I. Decision trees II. Ensamble methods: Mixtures of experts
|
|
|
- Lora Harris
- 5 years ago
- Views:
Transcription
1 CS 75 Machne Learnn Lectre 4 I. Decson trees II. Ensamble methods: Mtres of eperts Mlos Hasrecht mlos@cs.ptt.ed 539 Sennott Sqare CS 75 Machne Learnn Eam: Aprl 8 7 Schedle Term proects & proect presentatons: Aprl 5 7 At :-4:pm n SNSQ 533 No class: on Aprl 3 7 CS 75 Machne Learnn
2 Decson trees An alternatve approach to classfcaton: Partton the npt space to reons Classf ndependentl n ever reon CS 75 Machne Learnn Decson trees The parttonn dea s sed n the decson tree model: Splt the space recrsvel accordn to npts n Classf assn class label at the bottom of the tree Eample: Bnar classfcaton {} Bnar attrbtes 3 3 t f t f t f CS 75 Machne Learnn
3 Decson trees How to constrct the decson tree? Top-bottom alorthm: Fnd the best splt condton qantfed based on the mprt measre on the trann set Stops when no mprovement possble Imprt measre: Measres how well are the two classes separated Ideall we wold le to separate all s and Splts of fnte vs. contnos vale attrbtes Contnos vale attrbtes condtons: 3. 5 CS 75 Machne Learnn Let Imprt measre D - Total nmber of data entres D - Nmber of data entres classfed as D p - rato of nstances classfed as D Imprt measre defnes how well are the classes n the trann dataset separated In eneral the mprt measre shold satsf: Larest when data are splt evenl to classes p nmber of classes Shold be when all data belon to the same class CS 75 Machne Learnn
4 Imprt measres There are varos mprt measres sed n the lteratre Entrop based measre Qnlan C4.5 I D Entrop D p lo p Eample for Gn measre Breman CART I D Gn D CS 75 Machne Learnn p Decson-tree bldn Gan de to splt epected redcton n the mprt measre entrop eample Gan D A Entrop v D t EntropD D v Vales A CS 75 Machne Learnn v D Entrop D D - a partton of D wth the vale of attrbte A v 3 f EntropD t t f t f Entrop D Entrop D v 3 f
5 Decson tree learnn Greed learnn alorthm: Repeat ntl no or small mprovement n the prt Fnd the attrbte wth the hhest an Add the attrbte to the tree and splt the set accordnl Blds the tree n the top-down fashon Gradall epands the leaves of the partall blt tree The method s reed It loos at a snle attrbte and an n each step Ma fal when the combnaton of attrbtes s needed to mprove the prt part fnctons CS 75 Machne Learnn Decson tree learnn Lmtatons of reed methods Cases n whch a combnaton of two or more attrbtes mproves the mprt CS 75 Machne Learnn
6 Decson tree learnn B redcn the mprt measre we can row ver lare trees Problem: Overfttn We ma splt and classf ver well the trann set bt we ma do worse n terms of the eneralzaton error Soltons to the overfttn problem: Solton. Prne branches of the tree blt n the frst phase Use nternal valdaton set to test for the overft Solton. Test for the overft n the tree bldn phase Stop bldn the tree when performance on the valdaton set deterorates CS 75 Machne Learnn Mtre of eperts model Ensamble methods: Use a combnaton of smpler learners to mprove predctons Mtre of epert model: Dfferent npt reons covered wth dfferent learners A soft swtchn between learners Mtre of eperts Epert learner CS 75 Machne Learnn
7 Mtre of eperts model Gatn networ : decdes what epert to se... - atn fnctons Gatn networ Epert Epert... Epert CS 75 Machne Learnn Learnn mtre of eperts Learnn conssts of two tass: Learn the parameters of ndvdal epert networs Learn the parameters of the atn networ Decdes where to mae a splt Assme: atn fnctons ve probabltes... Based on the probablt we partton the space parttons belons to dfferent eperts How to model the atn networ? A mltwa classfer model: softma model a eneratve classfer model CS 75 Machne Learnn
8 CS 75 Machne Learnn Learnn mtre of eperts Assme we have a set of lnear eperts Assme a softma atn networ Lelhood of assmed that errors for dfferent eperts are normall dstrbted wth the same varance T µ ep ep T T p Note: bas terms are hdden n Θ p P P Θ ep ep ep σ µ σ π T T CS 75 Machne Learnn Learnn mtre of eperts Gradent learnn. On-lne pdate rle for parameters of epert If we now the epert that s responsble for If we do not now the epert µ α + h µ α + h - responsblt of the th epert a nd of posteror p p h / ep / ep µ µ -a pror ep... - a lelhood
