Direct Learning: Linear Classification. Donglin Zeng, Department of Biostatistics, University of North Carolina

Size: px
Start display at page:

Download "Direct Learning: Linear Classification. Donglin Zeng, Department of Biostatistics, University of North Carolina"

Transcription

1 Direct Learning: Linear Classification

2 Logistic regression models for classification problem We consider two class problem: Y {0, 1}. The Bayes rule for the classification is I(P(Y = 1 X = x) > 1/2) so it is natural to estimate P(Y = 1 X = x) using training data. Standard statistical model is logistic regression model assuming P(Y = 1 X) = exp{β 0 + X T β} 1 + exp{β 0 + X T β} then the resulting prediction rule is I(β 0 + x T β > 0). Shrinkage can also be introduced into this regression, for example, Glasso min n i=1 { Y i (β 0 + X T i β) + log(1 + exp{β 0 + X T i β}) } +λ β L1.

3 Perceptron model for classification A more direct method (nothing to do with statistics) is to directly search a hyperplane separating two class data (perceptron model). This algorithm aims to find a separating hyperplane by minimizing the distance of misclassified points to the decision boundary: i M(2Y i 1)(β 0 + X T i β). Stochastic gradient decent is used to find the solution ( ) ( ) ( ) β β (2Yi 1)X + ρ i, 2Y i 1 β 0 β 0 where ρ is the learning and i is taken from M one by one. The algorithm may not have unique solutions and may cycle solutions when data are not separable. A more reliable algorithm is warranted.

4 Discriminant analysis The idea behind discriminant analysis is to compare the distributions of feature variables for each class and identify the best rule to discriminate these distributions. Let f k (x) be the density of X in Y = k and π k is the prevalence of Y = k, k {0, 1}. Clearly, P(Y = 1 X = x) = f 1 (x)π 1 f 1 (x)π 1 + f 0 (x)π 0. Thus, the Bayes rule is { I log f 1 (x) log f 0 (x) + log π } 1 > 0. π 0

5 Discriminant analysis under multivariate normality Assume f k (x) N(µ k, Σ) for k {0, 1} (homogeneous variance). The prediction rule becomes I(x T Σ 1 (µ 1 µ 0 ) 1 2 µt 1 Σ 1 µ µt 0 Σ 1 µ 0 + log π 1 π 0 > 0). The decision boundary is a linear function of X so this analysis is called linear discriminant analysis. Assume f k (x) N(µ k, Σ k ). Then prediction rule becomes ( I 1 2 xt Σ 1 1 x xt Σ 1 0 x + xt Σ 1 1 µ 1 x T Σ 1 0 µ 0) 1 2 µt 1 Σ 1 µ µt 0 Σ 1 µ 0 + log π ) 1 > 0. π 0 The decision boundary is a quadractic function of X so this analysis is called quadractic discriminant analysis.

6 Example of LDA

7 Example of QDA

8 Direct Learning: Nonlinear Prediction

9 Nonlinear prediction rules Nonlinear prediction rules can be obtained if we include high-order interactions in addition to X, or replace X with some basis functions in feature space. Commonly used and flexible basis functions includes splines and wavelets. Splines are piecewise polynomials and can consist of B-splines and natural cubic splines etc. For splines, we need to define a sequence of knots, degrees of polynomials and smoothness at knots. Splines can also be represented by polynomials and truncated polynomials, where truncations occur at interior knots: x k, (x ξ 1 ) k +, (x ξ 2 ) k +,...

10 Recursive algorithm for B-spline construction

11 R code of constructing B-splines

12 South African heart disease data

13 Smooth splines They are also piecewise polynomials. However, instead of specifying knots in advance, smoothing spline approximation is obtained by minimizing the following penalized least squares: n (Y i f (X i )) 2 + λ {f (t)} 2 dt. i=1 Note that λ regularizes the smoothness of f (t). The solution is a linear combination of natural cubic splines whose knots are placed on X 1,..., X n. Tuning is performed via Cp, CV or GCV: GCV = 1 n ( n i=1 Y i f λ (X i ) 1 S λ (i, i) ) 2, where S λ is some projection matrix on the space spanned by the cubic splines.

14 Approximation in Reproducing Kernel Hilbert Space (RKHS) Smoothing splines can be treated as a special case of the approximation in RKHS. For a RKHS, denoted by H K, a kernel function K(x, y) (x, y R p ) is essential (satisfying symmetry and positivity conditions). Then the RKHS consists of any linear combinations of the form { } H k = f (x) = m α m K(x, y m ) Key result: there exists a sequence of basis functions, φ j (x), such that K(x, y) = j=1 γ jφ j (x)φ j (y) (eigen-expansion of K(x, y)) for γ j 0, j γ2 j <..

