Generative classification models
|
|
- Annabella Stone
- 5 years ago
- Views:
Transcription
1 CS 75 Mache Learg Lecture Geeratve classfcato models Mlos Hauskrecht 539 Seott Square Data: D { d, d,.., d} d, Classfcato represets a dscrete class value Goal: lear f : X Y Bar classfcato A specal case whe Y {,} Frst step: we eed to devse a model of the fucto f
2 Dscrmat fuctos A commo wa to represet a classfer s b usg Dscrmat fuctos Works for both the bar ad mult-wa classfcato Idea: For ever class =,, k defe a fucto g () mappg X Whe the decso o put should be made choose the class wth the hghest value of () g * arg ma ( ) g Logstc regresso model Dscrmat fuctos: g ( ) g( w ) g ( ) g( w ) Values of dscrmat fuctos var terval [,] Probablstc terpretato f (,w) w, ) g( ) g( w ) w w w z, w) Iput vector w d d
3 Whe does the logstc regresso fal? Nolear decso boudar 3 Decso boudar Whe does the logstc regresso fal? Aother eample of a o-lear decso boudar
4 No-lear eteso of logstc regresso use feature (bass) fuctos to model oleartes the same trck as used for the lear regresso Lear regresso f ( ) w w j j ( ) m j j () ( ) ( ) - a arbtrar fucto of w w w Logstc regresso ) g( w w j j ( )) m j d m () w m CS 75 Mache Learg Regularzed logstc regresso If the model s too comple ad ca cause overfttg, ts predcto accurac ca be mproved b removg some puts from the model = settg ther coeffcets to zero Recall the lear model: f ( ) w w w w w 3 3 d d w Iput vector w w w w d d f ( ) w w w w 3 3 d d w 4
5 Regularzed logstc regresso If the model s too comple ad ca cause overfttg, ts predcto accurac ca be mproved b removg some puts from the model = settg ther coeffcets to zero We ca appl the same dea to the logstc regresso: ) g( w ) Iput vector w w, w, w w w w d k - parameters (weghts) d ) g( w w w3 3 wd d ) g( w ) J Rdge (L) pealt Lear regresso Rdge pealt: J ( w) ( w ) w L,.. w Logstc regresso: J L d Ft to data Model complet pealt w w w ad ( w) log P( D w) w L Ft to data Model complet pealt ( w) log g( w ) ( )log( g( w )) w L Ft to data measured usg the egatve log lkelhood 5
6 J Lasso (L) pealt Lear regresso Lasso pealt: J Logstc regresso: ( w) ( w ) w L,.. J Ft to data Model complet pealt d w L w ad ( w) log P( D w) w L Ft to data Model complet pealt ( w) log g( w ) ( )log( g( w )) w L Ft to data measured usg the egatve log lkelhood Geeratve approach to classfcato Logstc regresso: Represets ad lears a model of ) A eample of a dscrmatve classfcato approach Model s uable to sample (geerate) data staces (, ) Geeratve approach: Represets ad lears the jot dstrbuto, ) Model s able to sample (geerate) data staces (, ) he jot model defes probablstc dscrmat fuctos How? (, ) ) ) g ) ) ) ), ) ) ) g o ( ) ) ) ) ) ) 6
7 Geeratve approach to classfcato pcal jot model, ) ) ) ) = Class-codtoal dstrbutos (destes) bar classfcato: two class-codtoal dstrbutos ) ) ) = Prors o classes probablt of class for bar classfcato: Beroull dstrbuto ) ) ) ) Quadratc dscrmat aalss (QDA) Model: Class-codtoal dstrbutos are multvarate ormal dstrbutos μ,σ) d / ( ) Σ ~ ~ N( μ, Σ ) for N( μ, Σ ) for Multvarate ormal ~ N( μ, Σ) / ep Prors o classes (class,) Beroull dstrbuto, ) ( ) ( μ) Σ ~ Beroull {,} ( μ) 7
8 Learg of parameters of the QDA model Dest estmato statstcs We see eamples we do ot kow the parameters of Gaussas (class-codtoal destes) μ, Σ) d / ( ) Σ ML estmate of parameters of a multvarate ormal for a set of eamples of Optmze log-lkelhood: l( D, μ, Σ) log μˆ How about class prors? / ep Σˆ ( μ) ( Σ μˆ)( ( μ) μˆ) N( μ, Σ) μ, Σ) Learg Quadratc dscrmat aalss (QDA) Learg Class-codtoal dstrbutos Lear parameters of multvarate ormal dstrbutos ~ ~ N( μ, Σ ) for N( μ, Σ ) for Use the dest estmato methods Learg Prors o classes (class,) ~ Beroull Lear the parameter of the Beroull dstrbuto Aga use the dest estmato methods, ) ( ) {,} 8
