CS 688 Pattern Recognition Lecture 4. Linear Models for Classification

Size: px
Start display at page:

Download "CS 688 Pattern Recognition Lecture 4. Linear Models for Classification"

Transcription

1 CS 688 Pattern Recognition Lecture 4 Linear Models for Classification Probabilistic generative models Probabilistic discriminative models 1

2 Generative Approach ( x ) p C k p( C k ) Ck p ( ) ( x Ck ) p( Ck ) Ck x = p( x) ( x) p( x C ) p( ) p p = k k C k Two class case: logistic sigmoid function 2

3 K >2 classes: softmax function Lets assume the class-conditional densities are Gaussian with the same covariance matrix: Two class case first. We can show the following result: 3

4 4

5 We have shown: Decision boundary: 5

6 K >2 classes: We can show the following result: The cancellation of the quadratic terms is due to the assumption of shared covariances. If we allow each class conditional density to have its own covariance matrix, the cancellations no longer occur, and we obtain a quadratic function of x, i.e. a quadratic discriminant. 6

7 Maximum likelihood solution We have a parametric functional form for the class-conditional densities: We can estimate the parameters and the prior class probabilities using maximum likelihood. Two class case with shared covariance matrix. Training data: 7

8 Maximum likelihood solution For a data point from class we have and therefore For a data point from class we have and therefore Maximum likelihood solution Assuming observations are drawn independently, we can write the likelihood function as follows: N p( t π,µ 1,µ 2,Σ) = p( t n π,µ 1,µ 2,Σ) = N n=1 [ p( x n,c 1 )] t n p x n,c 2 n=1 N n=1 [ πn ( x n µ 1,Σ)] t n [ ( )] 1 t n = [( 1 π)n ( x n µ 2,Σ)] 1 t n t = ( t 1,,t N ) T 8

9 Maximum likelihood solution We want to find the values of the parameters that maximize the likelihood function, i.e., fit a model that best describes the observed data. p t π,µ 1,µ 2,Σ N n=1 ( ) = [ πn ( x n µ 1,Σ)] t n [( 1 π)n ( x n µ 2,Σ)] 1 t n As usual, we consider the log of the likelihood: Maximum likelihood solution We first maximize the log likelihood with respect to terms that depend on are. The 9

10 Maximum likelihood solution Thus, the maximum likelihood estimate of of points in class is the fraction The result can be generalized to the multiclass case: the maximum likelihood estimate of is given by the fraction of points in the training set that belong to Maximum likelihood solution We now maximize the log likelihood with respect to terms that depend on are. The 10

11 Maximum likelihood solution Thus, the maximum likelihood estimate of is the sample mean of all the input vectors assigned to class By maximizing the log likelihood with respect to obtain a similar result for we Maximum likelihood solution Maximizing the log likelihood with respect to the maximum likelihood estimate we obtain Thus: the maximum likelihood estimate of the covariance is given by the weighted average of the sample covariance matrices associated with each of the classes. This results extend to K classes. 11

12 Probabilistic Discriminative Models Two-class case: Multiclass case: Discriminative approach: use the functional form of the generalized linear model for the posterior probabilities and determine its parameters directly using maximum likelihood. Probabilistic Discriminative Models Advantages: Fewer parameters to be determined Improved predictive performance, especially when the class-conditional density assumptions give a poor approximation of the true distributions. 12

13 Probabilistic Discriminative Models Two-class case: In the terminology of statistics, this model is known as logistic regression. Assuming estimate? how many parameters do we need to Probabilistic Discriminative Models How many parameters did we estimate to fit Gaussian class-conditional densities (generative approach)? 13

14 Logistic Regression We use maximum likelihood to determine the parameters of the logistic regression model. { x n,t n } n =1,,N t n =1 denotes class C 1 t n = 0 denotes class C 2 We want to find the values of w that maximize the posterior probabilities associated to the observed data Likelihood function : L w N ( ) = P( C 1 x n ) t n n=1 ( 1 P( C 1 x n )) 1 t n N ( ) t n = y x n n=1 ( 1 y( x n )) 1 t n Logistic Regression We consider the negative logarithm of the likelihood: 14

15 Logistic Regression We compute the derivative of the error function with respect to w (gradient): We need to compute the derivative of the logistic sigmoid function: a σ ( a ) = a 1 1+ e = e a a 1+ e a = σ a 1+ e a 1+ e a ( ) 2 = 1 ( ) ( ) 1 σ( a) e a 1+ e a 1+ e = a Logistic Regression 15