15 RKHS: continue For f H K, f (x) = j c jφ j (x) and f (x) = j c jφ j (x). We define an inner product as < f, f > HK = j c j c j /γ j. Important property of RKHS: < K(x, ), K(y, ) > HK = K(x, y) (reproducing)

16 Practical usefulness of RKHS Regularization problem n min L(Y i, f (X i )) + λ f 2 H K i=1 has a solution with form f (x) = n i=1 α ik(x, X i ). Thus, we need to solve min α n L(Y i, j i=1 K(X i, X j )α j ) + λα T Kα, where K is matrix of (K(X i, X j )). Advantages are (1) no need of knowledge about basis functions; (2) the optimization is independent of the dimensionality of X. How large is RKHS: for a Gaussian kernel, K(x, y) = exp{ x y /σ 2 }, the resulting RKHS can approximate any function if σ is close to 0 enough.

17 Wavelet approximation Wavelets can be used to approximate possibly irregular function/surfaces; to localize and identify such accumulation of wavelets (small waves) through both frequency and spatial resolution. They are extensively used in data compression, turbulence analysis, image & signal processing and statistical estimation. Comparatively, Fourier spectrum analysis is a global frequency decomposition approach. Alternative approximation such as blocked Fourier analysis, Splines, local polynomial approximation etc. also have local adaptivity; however, wavelet analysis is more elegant and mathematically consistent.

18 Wavelet construction We start with some father wavelet φ(x) = I(x (0, 1]). Example: Haar system: Consider V 0 = {f L 2 (R) : f = k c ki (k,k+1] (x)} then {φ 0k = φ(x k)} is ONB of V 0. Let V 1 = {f (x) : f (x) = h(2x) for some h V 0 } then {φ 1k = 2φ 0k (2x) = 2φ(2x k)} is ONB of V 1. Generally, let V j = {f (x) : f (x) = h(2 j x) for some h V 0 } then {φ jk = 2 j/2 φ(2 j x k)} is ONB of V j for any j....v 1 V 0 V 1...; j=0 V j is dense in L 2 (R); V j+1 = V j W j, where {ψ jk = 2 j/2 ψ(2 j x k)} is ONB of W j with ψ(x) = I [0,1/2] (x) + I [1/2,1] (x) (mother wavelet).

19 Multiresolution analysis based on wavelets First, Second, L 2 (R) = V j W j W j+1 W j+2. V 0 = W 1 W 2 W j V j. In other words, any f (x) L 2 (R) can be recovered using lower resolution projection (V j ) and additional details from higher resolutions (W j,, W 1 ). This is called the multiresolution analysis of f. Other choices of father wavelets: Daubechies wavelets, Coiflet wavelets, Symmlets wavelets (smoother than Haart wavelets).

20 Wavelet 1D demo 0.02 Analyzed signal Discrete Transform, absolute coefficients. 5 4 level Absolute Values of Ca,b Coefficients for a = scales a time (or space) b

21 Wavelet 2D demo X Reconstructed image a0. a1 h1 v1 d1 a2 h2 v2 d2

Indirect Rule Learning: Support Vector Machines. Donglin Zeng, Department of Biostatistics, University of North Carolina

Indirect Rule Learning: Support Vector Machines. Donglin Zeng, Department of Biostatistics, University of North Carolina Indirect Rule Learning: Support Vector Machines Indirect learning: loss optimization It doesn t estimate the prediction rule f (x) directly, since most loss functions do not have explicit optimizers. Indirection

More information

Data Mining Stat 588

Data Mining Stat 588 Data Mining Stat 588 Lecture 9: Basis Expansions Department of Statistics & Biostatistics Rutgers University Nov 01, 2011 Regression and Classification Linear Regression. E(Y X) = f(x) We want to learn

More information

Introduction to machine learning and pattern recognition Lecture 2 Coryn Bailer-Jones

Introduction to machine learning and pattern recognition Lecture 2 Coryn Bailer-Jones Introduction to machine learning and pattern recognition Lecture 2 Coryn Bailer-Jones http://www.mpia.de/homes/calj/mlpr_mpia2008.html 1 1 Last week... supervised and unsupervised methods need adaptive

More information

Support Vector Machines

Support Vector Machines Wien, June, 2010 Paul Hofmarcher, Stefan Theussl, WU Wien Hofmarcher/Theussl SVM 1/21 Linear Separable Separating Hyperplanes Non-Linear Separable Soft-Margin Hyperplanes Hofmarcher/Theussl SVM 2/21 (SVM)

More information

A Magiv CV Theory for Large-Margin Classifiers

A Magiv CV Theory for Large-Margin Classifiers A Magiv CV Theory for Large-Margin Classifiers Hui Zou School of Statistics, University of Minnesota June 30, 2018 Joint work with Boxiang Wang Outline 1 Background 2 Magic CV formula 3 Magic support vector

More information

Chap 2. Linear Classifiers (FTH, ) Yongdai Kim Seoul National University

Chap 2. Linear Classifiers (FTH, ) Yongdai Kim Seoul National University Chap 2. Linear Classifiers (FTH, 4.1-4.4) Yongdai Kim Seoul National University Linear methods for classification 1. Linear classifiers For simplicity, we only consider two-class classification problems

More information

Support Vector Machines for Classification: A Statistical Portrait

Support Vector Machines for Classification: A Statistical Portrait Support Vector Machines for Classification: A Statistical Portrait Yoonkyung Lee Department of Statistics The Ohio State University May 27, 2011 The Spring Conference of Korean Statistical Society KAIST,