9 QDA.5.5 g( ) g( ) Gaussa class-codtoal destes. 9
10 QDA: Makg class decso Bascall we eed to desg dscrmat fuctos Posteror of a class choose the class wth better posteror probablt ) ) the = g ( ) else = ), Σ Notce t s suffcet to compare:, Σ) ) ) ), Σ ) ), Σ) ), Σ) ) QDA: Quadratc decso boudar Cotours of class-codtoal destes
11 QDA: Quadratc decso boudar 3 Decso boudar Lear dscrmat aalss (LDA) Assumes covaraces are the same ~ N( μ, Σ), ~ N( μ, Σ),
12 LDA: Lear decso boudar Cotours of class-codtoal destes LDA: lear decso boudar Decso boudar
13 Geeratve classfcato models Idea:. Represet ad lear the dstrbuto, ). Model s able to sample (geerate) data staces (, ) 3. he model s used to get probablstc dscrmat fuctos g o ( ) ) g( ) ) pcal model, ) ) ) ) = Class-codtoal dstrbutos (destes) bar classfcato: two class-codtoal dstrbutos ) ) ) = Prors o classes - probablt of class bar classfcato: Beroull dstrbuto ) ) Naïve Baes classfer A geeratve classfer model wth a addtoal smplfg assumpto: All put attrbutes are codtoall depedet of each other gve the class. Oe of the basc ML classfcato models (ofte performs ver well practce) ) So we have:, ) ) ) ) d ) p ( ) p ( ) d ) d 3
14 Learg parameters of the model Much smpler dest estmato problems We eed to lear: ) ad ) ad ) Because of the assumpto of the codtoal depedece we eed to lear: for ever put varable : ) ad ) Much easer f the umber of put attrbutes s large Also, the model gves us a fleblt to represet put attrbutes of dfferet forms!!! E.g. oe attrbute ca be modeled usg the Beroull, the other usg Gaussa dest, or a Posso dstrbuto Makg a class decso for the Naïve Baes Dscrmat fuctos Posteror of a class choose the class wth better posteror probablt ) ) ) the = else = d d, ) ) d, ) ), ) ) 4
15 Net: two terestg questos () wo models wth lear decso boudares: Logstc regresso LDA model ( Gaussas wth the same covarace matrces ~ N(, ) for ~ N(, ) for Questo: Is there a relato betwee the two models? () wo models wth the same gradet: Lear model for regresso Logstc regresso model for classfcato have the same gradet update w w ( f ( )) Questo: Wh s the gradet the same? Logstc regresso ad geeratve models wo models wth lear decso boudares: Logstc regresso Geeratve model wth Gaussas wth the same covarace matrces ~ N(, ) for ~ N(, ) for Questo: Is there a relato betwee the two models? Aswer: Yes, the two models are related!!! Whe we have Gaussas wth the same covarace matr the probablt of gve has the form of a logstc regresso model!!!, μ, μ, Σ) g( w ) CS 75 Mache Learg 5
16 Logstc regresso ad geeratve models Members of the epoetal faml ca be ofte more aturall descrbed as θ f ( θ,φ) h(, φ)ep θ - A locato parameter A( θ) a( φ) Clam: A logstc regresso s a correct model whe class codtoal destes are from the same dstrbuto the epoetal faml ad have the same scale factor φ Ver powerful result!!!! We ca represet posterors of ma dstrbutos wth the same small logstc regresso model φ - A scale parameter CS 75 Mache Learg Lear regresso w w w he gradet puzzle f () Logstc regresso f ( ) w f ( ), w) g( w ) w w w z f () ) w d w d d d Gradet update: w w ( f ( )) Ole: CS 75 Mache Learg Gradet update: w w ( f ( )) he same w w ( f ( )) Ole: w w ( f ( )) 6
17 he gradet puzzle he same smple gradet update rule derved for both the lear ad logstc regresso models Where the magc comes from? Uder the log-lkelhood measure the fucto models ad the models for the output selecto ft together: Lear model + Gaussa ose Gaussa ose w ~ N(, ) w w w w d w Logstc + Beroull Beroull() ) g( w ) d w w w w d z g( w ) Beroull tral d Geeralzed lear models (GLIMs) Assumptos: he codtoal mea (epectato) s: f ( w ) Where f (.) s a respose fucto Output s characterzed b a epoetal faml dstrbuto wth a codtoal mea Gaussa ose w Eamples: Lear model + Gaussa ose w ~ N(, ) Logstc + Beroull Beroull() g( w ) e w d d w w w w d w w w d z w g( w ) Beroull tral 7
