Logistic Regression Maximum Likelihood Estimation
|
|
- Vernon Hodge
- 6 years ago
- Views:
Transcription
1 Harvard-MIT Dvson of Health Scences and Technology HST.951J: Medcal Decson Support, Fall 2005 Instructors: Professor Lucla Ohno-Machado and Professor Staal Vnterbo 6.873/HST.951 Medcal Decson Support Fall 2005 Logstc Regresson Mamum Lkelhood Estmaton Lucla Ohno-Machado
2 Rsk Score of Death from Angoplasty Unadjusted Overall Mortalty Rate = 2.1% % Number of Cases % Number of Cases 26% Mortalty Rsk 21.5% 53.6% 50% 40% 30% 20% 12.4% % 0.4% 1.4% 2.2% 2.9% 1.6% 1.3% 0 to 2 3 to 4 5 to 6 7 to 8 9 to 10 >10 Rsk Score Category 10% 0%
3 Lnear Regresson Ordnary Least Squares (OLS) Mnmze Sum of Squared Errors (SSE) y 3 n data ponts s the subscrpt for each pont ŷ = β 0 + β 1 n n =1 =1 SSE = (y ŷ ) = [y (β + β )]
4 Logt 1 p = 1+ e (β +β ) 0 1 y p = β 0 +β 1 e e β 0 +β 1 +1 p log 1 p = β 01+ β 1 logt
5 Increasng β Seres1 0.6 Seres1 Seres
6 Fndng β 0 Baselne case 1 p = 1+ e (β ) 0 Death Blue(1) Green(0) Lfe Total = 1+ e (β ) 0 β 0 =
7 Odds rato Odds: p/(1-p) Odds-rato Death Blue Green OR = p death blue 1 p death p blue death green 1 p death green 28/ OR = = / 52 Lfe Total
8 What do coeffcents mean? e β color = ORcolor Death OR = 28/ 45 = / 52 e β color = Blue Green β color = Lfe p blue = 1+ e ( Total = ) 1 p green = = e
9 What do coeffcents mean? e β age = ORage Death Age49 Age50 OR = p 1 p death age =50 death age = p death age =49 Lfe Total p death age =49
10 Why not search usng OLS? ŷ = β 0 + β 1 y 3 n SSE = (y ŷ ) 1 =1 logt p log 1 p = β 01+ β 1
11 P(model data)? p = 1+ e ( If only ntercept s allowed, whch 1 value would t have? β β 1 ) 0 + y y
12 P (data model)? P(data model) = [P(model data) P(data)] / P(model) When comparng models: P(model): assume all the same (e, chances of beng a model wth hgh coeffcents the same as low, etc) P(data): assume t s the same Then, P(data model) α P(model data)
13 Mamum Lkelhood Estmaton Mamze P(data model) Mamze the probablty that we would observe what we observed (gven assumpton of a partcular model) Choose the best parameters from the partcular model logt
14 Mamum Lkelhood Estmaton Steps: Defne epresson for the probablty of data as a functon of the parameters Fnd the values of the parameters that mamze ths epresson
15 Lkelhood Functon L = Pr(Y ) L = Pr( y 1, y 2,..., y n ) L = Pr( y ) Pr(y n 1 2 )... Pr( y n )= = 1 Pr( y )
16 L = Pr(Y ) L = Pr( Lkelhood Functon y, 1 y 2,..., y n ) L = Pr( y ) Pr( y Bnomal )... Pr( y ) = Pr( y ) 1 2 n =1 n Pr( y = 1) = p Pr( y = 0) = (1 p ) Pr( y ) = p y (1 p ) 1 y
17 Lkelhood Functon log L log L n L = Pr( y ) = p (1 p ) = 1 = 1 n p L = = 1 (1 p ) = = y log n y p (1 p ) + y ( β ) log(1 + e y (1 p ) 1 y log(1 β ) p ) Snce model s the logt
18 Log Lkelhood Functon n L = Pr( y ) = p (1 p ) = 1 = 1 n p y L = = 1 (1 p ) n y (1 p ) p 1 y log L = y log + log(1 p ) (1 p )
19 Log Lkelhood Functon p log L = y log + log(1 p ) (1 p ) log L = y (β ) log(1+ e β ) Snce model s the logt
20 Mamze log L = y (β ) log(1+ e β ) log L = β yˆ = 1 1+ e β y yˆ = 0 Not easy to solve because y-hat s non-lnear, need to use teratve methods: most popular s Newton-Raphson
21 Mamze log L = y (β ) log(1+ e β ) log L = y y β ˆ = 0 ŷ = 1 Not easy to solve because y-hat s non-lnear, need to use teratve methods: most popular s Newton-Raphson 1+ e β
22 Newton-Raphson Start wth random or zero βs walk n the drecton that mamzes MLE how bg a step (Gradent or Score) drecton
23 Mamzng the LogLkelhood Log Lkelhood β +1 Frst teraton LL β Intal LL
24 Mamzng the LogLkelhood Log Lkelhood β +1 Second teraton LL β New Intal LL
25 Smlar teratve method to Mnmzng the Error n Gradent Descent (neural nets) Error surface ntal error negatve dervatve fnal error local mnmum w ntal w traned postve change
