Topics in Machine Learning

Size: px
Start display at page:

Download "Topics in Machine Learning"

Transcription

1 Topics in Machine Learning Gilad Lerman School of Mathematics University of Minnesota Text/slides stolen from G. James, D. Witten, T. Hastie, R. Tibshirani and A. Ng

2 Machine Learning - Motivation Arthur Samuel (1959): Field of study that gives computers the ability to learn without being explicitly programmed

3 Machine Learning - Motivation Arthur Samuel (1959): Field of study that gives computers the ability to learn without being explicitly programmed In between, computer science, statistics, optimization, Three categories (soft dichotomy) Supervised learning Unsupervised learning Reinforcement learning

4 Difficulties Understanding the methods (requires knowledge of various areas) Understanding data and application areas Sometimes hard to establish mathematical guarantees Sometimes hard to code and test Fast developing area of research

5 Simplification To avoid such difficulties, but obtain a fine level of knowledge in 2 days, we ll follow: Book is available online Plan: last 3 chapters (8-10) and a bit more.

6 Review Supervised learning (training and test sets) vs. unsupervised learning Examples of supervised learning: regression, classification Examples of unsupervised learning: density/function estimation, clustering, dimension reduction Recall: regression, bias-variance tradeoff, resampling (e.g., cross validation), linear and non-linear models

7 Quick Review of Regression and Nearest Neighbors Regression predicts a response variable Y (quantitative variable) in terms of input variables (predictors) X 1,,X p given n samples in p ; denote X=(X 1,,X p ) The regression function f(x)=e(y X=x) is the minimizer of the mean square prediction error We cannot precisely compute f, since we have few if any values of given x

8 Estimating f by NN

9 Remarks on NN and Classification Need p 4 and sufficiently large n Nearest neighbors tend to be far away in high dimensions Can use kernel or spline smoothing Other common methods: parametric and structure models

10 Neighborhoods in Increasing Dimensions

11 More on Regression Assessing model accuracy:

12 More on Regression Dashed line explained later (irreducible error) Flexibility = degrees of freedom (each square represents method with same color),

13 More on Regression

14 More on Regression

15 More on Regression

16 On Regression Error For an estimator f learned on training set the mean squared error is E(Y f X X = x) 2 Assume Y = f X + ε, wherε is independent noise with mean zero, then E(Y f X X = x) 2 = E(f X + ε f X X = x) 2 = E(f X f X X = x) 2 + Var(ε) Var(ε) is the irreducible error E(f X f X X = x) 2 is the reducible error ( f X depends on random training sample)

17 Regression Error: Bias and Variance E(f X f X X = x) 2 = E( f X E( f X ) X = x) 2 + (E( f X X = x) f x ) 2 = Var( f X X = x)+bias 2 (( f X X = x) E(Y f X X = x) 2 = Var( f X X = x)+bias 2 (( f X X = x)+var(ε)

18 Bias-Variance Tradeoff Two other tradeoffs:

19 Bias-Variance Tradeoff

20 Quick Review of Classification and Nearest Neighbors Classification:

21 Quick Review of Classification and Nearest Neighbors Example:

22 Quick Review of Classification and Nearest Neighbors

23 Quick Review of Classification and Nearest Neighbors

24 Quick Review of Classification and Nearest Neighbors

25 Quick Review of Classification and Nearest Neighbors

26 Quick Review of Classification and Nearest Neighbors

27 Chapter 9: SVM

28 Separation of 2 Classes by a hyperplane Training set: n points (x i,1,, x i,p ), 1 i n, with n labels y i 1,1, 1 i n Separating hyperplane (if exists) satisfies:

29 Separation of 2 Classes by a Example: hyperplane

30 Separation of 2 Classes by a hyperplane If a separating hyperplane exists, then for a test observation x*, a classifier is obtained by the sign of (negative (positive) sign -1/1) The magnitude of f x * provides confidence on class assignment p p * * 2 i i 1 i 1 d( x,hyp.) β0 βix / β

31 Maximal Margin Classifier

32 Maximal Margin Classifier MMC is the solution of No explanation in book, but immediate for a math student Actual algorithm is not discussed