18 8 Geeralzed lear models (GLIMs) A caocal respose fuctos : ecoded the samplg dstrbuto Leads to a smple gradet form Eample: Beroull dstrbuto Logstc fucto matches the Beroull ) ( ) ( )ep, ( ) ( φ θ θ φ θ,φ a A h p (.) f p ) ( ) ( ) log( log ep log e
Classification : Logistic regression. Generative classification model.
CS 75 Mache Lear Lecture 8 Classfcato : Lostc reresso. Geeratve classfcato model. Mlos Hausrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Lear Bar classfcato o classes Y {} Our oal s to lear to classf
More informationGenerative classification models
CS 675 Intro to Machne Learnng Lecture Generatve classfcaton models Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Data: D { d, d,.., dn} d, Classfcaton represents a dscrete class value Goal: learn
More informationSupervised learning: Linear regression Logistic regression
CS 57 Itroducto to AI Lecture 4 Supervsed learg: Lear regresso Logstc regresso Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 57 Itro to AI Data: D { D D.. D D Supervsed learg d a set of eamples s
More informationCS 2750 Machine Learning. Lecture 8. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x
CS 75 Mache Learg Lecture 8 Lear regresso Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Learg Lear regresso Fucto f : X Y s a lear combato of put compoets f + + + K d d K k - parameters
More informationLinear regression (cont.) Linear methods for classification
CS 75 Mache Lear Lecture 7 Lear reresso cot. Lear methods for classfcato Mlos Hausrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Lear Coeffcet shrae he least squares estmates ofte have lo bas but hh
More informationCS 2750 Machine Learning. Lecture 7. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x
CS 75 Mache Learg Lecture 7 Lear regresso Mlos Hauskrecht los@cs.ptt.edu 59 Seott Square CS 75 Mache Learg Lear regresso Fucto f : X Y s a lear cobato of put copoets f + + + K d d K k - paraeters eghts
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 informationBayes (Naïve or not) Classifiers: Generative Approach
Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg
More informationBinary classification: Support Vector Machines
CS 57 Itroducto to AI Lecture 6 Bar classfcato: Support Vector Maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 57 Itro to AI Supervsed learg Data: D { D, D,.., D} a set of eamples D, (,,,,,
More informationOverview. Basic concepts of Bayesian learning. Most probable model given data Coin tosses Linear regression Logistic regression
Overvew Basc cocepts of Bayesa learg Most probable model gve data Co tosses Lear regresso Logstc regresso Bayesa predctos Co tosses Lear regresso 30 Recap: regresso problems Iput to learg problem: trag
More informationCS 1675 Introduction to Machine Learning Lecture 12 Support vector machines
CS 675 Itroducto to Mache Learg Lecture Support vector maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Mdterm eam October 9, 7 I-class eam Closed book Stud materal: Lecture otes Correspodg chapters
More informationLinear regression (cont) Logistic regression
CS 7 Fouatos of Mache Lear Lecture 4 Lear reresso cot Lostc reresso Mlos Hausrecht mlos@cs.ptt.eu 539 Seott Square Lear reresso Vector efto of the moel Iclue bas costat the put vector f - parameters ehts
More information( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model
Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch
More informationSupport vector machines II
CS 75 Mache Learg Lecture Support vector maches II Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Learl separable classes Learl separable classes: here s a hperplae that separates trag staces th o error