26 Newton-Raphson Algorthm log L = y (β ) log(1+ e β ) U (β ) = log L = y β ˆ y Gradent I (β ) = 2 log L = β β ' ' y ˆ (1 y ˆ ) Hessan 1 β j + 1 = β j I (β j )U (β j ) a step
27 Convergence Crteron β β j+1 j < β j Convergence problems: complete and quascomplete separaton
28 Complete separaton MLE does not est (e, t s nfnte) β β +1 y y
29 Quas-complete separaton Same values for predctors, dfferent outcomes y
30 No (quas)complete separaton s fne to fnd MLE y
31 How good s the model? Is t better than predctng the same pror probablty for everyone? (e, model wth just β 0 ) How well do the tranng data ft? How well does s generalze?
32 Generalzed lkelhood-rato test Are β 1, β 2,, β dfferent from 0? n n y n L = Pr( y ) = p (1 p ) = 1 = 1 1 y log L = [y log p + (1 y ) log( 1 p )] G = 2 log L + 2 log L G has χ 2 dstrbuton o 1 cross entropy _ error = [y log p + (1 y ) log( 1 p )]
33 AIC, SC, BIC To compare models Akake s Informaton Crteron, k parameters AIC= 2 log L+2 k Schwartz Crteron, Bayesan Informaton Crteron, n cases BIC= 2 log L +k logn
34 Summary Mamum Lkelhood Estmaton s used n fndng parameters for models MLE mamzes the probablty that the data obtaned would have been generated by the model Comng up: goodness-of-ft (how good are the predctons?) How well do the tranng data ft? How well does s generalze?
Decision Analysis (part 2 of 2) Review Linear Regression
Harvard-MIT Dvson of Health Scences and Technology HST.951J: Medcal Decson Support, Fall 2005 Instructors: Professor Lucla Ohno-Machado and Professor Staal Vnterbo 6.873/HST.951 Medcal Decson Support Fall
More informationINF 5860 Machine learning for image classification. Lecture 3 : Image classification and regression part II Anne Solberg January 31, 2018
INF 5860 Machne learnng for mage classfcaton Lecture 3 : Image classfcaton and regresson part II Anne Solberg January 3, 08 Today s topcs Multclass logstc regresson and softma Regularzaton Image classfcaton
More informationLogistic Regression. CAP 5610: Machine Learning Instructor: Guo-Jun QI
Logstc Regresson CAP 561: achne Learnng Instructor: Guo-Jun QI Bayes Classfer: A Generatve model odel the posteror dstrbuton P(Y X) Estmate class-condtonal dstrbuton P(X Y) for each Y Estmate pror dstrbuton
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationMLE and Bayesian Estimation. Jie Tang Department of Computer Science & Technology Tsinghua University 2012
MLE and Bayesan Estmaton Je Tang Department of Computer Scence & Technology Tsnghua Unversty 01 1 Lnear Regresson? As the frst step, we need to decde how we re gong to represent the functon f. One example:
More information8/25/17. Data Modeling. Data Modeling. Data Modeling. Patrice Koehl Department of Biological Sciences National University of Singapore
8/5/17 Data Modelng Patrce Koehl Department of Bologcal Scences atonal Unversty of Sngapore http://www.cs.ucdavs.edu/~koehl/teachng/bl59 koehl@cs.ucdavs.edu Data Modelng Ø Data Modelng: least squares Ø
More informationCIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M
CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationDiscriminative classifier: Logistic Regression. CS534-Machine Learning
Dscrmnatve classfer: Logstc Regresson CS534-Machne Learnng 2 Logstc Regresson Gven tranng set D stc regresson learns the condtonal dstrbuton We ll assume onl to classes and a parametrc form for here s
More informationPattern Classification
Pattern Classfcaton All materals n these sldes ere taken from Pattern Classfcaton (nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wley & Sons, 000 th the permsson of the authors and the publsher
More informationTopic 5: Non-Linear Regression
Topc 5: Non-Lnear Regresson The models we ve worked wth so far have been lnear n the parameters. They ve been of the form: y = Xβ + ε Many models based on economc theory are actually non-lnear n the parameters.