16 Logistic Regression The gradient of E at w gives the direction of the steepest increase of E at w. We need to minimize E. Thus we need to update w so that we move along the opposite direction of the gradient: This technique is called gradient descent It can be shown that E is a concave function of w. Thus, it has a unique minimum. An efficient iterative technique exists to find the optimal w parameters (Newton-Raphson optimization). Batch vs. on-line learning The computation of the above gradient requires the processing of the entire training set (batch technique) If the data set is large, the above technique can be costly; For real time applications in which data become available as continuous streams, we may want to update the parameters as data points are presented to us (on-line technique). 16

17 On-line learning After the presentation of each data point n, we compute the contribution of that data point to the gradient (stochastic gradient): The on-line updating rule for the parameters becomes: η > 0 is called learning rate. It's value needs to be chosen carefully to ensure convergence Multiclass Logistic Regression Multiclass case: We use maximum likelihood to determine the parameters of the logistic regression model. { x n,t n } n =1,, N ( ) denotes class C k t n = 0,, 1,,0 We want to find the values of w 1,,w K that maximize the posterior probabilities associated to the observed data Likelihood function : ( ) = P( C k x n ) t nk L w 1,,w K N K = y k x n n=1 k =1 n=1 k =1 N K ( ) t nk 17

18 Multiclass Logistic Regression We consider the negative logarithm of the likelihood: Multiclass Logistic Regression We compute the gradient of the error function with respect to one of the parameter vectors: 18

19 Multiclass Logistic Regression Thus, we need to compute the derivatives of the softmax function: y k = a k a k e ak e a j = j e a k j e a j j e a j e a k ea k ( ) 2 = y k y 2 k = y k 1 y k Multiclass Logistic Regression Thus, we need to compute the derivatives of the softmax function: for j k, y k = a j a j e ak j e a j = e a k e a j = y 2 k y j e a j j 19

20 Multiclass Logistic Regression Compact expression: Multiclass Logistic Regression 20

21 Multiclass Logistic Regression It can be shown that E is a concave function of w. Thus, it has a unique minimum. For a batch solution, we can use the Newton-Raphson optimization technique. On-line solution (stochastic gradient descent): 21

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical

More information

Linear Models for Classification

Linear Models for Classification Linear Models for Classification Sumeet Agarwal, EEL709 (Most figures from Bishop, PRML) Approaches to classification Discriminant function: Directly assigns each data point x to a particular class Ci

More information

Christfried Webers. Canberra February June 2015

Christfried Webers. Canberra February June 2015 c Statistical Group and College of Engineering and Computer Science Canberra February June (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 829 c Part VIII Linear Classification 2 Logistic

More information

These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher Bishop

These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher Bishop Music and Machine Learning (IFT6080 Winter 08) Prof. Douglas Eck, Université de Montréal These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct

More information

Pattern Analysis. Logistic Regression. 12. Mai 2009. Joachim Hornegger. Chair of Pattern Recognition Erlangen University

Pattern Analysis. Logistic Regression. 12. Mai 2009. Joachim Hornegger. Chair of Pattern Recognition Erlangen University Pattern Analysis Logistic Regression 12. Mai 2009 Joachim Hornegger Chair of Pattern Recognition Erlangen University Pattern Analysis 2 / 43 1 Logistic Regression Posteriors and the Logistic Function Decision

More information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

Pa8ern Recogni6on. and Machine Learning. Chapter 4: Linear Models for Classifica6on

Pa8ern Recogni6on. and Machine Learning. Chapter 4: Linear Models for Classifica6on Pa8ern Recogni6on and Machine Learning Chapter 4: Linear Models for Classifica6on Represen'ng the target values for classifica'on If there are only two classes, we typically use a single real valued output

More information

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max

More information

Linear Discrimination. Linear Discrimination. Linear Discrimination. Linearly Separable Systems Pairwise Separation. Steven J Zeil.

Linear Discrimination. Linear Discrimination. Linear Discrimination. Linearly Separable Systems Pairwise Separation. Steven J Zeil. Steven J Zeil Old Dominion Univ. Fall 200 Discriminant-Based Classification Linearly Separable Systems Pairwise Separation 2 Posteriors 3 Logistic Discrimination 2 Discriminant-Based Classification Likelihood-based:

More information

Classification Problems

Classification Problems Classification Read Chapter 4 in the text by Bishop, except omit Sections 4.1.6, 4.1.7, 4.2.4, 4.3.3, 4.3.5, 4.3.6, 4.4, and 4.5. Also, review sections 1.5.1, 1.5.2, 1.5.3, and 1.5.4. Classification Problems

More information

CSCI567 Machine Learning (Fall 2014)