9 Learnn mtres of eperts Gradent methods On-lne learnn of atn networ parameters + β h The learnn wth condtoned mtres can be etended to learnn of parameters of an arbtrar epert networ e.. lostc reresson mltlaer neral networ l + β l µ µ l h µ CS 75 Machne Learnn Learnn mtre of eperts EM alorthm offers an alternatve wa to learn the mtre Alorthm: Intalze parameters Θ Repeat Set Θ ' Θ. Epectaton step Q Θ Θ' EH X YΘ' lo P H Y X Θ ξ. Mamzaton step Θ ar ma Q Θ Θ ' Θ ntl no or small mprovement n Q Θ Θ' Hdden varables are denttes of epert networs responsble for data ponts CS 75 Machne Learnn
Generative 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
Classification 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
CS 3750 Machine Learning Lecture 6. Monte Carlo methods. CS 3750 Advanced Machine Learning. Markov chain Monte Carlo
CS 3750 Machne Learnng Lectre 6 Monte Carlo methods Mlos Haskrecht mlos@cs.ptt.ed 5329 Sennott Sqare Markov chan Monte Carlo Importance samplng: samples are generated accordng to Q and every sample from
Discriminative 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
Evaluation of classifiers MLPs
Lecture Evaluaton of classfers MLPs Mlos Hausrecht mlos@cs.ptt.edu 539 Sennott Square Evaluaton For any data set e use to test the model e can buld a confuson matrx: Counts of examples th: class label
Multilayer 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
INF 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
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
The 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
Admin NEURAL NETWORKS. Perceptron learning algorithm. Our Nervous System 10/25/16. Assignment 7. Class 11/22. Schedule for the rest of the semester
0/25/6 Admn Assgnment 7 Class /22 Schedule for the rest of the semester NEURAL NETWORKS Davd Kauchak CS58 Fall 206 Perceptron learnng algorthm Our Nervous System repeat untl convergence (or for some #
Discriminative 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,
ESS 265 Spring Quarter 2005 Time Series Analysis: Error Analysis
ESS 65 Sprng Qarter 005 Tme Seres Analyss: Error Analyss Lectre 9 May 3, 005 Some omenclatre Systematc errors Reprodcbly errors that reslt from calbraton errors or bas on the part of the obserer. Sometmes
Multi-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
Expectation 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
Multi-Layer Perceptrons
Mlt-Layer Perceptrons Mlt-Layer Perceptrons Wth hdden layers One hdden layer - any Boolean fncton or conve decson reons To hdden layers - arbtrary decson reons PR, ANN, & ML Decson bondares 1 1 1 1 1 PR,
Logistic 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
Ensemble 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
Multilayer 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
CIS526: 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
Correlation 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,
Boostrapaggregating (Bagging)
Boostrapaggregatng (Baggng) An ensemble meta-algorthm desgned to mprove the stablty and accuracy of machne learnng algorthms Can be used n both regresson and classfcaton Reduces varance and helps to avod
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
10-701/ 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
For 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
CS 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
Which 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
Kernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
Solutions to selected problems from homework 1.
Jan Hagemejer 1 Soltons to selected problems from homeork 1. Qeston 1 Let be a tlty fncton hch generates demand fncton xp, ) and ndrect tlty fncton vp, ). Let F : R R be a strctly ncreasng fncton. If the
CHAPTER 3: BAYESIAN DECISION THEORY
HATER 3: BAYESIAN DEISION THEORY Decson mang under uncertanty 3 Data comes from a process that s completely not nown The lac of nowledge can be compensated by modelng t as a random process May be the underlyng
Linear classification models: Perceptron. CS534-Machine learning
Lnear classfcaton odels: Perceptron CS534-Machne learnng Classfcaton proble Gven npt, the goal s to predct, hch s a categorcal varable s called the class label s the featre vector Eaple: : onthl ncoe and
Department of Computer Science Artificial Intelligence Research Laboratory. Iowa State University MACHINE LEARNING
MACHINE LEANING Vasant Honavar Bonformatcs and Computatonal Bology rogram Center for Computatonal Intellgence, Learnng, & Dscovery Iowa State Unversty honavar@cs.astate.edu www.cs.astate.edu/~honavar/
SDMML 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
Lecture 10: Dimensionality reduction