More information10-701/ Machine Learning, Fall 2005 Homework 3
10-701/15-781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons-10701@autonlaborg for queston Problem 1 Regresson and Cross-valdaton [40
More informationSTAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression
STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,
More informationClassification as a Regression Problem
Target varable y C C, C,, ; Classfcaton as a Regresson Problem { }, 3 L C K To treat classfcaton as a regresson problem we should transform the target y nto numercal values; The choce of numercal class
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informationDiscriminative classifier: Logistic Regression. CS534-Machine Learning
Dscrmnatve classfer: Logstc Regresson CS534-Machne Learnng robablstc Classfer Gven an nstance, hat does a probablstc classfer do dfferentl compared to, sa, perceptron? It does not drectl predct Instead,
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationMultilayer Perceptron (MLP)
Multlayer Perceptron (MLP) Seungjn Cho Department of Computer Scence and Engneerng Pohang Unversty of Scence and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjn@postech.ac.kr 1 / 20 Outlne
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationMaximum Likelihood Estimation (MLE)
Maxmum Lkelhood Estmaton (MLE) Ken Kreutz-Delgado (Nuno Vasconcelos) ECE 175A Wnter 01 UCSD Statstcal Learnng Goal: Gven a relatonshp between a feature vector x and a vector y, and d data samples (x,y
More informationLinear regression. Regression Models. Chapter 11 Student Lecture Notes Regression Analysis is the
Chapter 11 Student Lecture Notes 11-1 Lnear regresson Wenl lu Dept. Health statstcs School of publc health Tanjn medcal unversty 1 Regresson Models 1. Answer What Is the Relatonshp Between the Varables?.
More informationThe Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More informationIV. Performance Optimization
IV. Performance Optmzaton A. Steepest descent algorthm defnton how to set up bounds on learnng rate mnmzaton n a lne (varyng learnng rate) momentum learnng examples B. Newton s method defnton Gauss-Newton
More informationsince [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation
Econ 388 R. Butler 204 revsons Lecture 4 Dummy Dependent Varables I. Lnear Probablty Model: the Regresson model wth a dummy varables as the dependent varable assumpton, mplcaton regular multple regresson
More informationLINEAR REGRESSION MODELS W4315
LINEAR REGRESSION MODELS W4315 HOMEWORK ANSWERS February 15, 010 Instructor: Frank Wood 1. (0 ponts) In the fle problem1.txt (accessble on professor s webste), there are 500 pars of data, where the frst
More informationSDMML HT MSc Problem Sheet 4
SDMML HT 06 - MSc Problem Sheet 4. The recever operatng characterstc ROC curve plots the senstvty aganst the specfcty of a bnary classfer as the threshold for dscrmnaton s vared. Let the data space be
More informationThe EM Algorithm (Dempster, Laird, Rubin 1977) The missing data or incomplete data setting: ODL(φ;Y ) = [Y;φ] = [Y X,φ][X φ] = X
The EM Algorthm (Dempster, Lard, Rubn 1977 The mssng data or ncomplete data settng: An Observed Data Lkelhood (ODL that s a mxture or ntegral of Complete Data Lkelhoods (CDL. (1a ODL(;Y = [Y;] = [Y,][
More informationSupport Vector Machines
Support Vector Machnes Konstantn Tretyakov (kt@ut.ee) MTAT.03.227 Machne Learnng So far So far Supervsed machne learnng Lnear models Non-lnear models Unsupervsed machne learnng Generc scaffoldng So far