More information

Machine Learning. Regression-Based Classification & Gaussian Discriminant Analysis. Manfred Huber

Machine Learning. Regression-Based Classification & Gaussian Discriminant Analysis. Manfred Huber Machine Learning Regression-Based Classification & Gaussian Discriminant Analysis Manfred Huber 2015 1 Logistic Regression Linear regression provides a nice representation and an efficient solution to

More information

Chapter 9. Support Vector Machine. Yongdai Kim Seoul National University

Chapter 9. Support Vector Machine. Yongdai Kim Seoul National University Chapter 9. Support Vector Machine Yongdai Kim Seoul National University 1. Introduction Support Vector Machine (SVM) is a classification method developed by Vapnik (1996). It is thought that SVM improved

More information

Fractal functional regression for classification of gene expression data by wavelets

Fractal functional regression for classification of gene expression data by wavelets Fractal functional regression for classification of gene expression data by wavelets Margarita María Rincón 1 and María Dolores Ruiz-Medina 2 1 University of Granada Campus Fuente Nueva 18071 Granada,

More information

Linear Regression and Discrimination

Linear Regression and Discrimination Linear Regression and Discrimination Kernel-based Learning Methods Christian Igel Institut für Neuroinformatik Ruhr-Universität Bochum, Germany http://www.neuroinformatik.rub.de July 16, 2009 Christian

More information

Statistical Methods for SVM

Statistical Methods for SVM Statistical Methods for SVM Support Vector Machines Here we approach the two-class classification problem in a direct way: We try and find a plane that separates the classes in feature space. If we cannot,

More information

MLISP: Machine Learning in Signal Processing Spring Lecture 10 May 11

MLISP: Machine Learning in Signal Processing Spring Lecture 10 May 11 MLISP: Machine Learning in Signal Processing Spring 2018 Lecture 10 May 11 Prof. Venia Morgenshtern Scribe: Mohamed Elshawi Illustrations: The elements of statistical learning, Hastie, Tibshirani, Friedman

More information

CIS 520: Machine Learning Oct 09, Kernel Methods

CIS 520: Machine Learning Oct 09, Kernel Methods CIS 520: Machine Learning Oct 09, 207 Kernel Methods Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture They may or may not cover all the material discussed

More information

Linear Models for Regression

Linear Models for Regression Linear Models for Regression Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr

More information

Computing regularization paths for learning multiple kernels

Computing regularization paths for learning multiple kernels Computing regularization paths for learning multiple kernels Francis Bach Romain Thibaux Michael Jordan Computer Science, UC Berkeley December, 24 Code available at www.cs.berkeley.edu/~fbach Computing

More information

Theoretical Exercises Statistical Learning, 2009

Theoretical Exercises Statistical Learning, 2009 Theoretical Exercises Statistical Learning, 2009 Niels Richard Hansen April 20, 2009 The following exercises are going to play a central role in the course Statistical learning, block 4, 2009. The exercises

More information

Support Vector Machines

Support Vector Machines Support Vector Machines Here we approach the two-class classification problem in a direct way: We try and find a plane that separates the classes in feature space. If we cannot, we get creative in two

More information

5.6 Nonparametric Logistic Regression

5.6 Nonparametric Logistic Regression 5.6 onparametric Logistic Regression Dmitri Dranishnikov University of Florida Statistical Learning onparametric Logistic Regression onparametric? Doesnt mean that there are no parameters. Just means that

More information

Kernel Principal Component Analysis

Kernel Principal Component Analysis Kernel Principal Component Analysis Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr

More information

Stat542 (F11) Statistical Learning. First consider the scenario where the two classes of points are separable.

Stat542 (F11) Statistical Learning. First consider the scenario where the two classes of points are separable. Linear SVM (separable case) First consider the scenario where the two classes of points are separable. It s desirable to have the width (called margin) between the two dashed lines to be large, i.e., have

More information

Statistical Methods for Data Mining

Statistical Methods for Data Mining Statistical Methods for Data Mining Kuangnan Fang Xiamen University Email: xmufkn@xmu.edu.cn Support Vector Machines Here we approach the two-class classification problem in a direct way: We try and find

More information

Basis Expansion and Nonlinear SVM. Kai Yu

Basis Expansion and Nonlinear SVM. Kai Yu Basis Expansion and Nonlinear SVM Kai Yu Linear Classifiers f(x) =w > x + b z(x) = sign(f(x)) Help to learn more general cases, e.g., nonlinear models 8/7/12 2 Nonlinear Classifiers via Basis Expansion

More information

Function Spaces. 1 Hilbert Spaces

Function Spaces. 1 Hilbert Spaces Function Spaces A function space is a set of functions F that has some structure. Often a nonparametric regression function or classifier is chosen to lie in some function space, where the assume structure

More information

Support Vector Machine (SVM) and Kernel Methods

Support Vector Machine (SVM) and Kernel Methods Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2016 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