CSCI567 Machine Learning (Fall 2014) CSCI567 Machine Learning (Fall 2014) Drs. Sha & Liu {feisha,yanliu.cs}@usc.edu September 22, 2014 Drs. Sha & Liu ({feisha,yanliu.cs}@usc.edu) CSCI567 Machine Learning (Fall 2014) September 22, 2014 1 /

More information

Lecture 3: Linear methods for classification

Lecture 3: Linear methods for classification Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,

More information

Linear Classification. Volker Tresp Summer 2015

Linear Classification. Volker Tresp Summer 2015 Linear Classification Volker Tresp Summer 2015 1 Classification Classification is the central task of pattern recognition Sensors supply information about an object: to which class do the object belong

More information

Linear Threshold Units

Linear Threshold Units Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

More information

Probabilistic Linear Classification: Logistic Regression. Piyush Rai IIT Kanpur

Probabilistic Linear Classification: Logistic Regression. Piyush Rai IIT Kanpur Probabilistic Linear Classification: Logistic Regression Piyush Rai IIT Kanpur Probabilistic Machine Learning (CS772A) Jan 18, 2016 Probabilistic Machine Learning (CS772A) Probabilistic Linear Classification:

More information

Logistic Regression. Vibhav Gogate The University of Texas at Dallas. Some Slides from Carlos Guestrin, Luke Zettlemoyer and Dan Weld.

Logistic Regression. Vibhav Gogate The University of Texas at Dallas. Some Slides from Carlos Guestrin, Luke Zettlemoyer and Dan Weld. Logistic Regression Vibhav Gogate The University of Texas at Dallas Some Slides from Carlos Guestrin, Luke Zettlemoyer and Dan Weld. Generative vs. Discriminative Classifiers Want to Learn: h:x Y X features

More information

Machine Learning and Pattern Recognition Logistic Regression

Machine Learning and Pattern Recognition Logistic Regression Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,

More information

Lecture 8 February 4

Lecture 8 February 4 ICS273A: Machine Learning Winter 2008 Lecture 8 February 4 Scribe: Carlos Agell (Student) Lecturer: Deva Ramanan 8.1 Neural Nets 8.1.1 Logistic Regression Recall the logistic function: g(x) = 1 1 + e θt

More information

Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation

Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr CS 551, Fall 2015 CS 551, Fall 2015

More information

CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York

CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal - the stuff biology is not

More information

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not. Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

More information

3F3: Signal and Pattern Processing

3F3: Signal and Pattern Processing 3F3: Signal and Pattern Processing Lecture 3: Classification Zoubin Ghahramani zoubin@eng.cam.ac.uk Department of Engineering University of Cambridge Lent Term Classification We will represent data by

More information

CSC 411: Lecture 07: Multiclass Classification

CSC 411: Lecture 07: Multiclass Classification CSC 411: Lecture 07: Multiclass Classification Class based on Raquel Urtasun & Rich Zemel s lectures Sanja Fidler University of Toronto Feb 1, 2016 Urtasun, Zemel, Fidler (UofT) CSC 411: 07-Multiclass

More information

MVA ENS Cachan. Lecture 2: Logistic regression & intro to MIL Iasonas Kokkinos Iasonas.kokkinos@ecp.fr

MVA ENS Cachan. Lecture 2: Logistic regression & intro to MIL Iasonas Kokkinos Iasonas.kokkinos@ecp.fr Machine Learning for Computer Vision 1 MVA ENS Cachan Lecture 2: Logistic regression & intro to MIL Iasonas Kokkinos Iasonas.kokkinos@ecp.fr Department of Applied Mathematics Ecole Centrale Paris Galen

More information

Neural Networks. CAP5610 Machine Learning Instructor: Guo-Jun Qi

Neural Networks. CAP5610 Machine Learning Instructor: Guo-Jun Qi Neural Networks CAP5610 Machine Learning Instructor: Guo-Jun Qi Recap: linear classifier Logistic regression Maximizing the posterior distribution of class Y conditional on the input vector X Support vector

More information

Introduction to Machine Learning

Introduction to Machine Learning Introduction to Machine Learning Prof. Alexander Ihler Prof. Max Welling icamp Tutorial July 22 What is machine learning? The ability of a machine to improve its performance based on previous results:

More information

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as LOGISTIC REGRESSION Nitin R Patel Logistic regression extends the ideas of multiple linear regression to the situation where the dependent variable, y, is binary (for convenience we often code these values

More information

Introduction to Online Learning Theory

Introduction to Online Learning Theory Introduction to Online Learning Theory Wojciech Kot lowski Institute of Computing Science, Poznań University of Technology IDSS, 04.06.2013 1 / 53 Outline 1 Example: Online (Stochastic) Gradient Descent