More informationBIO Lab 2: TWO-LEVEL NORMAL MODELS with school children popularity data
Lab : TWO-LEVEL NORMAL MODELS wth school chldren popularty data Purpose: Introduce basc two-level models for normally dstrbuted responses usng STATA. In partcular, we dscuss Random ntercept models wthout
More informationLaboratory 1c: Method of Least Squares
Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly
More informationSupport Vector Machines
Support Vector Machnes Konstantn Tretyakov (kt@ut.ee) MTAT.03.227 Machne Learnng So far Supervsed machne learnng Lnear models Least squares regresson Fsher s dscrmnant, Perceptron, Logstc model Non-lnear
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
More informationMACHINE APPLIED MACHINE LEARNING LEARNING. Gaussian Mixture Regression
11 MACHINE APPLIED MACHINE LEARNING LEARNING MACHINE LEARNING Gaussan Mture Regresson 22 MACHINE APPLIED MACHINE LEARNING LEARNING Bref summary of last week s lecture 33 MACHINE APPLIED MACHINE LEARNING
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationINTRODUCTION TO MACHINE LEARNING 3RD EDITION
ETHEM ALPAYDIN The MIT Press, 2014 Lecture Sldes for INTRODUCTION TO MACHINE LEARNING 3RD EDITION alpaydn@boun.edu.tr http://www.cmpe.boun.edu.tr/~ethem/2ml3e CHAPTER 3: BAYESIAN DECISION THEORY Probablty
More informationChapter 3. Two-Variable Regression Model: The Problem of Estimation
Chapter 3. Two-Varable Regresson Model: The Problem of Estmaton Ordnary Least Squares Method (OLS) Recall that, PRF: Y = β 1 + β X + u Thus, snce PRF s not drectly observable, t s estmated by SRF; that
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More informationHydrological statistics. Hydrological statistics and extremes
5--0 Stochastc Hydrology Hydrologcal statstcs and extremes Marc F.P. Berkens Professor of Hydrology Faculty of Geoscences Hydrologcal statstcs Mostly concernes wth the statstcal analyss of hydrologcal
More informationWhich Separator? Spring 1
Whch Separator? 6.034 - Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng 3 Margn of a pont " # y (w $ + b) proportonal
More informationChapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of
Chapter 7 Generalzed and Weghted Least Squares Estmaton The usual lnear regresson model assumes that all the random error components are dentcally and ndependently dstrbuted wth constant varance. When
More informationBasic R Programming: Exercises
Basc R Programmng: Exercses RProgrammng John Fox ICPSR, Summer 2009 1. Logstc Regresson: Iterated weghted least squares (IWLS) s a standard method of fttng generalzed lnear models to data. As descrbed
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationIntroduction to the R Statistical Computing Environment R Programming
Introducton to the R Statstcal Computng Envronment R Programmng John Fox McMaster Unversty ICPSR 2018 John Fox (McMaster Unversty) R Programmng ICPSR 2018 1 / 14 Programmng Bascs Topcs Functon defnton
More informationTime to dementia onset: competing risk analysis with Laplace regression
Tme to dementa onset: competng rsk analyss wth Laplace regresson Gola Santon, Debora Rzzuto, Laura Fratglon 4 th Nordc and Baltc STATA Users group meetng, Stockholm, November 20 Agng Research Center (ARC),
More informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationOutline. Zero Conditional mean. I. Motivation. 3. Multiple Regression Analysis: Estimation. Read Wooldridge (2013), Chapter 3.