More information

Oslo Class 2 Tikhonov regularization and kernels

Oslo Class 2 Tikhonov regularization and kernels RegML2017@SIMULA Oslo Class 2 Tikhonov regularization and kernels Lorenzo Rosasco UNIGE-MIT-IIT May 3, 2017 Learning problem Problem For H {f f : X Y }, solve min E(f), f H dρ(x, y)l(f(x), y) given S n

More information

An Introduction to Wavelets and some Applications

An Introduction to Wavelets and some Applications An Introduction to Wavelets and some Applications Milan, May 2003 Anestis Antoniadis Laboratoire IMAG-LMC University Joseph Fourier Grenoble, France An Introduction to Wavelets and some Applications p.1/54

More information

Cheng Soon Ong & Christian Walder. Canberra February June 2018

Cheng Soon Ong & Christian Walder. Canberra February June 2018 Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 254 Part V

More information

Machine Learning Support Vector Machines. Prof. Matteo Matteucci

Machine Learning Support Vector Machines. Prof. Matteo Matteucci Machine Learning Support Vector Machines Prof. Matteo Matteucci Discriminative vs. Generative Approaches 2 o Generative approach: we derived the classifier from some generative hypothesis about the way

More information

Wavelets and Image Compression Augusta State University April, 27, Joe Lakey. Department of Mathematical Sciences. New Mexico State University

Wavelets and Image Compression Augusta State University April, 27, Joe Lakey. Department of Mathematical Sciences. New Mexico State University Wavelets and Image Compression Augusta State University April, 27, 6 Joe Lakey Department of Mathematical Sciences New Mexico State University 1 Signals and Images Goal Reduce image complexity with little

More information

Introduction to Machine Learning

Introduction to Machine Learning 1, DATA11002 Introduction to Machine Learning Lecturer: Teemu Roos TAs: Ville Hyvönen and Janne Leppä-aho Department of Computer Science University of Helsinki (based in part on material by Patrik Hoyer

More information

CS6375: Machine Learning Gautam Kunapuli. Support Vector Machines

CS6375: Machine Learning Gautam Kunapuli. Support Vector Machines Gautam Kunapuli Example: Text Categorization Example: Develop a model to classify news stories into various categories based on their content. sports politics Use the bag-of-words representation for this

More information

Lecture 3: Statistical Decision Theory (Part II)

Lecture 3: Statistical Decision Theory (Part II) Lecture 3: Statistical Decision Theory (Part II) Hao Helen Zhang Hao Helen Zhang Lecture 3: Statistical Decision Theory (Part II) 1 / 27 Outline of This Note Part I: Statistics Decision Theory (Classical

More information

Lecture 10: Support Vector Machine and Large Margin Classifier

Lecture 10: Support Vector Machine and Large Margin Classifier Lecture 10: Support Vector Machine and Large Margin Classifier Applied Multivariate Analysis Math 570, Fall 2014 Xingye Qiao Department of Mathematical Sciences Binghamton University E-mail: qiao@math.binghamton.edu

More information

Linear Decision Boundaries

Linear Decision Boundaries Linear Decision Boundaries A basic approach to classification is to find a decision boundary in the space of the predictor variables. The decision boundary is often a curve formed by a regression model:

More information

Diffeomorphic Warping. Ben Recht August 17, 2006 Joint work with Ali Rahimi (Intel)

Diffeomorphic Warping. Ben Recht August 17, 2006 Joint work with Ali Rahimi (Intel) Diffeomorphic Warping Ben Recht August 17, 2006 Joint work with Ali Rahimi (Intel) What Manifold Learning Isn t Common features of Manifold Learning Algorithms: 1-1 charting Dense sampling Geometric Assumptions

More information

A Bahadur Representation of the Linear Support Vector Machine

A Bahadur Representation of the Linear Support Vector Machine A Bahadur Representation of the Linear Support Vector Machine Yoonkyung Lee Department of Statistics The Ohio State University October 7, 2008 Data Mining and Statistical Learning Study Group Outline Support

More information

Support Vector Machine

Support Vector Machine Support Vector Machine Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Linear Support Vector Machine Kernelized SVM Kernels 2 From ERM to RLM Empirical Risk Minimization in the binary

More information

MIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications. Class 04: Features and Kernels. Lorenzo Rosasco

MIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications. Class 04: Features and Kernels. Lorenzo Rosasco MIT 9.520/6.860, Fall 2018 Statistical Learning Theory and Applications Class 04: Features and Kernels Lorenzo Rosasco Linear functions Let H lin be the space of linear functions f(x) = w x. f w is one

More information

Lecture 10: A brief introduction to Support Vector Machine

Lecture 10: A brief introduction to Support Vector Machine Lecture 10: A brief introduction to Support Vector Machine Advanced Applied Multivariate Analysis STAT 2221, Fall 2013 Sungkyu Jung Department of Statistics, University of Pittsburgh Xingye Qiao Department

More information

Chap 1. Overview of Statistical Learning (HTF, , 2.9) Yongdai Kim Seoul National University

Chap 1. Overview of Statistical Learning (HTF, , 2.9) Yongdai Kim Seoul National University Chap 1. Overview of Statistical Learning (HTF, 2.1-2.6, 2.9) Yongdai Kim Seoul National University 0. Learning vs Statistical learning Learning procedure Construct a claim by observing data or using logics

