Dimensionality reduction Feature selection

Size: px
Start display at page:

Download "Dimensionality reduction Feature selection"

Transcription

1 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 { 1,,.., N } such that 1 d = (,,.., ) ad a set of correspodg output labels { y1, y,.., y N } Assume the dmeso d of the data pot s very large We wat to classfy Problems wth hgh dmesoal put vectors A large umber of parameters to lear, f a dataset s small ths ca result : Large varace of estmates ad overft t becomes hard to epla what features are mportat the model (too may choces some ca be substtutable) CS 750 Mache Learg

2 Dmesoalty reducto. Solutos: Selecto of a smaller subset of puts (features) from a large set of puts; tra classfer o the reduced put set Combato of hgh dmesoal puts to a smaller set of features () ; tra classfer o ew features φ k selecto combato CS 750 Mache Learg Feature selecto How to fd a good subset of puts/features? We eed: A crtero for rakg good puts/features Search procedure for fdg a good set of features Feature selecto process ca be: Depedet o the learg task e.g. classfcato Selecto of features affected by what we wat to predct Idepedet of the learg task puts are reduced wthout lookg at the output PCA, compoet aalyss, clusterg of puts may lack the accuracy for classfcato/regresso tasks CS 750 Mache Learg

3 Task-depedet feature selecto Assume: Classfcato problem: put vector, y - output Feature mappgs φ = { φ1( ), φ ( ), K φ k ( ), K} Objectve: Fd a subset of features that gves/preserves most of the output predcto capabltes Selecto approaches: Flterg approaches Flter out features wth small predctve potetal doe before classfcato; typcally uses uvarate aalyss Wrapper approaches Select features that drectly optmze the accuracy of the multvarate classfer Embedded methods Feature selecto ad learg closely ted the method CS 750 Mache Learg Feature selecto through flterg Assume: Classfcato problem: put vector, y - output Iputs or feature mappgs () φ k How to select the feature: Uvarate aalyss Preted that oly oe varable, k, ests See how well t predcts the output y aloe Eample: dfferetally epressed features (or puts) Good separato bary (case/cotrol settgs) CS 750 Mache Learg

4 Dfferetally epressed features Crtera for measurg the dfferetal epresso T-Test score (Bald & Log) Based o the test that two groups come from the same populato ( + ) ( ) Fsher Score µ µ Fsher ( ) = ( + ) ( ) σ + σ Area uder Recever Operatg Characterstc (AUC) score Problems: f may radom features, the features wth a good dfferetally epressed score must arse Techques to reduce FDR (False dscovery rate) ad FWER (Famly wse error). CS 750 Mache Learg Feature flterg Other uvarate scores: Cov ( φk, y) Correlato coeffcets ρ ( φk, y) = Var ( φk ) Var ( y ) Measures lear depedeces Mutual formato ~ ~ ~ P( φk = j, y = ) I ( φk, y) = P( φk = j, y = ) log ~ P( φ = j) P( y = ) j Uvarate assumptos: Oly oe feature ad ts effect o y s corporated the mutual formato score Effects of two features o y are depedet What to do f the combato of features gves the best predcto? k CS 750 Mache Learg

5 Feature selecto: depedet features Flterg wth depedet features Let φ be a curret set of features (startg from complete set) We ca remove feature ~ φ k () from t whe: ~ P ( y φ \ φ k ) P ( y φ) for all values of φ k, y Repeat removals utl the probabltes dffer too much. Problem: how to compute/estmate P ~ ( y φ \ φ ~ k ), P ( y φ)? Soluto: make some smplfyg assumpto about the uderlyg probablstc model Eample: use a Naïve Bayes Advatage: speed, modularty, appled before classfcato Dsadvatage: may ot be as accurate CS 750 Mache Learg Feature selecto: wrappers Wrapper approach: The feature selecto s drve by the predcto accuracy of the classfer (regressor) actually bult How to fd the approprate feature set? Idea: Greedy search the space of classfers Gradually add features mprovg most the qualty score Gradually remove features that effect the accuracy the least Score should reflect the accuracy of the classfer (error) ad also prevet overft Stadard way to measure the qualty: Iteral cross-valdato (m-fold cross valdato) CS 750 Mache Learg