Outlne 3. Multple Regresson Analyss: Estmaton I. Motvaton II. Mechancs and Interpretaton of OLS Read Wooldrdge (013), Chapter 3. III. Expected Values of the OLS IV. Varances of the OLS V. The Gauss Markov
More informationis the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors
Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson
More informationβ0 + β1xi. You are interested in estimating the unknown parameters β
Ordnary Least Squares (OLS): Smple Lnear Regresson (SLR) Analytcs The SLR Setup Sample Statstcs Ordnary Least Squares (OLS): FOCs and SOCs Back to OLS and Sample Statstcs Predctons (and Resduals) wth OLS
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationIssues To Consider when Estimating Health Care Costs with Generalized Linear Models (GLMs): To Gamma/Log Or Not To Gamma/Log? That Is The New Question
Issues To Consder when Estmatng Health Care Costs wth Generalzed Lnear Models (GLMs): To Gamma/Log Or Not To Gamma/Log? That Is The New Queston ISPOR 20th Annual Internatonal Meetng May 19, 2015 Jalpa
More informationAdvanced Statistical Methods: Beyond Linear Regression
Advanced Statstcal Methods: Beyond Lnear Regresson John R. Stevens Utah State Unversty Notes 2. Statstcal Methods I Mathematcs Educators Workshop 28 March 2009 1 http://www.stat.usu.edu/~rstevens/pcm 2
More informationLearning Objectives for Chapter 11
Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method
More informationMultilayer Perceptrons and Backpropagation. Perceptrons. Recap: Perceptrons. Informatics 1 CG: Lecture 6. Mirella Lapata
Multlayer Perceptrons and Informatcs CG: Lecture 6 Mrella Lapata School of Informatcs Unversty of Ednburgh mlap@nf.ed.ac.uk Readng: Kevn Gurney s Introducton to Neural Networks, Chapters 5 6.5 January,
More informationLinear Feature Engineering 11
Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19
More informationLab 4: Two-level Random Intercept Model
BIO 656 Lab4 009 Lab 4: Two-level Random Intercept Model Data: Peak expratory flow rate (pefr) measured twce, usng two dfferent nstruments, for 17 subjects. (from Chapter 1 of Multlevel and Longtudnal
More informationThe Ordinary Least Squares (OLS) Estimator
The Ordnary Least Squares (OLS) Estmator 1 Regresson Analyss Regresson Analyss: a statstcal technque for nvestgatng and modelng the relatonshp between varables. Applcatons: Engneerng, the physcal and chemcal
More informationCS 3710: Visual Recognition Classification and Detection. Adriana Kovashka Department of Computer Science January 13, 2015
CS 3710: Vsual Recognton Classfcaton and Detecton Adrana Kovashka Department of Computer Scence January 13, 2015 Plan for Today Vsual recognton bascs part 2: Classfcaton and detecton Adrana s research
More informationOptimization. September 4, 2018
Optmzaton September 4, 2018 Optmzaton problem 1/34 An optmzaton problem s the problem of fndng the best soluton for an objectve functon. Optmzaton method plays an mportant role n statstcs, for example,
More information4.3 Poisson Regression
of teratvely reweghted least squares regressons (the IRLS algorthm). We do wthout gvng further detals, but nstead focus on the practcal applcaton. > glm(survval~log(weght)+age, famly="bnomal", data=baby)
More informationLogistic regression models 1/12
Logstc regresson models 1/12 2/12 Example 1: dogs look lke ther owners? Some people beleve that dogs look lke ther owners. Is ths true? To test the above hypothess, The New York Tmes conducted a quz onlne.
More informationMaximum Likelihood Estimation
Maxmum Lkelhood Estmaton INFO-2301: Quanttatve Reasonng 2 Mchael Paul and Jordan Boyd-Graber MARCH 7, 2017 INFO-2301: Quanttatve Reasonng 2 Paul and Boyd-Graber Maxmum Lkelhood Estmaton 1 of 9 Why MLE?
More informationLogistic Classifier CISC 5800 Professor Daniel Leeds
lon 9/7/8 Logstc Classfer CISC 58 Professor Danel Leeds Classfcaton strategy: generatve vs. dscrmnatve Generatve, e.g., Bayes/Naïve Bayes: 5 5 Identfy probablty dstrbuton for each class Determne class
More informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationMultilayer neural networks
Lecture Multlayer neural networks Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Mdterm exam Mdterm Monday, March 2, 205 In-class (75 mnutes) closed book materal covered by February 25, 205 Multlayer
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationMulti-layer neural networks
Lecture 0 Mult-layer neural networks Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Lnear regresson w Lnear unts f () Logstc regresson T T = w = p( y =, w) = g( w ) w z f () = p ( y = ) w d w d Gradent
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationOptimization. August 30, 2016
Optmzaton August 30, 2016 Optmzaton problem 1/31 An optmzaton problem s the problem of fndng the best soluton for an objectve functon. Optmzaton method plays an mportant role n statstcs, for example, to