Lecture : Dmensonalt reducton g The curse of dmensonalt g Feature etracton s. feature selecton g Prncpal Components Analss g Lnear Dscrmnant Analss Intellgent Sensor Sstems Rcardo Guterrez-Osuna Wrght
CSC 411 / CSC D11 / CSC C11
18 Boostng s a general strategy for learnng classfers by combnng smpler ones. The dea of boostng s to take a weak classfer that s, any classfer that wll do at least slghtly better than chance and use t
CHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
Support Vector Machines
/14/018 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
Generative and Discriminative Models. Jie Tang Department of Computer Science & Technology Tsinghua University 2012
Generatve and Dscrmnatve Models Je Tang Department o Computer Scence & Technolog Tsnghua Unverst 202 ML as Searchng Hpotheses Space ML Methodologes are ncreasngl statstcal Rule-based epert sstems beng
Neural networks. Nuno Vasconcelos ECE Department, UCSD
Neural networs Nuno Vasconcelos ECE Department, UCSD Classfcaton a classfcaton problem has two types of varables e.g. X - vector of observatons (features) n the world Y - state (class) of the world x X
Experimental Errors and Error Analysis
Expermental Errors and Error Analss Rajee Prabhakar Unerst of Texas at Astn, Astn, TX Freeman Research Grop Meetng March 10, 004 Topcs coered Tpes of expermental errors Redcng errors Descrbng errors qanttatel
Support Vector Machines
CS 2750: Machne Learnng Support Vector Machnes Prof. Adrana Kovashka Unversty of Pttsburgh February 17, 2016 Announcement Homework 2 deadlne s now 2/29 We ll have covered everythng you need today or at
We present the algorithm first, then derive it later. Assume access to a dataset {(x i, y i )} n i=1, where x i R d and y i { 1, 1}.
CS 189 Introducton to Machne Learnng Sprng 2018 Note 26 1 Boostng We have seen that n the case of random forests, combnng many mperfect models can produce a snglodel that works very well. Ths s the dea
Bayesian decision theory. Nuno Vasconcelos ECE Department, UCSD
Bayesan decson theory Nuno Vasconcelos ECE Department UCSD Notaton the notaton n DHS s qute sloppy e.. show that error error z z dz really not clear what ths means we wll use the follown notaton subscrpts
Evaluation for sets of classes
Evaluaton for Tet Categorzaton Classfcaton accuracy: usual n ML, the proporton of correct decsons, Not approprate f the populaton rate of the class s low Precson, Recall and F 1 Better measures 21 Evaluaton
Hidden Markov Models
CM229S: Machne Learnng for Bonformatcs Lecture 12-05/05/2016 Hdden Markov Models Lecturer: Srram Sankararaman Scrbe: Akshay Dattatray Shnde Edted by: TBD 1 Introducton For a drected graph G we can wrte
AE/ME 339. K. M. Isaac. 8/31/2004 topic4: Implicit method, Stability, ADI method. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Comptatonal Fld Dynamcs (CFD) Comptatonal Fld Dynamcs (AE/ME 339) Implct form of dfference eqaton In the prevos explct method, the solton at tme level n,,n, depended only on the known vales of,
Support 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
Support 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
Logistic 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
Finite Mixture Models and Expectation Maximization. Most slides are from: Dr. Mario Figueiredo, Dr. Anil Jain and Dr. Rong Jin
Fnte Mxture Models and Expectaton Maxmzaton Most sldes are from: Dr. Maro Fgueredo, Dr. Anl Jan and Dr. Rong Jn Recall: The Supervsed Learnng Problem Gven a set of n samples X {(x, y )},,,n Chapter 3 of
Support 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
Clustering gene expression data & the EM algorithm
CG, Fall 2011-12 Clusterng gene expresson data & the EM algorthm CG 08 Ron Shamr 1 How Gene Expresson Data Looks Entres of the Raw Data matrx: Rato values Absolute values Row = gene s expresson pattern
Mixture o f of Gaussian Gaussian clustering Nov
Mture of Gaussan clusterng Nov 11 2009 Soft vs hard lusterng Kmeans performs Hard clusterng: Data pont s determnstcally assgned to one and only one cluster But n realty clusters may overlap Soft-clusterng:
Learning Theory: Lecture Notes
Learnng Theory: Lecture Notes Lecturer: Kamalka Chaudhur Scrbe: Qush Wang October 27, 2012 1 The Agnostc PAC Model Recall that one of the constrants of the PAC model s that the data dstrbuton has to be
MIMA 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
Lecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
Feature Selection: Part 1
CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?
