 High Schools and K-means on Clustering Schools Abstract In this project, we study 374 public high schools in New York City. The project seeks to use regression techniques
More informationPackage ridge. R topics documented: February 15, Title Ridge Regression with automatic selection of the penalty parameter. Version 2.
Package ridge February 15, 2013 Title Ridge Regression with automatic selection of the penalty parameter Version 2.1-2 Date 2012-25-09 Author Erika Cule Linear and logistic ridge regression for small data
More informationTwo-Stage Least Squares
Chapter 316 Two-Stage Least Squares Introduction This procedure calculates the two-stage least squares (2SLS) estimate. This method is used fit models that include instrumental variables. 2SLS includes
More informationSection 2.2: Covariance, Correlation, and Least Squares
Section 2.2: Covariance, Correlation, and Least Squares Jared S. Murray The University of Texas at Austin McCombs School of Business Suggested reading: OpenIntro Statistics, Chapter 7.1, 7.2 1 A Deeper
More information