More informationCS 2750 Machine Learning Lecture 5. Density estimation. Density estimation
CS 750 Mache Learg Lecture 5 esty estmato Mlos Hausrecht mlos@tt.edu 539 Seott Square esty estmato esty estmato: s a usuervsed learg roblem Goal: Lear a model that rereset the relatos amog attrbutes the
More informationBayesian belief networks
Lecture 14 ayesa belef etworks los Hauskrecht mlos@cs.ptt.edu 5329 Seott Square Desty estmato Data: D { D1 D2.. D} D x a vector of attrbute values ttrbutes: modeled by radom varables { 1 2 d} wth: otuous
More informationDimensionality reduction Feature selection
CS 750 Mache Learg Lecture 3 Dmesoalty reducto Feature selecto Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 750 Mache Learg Dmesoalty reducto. Motvato. Classfcato problem eample: We have a put data
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationPart I: Background on the Binomial Distribution
Part I: Bacgroud o the Bomal Dstrbuto A radom varable s sad to have a Beroull dstrbuto f t taes o the value wth probablt "p" ad the value wth probablt " - p". The umber of "successes" "" depedet Beroull
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationLinear models for classification
CS 75 Mache Lear Lecture 9 Lear modes for cassfcato Mos Hausrecht mos@cs.ptt.edu 539 Seott Square ata: { d d.. d} d Cassfcato represets a dscrete cass vaue Goa: ear f : X Y Bar cassfcato A speca case he
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationCS 3710 Advanced Topics in AI Lecture 17. Density estimation. CS 3710 Probabilistic graphical models. Administration
CS 37 Avace Topcs AI Lecture 7 esty estmato Mlos Hauskrecht mlos@cs.ptt.eu 539 Seott Square CS 37 robablstc graphcal moels Amstrato Mterm: A take-home exam week ue o Weesay ovember 5 before the class epes
More informationBayesian Classification. CS690L Data Mining: Classification(2) Bayesian Theorem: Basics. Bayesian Theorem. Training dataset. Naïve Bayes Classifier
Baa Classfcato CS6L Data Mg: Classfcato() Referece: J. Ha ad M. Kamber, Data Mg: Cocepts ad Techques robablstc learg: Calculate explct probabltes for hypothess, amog the most practcal approaches to certa
More informationClassification with linear models
Lecture 8 Classificatio with liear models Milos Hauskrecht milos@cs.pitt.edu 539 Seott Square Geerative approach to classificatio Idea:. Represet ad lear the distributio, ). Use it to defie probabilistic
More informationMachine Learning. Introduction to Regression. Le Song. CSE6740/CS7641/ISYE6740, Fall 2012
Mache Learg CSE6740/CS764/ISYE6740, Fall 0 Itroducto to Regresso Le Sog Lecture 4, August 30, 0 Based o sldes from Erc g, CMU Readg: Chap. 3, CB Mache learg for apartmet hutg Suppose ou are to move to
More informationSupport vector machines
CS 75 Mache Learg Lecture Support vector maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Learg Outle Outle: Algorthms for lear decso boudary Support vector maches Mamum marg hyperplae.
More information7. Joint Distributions
7. Jot Dstrbutos Chrs Pech ad Mehra Saham Ma 2017 Ofte ou wll work o problems where there are several radom varables (ofte teractg wth oe aother. We are gog to start to formall look at how those teractos
More informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationLinear Regression with One Regressor
Lear Regresso wth Oe Regressor AIM QA.7. Expla how regresso aalyss ecoometrcs measures the relatoshp betwee depedet ad depedet varables. A regresso aalyss has the goal of measurg how chages oe varable,
More informationNaïve Bayes MIT Course Notes Cynthia Rudin
Thaks to Şeyda Ertek Credt: Ng, Mtchell Naïve Bayes MIT 5.097 Course Notes Cytha Rud The Naïve Bayes algorthm comes from a geeratve model. There s a mportat dstcto betwee geeratve ad dscrmatve models.