Retrieval Models: Language models
CS-590I Informaton Retreval Retreval Models: Language models Luo S Department of Computer Scence Purdue Unversty Introducton to language model Ungram language model Document language model estmaton Maxmum
Linear 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
Week 5: Neural Networks
Week 5: Neural Networks Instructor: Sergey Levne Neural Networks Summary In the prevous lecture, we saw how we can construct neural networks by extendng logstc regresson. Neural networks consst of multple
MLE 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:
C4B Machine Learning Answers II. = σ(z) (1 σ(z)) 1 1 e z. e z = σ(1 σ) (1 + e z )
C4B Machne Learnng Answers II.(a) Show that for the logstc sgmod functon dσ(z) dz = σ(z) ( σ(z)) A. Zsserman, Hlary Term 20 Start from the defnton of σ(z) Note that Then σ(z) = σ = dσ(z) dz = + e z e z
S Advanced Digital Communication (4 cr) Targets today
S.72-3320 Advanced Dtal Communcaton (4 cr) Convolutonal Codes Tarets today Why to apply convolutonal codn? Defnn convolutonal codes Practcal encodn crcuts Defnn qualty of convolutonal codes Decodn prncples
Pattern Classification
attern Classfcaton All materals n these sldes were taken from attern Classfcaton nd ed by R. O. Duda,. E. Hart and D. G. Stork, John Wley & Sons, 000 wth the ermsson of the authors and the ublsher Chater
CS 2750 Machine Learning. Lecture 5. Density estimation. CS 2750 Machine Learning. Announcements
CS 750 Machne Learnng Lecture 5 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square CS 750 Machne Learnng Announcements Homework Due on Wednesday before the class Reports: hand n before
IV. 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
The 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.
Machine Learning for Signal Processing Linear Gaussian Models
Machne Learnng for Sgnal rocessng Lnear Gaussan Models lass 2. 2 Nov 203 Instructor: Bhsha Raj 2 Nov 203 755/8797 HW3 s up. Admnstrva rojects please send us an update 2 Nov 203 755/8797 2 Recap: MA stmators
Learning with Partially Observed Data
Readngs K&F 8.6 9. 9.2 Learnng wth artall Observed ata Lecture 2 Ma 4 2 CSE 55 Statstcal Methods Sprng 2 Instructor Su-In Lee nverst of Washngton Seattle Model Selecton So far we focused on sngle model
Homework 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.
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
1. 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
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE School of Computer and Communcaton Scences Handout 0 Prncples of Dgtal Communcatons Solutons to Problem Set 4 Mar. 6, 08 Soluton. If H = 0, we have Y = Z Z = Y
Cluster Validation Determining Number of Clusters. Umut ORHAN, PhD.
Cluster Analyss Cluster Valdaton Determnng Number of Clusters 1 Cluster Valdaton The procedure of evaluatng the results of a clusterng algorthm s known under the term cluster valdty. How do we evaluate
Lecture Nov
Lecture 18 Nov 07 2008 Revew Clusterng Groupng smlar obects nto clusters Herarchcal clusterng Agglomeratve approach (HAC: teratvely merge smlar clusters Dfferent lnkage algorthms for computng dstances
Generalized 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
Bayesian decision theory. Nuno Vasconcelos ECE Department, UCSD
Bayesan decson theory Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world observatons decson functon L[,y] loss of predctn y wth the epected value of the
Gaussian process classification: a message-passing viewpoint
Gaussan process classfcaton: a message-passng vewpont Flpe Rodrgues fmpr@de.uc.pt November 014 Abstract The goal of ths short paper s to provde a message-passng vewpont of the Expectaton Propagaton EP
Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
Linear Correlation. Many research issues are pursued with nonexperimental studies that seek to establish relationships among 2 or more variables
Lnear Correlaton Many research ssues are pursued wth nonexpermental studes that seek to establsh relatonshps among or more varables E.g., correlates of ntellgence; relaton between SAT and GPA; relaton
Statistical Foundations of Pattern Recognition
Statstcal Foundatons of Pattern Recognton Learnng Objectves Bayes Theorem Decson-mang Confdence factors Dscrmnants The connecton to neural nets Statstcal Foundatons of Pattern Recognton NDE measurement
Lecture 3 January 31, 2017
CS 224: Advanced Algorthms Sprng 207 Prof. Jelan Nelson Lecture 3 January 3, 207 Scrbe: Saketh Rama Overvew In the last lecture we covered Y-fast tres and Fuson Trees. In ths lecture we start our dscusson