33 Numerical Solution (following A. Ng s Cs229 notes) Change of notation: y (i) =y i, x (i) =(x i,1,, x i,p ) Recall Distance of (x (i),y (i) ) to a hyperplane w T X +b=0 is w T x (i) +b / w

34 Numerical Solution (following A. Ng s Cs229 notes) Original Problem (non-convex): Equivalent non-convex problem via

35 Numerical Solution (following A. Ng s Cs229 notes) Scale w and b by the same constant so that (no effect on problem) and change to the convex problem (quadratic program)

36 Equivalent Formulation (following A. Ng s Cs229 notes) Lagrangian: Dual: Solution: Hence: (used later)

37 A Non-separable Example

38 Non-robustness of the Maximal Margin Classifier

39 The Support Vector Classifier If ε i =0 correct side of boundary If ε i >0 wrong side of margin If ε i >1 wrong side of hyperplane Solution is effected only by support vectors, i.e., observations on wrong side of margins or boundary.

40 Concept Demonstration

41 More on the Optimization Problem C controls # observations on wrong side of margin C controls the bias-variance trade-off Optimizer is effected only by support vectors Increasing C in clock-wise order:

42 Equivalent Formulation (following A. Ng s Cs229 notes) Dual: Similarly as before w T x is a linear combination of <x,x (i) >

43 Support Vector Machine (SVM) From linear to nonlinear boundaries by embedding to a higher-dimensional space The algorithm can be written in terms of a dot product Instead of embed to a very high-dimen. space, replace dot products with kernels

44 Clarification

45 Clarification

46 More (following book) By solution of SVC (recall earlier comment) Can use only support vectors for SVC For SVM replace dot products with kernels

47 Demonstration

48 SVM for K>2 Classes OVO (One vs. One): For training data, K construct 1/-1 classifiers (2 classes 2 out of K classes). For test point, use voting (class with most pairwise assignments) OVA (One vs. All): For training, construct K classifiers (class with 1 vs. rest of classes with -1). For test x*, classify according to largest estimated f(x*) OVO is better for K not too large

49 Chapter 8: Tree-based Methods (or CART) Decision Trees for Regression Demonstration of predicting log(salary/1000) as a func. of # of years in major leagues and hits in previous year Terminology: leaf/terminal node, internal node, branch

50 Chapter 8: Tree-based Methods (or CART)

51 Building a Decision Tree We wish to minimize the RSS (residual sum of squares): Computationally infeasible. Use instead recursive binary splitting (top-down greedy procedure)

52 Recursive Binary Splitting At each node (top to bottom) determine predictor X j and cutoff s minimizing 2 y i yi 2 i: xi R1( j, s ) 2 i: xi R2( j, s ) yi yi i: x 1(, ) 1(, ) : 2(, ) 2(, ) i R j s R j s i xi R j s R j s 2

53 Recursive Binary Splitting For j = 1,, p, determine s that maximize 2 y i yi i: xi R1 ( j, s ) i: x i R2( j, s ) R1( j, s) R2( j, s) Can be done by sorting the j-values and checking all n-1 pairs (x i,x i+1 ) (O(1) operations for each) and reporting average of x i and x i+1, for max. i. Total cost is O(pn). We assumed continuous random variables (can modify for discrete ones) 2

54 More on Recursive Binary Splitting The previous process is repeated until a stopping criteria is met Predict response by mean of training observations in region the test sample belong to

55 Tree Pruning Continue page 17 of books slides trees.pdf

Module 4. Non-linear machine learning econometrics: Support Vector Machine

Module 4. Non-linear machine learning econometrics: Support Vector Machine Module 4. Non-linear machine learning econometrics: Support Vector Machine THE CONTRACTOR IS ACTING UNDER A FRAMEWORK CONTRACT CONCLUDED WITH THE COMMISSION Introduction When the assumption of linearity

More information

Supervised vs unsupervised clustering

Supervised vs unsupervised clustering Classification Supervised vs unsupervised clustering Cluster analysis: Classes are not known a- priori. Classification: Classes are defined a-priori Sometimes called supervised clustering Extract useful