More informationRegression and the LMS Algorithm
CSE 556: Itroducto to Neural Netorks Regresso ad the LMS Algorthm CSE 556: Regresso 1 Problem statemet CSE 556: Regresso Lear regresso th oe varable Gve a set of N pars of data {, d }, appromate d b a
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationSTK3100 and STK4100 Autumn 2017
SK3 ad SK4 Autum 7 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Sectos 4..5, 4.3.5, 4.3.6, 4.4., 4.4., ad 4.4.3 Sectos 5.., 5.., ad 5.5. Ørulf Borga Deartmet of Mathematcs
More informationChapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
More informationSTK3100 and STK4100 Autumn 2018
SK3 ad SK4 Autum 8 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Cofdece tervals by vertg tests Cosder a model wth a sgle arameter β We may obta a ( α% cofdece terval for
More informationε. Therefore, the estimate
Suggested Aswers, Problem Set 3 ECON 333 Da Hugerma. Ths s ot a very good dea. We kow from the secod FOC problem b) that ( ) SSE / = y x x = ( ) Whch ca be reduced to read y x x = ε x = ( ) The OLS model
More information9.1 Introduction to the probit and logit models
EC3000 Ecoometrcs Lecture 9 Probt & Logt Aalss 9. Itroducto to the probt ad logt models 9. The logt model 9.3 The probt model Appedx 9. Itroducto to the probt ad logt models These models are used regressos
More informationLecture 7: Linear and quadratic classifiers
Lecture 7: Lear ad quadratc classfers Bayes classfers for ormally dstrbuted classes Case : Σ σ I Case : Σ Σ (Σ daoal Case : Σ Σ (Σ o-daoal Case 4: Σ σ I Case 5: Σ Σ j eeral case Lear ad quadratc classfers:
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationLinear Regression Linear Regression with Shrinkage. Some slides are due to Tommi Jaakkola, MIT AI Lab
Lear Regresso Lear Regresso th Shrkage Some sldes are due to Tomm Jaakkola, MIT AI Lab Itroducto The goal of regresso s to make quattatve real valued predctos o the bass of a vector of features or attrbutes.
More informationObjectives of Multiple Regression
Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of
More informationUnsupervised Learning and Other Neural Networks
CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationSection 2 Notes. Elizabeth Stone and Charles Wang. January 15, Expectation and Conditional Expectation of a Random Variable.
Secto Notes Elzabeth Stoe ad Charles Wag Jauar 5, 9 Jot, Margal, ad Codtoal Probablt Useful Rules/Propertes. P ( x) P P ( x; ) or R f (x; ) d. P ( xj ) P (x; ) P ( ) 3. P ( x; ) P ( xj ) P ( ) 4. Baes
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationRadial Basis Function Networks
Radal Bass Fucto Netorks Radal Bass Fucto Netorks A specal types of ANN that have three layers Iput layer Hdde layer Output layer Mappg from put to hdde layer s olear Mappg from hdde to output layer s
More informationKernel-based Methods and Support Vector Machines
Kerel-based Methods ad Support Vector Maches Larr Holder CptS 570 Mache Learg School of Electrcal Egeerg ad Computer Scece Washgto State Uverst Refereces Muller et al. A Itroducto to Kerel-Based Learg
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationAn Introduction to. Support Vector Machine
A Itroducto to Support Vector Mache Support Vector Mache (SVM) A classfer derved from statstcal learg theory by Vapk, et al. 99 SVM became famous whe, usg mages as put, t gave accuracy comparable to eural-etwork
More informationCSE 5526: Introduction to Neural Networks Linear Regression
CSE 556: Itroducto to Neural Netorks Lear Regresso Part II 1 Problem statemet Part II Problem statemet Part II 3 Lear regresso th oe varable Gve a set of N pars of data , appromate d by a lear fucto
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationParametric Density Estimation: Bayesian Estimation. Naïve Bayes Classifier
arametrc Dest Estmato: Baesa Estmato. Naïve Baes Classfer Baesa arameter Estmato Suppose we have some dea of the rage where parameters θ should be Should t we formalze such pror owledge hopes that t wll
More informationLikelihood Ratio, Wald, and Lagrange Multiplier (Score) Tests. Soccer Goals in European Premier Leagues
Lkelhood Rato, Wald, ad Lagrage Multpler (Score) Tests Soccer Goals Europea Premer Leagues - 4 Statstcal Testg Prcples Goal: Test a Hpothess cocerg parameter value(s) a larger populato (or ature), based
More informationParameter, Statistic and Random Samples
Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationå 1 13 Practice Final Examination Solutions - = CS109 Dec 5, 2018
Chrs Pech Fal Practce CS09 Dec 5, 08 Practce Fal Examato Solutos. Aswer: 4/5 8/7. There are multle ways to obta ths aswer; here are two: The frst commo method s to sum over all ossbltes for the rak of
More informationENGI 4421 Propagation of Error Page 8-01
ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b
CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package
More informationLecture 3. Least Squares Fitting. Optimization Trinity 2014 P.H.S.Torr. Classic least squares. Total least squares.