More information

11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial Least Squares Regression

11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial Least Squares Regression Frank C Porter and Ilya Narsky: Statistical Analysis Techniques in Particle Physics Chap. c11 2013/9/9 page 221 le-tex 221 11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial

More information

CS229 Lecture notes. Andrew Ng

CS229 Lecture notes. Andrew Ng CS229 Lecture notes Andrew Ng Supervised learning Let s start by talking about a few examples of supervised learning problems Suppose we have a dataset giving the living areas and prices of 47 houses from

More information

Linear smoother. ŷ = S y. where s ij = s ij (x) e.g. s ij = diag(l i (x)) To go the other way, you need to diagonalize S

Linear smoother. ŷ = S y. where s ij = s ij (x) e.g. s ij = diag(l i (x)) To go the other way, you need to diagonalize S Linear smoother ŷ = S y where s ij = s ij (x) e.g. s ij = diag(l i (x)) To go the other way, you need to diagonalize S 2 Online Learning: LMS and Perceptrons Partially adapted from slides by Ryan Gabbard

More information

Predict Influencers in the Social Network

Predict Influencers in the Social Network Predict Influencers in the Social Network Ruishan Liu, Yang Zhao and Liuyu Zhou Email: rliu2, yzhao2, lyzhou@stanford.edu Department of Electrical Engineering, Stanford University Abstract Given two persons

More information

University of Cambridge Engineering Part IIB Module 4F10: Statistical Pattern Processing Handout 8: Multi-Layer Perceptrons

University of Cambridge Engineering Part IIB Module 4F10: Statistical Pattern Processing Handout 8: Multi-Layer Perceptrons University of Cambridge Engineering Part IIB Module 4F0: Statistical Pattern Processing Handout 8: Multi-Layer Perceptrons x y (x) Inputs x 2 y (x) 2 Outputs x d First layer Second Output layer layer y

More information

Logistic Regression (1/24/13)

Logistic Regression (1/24/13) STA63/CBB540: Statistical methods in computational biology Logistic Regression (/24/3) Lecturer: Barbara Engelhardt Scribe: Dinesh Manandhar Introduction Logistic regression is model for regression used

More information

Machine Learning Logistic Regression

Machine Learning Logistic Regression Machine Learning Logistic Regression Jeff Howbert Introduction to Machine Learning Winter 2012 1 Logistic regression Name is somewhat misleading. Really a technique for classification, not regression.

More information

An Introduction to Machine Learning

An Introduction to Machine Learning An Introduction to Machine Learning L5: Novelty Detection and Regression Alexander J. Smola Statistical Machine Learning Program Canberra, ACT 0200 Australia Alex.Smola@nicta.com.au Tata Institute, Pune,

More information

Logistic Regression for Spam Filtering

Logistic Regression for Spam Filtering Logistic Regression for Spam Filtering Nikhila Arkalgud February 14, 28 Abstract The goal of the spam filtering problem is to identify an email as a spam or not spam. One of the classic techniques used

More information

Classification. Chapter 3

Classification. Chapter 3 Chapter 3 Classification In chapter we have considered regression problems, where the targets are real valued. Another important class of problems is classification problems, where we wish to assign an

More information

Lecture 6: Logistic Regression

Lecture 6: Logistic Regression Lecture 6: CS 194-10, Fall 2011 Laurent El Ghaoui EECS Department UC Berkeley September 13, 2011 Outline Outline Classification task Data : X = [x 1,..., x m]: a n m matrix of data points in R n. y { 1,

More information

Efficient Streaming Classification Methods

Efficient Streaming Classification Methods 1/44 Efficient Streaming Classification Methods Niall M. Adams 1, Nicos G. Pavlidis 2, Christoforos Anagnostopoulos 3, Dimitris K. Tasoulis 1 1 Department of Mathematics 2 Institute for Mathematical Sciences

More information

Classification using Logistic Regression

Classification using Logistic Regression Classification using Logistic Regression Ingmar Schuster Patrick Jähnichen using slides by Andrew Ng Institut für Informatik This lecture covers Logistic regression hypothesis Decision Boundary Cost function

More information

Lecture 2: The SVM classifier

Lecture 2: The SVM classifier Lecture 2: The SVM classifier C19 Machine Learning Hilary 2015 A. Zisserman Review of linear classifiers Linear separability Perceptron Support Vector Machine (SVM) classifier Wide margin Cost function