6 Feature selecto: wrappers Eample of a greedy (forward) search: logstc regresso model wth features Start wth p ( y = 1, w) = g( wo ) Choose the feature φ () wth the best score p( y = 1, w) = g( w o + w φ ( )) Choose the feature φ j () wth the best score p( y = 1, w) = g( w + w φ ( ) + w φ ( )) Etc. Whe to stop? o j j CS 750 Mache Learg Iteral cross-valdato Goal: Stop the learg whe smallest geeralzato error (performace o the populato from whch data were draw) Test set ca be used to estmate geeralzato error Data dfferet from the trag set Iteral valdato set = test set used to stop the learg process E.g. feature selecto process Cross-valdato (m-fold): Dvde the data to m equal parttos (of sze N/m) Hold out oe partto for valdato, tra the classfer o the rest of data Repeat such that every partto s held out oce The estmate of the geeralzato error of the learer s the mea of errors of all classfers CS 750 Mache Learg

7 Embedded methods Feature selecto + classfcato model learg doe together Embedded models: Regularzed models Models of hgher complety are eplctly pealzed leadg to vrtual removal of puts from the model Regularzed logstc/lear regresso Support vector maches Optmzato of margs pealzes ozero weghts CART/Decso trees CS 750 Mache Learg Prcpal compoet aalyss (PCA) Objectve: We wat to replace a hgh dmesoal put wth a small set of features (obtaed by combg puts) Dfferet from the feature subset selecto!!! PCA: A lear trasformato of d dmesoal put to M dmesoal feature vector z such that M < d uder whch the retaed varace s mamal. Equvaletly t s the lear projecto for whch the sum of squares recostructo cost s mmzed. CS 750 Mache Learg

8 PCA CS 750 Mache Learg PCA CS 750 Mache Learg

9 PCA CS 750 Mache Learg PCA Xprm= y- 0.99z Yprm= y+0.07z 97% varace retaed 0 10 Yprm Xprm CS 750 Mache Learg

10 Prcpal compoet aalyss (PCA) PCA: lear trasformato of d dmesoal put to M dmesoal feature vector z such that M < d uder whch the retaed varace s mamal. Task depedet Fact: A vector ca be represeted usg a set of orthoormal d vectors u = z u = 1 Leads to trasformato of coordates (from to z usg u s) z u T = CS 750 Mache Learg PCA Idea: replace d coordates wth M of z coordates to represet. We wat to fd the subset M of bass vectors. M ~ = z u + b u = 1 d = M + 1 b - costat ad fed How to choose the best set of bass vectors? We wat the subset that gves the best appromato of data the dataset o average (we use least squares ft) Error for data etry E M = 1 N = 1 ~ = CS 750 Mache Learg 1 d ~ = ( z b ) u N = 1 d = M + 1 = M + 1 ( z b )

11 PCA Dfferetate the error fucto wth regard to all b ad set equal to 0 we get: N 1 b u T = z = N =1 The we ca rewrte: T E M = u Σu Σ = ( )( ) = M + 1 = 1 The error fucto s optmzed whe bass vectors satsfy: d 1 Σu = λ u E M = λ The best M bass vectors: dscard vectors wth d-m smallest egevalues (or keep vectors wth M largest egevalues) Egevector s called a prcpal compoet CS 750 Mache Learg = 1 N N = 1 1 d N T u = M + 1 u PCA Oce egevectors wth largest egevalues are detfed, they are used to trasform the orgal d-dmesoal data to M dmesos u u 1 To fd the true dmesoalty of the data d we ca just look at egevalues that cotrbute the most (small egevalues are dsregarded) Problem: PCA s a lear method. The true dmesoalty ca be overestmated. There ca be o-lear correlatos. 1 CS 750 Mache Learg

12 Dmesoalty reducto wth eural ets PCA s lmted to lear dmesoalty reducto To do o-lear reductos we ca use eural ets Auto-assocatve etwork: a eural etwork wth the same puts ad outputs ( ) 1 d z = ( z 1, z ) 1 d The mddle layer correspods to the reduced dmesos CS 750 Mache Learg Dmesoalty reducto wth eural ets Error crtero: 1 E = N d ( y ( ) ) = 1 = 1 Error measure tres to recover the orgal data through lmted umber of dmesos the mddle layer No-leartes modeled through termedate layers betwee the mddle layer ad put/output If o termedate layers are used the model replcates PCA optmzato through learg 1 d z = ( z 1, z 1 d ) CS 750 Mache Learg