More information

CS6375: Machine Learning Gautam Kunapuli. Mid-Term Review

CS6375: Machine Learning Gautam Kunapuli. Mid-Term Review Gautam Kunapuli Machine Learning Data is identically and independently distributed Goal is to learn a function that maps to Data is generated using an unknown function Learn a hypothesis that minimizes

More information

CS 229 Midterm Review

CS 229 Midterm Review CS 229 Midterm Review Course Staff Fall 2018 11/2/2018 Outline Today: SVMs Kernels Tree Ensembles EM Algorithm / Mixture Models [ Focus on building intuition, less so on solving specific problems. Ask

More information

Model Assessment and Selection. Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer

Model Assessment and Selection. Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer Model Assessment and Selection Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer 1 Model Training data Testing data Model Testing error rate Training error

More information

Data Mining: Concepts and Techniques. Chapter 9 Classification: Support Vector Machines. Support Vector Machines (SVMs)

Data Mining: Concepts and Techniques. Chapter 9 Classification: Support Vector Machines. Support Vector Machines (SVMs) Data Mining: Concepts and Techniques Chapter 9 Classification: Support Vector Machines 1 Support Vector Machines (SVMs) SVMs are a set of related supervised learning methods used for classification Based

More information

Supervised Learning (contd) Linear Separation. Mausam (based on slides by UW-AI faculty)

Supervised Learning (contd) Linear Separation. Mausam (based on slides by UW-AI faculty) Supervised Learning (contd) Linear Separation Mausam (based on slides by UW-AI faculty) Images as Vectors Binary handwritten characters Treat an image as a highdimensional vector (e.g., by reading pixel

More information

The Curse of Dimensionality

The Curse of Dimensionality The Curse of Dimensionality ACAS 2002 p1/66 Curse of Dimensionality The basic idea of the curse of dimensionality is that high dimensional data is difficult to work with for several reasons: Adding more

More information

Machine Learning: Think Big and Parallel

Machine Learning: Think Big and Parallel Day 1 Inderjit S. Dhillon Dept of Computer Science UT Austin CS395T: Topics in Multicore Programming Oct 1, 2013 Outline Scikit-learn: Machine Learning in Python Supervised Learning day1 Regression: Least

More information

Support vector machines

Support vector machines Support vector machines When the data is linearly separable, which of the many possible solutions should we prefer? SVM criterion: maximize the margin, or distance between the hyperplane and the closest

More information

Lecture 27: Review. Reading: All chapters in ISLR. STATS 202: Data mining and analysis. December 6, 2017

Lecture 27: Review. Reading: All chapters in ISLR. STATS 202: Data mining and analysis. December 6, 2017 Lecture 27: Review Reading: All chapters in ISLR. STATS 202: Data mining and analysis December 6, 2017 1 / 16 Final exam: Announcements Tuesday, December 12, 8:30-11:30 am, in the following rooms: Last

More information

Naïve Bayes for text classification

Naïve Bayes for text classification Road Map Basic concepts Decision tree induction Evaluation of classifiers Rule induction Classification using association rules Naïve Bayesian classification Naïve Bayes for text classification Support

More information

Using Machine Learning to Optimize Storage Systems

Using Machine Learning to Optimize Storage Systems Using Machine Learning to Optimize Storage Systems Dr. Kiran Gunnam 1 Outline 1. Overview 2. Building Flash Models using Logistic Regression. 3. Storage Object classification 4. Storage Allocation recommendation

More information

Data Mining Practical Machine Learning Tools and Techniques. Slides for Chapter 6 of Data Mining by I. H. Witten and E. Frank

Data Mining Practical Machine Learning Tools and Techniques. Slides for Chapter 6 of Data Mining by I. H. Witten and E. Frank Data Mining Practical Machine Learning Tools and Techniques Slides for Chapter 6 of Data Mining by I. H. Witten and E. Frank Implementation: Real machine learning schemes Decision trees Classification

More information

Kernels + K-Means Introduction to Machine Learning. Matt Gormley Lecture 29 April 25, 2018

Kernels + K-Means Introduction to Machine Learning. Matt Gormley Lecture 29 April 25, 2018 10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Kernels + K-Means Matt Gormley Lecture 29 April 25, 2018 1 Reminders Homework 8:

More information

Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), University of Hildesheim, Germany

Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), University of Hildesheim, Germany Syllabus Fri. 27.10. (1) 0. Introduction A. Supervised Learning: Linear Models & Fundamentals Fri. 3.11. (2) A.1 Linear Regression Fri. 10.11. (3) A.2 Linear Classification Fri. 17.11. (4) A.3 Regularization

More information

Network Traffic Measurements and Analysis

Network Traffic Measurements and Analysis DEIB - Politecnico di Milano Fall, 2017 Sources Hastie, Tibshirani, Friedman: The Elements of Statistical Learning James, Witten, Hastie, Tibshirani: An Introduction to Statistical Learning Andrew Ng:

More information

Slides for Data Mining by I. H. Witten and E. Frank

Slides for Data Mining by I. H. Witten and E. Frank Slides for Data Mining by I. H. Witten and E. Frank 7 Engineering the input and output Attribute selection Scheme-independent, scheme-specific Attribute discretization Unsupervised, supervised, error-

More information

Machine Learning / Jan 27, 2010

Machine Learning / Jan 27, 2010 Revisiting Logistic Regression & Naïve Bayes Aarti Singh Machine Learning 10-701/15-781 Jan 27, 2010 Generative and Discriminative Classifiers Training classifiers involves learning a mapping f: X -> Y,

More information

More on Classification: Support Vector Machine

More on Classification: Support Vector Machine More on Classification: Support Vector Machine The Support Vector Machine (SVM) is a classification method approach developed in the computer science field in the 1990s. It has shown good performance in

More information

Machine Learning. A. Supervised Learning A.7. Decision Trees. Lars Schmidt-Thieme

Machine Learning. A. Supervised Learning A.7. Decision Trees. Lars Schmidt-Thieme Machine Learning A. Supervised Learning A.7. Decision Trees Lars Schmidt-Thieme Information Systems and Machine Learning Lab (ISMLL) Institute for Computer Science University of Hildesheim, Germany 1 /

More information

Support Vector Machines

Support Vector Machines Support Vector Machines . Importance of SVM SVM is a discriminative method that brings together:. computational learning theory. previously known methods in linear discriminant functions 3. optimization

More information

Random Forest A. Fornaser

Random Forest A. Fornaser Random Forest A. Fornaser alberto.fornaser@unitn.it Sources Lecture 15: decision trees, information theory and random forests, Dr. Richard E. Turner Trees and Random Forests, Adele Cutler, Utah State University

More information

Business Club. Decision Trees

Business Club. Decision Trees Business Club Decision Trees Business Club Analytics Team December 2017 Index 1. Motivation- A Case Study 2. The Trees a. What is a decision tree b. Representation 3. Regression v/s Classification 4. Building

More information

Support Vector Machines

Support Vector Machines Support Vector Machines Chapter 9 Chapter 9 1 / 50 1 91 Maximal margin classifier 2 92 Support vector classifiers 3 93 Support vector machines 4 94 SVMs with more than two classes 5 95 Relationshiop to

More information

MTTTS17 Dimensionality Reduction and Visualization. Spring 2018 Jaakko Peltonen. Lecture 11: Neighbor Embedding Methods continued

MTTTS17 Dimensionality Reduction and Visualization. Spring 2018 Jaakko Peltonen. Lecture 11: Neighbor Embedding Methods continued MTTTS17 Dimensionality Reduction and Visualization Spring 2018 Jaakko Peltonen Lecture 11: Neighbor Embedding Methods continued This Lecture Neighbor embedding by generative modeling Some supervised neighbor

More information

Nonparametric Regression

Nonparametric Regression Nonparametric Regression John Fox Department of Sociology McMaster University 1280 Main Street West Hamilton, Ontario Canada L8S 4M4 jfox@mcmaster.ca February 2004 Abstract Nonparametric regression analysis

More information

Classification and Regression Trees

Classification and Regression Trees Classification and Regression Trees Matthew S. Shotwell, Ph.D. Department of Biostatistics Vanderbilt University School of Medicine Nashville, TN, USA March 16, 2018 Introduction trees partition feature

More information

Feature Extractors. CS 188: Artificial Intelligence Fall Some (Vague) Biology. The Binary Perceptron. Binary Decision Rule.