Lecture 3 Optmzato Trt 04 P.H.S.Torr Least Squares Fttg Classc least squares Total least squares Robust Estmato Fttg: Cocepts ad recpes Least squares le fttg Data:,,,, Le equato: = m + b Fd m, b to mmze
More informationMultivariate Transformation of Variables and Maximum Likelihood Estimation
Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty
More informationStudy of Correlation using Bayes Approach under bivariate Distributions
Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 Stud of Correlato usg Baes Approach uder bvarate Dstrbutos N.S.Padharkar* ad. M.N.Deshpade** *Govt.Vdarbha Isttute of
More informationECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model
ECON 48 / WH Hog The Smple Regresso Model. Defto of the Smple Regresso Model Smple Regresso Model Expla varable y terms of varable x y = β + β x+ u y : depedet varable, explaed varable, respose varable,
More informationRecall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I
Chapter 8 Heterosedastcty Recall MLR 5 Homsedastcty error u has the same varace gve ay values of the eplaatory varables Varu,..., = or EUU = I Suppose other GM assumptos hold but have heterosedastcty.
More information6.867 Machine Learning
6.867 Mache Learg Problem set Due Frday, September 9, rectato Please address all questos ad commets about ths problem set to 6.867-staff@a.mt.edu. You do ot eed to use MATLAB for ths problem set though
More informationTHE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5
THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should
More informationSpecial Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
More informationAdvanced Introduction to Machine Learning
Advaced Itroducto to Mache Learg 075, Fall 04 Lear Regresso ad Sparst Erc g Lecture, September 0, 04 Readg: Erc g @ CMU, 04 Mache learg for apartmet hutg ow ou've moved to Pttsburgh!! Ad ou wat to fd the
More informationMaximum Likelihood Estimation
Marquette Uverst Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Coprght 08 b Marquette Uverst Maxmum Lkelhood Estmato We have bee sag that ~
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationMathematics HL and Further mathematics HL Formula booklet
Dploma Programme Mathematcs HL ad Further mathematcs HL Formula booklet For use durg the course ad the eamatos Frst eamatos 04 Mathematcal Iteratoal Baccalaureate studes SL: Formula Orgazato booklet 0
More informationLecture 3 Naïve Bayes, Maximum Entropy and Text Classification COSI 134
Lecture 3 Naïve Baes, Mamum Etro ad Tet Classfcato COSI 34 Codtoal Parameterzato Two RVs: ItellgeceI ad SATS ValI = {Hgh,Low}, ValS={Hgh,Low} A ossble jot dstrbuto Ca descrbe usg cha rule as PI,S PIPS
More informationFeature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)
CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.