More information

Introduction to Logistic Regression

Introduction to Logistic Regression OpenStax-CNX module: m42090 1 Introduction to Logistic Regression Dan Calderon This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract Gives introduction

More information

Principled Hybrids of Generative and Discriminative Models

Principled Hybrids of Generative and Discriminative Models Principled Hybrids of Generative and Discriminative Models Julia A. Lasserre University of Cambridge Cambridge, UK jal62@cam.ac.uk Christopher M. Bishop Microsoft Research Cambridge, UK cmbishop@microsoft.com

More information

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed

More information

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014

More information

Parametric Statistical Modeling

Parametric Statistical Modeling Parametric Statistical Modeling ECE 275A Statistical Parameter Estimation Ken Kreutz-Delgado ECE Department, UC San Diego Ken Kreutz-Delgado (UC San Diego) ECE 275A SPE Version 1.1 Fall 2012 1 / 12 Why

More information

Probabilistic Models for Big Data. Alex Davies and Roger Frigola University of Cambridge 13th February 2014

Probabilistic Models for Big Data. Alex Davies and Roger Frigola University of Cambridge 13th February 2014 Probabilistic Models for Big Data Alex Davies and Roger Frigola University of Cambridge 13th February 2014 The State of Big Data Why probabilistic models for Big Data? 1. If you don t have to worry about

More information

MACHINE LEARNING IN HIGH ENERGY PHYSICS

MACHINE LEARNING IN HIGH ENERGY PHYSICS MACHINE LEARNING IN HIGH ENERGY PHYSICS LECTURE #1 Alex Rogozhnikov, 2015 INTRO NOTES 4 days two lectures, two practice seminars every day this is introductory track to machine learning kaggle competition!

More information

Regression III: Advanced Methods

Regression III: Advanced Methods Lecture 4: Transformations Regression III: Advanced Methods William G. Jacoby Michigan State University Goals of the lecture The Ladder of Roots and Powers Changing the shape of distributions Transforming

More information

Lecture 6. Artificial Neural Networks

Lecture 6. Artificial Neural Networks Lecture 6 Artificial Neural Networks 1 1 Artificial Neural Networks In this note we provide an overview of the key concepts that have led to the emergence of Artificial Neural Networks as a major paradigm

More information

Gaussian Classifiers CS498

Gaussian Classifiers CS498 Gaussian Classifiers CS498 Today s lecture The Gaussian Gaussian classifiers A slightly more sophisticated classifier Nearest Neighbors We can classify with nearest neighbors x m 1 m 2 Decision boundary

More information

Natural Language Processing. Today. Logistic Regression Models. Lecture 13 10/6/2015. Jim Martin. Multinomial Logistic Regression

Natural Language Processing. Today. Logistic Regression Models. Lecture 13 10/6/2015. Jim Martin. Multinomial Logistic Regression Natural Language Processing Lecture 13 10/6/2015 Jim Martin Today Multinomial Logistic Regression Aka log-linear models or maximum entropy (maxent) Components of the model Learning the parameters 10/1/15

More information

Classification by Pairwise Coupling

Classification by Pairwise Coupling Classification by Pairwise Coupling TREVOR HASTIE * Stanford University and ROBERT TIBSHIRANI t University of Toronto Abstract We discuss a strategy for polychotomous classification that involves estimating

More information

Bayesian Machine Learning (ML): Modeling And Inference in Big Data. Zhuhua Cai Google, Rice University caizhua@gmail.com

Bayesian Machine Learning (ML): Modeling And Inference in Big Data. Zhuhua Cai Google, Rice University caizhua@gmail.com Bayesian Machine Learning (ML): Modeling And Inference in Big Data Zhuhua Cai Google Rice University caizhua@gmail.com 1 Syllabus Bayesian ML Concepts (Today) Bayesian ML on MapReduce (Next morning) Bayesian

More information

Azure Machine Learning, SQL Data Mining and R

Azure Machine Learning, SQL Data Mining and R Azure Machine Learning, SQL Data Mining and R Day-by-day Agenda Prerequisites No formal prerequisites. Basic knowledge of SQL Server Data Tools, Excel and any analytical experience helps. Best of all:

More information

Likelihood, MLE & EM for Gaussian Mixture Clustering. Nick Duffield Texas A&M University

Likelihood, MLE & EM for Gaussian Mixture Clustering. Nick Duffield Texas A&M University Likelihood, MLE & EM for Gaussian Mixture Clustering Nick Duffield Texas A&M University Probability vs. Likelihood Probability: predict unknown outcomes based on known parameters: P(x θ) Likelihood: eskmate