13 Dmesoalty reducto through clusterg Clusterg algorthms group together smlar staces the data sample Dmesoalty reducto based o clusterg: Replace a hgh dmesoal data etry wth a cluster label Problem: Determstc clusterg gves oly oe label per put May ot be eough to represet the data for predcto Solutos: Clusterg over subsets of put data Soft clusterg (probablty of a cluster s used drectly) CS 750 Mache Learg Dmesoalty reducto through clusterg Soft clusterg (e.g. mture of Gaussas) attempts to cover all staces the data sample wth a small umber of groups Each group s more or less resposble for a data etry (resposblty a posteror of a group gve the data etry) Mture of G. resposblty = k Dmesoalty reducto based o soft clusterg Replace a hgh dmesoal data wth the set of group posterors Feed all posterors to the learer e.g. lear regressor, classfer h l π p ( u = 1 π p ( u l y l l y = ) l = u ) CS 750 Mache Learg

14 Dmesoalty reducto through clusterg We ca use the dea of soft clusterg before applyg regresso/classfcato learg Two stage algorthms Lear the clusterg Lear the classfcato Iput clusterg: (hgh dmesoal) Output clusterg (Iput classfer): p ( c = ) Output classfer: y Eample: Networks wth Radal Bass Fuctos (RBFs) Problem: Clusterg lears based o p() (dsregards the target) Predcto based o p( y ) CS 750 Mache Learg Networks wth radal bass fuctos A alteratve to multlayer NN for o-leartes k Radal bass fuctos: f ( ) = w + φ ( ) Based o terpolatos of prototype pots (meas) Affected by the dstace betwee the ad the mea Ft the outputs of bass fuctos through the lear model Choce of bass fuctos: µ j Gaussa φ j ( ) = ep σ j Learg: I practce seem to work OK for up to 10 dmesos For hgher dmesos (rdge fuctos logstc) combg multple learers seem to do better job CS 750 Mache Learg 0 w j j j= 1

Dimensionality reduction Feature selection

Dimensionality reduction Feature selection CS 675 Itroucto to ache Learg Lecture Dmesoalty reucto Feature selecto los Hauskrecht mlos@cs.ptt.eu 539 Seott Square Dmesoalty reucto. otvato. L methos are sestve to the mesoalty of ata Questo: Is there

More information

CS 2750 Machine Learning. Lecture 8. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x

CS 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 information

Generative classification models

Generative classification models CS 75 Mache Learg Lecture Geeratve classfcato models Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Data: D { d, d,.., d} d, Classfcato represets a dscrete class value Goal: lear f : X Y Bar classfcato

More information

Kernel-based Methods and Support Vector Machines

Kernel-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 information

Binary classification: Support Vector Machines

Binary 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 information

CS 1675 Introduction to Machine Learning Lecture 12 Support vector machines

CS 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 information

Support vector machines II

Support 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 information

Supervised learning: Linear regression Logistic regression

Supervised 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 information

Support vector machines

Support 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 information

Bayes (Naïve or not) Classifiers: Generative Approach

Bayes (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 information

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 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 information

An Introduction to. Support Vector Machine

An 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 information

Naïve Bayes MIT Course Notes Cynthia Rudin

Naïve 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 information

Unsupervised Learning and Other Neural Networks

Unsupervised 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 information

CS 2750 Machine Learning. Lecture 7. Linear regression. CS 2750 Machine Learning. Linear regression. is a linear combination of input components x

CS 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 information

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)

Feature 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 information

Principal Components. Analysis. Basic Intuition. A Method of Self Organized Learning

Principal Components. Analysis. Basic Intuition. A Method of Self Organized Learning Prcpal Compoets Aalss A Method of Self Orgazed Learg Prcpal Compoets Aalss Stadard techque for data reducto statstcal patter matchg ad sgal processg Usupervsed learg: lear from examples wthout a teacher