More informationLecture 2: Prelude to the big shrink
Lecture 2: Prelude to the bg shrnk Last tme A slght detour wth vsualzaton tools (hey, t was the frst day... why not start out wth somethng pretty to look at?) Then, we consdered a smple 120a-style regresson
More informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationxp(x µ) = 0 p(x = 0 µ) + 1 p(x = 1 µ) = µ
CSE 455/555 Sprng 2013 Homework 7: Parametrc Technques Jason J. Corso Computer Scence and Engneerng SUY at Buffalo jcorso@buffalo.edu Solutons by Yngbo Zhou Ths assgnment does not need to be submtted and
More informationLaboratory 3: Method of Least Squares
Laboratory 3: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly they are correlated wth
More informationProperties of Least Squares
Week 3 3.1 Smple Lnear Regresson Model 3. Propertes of Least Squares Estmators Y Y β 1 + β X + u weekly famly expendtures X weekly famly ncome For a gven level of x, the expected level of food expendtures
More informationDesigning a Pseudo R-Squared Goodness-of-Fit Measure in Generalized Linear Models
Desgnng a Pseudo R-Squared Goodness-of-Ft Measure n Generalzed Lnear Models H. I. Mbachu Dept. of Mathematcs/Statstcs, Unversty of Port Harcourt, Port Harcourt E. C. Nduka Dept. of Mathematcs/Statstcs,
More informationMarginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients
ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It
More informationHomework Assignment 3 Due in class, Thursday October 15
Homework Assgnment 3 Due n class, Thursday October 15 SDS 383C Statstcal Modelng I 1 Rdge regresson and Lasso 1. Get the Prostrate cancer data from http://statweb.stanford.edu/~tbs/elemstatlearn/ datasets/prostate.data.
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More information1 Convex Optimization
Convex Optmzaton We wll consder convex optmzaton problems. Namely, mnmzaton problems where the objectve s convex (we assume no constrants for now). Such problems often arse n machne learnng. For example,
More information[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.
PPOL 59-3 Problem Set Exercses n Smple Regresson Due n class /8/7 In ths problem set, you are asked to compute varous statstcs by hand to gve you a better sense of the mechancs of the Pearson correlaton
More informationExpectation Maximization Mixture Models HMMs
-755 Machne Learnng for Sgnal Processng Mture Models HMMs Class 9. 2 Sep 200 Learnng Dstrbutons for Data Problem: Gven a collecton of eamples from some data, estmate ts dstrbuton Basc deas of Mamum Lelhood
More informationEnsemble Methods: Boosting
Ensemble Methods: Boostng Ncholas Ruozz Unversty of Texas at Dallas Based on the sldes of Vbhav Gogate and Rob Schapre Last Tme Varance reducton va baggng Generate new tranng data sets by samplng wth replacement
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationMIMA Group. Chapter 2 Bayesian Decision Theory. School of Computer Science and Technology, Shandong University. Xin-Shun SDU
Group M D L M Chapter Bayesan Decson heory Xn-Shun Xu @ SDU School of Computer Scence and echnology, Shandong Unversty Bayesan Decson heory Bayesan decson theory s a statstcal approach to data mnng/pattern
More informationClassification learning II
Lecture 8 Classfcaton learnng II Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Logstc regresson model Defnes a lnear decson boundar Dscrmnant functons: g g g g here g z / e z f, g g - s a logstc functon
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationSingle Variable Optimization
8/4/07 Course Instructor Dr. Raymond C. Rump Oce: A 337 Phone: (95) 747 6958 E Mal: rcrump@utep.edu Topc 8b Sngle Varable Optmzaton EE 4386/530 Computatonal Methods n EE Outlne Mathematcal Prelmnares Sngle
More informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More informationLinear Regression Introduction to Machine Learning. Matt Gormley Lecture 5 September 14, Readings: Bishop, 3.1
School of Computer Scence 10-601 Introducton to Machne Learnng Lnear Regresson Readngs: Bshop, 3.1 Matt Gormle Lecture 5 September 14, 016 1 Homework : Remnders Extenson: due Frda (9/16) at 5:30pm Rectaton
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationDiscrete Dependent Variable Models James J. Heckman
Dscrete Dependent Varable Models James J. Heckman Here s the general approach of ths lecture: Economc model (e.g. utlty maxmzaton) Decson rule (e.g. FOC) }{{} Sec. 1 Motvaton: Index functon and random
More informationSupport Vector Machines
Separatng boundary, defned by w Support Vector Machnes CISC 5800 Professor Danel Leeds Separatng hyperplane splts class 0 and class 1 Plane s defned by lne w perpendcular to plan Is data pont x n class
More informationStatistics MINITAB - Lab 2
Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that
More information