More information

Loss Functions for Preference Levels: Regression with Discrete Ordered Labels

Loss Functions for Preference Levels: Regression with Discrete Ordered Labels Loss Functions for Preference Levels: Regression with Discrete Ordered Labels Jason D. M. Rennie Massachusetts Institute of Technology Comp. Sci. and Artificial Intelligence Laboratory Cambridge, MA 9,

More information

Designing a learning system

Designing a learning system Lecture Designing a learning system Milos Hauskrecht milos@cs.pitt.edu 539 Sennott Square, x4-8845 http://.cs.pitt.edu/~milos/courses/cs750/ Design of a learning system (first vie) Application or Testing

More information

Scalable Machine Learning - or what to do with all that Big Data infrastructure

Scalable Machine Learning - or what to do with all that Big Data infrastructure - or what to do with all that Big Data infrastructure TU Berlin blog.mikiobraun.de Strata+Hadoop World London, 2015 1 Complex Data Analysis at Scale Click-through prediction Personalized Spam Detection

More information

Statistical Machine Learning from Data

Statistical Machine Learning from Data Samy Bengio Statistical Machine Learning from Data 1 Statistical Machine Learning from Data Gaussian Mixture Models Samy Bengio IDIAP Research Institute, Martigny, Switzerland, and Ecole Polytechnique

More information

Sampling via Moment Sharing: A New Framework for Distributed Bayesian Inference for Big Data

Sampling via Moment Sharing: A New Framework for Distributed Bayesian Inference for Big Data Sampling via Moment Sharing: A New Framework for Distributed Bayesian Inference for Big Data (Oxford) in collaboration with: Minjie Xu, Jun Zhu, Bo Zhang (Tsinghua) Balaji Lakshminarayanan (Gatsby) Bayesian

More information

Lecture 9: Introduction to Pattern Analysis

Lecture 9: Introduction to Pattern Analysis Lecture 9: Introduction to Pattern Analysis g Features, patterns and classifiers g Components of a PR system g An example g Probability definitions g Bayes Theorem g Gaussian densities Features, patterns

More information

Statistical Models in Data Mining

Statistical Models in Data Mining Statistical Models in Data Mining Sargur N. Srihari University at Buffalo The State University of New York Department of Computer Science and Engineering Department of Biostatistics 1 Srihari Flood of

More information

CSC 321 H1S Study Guide (Last update: April 3, 2016) Winter 2016

CSC 321 H1S Study Guide (Last update: April 3, 2016) Winter 2016 1. Suppose our training set and test set are the same. Why would this be a problem? 2. Why is it necessary to have both a test set and a validation set? 3. Images are generally represented as n m 3 arrays,

More information

Revenue Management with Correlated Demand Forecasting

Revenue Management with Correlated Demand Forecasting Revenue Management with Correlated Demand Forecasting Catalina Stefanescu Victor DeMiguel Kristin Fridgeirsdottir Stefanos Zenios 1 Introduction Many airlines are struggling to survive in today's economy.

More information

Generalized Linear Models. Today: definition of GLM, maximum likelihood estimation. Involves choice of a link function (systematic component)

Generalized Linear Models. Today: definition of GLM, maximum likelihood estimation. Involves choice of a link function (systematic component) Generalized Linear Models Last time: definition of exponential family, derivation of mean and variance (memorize) Today: definition of GLM, maximum likelihood estimation Include predictors x i through

More information

Probabilistic Discriminative Kernel Classifiers for Multi-class Problems

Probabilistic Discriminative Kernel Classifiers for Multi-class Problems c Springer-Verlag Probabilistic Discriminative Kernel Classifiers for Multi-class Problems Volker Roth University of Bonn Department of Computer Science III Roemerstr. 164 D-53117 Bonn Germany roth@cs.uni-bonn.de

More information

Machine Learning. CUNY Graduate Center, Spring 2013. Professor Liang Huang. huang@cs.qc.cuny.edu

Machine Learning. CUNY Graduate Center, Spring 2013. Professor Liang Huang. huang@cs.qc.cuny.edu Machine Learning CUNY Graduate Center, Spring 2013 Professor Liang Huang huang@cs.qc.cuny.edu http://acl.cs.qc.edu/~lhuang/teaching/machine-learning Logistics Lectures M 9:30-11:30 am Room 4419 Personnel

More information

The Probit Link Function in Generalized Linear Models for Data Mining Applications

The Probit Link Function in Generalized Linear Models for Data Mining Applications Journal of Modern Applied Statistical Methods Copyright 2013 JMASM, Inc. May 2013, Vol. 12, No. 1, 164-169 1538 9472/13/$95.00 The Probit Link Function in Generalized Linear Models for Data Mining Applications