Feature Extractors. CS 188: Artificial Intelligence Fall Some (Vague) Biology. The Binary Perceptron. Binary Decision Rule. CS 188: Artificial Intelligence Fall 2008 Lecture 24: Perceptrons II 11/24/2008 Dan Klein UC Berkeley Feature Extractors A feature extractor maps inputs to feature vectors Dear Sir. First, I must solicit

More information

Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1

Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1 Preface to the Second Edition Preface to the First Edition vii xi 1 Introduction 1 2 Overview of Supervised Learning 9 2.1 Introduction... 9 2.2 Variable Types and Terminology... 9 2.3 Two Simple Approaches

More information

Data Analysis 3. Support Vector Machines. Jan Platoš October 30, 2017

Data Analysis 3. Support Vector Machines. Jan Platoš October 30, 2017 Data Analysis 3 Support Vector Machines Jan Platoš October 30, 2017 Department of Computer Science Faculty of Electrical Engineering and Computer Science VŠB - Technical University of Ostrava Table of

More information

Classification/Regression Trees and Random Forests

Classification/Regression Trees and Random Forests Classification/Regression Trees and Random Forests Fabio G. Cozman - fgcozman@usp.br November 6, 2018 Classification tree Consider binary class variable Y and features X 1,..., X n. Decide Ŷ after a series

More information

Linear Regression and K-Nearest Neighbors 3/28/18

Linear Regression and K-Nearest Neighbors 3/28/18 Linear Regression and K-Nearest Neighbors 3/28/18 Linear Regression Hypothesis Space Supervised learning For every input in the data set, we know the output Regression Outputs are continuous A number,

More information

Lecture 25: Review I

Lecture 25: Review I Lecture 25: Review I Reading: Up to chapter 5 in ISLR. STATS 202: Data mining and analysis Jonathan Taylor 1 / 18 Unsupervised learning In unsupervised learning, all the variables are on equal standing,

More information

Applying Supervised Learning

Applying Supervised Learning Applying Supervised Learning When to Consider Supervised Learning A supervised learning algorithm takes a known set of input data (the training set) and known responses to the data (output), and trains

More information

What is machine learning?

What is machine learning? Machine learning, pattern recognition and statistical data modelling Lecture 12. The last lecture Coryn Bailer-Jones 1 What is machine learning? Data description and interpretation finding simpler relationship

More information

Performance Evaluation of Various Classification Algorithms

Performance Evaluation of Various Classification Algorithms Performance Evaluation of Various Classification Algorithms Shafali Deora Amritsar College of Engineering & Technology, Punjab Technical University -----------------------------------------------------------***----------------------------------------------------------

More information

Lecture 7: Support Vector Machine

Lecture 7: Support Vector Machine Lecture 7: Support Vector Machine Hien Van Nguyen University of Houston 9/28/2017 Separating hyperplane Red and green dots can be separated by a separating hyperplane Two classes are separable, i.e., each

More information

Support Vector Machines

Support Vector Machines Support Vector Machines About the Name... A Support Vector A training sample used to define classification boundaries in SVMs located near class boundaries Support Vector Machines Binary classifiers whose

More information

6 Model selection and kernels

6 Model selection and kernels 6. Bias-Variance Dilemma Esercizio 6. While you fit a Linear Model to your data set. You are thinking about changing the Linear Model to a Quadratic one (i.e., a Linear Model with quadratic features φ(x)

More information

Introduction to machine learning, pattern recognition and statistical data modelling Coryn Bailer-Jones

Introduction to machine learning, pattern recognition and statistical data modelling Coryn Bailer-Jones Introduction to machine learning, pattern recognition and statistical data modelling Coryn Bailer-Jones What is machine learning? Data interpretation describing relationship between predictors and responses

More information

CS570: Introduction to Data Mining

CS570: Introduction to Data Mining CS570: Introduction to Data Mining Classification Advanced Reading: Chapter 8 & 9 Han, Chapters 4 & 5 Tan Anca Doloc-Mihu, Ph.D. Slides courtesy of Li Xiong, Ph.D., 2011 Han, Kamber & Pei. Data Mining.