More information

CMSC858P Supervised Learning Methods

CMSC858P Supervised Learning Methods CMSC858P Supervised Learning Methods Hector Corrada Bravo March, 2010 Introduction Today we discuss the classification setting in detail. Our setting is that we observe for each subject i a set of p predictors

More information

DEPARTMENT OF COMPUTER SCIENCE Autumn Semester MACHINE LEARNING AND ADAPTIVE INTELLIGENCE

DEPARTMENT OF COMPUTER SCIENCE Autumn Semester MACHINE LEARNING AND ADAPTIVE INTELLIGENCE Data Provided: None DEPARTMENT OF COMPUTER SCIENCE Autumn Semester 203 204 MACHINE LEARNING AND ADAPTIVE INTELLIGENCE 2 hours Answer THREE of the four questions. All questions carry equal weight. Figures

More information

Support Vector Machines

Support Vector Machines Support Vector Machines Le Song Machine Learning I CSE 6740, Fall 2013 Naïve Bayes classifier Still use Bayes decision rule for classification P y x = P x y P y P x But assume p x y = 1 is fully factorized

More information

Multiscale Frame-based Kernels for Image Registration

Multiscale Frame-based Kernels for Image Registration Multiscale Frame-based Kernels for Image Registration Ming Zhen, Tan National University of Singapore 22 July, 16 Ming Zhen, Tan (National University of Singapore) Multiscale Frame-based Kernels for Image

More information

RegML 2018 Class 2 Tikhonov regularization and kernels

RegML 2018 Class 2 Tikhonov regularization and kernels RegML 2018 Class 2 Tikhonov regularization and kernels Lorenzo Rosasco UNIGE-MIT-IIT June 17, 2018 Learning problem Problem For H {f f : X Y }, solve min E(f), f H dρ(x, y)l(f(x), y) given S n = (x i,

More information

MLISP: Machine Learning in Signal Processing Spring Lecture 8-9 May 4-7

MLISP: Machine Learning in Signal Processing Spring Lecture 8-9 May 4-7 MLISP: Machine Learning in Signal Processing Spring 2018 Prof. Veniamin Morgenshtern Lecture 8-9 May 4-7 Scribe: Mohamed Solomon Agenda 1. Wavelets: beyond smoothness 2. A problem with Fourier transform

More information

Math for Machine Learning Open Doors to Data Science and Artificial Intelligence. Richard Han

Math for Machine Learning Open Doors to Data Science and Artificial Intelligence. Richard Han Math for Machine Learning Open Doors to Data Science and Artificial Intelligence Richard Han Copyright 05 Richard Han All rights reserved. CONTENTS PREFACE... - INTRODUCTION... LINEAR REGRESSION... 4 LINEAR

More information

Linear discriminant functions

Linear discriminant functions Andrea Passerini passerini@disi.unitn.it Machine Learning Discriminative learning Discriminative vs generative Generative learning assumes knowledge of the distribution governing the data Discriminative

More information

Outline. Supervised Learning. Hong Chang. Institute of Computing Technology, Chinese Academy of Sciences. Machine Learning Methods (Fall 2012)

Outline. Supervised Learning. Hong Chang. Institute of Computing Technology, Chinese Academy of Sciences. Machine Learning Methods (Fall 2012) Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Linear Models for Regression Linear Regression Probabilistic Interpretation

More information

Ch 4. Linear Models for Classification

Ch 4. Linear Models for Classification Ch 4. Linear Models for Classification Pattern Recognition and Machine Learning, C. M. Bishop, 2006. Department of Computer Science and Engineering Pohang University of Science and echnology 77 Cheongam-ro,

More information

Machine Learning (CS 567) Lecture 5

Machine Learning (CS 567) Lecture 5 Machine Learning (CS 567) Lecture 5 Time: T-Th 5:00pm - 6:20pm Location: GFS 118 Instructor: Sofus A. Macskassy (macskass@usc.edu) Office: SAL 216 Office hours: by appointment Teaching assistant: Cheol

More information

COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017

COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017 COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University FEATURE EXPANSIONS FEATURE EXPANSIONS

More information

Linear Models for Regression

Linear Models for Regression Linear Models for Regression Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr

More information

Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen. Bayesian Learning. Tobias Scheffer, Niels Landwehr

Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen. Bayesian Learning. Tobias Scheffer, Niels Landwehr Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning Tobias Scheffer, Niels Landwehr Remember: Normal Distribution Distribution over x. Density function with parameters

More information

Statistical Data Mining and Machine Learning Hilary Term 2016

Statistical Data Mining and Machine Learning Hilary Term 2016 Statistical Data Mining and Machine Learning Hilary Term 2016 Dino Sejdinovic Department of Statistics Oxford Slides and other materials available at: http://www.stats.ox.ac.uk/~sejdinov/sdmml Naïve Bayes

More information

Curve learning. p.1/35

Curve learning. p.1/35 Curve learning Gérard Biau UNIVERSITÉ MONTPELLIER II p.1/35 Summary The problem The mathematical model Functional classification 1. Fourier filtering 2. Wavelet filtering Applications p.2/35 The problem