More information

STATISTICA Formula Guide: Logistic Regression. Table of Contents

STATISTICA Formula Guide: Logistic Regression. Table of Contents : Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

More information

1. χ 2 minimization 2. Fits in case of of systematic errors

1. χ 2 minimization 2. Fits in case of of systematic errors Data fitting Volker Blobel University of Hamburg March 2005 1. χ 2 minimization 2. Fits in case of of systematic errors Keys during display: enter = next page; = next page; = previous page; home = first

More information

Introduction to Machine Learning. Speaker: Harry Chao Advisor: J.J. Ding Date: 1/27/2011

Introduction to Machine Learning. Speaker: Harry Chao Advisor: J.J. Ding Date: 1/27/2011 Introduction to Machine Learning Speaker: Harry Chao Advisor: J.J. Ding Date: 1/27/2011 1 Outline 1. What is machine learning? 2. The basic of machine learning 3. Principles and effects of machine learning

More information

large-scale machine learning revisited Léon Bottou Microsoft Research (NYC)

large-scale machine learning revisited Léon Bottou Microsoft Research (NYC) large-scale machine learning revisited Léon Bottou Microsoft Research (NYC) 1 three frequent ideas in machine learning. independent and identically distributed data This experimental paradigm has driven

More information

THE GOAL of this work is to learn discriminative components

THE GOAL of this work is to learn discriminative components 68 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 16, NO. 1, JANUARY 2005 Discriminative Components of Data Jaakko Peltonen and Samuel Kaski, Senior Member, IEEE Abstract A simple probabilistic model is introduced

More information

Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression

Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction

More information

Basics of Statistical Machine Learning

Basics of Statistical Machine Learning CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar

More information

A Log-Robust Optimization Approach to Portfolio Management

A Log-Robust Optimization Approach to Portfolio Management A Log-Robust Optimization Approach to Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Ban Kawas Research partially supported by the National Science Foundation Grant CMMI-0757983

More information

The Optimality of Naive Bayes

The Optimality of Naive Bayes The Optimality of Naive Bayes Harry Zhang Faculty of Computer Science University of New Brunswick Fredericton, New Brunswick, Canada email: hzhang@unbca E3B 5A3 Abstract Naive Bayes is one of the most

More information

Wes, Delaram, and Emily MA751. Exercise 4.5. 1 p(x; β) = [1 p(xi ; β)] = 1 p(x. y i [βx i ] log [1 + exp {βx i }].

Wes, Delaram, and Emily MA751. Exercise 4.5. 1 p(x; β) = [1 p(xi ; β)] = 1 p(x. y i [βx i ] log [1 + exp {βx i }]. Wes, Delaram, and Emily MA75 Exercise 4.5 Consider a two-class logistic regression problem with x R. Characterize the maximum-likelihood estimates of the slope and intercept parameter if the sample for

More information

A Game Theoretical Framework for Adversarial Learning

A Game Theoretical Framework for Adversarial Learning A Game Theoretical Framework for Adversarial Learning Murat Kantarcioglu University of Texas at Dallas Richardson, TX 75083, USA muratk@utdallas Chris Clifton Purdue University West Lafayette, IN 47907,

More information

Weighted Pseudo-Metric Discriminatory Power Improvement Using a Bayesian Logistic Regression Model Based on a Variational Method

Weighted Pseudo-Metric Discriminatory Power Improvement Using a Bayesian Logistic Regression Model Based on a Variational Method Weighted Pseudo-Metric Discriminatory Power Improvement Using a Bayesian Logistic Regression Model Based on a Variational Method R. Ksantini 1, D. Ziou 1, B. Colin 2, and F. Dubeau 2 1 Département d Informatique

More information

PMR5406 Redes Neurais e Lógica Fuzzy Aula 3 Multilayer Percetrons

PMR5406 Redes Neurais e Lógica Fuzzy Aula 3 Multilayer Percetrons PMR5406 Redes Neurais e Aula 3 Multilayer Percetrons Baseado em: Neural Networks, Simon Haykin, Prentice-Hall, 2 nd edition Slides do curso por Elena Marchiori, Vrie Unviersity Multilayer Perceptrons Architecture

More information

Practical Data Science with Azure Machine Learning, SQL Data Mining, and R

Practical Data Science with Azure Machine Learning, SQL Data Mining, and R Practical Data Science with Azure Machine Learning, SQL Data Mining, and R Overview This 4-day class is the first of the two data science courses taught by Rafal Lukawiecki. Some of the topics will be

More information

Statistical Data Mining. Practical Assignment 3 Discriminant Analysis and Decision Trees

Statistical Data Mining. Practical Assignment 3 Discriminant Analysis and Decision Trees Statistical Data Mining Practical Assignment 3 Discriminant Analysis and Decision Trees In this practical we discuss linear and quadratic discriminant analysis and tree-based classification techniques.