More information

Machine Learning Techniques for Data Mining

Machine Learning Techniques for Data Mining Machine Learning Techniques for Data Mining Eibe Frank University of Waikato New Zealand 10/25/2000 1 PART VII Moving on: Engineering the input and output 10/25/2000 2 Applying a learner is not all Already

More information

Evaluation Metrics. (Classifiers) CS229 Section Anand Avati

Evaluation Metrics. (Classifiers) CS229 Section Anand Avati Evaluation Metrics (Classifiers) CS Section Anand Avati Topics Why? Binary classifiers Metrics Rank view Thresholding Confusion Matrix Point metrics: Accuracy, Precision, Recall / Sensitivity, Specificity,

More information

Robot Learning. There are generally three types of robot learning: Learning from data. Learning by demonstration. Reinforcement learning

Robot Learning. There are generally three types of robot learning: Learning from data. Learning by demonstration. Reinforcement learning Robot Learning 1 General Pipeline 1. Data acquisition (e.g., from 3D sensors) 2. Feature extraction and representation construction 3. Robot learning: e.g., classification (recognition) or clustering (knowledge

More information

Big Data Methods. Chapter 5: Machine learning. Big Data Methods, Chapter 5, Slide 1

Big Data Methods. Chapter 5: Machine learning. Big Data Methods, Chapter 5, Slide 1 Big Data Methods Chapter 5: Machine learning Big Data Methods, Chapter 5, Slide 1 5.1 Introduction to machine learning What is machine learning? Concerned with the study and development of algorithms that

More information

Support Vector Machines + Classification for IR

Support Vector Machines + Classification for IR Support Vector Machines + Classification for IR Pierre Lison University of Oslo, Dep. of Informatics INF3800: Søketeknologi April 30, 2014 Outline of the lecture Recap of last week Support Vector Machines

More information

5 Learning hypothesis classes (16 points)

5 Learning hypothesis classes (16 points) 5 Learning hypothesis classes (16 points) Consider a classification problem with two real valued inputs. For each of the following algorithms, specify all of the separators below that it could have generated

More information

All lecture slides will be available at CSC2515_Winter15.html

All lecture slides will be available at  CSC2515_Winter15.html CSC2515 Fall 2015 Introduc3on to Machine Learning Lecture 9: Support Vector Machines All lecture slides will be available at http://www.cs.toronto.edu/~urtasun/courses/csc2515/ CSC2515_Winter15.html Many

More information

Supervised Learning Classification Algorithms Comparison

Supervised Learning Classification Algorithms Comparison Supervised Learning Classification Algorithms Comparison Aditya Singh Rathore B.Tech, J.K. Lakshmipat University -------------------------------------------------------------***---------------------------------------------------------

More information

CSE 417T: Introduction to Machine Learning. Lecture 22: The Kernel Trick. Henry Chai 11/15/18

CSE 417T: Introduction to Machine Learning. Lecture 22: The Kernel Trick. Henry Chai 11/15/18 CSE 417T: Introduction to Machine Learning Lecture 22: The Kernel Trick Henry Chai 11/15/18 Linearly Inseparable Data What can we do if the data is not linearly separable? Accept some non-zero in-sample

More information

Linear methods for supervised learning

Linear methods for supervised learning Linear methods for supervised learning LDA Logistic regression Naïve Bayes PLA Maximum margin hyperplanes Soft-margin hyperplanes Least squares resgression Ridge regression Nonlinear feature maps Sometimes

More information

Support Vector Machines

Support Vector Machines Support Vector Machines RBF-networks Support Vector Machines Good Decision Boundary Optimization Problem Soft margin Hyperplane Non-linear Decision Boundary Kernel-Trick Approximation Accurancy Overtraining

More information

Support Vector Machines

Support Vector Machines Support Vector Machines Xiaojin Zhu jerryzhu@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [ Based on slides from Andrew Moore http://www.cs.cmu.edu/~awm/tutorials] slide 1