More information

Classification objectives COMS 4771

Classification objectives COMS 4771 Classification objectives COMS 4771 1. Recap: binary classification Scoring functions Consider binary classification problems with Y = { 1, +1}. 1 / 22 Scoring functions Consider binary classification

More information

Notes on Discriminant Functions and Optimal Classification

Notes on Discriminant Functions and Optimal Classification Notes on Discriminant Functions and Optimal Classification Padhraic Smyth, Department of Computer Science University of California, Irvine c 2017 1 Discriminant Functions Consider a classification problem

More information

Review: Support vector machines. Machine learning techniques and image analysis

Review: Support vector machines. Machine learning techniques and image analysis Review: Support vector machines Review: Support vector machines Margin optimization min (w,w 0 ) 1 2 w 2 subject to y i (w 0 + w T x i ) 1 0, i = 1,..., n. Review: Support vector machines Margin optimization

More information

Machine Learning. Kernels. Fall (Kernels, Kernelized Perceptron and SVM) Professor Liang Huang. (Chap. 12 of CIML)

Machine Learning. Kernels. Fall (Kernels, Kernelized Perceptron and SVM) Professor Liang Huang. (Chap. 12 of CIML) Machine Learning Fall 2017 Kernels (Kernels, Kernelized Perceptron and SVM) Professor Liang Huang (Chap. 12 of CIML) Nonlinear Features x4: -1 x1: +1 x3: +1 x2: -1 Concatenated (combined) features XOR:

More information

Machine Learning Linear Classification. Prof. Matteo Matteucci

Machine Learning Linear Classification. Prof. Matteo Matteucci Machine Learning Linear Classification Prof. Matteo Matteucci Recall from the first lecture 2 X R p Regression Y R Continuous Output X R p Y {Ω 0, Ω 1,, Ω K } Classification Discrete Output X R p Y (X)

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 (IFT68 Winter 8) Prof. Douglas Eck, Université de Montréal These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher Bishop

More information

Approximation Theoretical Questions for SVMs

Approximation Theoretical Questions for SVMs Ingo Steinwart LA-UR 07-7056 October 20, 2007 Statistical Learning Theory: an Overview Support Vector Machines Informal Description of the Learning Goal X space of input samples Y space of labels, usually

More information

Contents Lecture 4. Lecture 4 Linear Discriminant Analysis. Summary of Lecture 3 (II/II) Summary of Lecture 3 (I/II)

Contents Lecture 4. Lecture 4 Linear Discriminant Analysis. Summary of Lecture 3 (II/II) Summary of Lecture 3 (I/II) Contents Lecture Lecture Linear Discriminant Analysis Fredrik Lindsten Division of Systems and Control Department of Information Technology Uppsala University Email: fredriklindsten@ituuse Summary of lecture

More information

BAYESIAN METHODS FOR VARIABLE SELECTION WITH APPLICATIONS TO HIGH-DIMENSIONAL DATA

BAYESIAN METHODS FOR VARIABLE SELECTION WITH APPLICATIONS TO HIGH-DIMENSIONAL DATA BAYESIAN METHODS FOR VARIABLE SELECTION WITH APPLICATIONS TO HIGH-DIMENSIONAL DATA Part 3: Functional Data & Wavelets Marina Vannucci Rice University, USA PASI-CIMAT 4/28-3/2 Marina Vannucci (Rice University,

More information

Midterm Review CS 6375: Machine Learning. Vibhav Gogate The University of Texas at Dallas

Midterm Review CS 6375: Machine Learning. Vibhav Gogate The University of Texas at Dallas Midterm Review CS 6375: Machine Learning Vibhav Gogate The University of Texas at Dallas Machine Learning Supervised Learning Unsupervised Learning Reinforcement Learning Parametric Y Continuous Non-parametric

More information

Hilbert Space Methods in Learning

Hilbert Space Methods in Learning Hilbert Space Methods in Learning guest lecturer: Risi Kondor 6772 Advanced Machine Learning and Perception (Jebara), Columbia University, October 15, 2003. 1 1. A general formulation of the learning problem

More information

Statistical Machine Learning Hilary Term 2018

Statistical Machine Learning Hilary Term 2018 Statistical Machine Learning Hilary Term 2018 Pier Francesco Palamara Department of Statistics University of Oxford Slide credits and other course material can be found at: http://www.stats.ox.ac.uk/~palamara/sml18.html

More information

9.2 Support Vector Machines 159

9.2 Support Vector Machines 159 9.2 Support Vector Machines 159 9.2.3 Kernel Methods We have all the tools together now to make an exciting step. Let us summarize our findings. We are interested in regularized estimation problems of

More information

Introduction to Machine Learning

Introduction to Machine Learning 1, DATA11002 Introduction to Machine Learning Lecturer: Antti Ukkonen TAs: Saska Dönges and Janne Leppä-aho Department of Computer Science University of Helsinki (based in part on material by Patrik Hoyer,