More information

ENGI 3423 Simple Linear Regression Page 12-01

ENGI 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 information

Overview. Basic concepts of Bayesian learning. Most probable model given data Coin tosses Linear regression Logistic regression

Overview. 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 information

Bayesian Classification. CS690L Data Mining: Classification(2) Bayesian Theorem: Basics. Bayesian Theorem. Training dataset. Naïve Bayes Classifier

Bayesian 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 information

Announcements. Recognition II. Computer Vision I. Example: Face Detection. Evaluating a binary classifier

Announcements. Recognition II. Computer Vision I. Example: Face Detection. Evaluating a binary classifier Aoucemets Recogto II H3 exteded to toght H4 to be aouced today. Due Frday 2/8. Note wll take a whle to ru some thgs. Fal Exam: hursday 2/4 at 7pm-0pm CSE252A Lecture 7 Example: Face Detecto Evaluatg a

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL 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 information

Introduction to local (nonparametric) density estimation. methods

Introduction 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 information

ρ < 1 be five real numbers. The

ρ < 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 information

Machine Learning. Introduction to Regression. Le Song. CSE6740/CS7641/ISYE6740, Fall 2012

Machine 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 information

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements

Chapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall

More information

Radial Basis Function Networks

Radial 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 information

Linear regression (cont.) Linear methods for classification

Linear 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 information

Chapter 4 Multiple Random Variables

Chapter 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 information

Dimensionality Reduction and Learning

Dimensionality 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 information

Linear Regression Linear Regression with Shrinkage. Some slides are due to Tommi Jaakkola, MIT AI Lab

Linear 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 information

Tema 5: Aprendizaje NO Supervisado: CLUSTERING Unsupervised Learning: CLUSTERING. Febrero-Mayo 2005

Tema 5: Aprendizaje NO Supervisado: CLUSTERING Unsupervised Learning: CLUSTERING. Febrero-Mayo 2005 Tema 5: Apredzae NO Supervsado: CLUSTERING Usupervsed Learg: CLUSTERING Febrero-Mayo 2005 SUPERVISED METHODS: LABELED Data Base Labeled Data Base Dvded to Tra ad Test Choose Algorthm: MAP, ML, K-Nearest

More information

6. Nonparametric techniques

6. 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

Regresso 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 information

Lecture 3 Probability review (cont d)

Lecture 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 information

Objectives of Multiple Regression

Objectives 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 information

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,

More information

CHAPTER VI Statistical Analysis of Experimental Data

CHAPTER VI Statistical Analysis of Experimental Data Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca

More information

Lecture 8: Linear Regression

Lecture 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 information

Principal Component Analysis (PCA)

Principal Component Analysis (PCA) BBM406 - Itroduc0o to ML Sprg 204 Prcpal Compoet Aalyss PCA Aykut Erdem Dept. of Computer Egeerg HaceDepe Uversty Today Mo0va0o PCA algorthms Applca0os PCA shortcomgs Kerel PCA Sldes adopted from Barabás

More information

Bayesian belief networks

Bayesian 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 information

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model

( ) = ( ) ( ) 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 information

Algebraic-Geometric and Probabilistic Approaches for Clustering and Dimension Reduction of Mixtures of Principle Component Subspaces

Algebraic-Geometric and Probabilistic Approaches for Clustering and Dimension Reduction of Mixtures of Principle Component Subspaces Algebrac-Geometrc ad Probablstc Approaches for Clusterg ad Dmeso Reducto of Mxtures of Prcple Compoet Subspaces ECE842 Course Project Report Chagfag Zhu Dec. 4, 2004 Algebrac-Geometrc ad Probablstc Approach

More information

Simple Linear Regression

Simple 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 information

Econometric Methods. Review of Estimation

Econometric 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 information

Classification : Logistic regression. Generative classification model.

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 information

Lecture 12: Multilayer perceptrons II

Lecture 12: Multilayer perceptrons II Lecture : Multlayer perceptros II Bayes dscrmats ad MLPs he role of hdde uts A eample Itroducto to Patter Recoto Rcardo Guterrez-Osua Wrht State Uversty Bayes dscrmats ad MLPs ( As we have see throuhout

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

QR Factorization and Singular Value Decomposition COS 323

QR Factorization and Singular Value Decomposition COS 323 QR Factorzato ad Sgular Value Decomposto COS 33 Why Yet Aother Method? How do we solve least-squares wthout currg codto-squarg effect of ormal equatos (A T A A T b) whe A s sgular, fat, or otherwse poorly-specfed?