More information

Kernels and Clustering

Kernels and Clustering Kernels and Clustering Robert Platt Northeastern University All slides in this file are adapted from CS188 UC Berkeley Case-Based Learning Non-Separable Data Case-Based Reasoning Classification from similarity

More information

A Short SVM (Support Vector Machine) Tutorial

A Short SVM (Support Vector Machine) Tutorial A Short SVM (Support Vector Machine) Tutorial j.p.lewis CGIT Lab / IMSC U. Southern California version 0.zz dec 004 This tutorial assumes you are familiar with linear algebra and equality-constrained optimization/lagrange

More information

Nonparametric Classification Methods

Nonparametric Classification Methods Nonparametric Classification Methods We now examine some modern, computationally intensive methods for regression and classification. Recall that the LDA approach constructs a line (or plane or hyperplane)

More information

Support Vector Machines

Support Vector Machines Support Vector Machines SVM Discussion Overview. Importance of SVMs. Overview of Mathematical Techniques Employed 3. Margin Geometry 4. SVM Training Methodology 5. Overlapping Distributions 6. Dealing

More information

Contents. Preface to the Second Edition

Contents. Preface to the Second Edition Preface to the Second Edition v 1 Introduction 1 1.1 What Is Data Mining?....................... 4 1.2 Motivating Challenges....................... 5 1.3 The Origins of Data Mining....................

More information

Introduction to object recognition. Slides adapted from Fei-Fei Li, Rob Fergus, Antonio Torralba, and others

Introduction to object recognition. Slides adapted from Fei-Fei Li, Rob Fergus, Antonio Torralba, and others Introduction to object recognition Slides adapted from Fei-Fei Li, Rob Fergus, Antonio Torralba, and others Overview Basic recognition tasks A statistical learning approach Traditional or shallow recognition

More information

Classifiers and Detection. D.A. Forsyth

Classifiers and Detection. D.A. Forsyth Classifiers and Detection D.A. Forsyth Classifiers Take a measurement x, predict a bit (yes/no; 1/-1; 1/0; etc) Detection with a classifier Search all windows at relevant scales Prepare features Classify

More information

Advanced Studies in Applied Statistics (WBL), ETHZ Applied Multivariate Statistics Spring 2018, Week 11

Advanced Studies in Applied Statistics (WBL), ETHZ Applied Multivariate Statistics Spring 2018, Week 11 Advanced Studies in Applied Statistics (WBL), ETHZ Applied Multivariate Statistics Spring 2018, Week 11 Lecturer: Beate Sick sickb@ethz.ch Remark: Much of the material have been developed together with

More information

Support Vector Machines

Support Vector Machines Support Vector Machines RBF-networks Support Vector Machines Good Decision Boundary Optimization Problem Soft margin Hyperplane Non-linear Decision Boundary Kernel-Trick Approximation Accurancy Overtraining

More information

The Basics of Decision Trees

The Basics of Decision Trees Tree-based Methods Here we describe tree-based methods for regression and classification. These involve stratifying or segmenting the predictor space into a number of simple regions. Since the set of splitting

More information

Semi-supervised Learning

Semi-supervised Learning Semi-supervised Learning Piyush Rai CS5350/6350: Machine Learning November 8, 2011 Semi-supervised Learning Supervised Learning models require labeled data Learning a reliable model usually requires plenty

More information

Support Vector Machines.

Support Vector Machines. Support Vector Machines srihari@buffalo.edu SVM Discussion Overview. Importance of SVMs. Overview of Mathematical Techniques Employed 3. Margin Geometry 4. SVM Training Methodology 5. Overlapping Distributions

More information

Linear Methods for Regression and Shrinkage Methods

Linear Methods for Regression and Shrinkage Methods Linear Methods for Regression and Shrinkage Methods Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer 1 Linear Regression Models Least Squares Input vectors

More information

Machine Learning Models for Pattern Classification. Comp 473/6731

Machine Learning Models for Pattern Classification. Comp 473/6731 Machine Learning Models for Pattern Classification Comp 473/6731 November 24th 2016 Prof. Neamat El Gayar Neural Networks Neural Networks Low level computational algorithms Learn by example (no required

More information

Case-Based Reasoning. CS 188: Artificial Intelligence Fall Nearest-Neighbor Classification. Parametric / Non-parametric.