More information

LMS Algorithm Summary

LMS Algorithm Summary LMS Algorithm Summary Step size tradeoff Other Iterative Algorithms LMS algorithm with variable step size: w(k+1) = w(k) + µ(k)e(k)x(k) When step size µ(k) = µ/k algorithm converges almost surely to optimal

More information

EXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING

EXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING EXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING DATE AND TIME: June 9, 2018, 09.00 14.00 RESPONSIBLE TEACHER: Andreas Svensson NUMBER OF PROBLEMS: 5 AIDING MATERIAL: Calculator, mathematical

More information

Midterm Review CS 7301: Advanced Machine Learning. Vibhav Gogate The University of Texas at Dallas

Midterm Review CS 7301: Advanced Machine Learning. Vibhav Gogate The University of Texas at Dallas Midterm Review CS 7301: Advanced Machine Learning Vibhav Gogate The University of Texas at Dallas Supervised Learning Issues in supervised learning What makes learning hard Point Estimation: MLE vs Bayesian

More information

Lecture 18: Kernels Risk and Loss Support Vector Regression. Aykut Erdem December 2016 Hacettepe University

Lecture 18: Kernels Risk and Loss Support Vector Regression. Aykut Erdem December 2016 Hacettepe University Lecture 18: Kernels Risk and Loss Support Vector Regression Aykut Erdem December 2016 Hacettepe University Administrative We will have a make-up lecture on next Saturday December 24, 2016 Presentations

More information

Wavelets in Scattering Calculations

Wavelets in Scattering Calculations Wavelets in Scattering Calculations W. P., Brian M. Kessler, Gerald L. Payne polyzou@uiowa.edu The University of Iowa Wavelets in Scattering Calculations p.1/43 What are Wavelets? Orthonormal basis functions.

More information

ECE 5984: Introduction to Machine Learning

ECE 5984: Introduction to Machine Learning ECE 5984: Introduction to Machine Learning Topics: Classification: Logistic Regression NB & LR connections Readings: Barber 17.4 Dhruv Batra Virginia Tech Administrativia HW2 Due: Friday 3/6, 3/15, 11:55pm

More information

Discriminative Models

Discriminative Models No.5 Discriminative Models Hui Jiang Department of Electrical Engineering and Computer Science Lassonde School of Engineering York University, Toronto, Canada Outline Generative vs. Discriminative models

More information

NONLINEAR CLASSIFICATION AND REGRESSION. J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition

NONLINEAR CLASSIFICATION AND REGRESSION. J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition NONLINEAR CLASSIFICATION AND REGRESSION Nonlinear Classification and Regression: Outline 2 Multi-Layer Perceptrons The Back-Propagation Learning Algorithm Generalized Linear Models Radial Basis Function

More information

RKHS, Mercer s theorem, Unbounded domains, Frames and Wavelets Class 22, 2004 Tomaso Poggio and Sayan Mukherjee

RKHS, Mercer s theorem, Unbounded domains, Frames and Wavelets Class 22, 2004 Tomaso Poggio and Sayan Mukherjee RKHS, Mercer s theorem, Unbounded domains, Frames and Wavelets 9.520 Class 22, 2004 Tomaso Poggio and Sayan Mukherjee About this class Goal To introduce an alternate perspective of RKHS via integral operators

More information

Kernel Method: Data Analysis with Positive Definite Kernels

Kernel Method: Data Analysis with Positive Definite Kernels Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University

More information

EXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING

EXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING EXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING DATE AND TIME: August 30, 2018, 14.00 19.00 RESPONSIBLE TEACHER: Niklas Wahlström NUMBER OF PROBLEMS: 5 AIDING MATERIAL: Calculator, mathematical

More information

Support Vector Machines.

Support Vector Machines. Support Vector Machines www.cs.wisc.edu/~dpage 1 Goals for the lecture you should understand the following concepts the margin slack variables the linear support vector machine nonlinear SVMs the kernel

More information

Wavelet Neural Networks for Nonlinear Time Series Analysis

Wavelet Neural Networks for Nonlinear Time Series Analysis Applied Mathematical Sciences, Vol. 4, 2010, no. 50, 2485-2495 Wavelet Neural Networks for Nonlinear Time Series Analysis K. K. Minu, M. C. Lineesh and C. Jessy John Department of Mathematics National

More information

10-701/ Recitation : Kernels

10-701/ Recitation : Kernels 10-701/15-781 Recitation : Kernels Manojit Nandi February 27, 2014 Outline Mathematical Theory Banach Space and Hilbert Spaces Kernels Commonly Used Kernels Kernel Theory One Weird Kernel Trick Representer

More information

Linear & nonlinear classifiers

Linear & nonlinear classifiers Linear & nonlinear classifiers Machine Learning Hamid Beigy Sharif University of Technology Fall 1394 Hamid Beigy (Sharif University of Technology) Linear & nonlinear classifiers Fall 1394 1 / 34 Table

More information

Learning gradients: prescriptive models

Learning gradients: prescriptive models Department of Statistical Science Institute for Genome Sciences & Policy Department of Computer Science Duke University May 11, 2007 Relevant papers Learning Coordinate Covariances via Gradients. Sayan