Machine learning: Density estimation
CS 70 Foundatons of AI Lecture 3 Machne learnng: ensty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square ata: ensty estmaton {.. n} x a vector of attrbute values Objectve: estmate the model of
Hidden Markov Models
Hdden Markov Models Namrata Vaswan, Iowa State Unversty Aprl 24, 204 Hdden Markov Model Defntons and Examples Defntons:. A hdden Markov model (HMM) refers to a set of hdden states X 0, X,..., X t,...,
Supporting Information
Supportng Informaton The neural network f n Eq. 1 s gven by: f x l = ReLU W atom x l + b atom, 2 where ReLU s the element-wse rectfed lnear unt, 21.e., ReLUx = max0, x, W atom R d d s the weght matrx to
Statistical performance analysis by loopy belief propagation in probabilistic image processing
Statstcal perormance analyss by loopy bele propaaton n probablstc mae processn Kazuyuk Tanaka raduate School o Inormaton Scences Tohoku Unversty Japan http://www.smapp.s.tohoku.ac.p/~kazu/ Collaborators
Bruce A. Draper & J. Ross Beveridge, January 25, Geometric Image Manipulation. Lecture #1 January 25, 2013
Brce A. Draper & J. Ross Beerdge, Janar 5, Geometrc Image Manplaton Lectre # Janar 5, Brce A. Draper & J. Ross Beerdge, Janar 5, Image Manplaton: Contet To start wth the obos, an mage s a D arra of pels
The Bellman Equation
The Bellman Eqaton Reza Shadmehr In ths docment I wll rovde an elanaton of the Bellman eqaton, whch s a method for otmzng a cost fncton and arrvng at a control olcy.. Eamle of a game Sose that or states
A random variable is a function which associates a real number to each element of the sample space
Introducton to Random Varables Defnton of random varable Defnton of of random varable Dscrete and contnuous random varable Probablty blt functon Dstrbuton functon Densty functon Sometmes, t s not enough
2E 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
Introduction to the Introduction to Artificial Neural Network
Introducton to the Introducton to Artfcal Neural Netork Vuong Le th Hao Tang s sldes Part of the content of the sldes are from the Internet (possbly th modfcatons). The lecturer does not clam any onershp
CLASSIFICATION OF HIGGS JETS AS DECAY PRODUCTS OF A RANDALL-SUNDRUM GRAVITON AT THE ATLAS EXPERIMENT
CLASSIFICATION OF HIGGS JETS AS DECAY PRODUCTS OF A RANDALL-SUNDRUM GRAVITON AT THE ATLAS EXPERIMENT AVIV CUKIERMAN, ZIHAO JIANG. Introducton In 202, physcsts at the Large Hadron Collder announced the
Support Vector Machines. Vibhav Gogate The University of Texas at dallas
Support Vector Machnes Vbhav Gogate he Unversty of exas at dallas What We have Learned So Far? 1. Decson rees. Naïve Bayes 3. Lnear Regresson 4. Logstc Regresson 5. Perceptron 6. Neural networks 7. K-Nearest
Expected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
Natural Language Processing and Information Retrieval
Natural Language Processng and Informaton Retreval Support Vector Machnes Alessandro Moschtt Department of nformaton and communcaton technology Unversty of Trento Emal: moschtt@ds.untn.t Summary Support
MDL-Based Unsupervised Attribute Ranking
MDL-Based Unsupervsed Attrbute Rankng Zdravko Markov Computer Scence Department Central Connectcut State Unversty New Brtan, CT 06050, USA http://www.cs.ccsu.edu/~markov/ markovz@ccsu.edu MDL-Based Unsupervsed
15-381: Artificial Intelligence. Regression and cross validation
15-381: Artfcal Intellgence Regresson and cross valdaton Where e are Inputs Densty Estmator Probablty Inputs Classfer Predct category Inputs Regressor Predct real no. Today Lnear regresson Gven an nput
Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. . For P such independent random variables (aka degrees of freedom): 1 =
Fall Analss of Epermental Measurements B. Esensten/rev. S. Errede More on : The dstrbuton s the.d.f. for a (normalzed sum of squares of ndependent random varables, each one of whch s dstrbuted as N (,.
Machine Learning & Data Mining CS/CNS/EE 155. Lecture 4: Regularization, Sparsity & Lasso
Machne Learnng Data Mnng CS/CS/EE 155 Lecture 4: Regularzaton, Sparsty Lasso 1 Recap: Complete Ppelne S = {(x, y )} Tranng Data f (x, b) = T x b Model Class(es) L(a, b) = (a b) 2 Loss Functon,b L( y, f
Assortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
Edinburgh Research Archive: Contact:
Z. H. Hang and Q. Shen. Transformaton based nterpolaton wth generalzed representatve vales. In Proc. of the Internatonal Conference on Fzzy Systems, pages 821 826, 2005. 2005 IEEE. Personal se of ths materal