More information

ESS Line Fitting

ESS Line Fitting ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here

More information

Sampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION

Sampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION Samplg Theor MODULE V LECTUE - 4 ATIO AND PODUCT METHODS OF ESTIMATION D. SHALABH DEPATMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOG KANPU A mportat objectve a statstcal estmato procedure

More information

4. Standard Regression Model and Spatial Dependence Tests

4. 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 information

Chapter 9 Jordan Block Matrices

Chapter 9 Jordan Block Matrices Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.

More information

TESTS BASED ON MAXIMUM LIKELIHOOD

TESTS 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 information

Rademacher Complexity. Examples

Rademacher Complexity. Examples Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed

More information

Lecture Notes 2. The ability to manipulate matrices is critical in economics.

Lecture Notes 2. The ability to manipulate matrices is critical in economics. Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets

More information

LECTURE - 4 SIMPLE RANDOM SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR

LECTURE - 4 SIMPLE RANDOM SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR amplg Theory MODULE II LECTURE - 4 IMPLE RADOM AMPLIG DR. HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR Estmato of populato mea ad populato varace Oe of the ma objectves after

More information

Model Fitting, RANSAC. Jana Kosecka

Model 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 information

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67. Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please

More information

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5

THE 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 information

Linear Regression with One Regressor

Linear 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 information

Generalized Linear Regression with Regularization

Generalized Linear Regression with Regularization Geeralze Lear Regresso wth Regularzato Zoya Bylsk March 3, 05 BASIC REGRESSION PROBLEM Note: I the followg otes I wll make explct what s a vector a what s a scalar usg vec t or otato, to avo cofuso betwee

More information

ECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model

ECON 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 information

Example: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger

Example: 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 information

Machine Learning. knowledge acquisition skill refinement. Relation between machine learning and data mining. P. Berka, /18

Machine Learning. knowledge acquisition skill refinement. Relation between machine learning and data mining. P. Berka, /18 Mache Learg The feld of mache learg s cocered wth the questo of how to costruct computer programs that automatcally mprove wth eperece. (Mtchell, 1997) Thgs lear whe they chage ther behavor a way that

More information

LINEAR REGRESSION ANALYSIS

LINEAR REGRESSION ANALYSIS LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for

More information

Mean is only appropriate for interval or ratio scales, not ordinal or nominal.

Mean is only appropriate for interval or ratio scales, not ordinal or nominal. Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot

More information

Bayesian belief networks

Bayesian 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 information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

Multiple Choice Test. Chapter Adequacy of Models for Regression

Multiple Choice Test. Chapter Adequacy of Models for Regression Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to

More information

Research on SVM Prediction Model Based on Chaos Theory

Research on SVM Prediction Model Based on Chaos Theory Advaced Scece ad Techology Letters Vol.3 (SoftTech 06, pp.59-63 http://dx.do.org/0.457/astl.06.3.3 Research o SVM Predcto Model Based o Chaos Theory Sog Lagog, Wu Hux, Zhag Zezhog 3, College of Iformato

More information

THE ROYAL STATISTICAL SOCIETY 2009 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULAR FORMAT MODULE 2 STATISTICAL INFERENCE

THE ROYAL STATISTICAL SOCIETY 2009 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULAR FORMAT MODULE 2 STATISTICAL INFERENCE THE ROYAL STATISTICAL SOCIETY 009 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULAR FORMAT MODULE STATISTICAL INFERENCE The Socety provdes these solutos to assst caddates preparg for the examatos future

More information

ENGI 4421 Propagation of Error Page 8-01

ENGI 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 information

Chapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:

Chapter 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 information

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory

C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?