More information

Nonparametric Regression. Badr Missaoui

Nonparametric Regression. Badr Missaoui Badr Missaoui Outline Kernel and local polynomial regression. Penalized regression. We are given n pairs of observations (X 1, Y 1 ),...,(X n, Y n ) where Y i = r(x i ) + ε i, i = 1,..., n and r(x) = E(Y

More information

LINEAR CLASSIFICATION, PERCEPTRON, LOGISTIC REGRESSION, SVC, NAÏVE BAYES. Supervised Learning

LINEAR CLASSIFICATION, PERCEPTRON, LOGISTIC REGRESSION, SVC, NAÏVE BAYES. Supervised Learning LINEAR CLASSIFICATION, PERCEPTRON, LOGISTIC REGRESSION, SVC, NAÏVE BAYES Supervised Learning Linear vs non linear classifiers In K-NN we saw an example of a non-linear classifier: the decision boundary

More information

Lecture 5: Linear models for classification. Logistic regression. Gradient Descent. Second-order methods.

Lecture 5: Linear models for classification. Logistic regression. Gradient Descent. Second-order methods. Lecture 5: Linear models for classification. Logistic regression. Gradient Descent. Second-order methods. Linear models for classification Logistic regression Gradient descent and second-order methods

More information

Spatial Process Estimates as Smoothers: A Review

Spatial Process Estimates as Smoothers: A Review Spatial Process Estimates as Smoothers: A Review Soutir Bandyopadhyay 1 Basic Model The observational model considered here has the form Y i = f(x i ) + ɛ i, for 1 i n. (1.1) where Y i is the observed

More information

Kernel Methods. Konstantin Tretyakov MTAT Machine Learning

Kernel Methods. Konstantin Tretyakov MTAT Machine Learning Kernel Methods Konstantin Tretyakov (kt@ut.ee) MTAT.03.227 Machine Learning So far Supervised machine learning Linear models Non-linear models Unsupervised machine learning Generic scaffolding So far Supervised

More information

Nearest Neighbor. Machine Learning CSE546 Kevin Jamieson University of Washington. October 26, Kevin Jamieson 2

Nearest Neighbor. Machine Learning CSE546 Kevin Jamieson University of Washington. October 26, Kevin Jamieson 2 Nearest Neighbor Machine Learning CSE546 Kevin Jamieson University of Washington October 26, 2017 2017 Kevin Jamieson 2 Some data, Bayes Classifier Training data: True label: +1 True label: -1 Optimal

More information

Regularization in Reproducing Kernel Banach Spaces

Regularization in Reproducing Kernel Banach Spaces .... Regularization in Reproducing Kernel Banach Spaces Guohui Song School of Mathematical and Statistical Sciences Arizona State University Comp Math Seminar, September 16, 2010 Joint work with Dr. Fred

More information

Supervised Learning: Linear Methods (1/2) Applied Multivariate Statistics Spring 2012

Supervised Learning: Linear Methods (1/2) Applied Multivariate Statistics Spring 2012 Supervised Learning: Linear Methods (1/2) Applied Multivariate Statistics Spring 2012 Overview Review: Conditional Probability LDA / QDA: Theory Fisher s Discriminant Analysis LDA: Example Quality control:

More information

Introduction Dual Representations Kernel Design RBF Linear Reg. GP Regression GP Classification Summary. Kernel Methods. Henrik I Christensen

Introduction Dual Representations Kernel Design RBF Linear Reg. GP Regression GP Classification Summary. Kernel Methods. Henrik I Christensen Kernel Methods Henrik I Christensen Robotics & Intelligent Machines @ GT Georgia Institute of Technology, Atlanta, GA 30332-0280 hic@cc.gatech.edu Henrik I Christensen (RIM@GT) Kernel Methods 1 / 37 Outline

More information

Neural networks and support vector machines

Neural networks and support vector machines Neural netorks and support vector machines Perceptron Input x 1 Weights 1 x 2 x 3... x D 2 3 D Output: sgn( x + b) Can incorporate bias as component of the eight vector by alays including a feature ith

More information

10/05/2016. Computational Methods for Data Analysis. Massimo Poesio SUPPORT VECTOR MACHINES. Support Vector Machines Linear classifiers

10/05/2016. Computational Methods for Data Analysis. Massimo Poesio SUPPORT VECTOR MACHINES. Support Vector Machines Linear classifiers Computational Methods for Data Analysis Massimo Poesio SUPPORT VECTOR MACHINES Support Vector Machines Linear classifiers 1 Linear Classifiers denotes +1 denotes -1 w x + b>0 f(x,w,b) = sign(w x + b) How

More information

Reproducing Kernel Hilbert Spaces

Reproducing Kernel Hilbert Spaces Reproducing Kernel Hilbert Spaces Lorenzo Rosasco 9.520 Class 03 February 11, 2009 About this class Goal To introduce a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert

More information

Support Vector Machines

Support Vector Machines Support Vector Machines Hypothesis Space variable size deterministic continuous parameters Learning Algorithm linear and quadratic programming eager batch SVMs combine three important ideas Apply optimization

More information