More informationRegresso What s a Model? 1. Ofte Descrbe Relatoshp betwee Varables 2. Types - Determstc Models (o radomess) - Probablstc Models (wth radomess) EPI 809/Sprg 2008 9 Determstc Models 1. Hypothesze
More information3. Basic Concepts: Consequences and Properties
: 3. Basc Cocepts: Cosequeces ad Propertes Markku Jutt Overvew More advaced cosequeces ad propertes of the basc cocepts troduced the prevous lecture are derved. Source The materal s maly based o Sectos.6.8
More informationExample: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger
Example: Multple lear regresso 5000,00 4000,00 Tro Aders Moger 0.0.007 brthweght 3000,00 000,00 000,00 0,00 50,00 00,00 50,00 00,00 50,00 weght pouds Repetto: Smple lear regresso We defe a model Y = β0
More informationMultiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades
STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationDimensionality Reduction and Learning
CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that
More information6. Nonparametric techniques
6. Noparametrc techques Motvato Problem: how to decde o a sutable model (e.g. whch type of Gaussa) Idea: just use the orgal data (lazy learg) 2 Idea 1: each data pot represets a pece of probablty P(x)
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationBig Data Analytics. Data Fitting and Sampling. Acknowledgement: Notes by Profs. R. Szeliski, S. Seitz, S. Lazebnik, K. Chaturvedi, and S.
Bg Data Aaltcs Data Fttg ad Samplg Ackowledgemet: Notes b Profs. R. Szelsk, S. Setz, S. Lazebk, K. Chaturved, ad S. Shah Fttg: Cocepts ad recpes A bag of techques If we kow whch pots belog to the le, how
More informationMultiple Linear Regression Analysis
LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple
More informationENGI 3423 Simple Linear Regression Page 12-01
ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable
More informationModel Fitting, RANSAC. Jana Kosecka
Model Fttg, RANSAC Jaa Kosecka Fttg: Issues Prevous strateges Le detecto Hough trasform Smple parametrc model, two parameters m, b m + b Votg strateg Hard to geeralze to hgher dmesos a o + a + a 2 2 +
More informationECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity
ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data
More informationStatistics. Correlational. Dr. Ayman Eldeib. Simple Linear Regression and Correlation. SBE 304: Linear Regression & Correlation 1/3/2018
/3/08 Sstems & Bomedcal Egeerg Departmet SBE 304: Bo-Statstcs Smple Lear Regresso ad Correlato Dr. Ama Eldeb Fall 07 Descrptve Orgasg, summarsg & descrbg data Statstcs Correlatoal Relatoshps Iferetal Geeralsg
More informationChapter Two. An Introduction to Regression ( )
ubject: A Itroducto to Regresso Frst tage Chapter Two A Itroducto to Regresso (018-019) 1 pg. ubject: A Itroducto to Regresso Frst tage A Itroducto to Regresso Regresso aalss s a statstcal tool for the
More informationChapter 13 Student Lecture Notes 13-1
Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato
More informationModule 7. Lecture 7: Statistical parameter estimation
Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato
More informationLecture 8: Linear Regression
Lecture 8: Lear egresso May 4, GENOME 56, Sprg Goals Develop basc cocepts of lear regresso from a probablstc framework Estmatg parameters ad hypothess testg wth lear models Lear regresso Su I Lee, CSE
More informationChapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:
Chapter 3 3- Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic. Chap 2- Learg Objectves I ths chapter, you lear:
More informationPrevious lecture. Lecture 8. Learning outcomes of this lecture. Today. Statistical test and Scales of measurement. Correlation
Lecture 8 Emprcal Research Methods I434 Quattatve Data aalss II Relatos Prevous lecture Idea behd hpothess testg Is the dfferece betwee two samples a reflecto of the dfferece of two dfferet populatos or
More informationLinear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan
Lear Regresso Hsao-Lug Cha Dept Electrcal Egeerg Chag Gug Uverst, Tawa chahl@mal.cgu.edu.tw Curve fttg Least-squares regresso Data ehbt a sgfcat degree of error or scatter A curve for the tred of the data
More informationBayesian belief networks
Lecture 19 ayesa belef etworks los Hauskrecht mlos@cs.ptt.edu 539 Seott Square Varous ferece tasks: robablstc ferece Dagostc task. from effect to cause eumoa Fever redcto task. from cause to effect Fever
More information4. Standard Regression Model and Spatial Dependence Tests
4. Stadard Regresso Model ad Spatal Depedece Tests Stadard regresso aalss fals the presece of spatal effects. I case of spatal depedeces ad/or spatal heterogeet a stadard regresso model wll be msspecfed.
More informationComparison of Dual to Ratio-Cum-Product Estimators of Population Mean
Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract
More information