More information

Application of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design

Application of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud Applcato of Calbrato Approach for Regresso Coeffcet Estmato uder Two-stage Samplg Desg Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud

More information

Chapter Two. An Introduction to Regression ( )

Chapter 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 information

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1 STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal

More information

X 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

X 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 information

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:

{ }{ ( )} (, ) = ( ) ( ) ( ) 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 information

Multivariate Transformation of Variables and Maximum Likelihood Estimation

Multivariate 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 information

Nonparametric Density Estimation Intro

Nonparametric Density Estimation Intro Noarametrc Desty Estmato Itro Parze Wdows No-Parametrc Methods Nether robablty dstrbuto or dscrmat fucto s kow Haes qute ofte All we have s labeled data a lot s kow easer salmo bass salmo salmo Estmate

More information

Investigation of Partially Conditional RP Model with Response Error. Ed Stanek

Investigation of Partially Conditional RP Model with Response Error. Ed Stanek Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a

More information

Lecture 1: Introduction to Regression

Lecture 1: Introduction to Regression Lecture : Itroducto to Regresso A Eample: Eplag State Homcde Rates What kds of varables mght we use to epla/predct state homcde rates? Let s cosder just oe predctor for ow: povert Igore omtted varables,

More information

Recall 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

Recall 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 information

Lecture Note to Rice Chapter 8

Lecture Note to Rice Chapter 8 ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,

More information

6.867 Machine Learning

6.867 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 information

Sampling Theory MODULE X LECTURE - 35 TWO STAGE SAMPLING (SUB SAMPLING)

Sampling Theory MODULE X LECTURE - 35 TWO STAGE SAMPLING (SUB SAMPLING) Samplg Theory ODULE X LECTURE - 35 TWO STAGE SAPLIG (SUB SAPLIG) DR SHALABH DEPARTET OF ATHEATICS AD STATISTICS IDIA ISTITUTE OF TECHOLOG KAPUR Two stage samplg wth uequal frst stage uts: Cosder two stage

More information

Special Instructions / Useful Data

Special 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 information

MATH 247/Winter Notes on the adjoint and on normal operators.

MATH 247/Winter Notes on the adjoint and on normal operators. MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say

More information

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades

Multiple 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 information

Analysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems

Analysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems Char for Network Archtectures ad Servces Prof. Carle Departmet of Computer Scece U Müche Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems Dr. Alexader Kle Prof. Dr.-Ig. Georg Carle

More information

L5 Polynomial / Spline Curves

L5 Polynomial / Spline Curves L5 Polyomal / Sple Curves Cotets Coc sectos Polyomal Curves Hermte Curves Bezer Curves B-Sples No-Uform Ratoal B-Sples (NURBS) Mapulato ad Represetato of Curves Types of Curve Equatos Implct: Descrbe a

More information

CS286.2 Lecture 4: Dinur s Proof of the PCP Theorem

CS286.2 Lecture 4: Dinur s Proof of the PCP Theorem CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat

More information

ε. Therefore, the estimate

ε. 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 information

STA302/1001-Fall 2008 Midterm Test October 21, 2008

STA302/1001-Fall 2008 Midterm Test October 21, 2008 STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from

More information

Gender Classification from ECG Signal Analysis using Least Square Support Vector Machine

Gender Classification from ECG Signal Analysis using Least Square Support Vector Machine Amerca Joural of Sgal Processg, (5): 45-49 DOI:.593/.asp.5.8 Geder Classfcato from ECG Sgal Aalyss usg Least Square Support Vector Mache Raesh Ku. rpathy,*, Ashutosh Acharya, Sumt Kumar Choudhary Departmet

More information

Handout #8. X\Y f(x) 0 1/16 1/ / /16 3/ / /16 3/16 0 3/ /16 1/16 1/8 g(y) 1/16 1/4 3/8 1/4 1/16 1

Handout #8. X\Y f(x) 0 1/16 1/ / /16 3/ / /16 3/16 0 3/ /16 1/16 1/8 g(y) 1/16 1/4 3/8 1/4 1/16 1 Hadout #8 Ttle: Foudatos of Ecoometrcs Course: Eco 367 Fall/05 Istructor: Dr. I-Mg Chu Lear Regresso Model So far we have focused mostly o the study of a sgle radom varable, ts correspodg theoretcal dstrbuto,

More information

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..

More information

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model 1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed

More information