More information

Response variables assume only two values, say Y j = 1 or = 0, called success and failure (spam detection, credit scoring, contracting.

Response variables assume only two values, say Y j = 1 or = 0, called success and failure (spam detection, credit scoring, contracting. Prof. Dr. J. Franke All of Statistics 1.52 Binary response variables - logistic regression Response variables assume only two values, say Y j = 1 or = 0, called success and failure (spam detection, credit

More information

i=1 In practice, the natural logarithm of the likelihood function, called the log-likelihood function and denoted by

i=1 In practice, the natural logarithm of the likelihood function, called the log-likelihood function and denoted by Statistics 580 Maximum Likelihood Estimation Introduction Let y (y 1, y 2,..., y n be a vector of iid, random variables from one of a family of distributions on R n and indexed by a p-dimensional parameter

More information

An Introduction to Statistical Machine Learning - Overview -

An Introduction to Statistical Machine Learning - Overview - An Introduction to Statistical Machine Learning - Overview - Samy Bengio bengio@idiap.ch Dalle Molle Institute for Perceptual Artificial Intelligence (IDIAP) CP 592, rue du Simplon 4 1920 Martigny, Switzerland

More information

Penalized Logistic Regression and Classification of Microarray Data

Penalized Logistic Regression and Classification of Microarray Data Penalized Logistic Regression and Classification of Microarray Data Milan, May 2003 Anestis Antoniadis Laboratoire IMAG-LMC University Joseph Fourier Grenoble, France Penalized Logistic Regression andclassification

More information

L4: Bayesian Decision Theory

L4: Bayesian Decision Theory L4: Bayesian Decision Theory Likelihood ratio test Probability of error Bayes risk Bayes, MAP and ML criteria Multi-class problems Discriminant functions CSCE 666 Pattern Analysis Ricardo Gutierrez-Osuna

More information

Pattern Recognition and Prediction in Equity Market

Pattern Recognition and Prediction in Equity Market Pattern Recognition and Prediction in Equity Market Lang Lang, Kai Wang 1. Introduction In finance, technical analysis is a security analysis discipline used for forecasting the direction of prices through

More information

171:290 Model Selection Lecture II: The Akaike Information Criterion

171:290 Model Selection Lecture II: The Akaike Information Criterion 171:290 Model Selection Lecture II: The Akaike Information Criterion Department of Biostatistics Department of Statistics and Actuarial Science August 28, 2012 Introduction AIC, the Akaike Information

More information

Chapter 4: Artificial Neural Networks

Chapter 4: Artificial Neural Networks Chapter 4: Artificial Neural Networks CS 536: Machine Learning Littman (Wu, TA) Administration icml-03: instructional Conference on Machine Learning http://www.cs.rutgers.edu/~mlittman/courses/ml03/icml03/

More information

Gaussian Conjugate Prior Cheat Sheet

Gaussian Conjugate Prior Cheat Sheet Gaussian Conjugate Prior Cheat Sheet Tom SF Haines 1 Purpose This document contains notes on how to handle the multivariate Gaussian 1 in a Bayesian setting. It focuses on the conjugate prior, its Bayesian

More information

Environmental Remote Sensing GEOG 2021

Environmental Remote Sensing GEOG 2021 Environmental Remote Sensing GEOG 2021 Lecture 4 Image classification 2 Purpose categorising data data abstraction / simplification data interpretation mapping for land cover mapping use land cover class

More information

IFT3395/6390. Machine Learning from linear regression to Neural Networks. Machine Learning. Training Set. t (3.5, -2,..., 127, 0,...

IFT3395/6390. Machine Learning from linear regression to Neural Networks. Machine Learning. Training Set. t (3.5, -2,..., 127, 0,... IFT3395/6390 Historical perspective: back to 1957 (Prof. Pascal Vincent) (Rosenblatt, Perceptron ) Machine Learning from linear regression to Neural Networks Computer Science Artificial Intelligence Symbolic

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

More information

Supervised Learning (Big Data Analytics)

Supervised Learning (Big Data Analytics) Supervised Learning (Big Data Analytics) Vibhav Gogate Department of Computer Science The University of Texas at Dallas Practical advice Goal of Big Data Analytics Uncover patterns in Data. Can be used

More information

Master s Theory Exam Spring 2006

Master s Theory Exam Spring 2006 Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem

More information