Case-Based Reasoning. CS 188: Artificial Intelligence Fall Nearest-Neighbor Classification. Parametric / Non-parametric. CS 188: Artificial Intelligence Fall 2008 Lecture 25: Kernels and Clustering 12/2/2008 Dan Klein UC Berkeley Case-Based Reasoning Similarity for classification Case-based reasoning Predict an instance

More information

CS 188: Artificial Intelligence Fall 2008

CS 188: Artificial Intelligence Fall 2008 CS 188: Artificial Intelligence Fall 2008 Lecture 25: Kernels and Clustering 12/2/2008 Dan Klein UC Berkeley 1 1 Case-Based Reasoning Similarity for classification Case-based reasoning Predict an instance

More information

CSE 573: Artificial Intelligence Autumn 2010

CSE 573: Artificial Intelligence Autumn 2010 CSE 573: Artificial Intelligence Autumn 2010 Lecture 16: Machine Learning Topics 12/7/2010 Luke Zettlemoyer Most slides over the course adapted from Dan Klein. 1 Announcements Syllabus revised Machine

More information

Support Vector Machines: Brief Overview" November 2011 CPSC 352

Support Vector Machines: Brief Overview November 2011 CPSC 352 Support Vector Machines: Brief Overview" Outline Microarray Example Support Vector Machines (SVMs) Software: libsvm A Baseball Example with libsvm Classifying Cancer Tissue: The ALL/AML Dataset Golub et

More information

Linear Models. Lecture Outline: Numeric Prediction: Linear Regression. Linear Classification. The Perceptron. Support Vector Machines

Linear Models. Lecture Outline: Numeric Prediction: Linear Regression. Linear Classification. The Perceptron. Support Vector Machines Linear Models Lecture Outline: Numeric Prediction: Linear Regression Linear Classification The Perceptron Support Vector Machines Reading: Chapter 4.6 Witten and Frank, 2nd ed. Chapter 4 of Mitchell Solving

More information

CSC411/2515 Tutorial: K-NN and Decision Tree

CSC411/2515 Tutorial: K-NN and Decision Tree CSC411/2515 Tutorial: K-NN and Decision Tree Mengye Ren csc{411,2515}ta@cs.toronto.edu September 25, 2016 Cross-validation K-nearest-neighbours Decision Trees Review: Motivation for Validation Framework:

More information

Semi-supervised learning and active learning

Semi-supervised learning and active learning Semi-supervised learning and active learning Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Combining classifiers Ensemble learning: a machine learning paradigm where multiple learners

More information

Generative and discriminative classification techniques

Generative and discriminative classification techniques Generative and discriminative classification techniques Machine Learning and Category Representation 013-014 Jakob Verbeek, December 13+0, 013 Course website: http://lear.inrialpes.fr/~verbeek/mlcr.13.14

More information

Lecture 3. Oct

Lecture 3. Oct Lecture 3 Oct 3 2008 Review of last lecture A supervised learning example spam filter, and the design choices one need to make for this problem use bag-of-words to represent emails linear functions as

More information

Machine Learning. Topic 5: Linear Discriminants. Bryan Pardo, EECS 349 Machine Learning, 2013

Machine Learning. Topic 5: Linear Discriminants. Bryan Pardo, EECS 349 Machine Learning, 2013 Machine Learning Topic 5: Linear Discriminants Bryan Pardo, EECS 349 Machine Learning, 2013 Thanks to Mark Cartwright for his extensive contributions to these slides Thanks to Alpaydin, Bishop, and Duda/Hart/Stork

More information

CS 343: Artificial Intelligence

CS 343: Artificial Intelligence CS 343: Artificial Intelligence Kernels and Clustering Prof. Scott Niekum The University of Texas at Austin [These slides based on those of Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley.

More information

DATA MINING LECTURE 10B. Classification k-nearest neighbor classifier Naïve Bayes Logistic Regression Support Vector Machines

DATA MINING LECTURE 10B. Classification k-nearest neighbor classifier Naïve Bayes Logistic Regression Support Vector Machines DATA MINING LECTURE 10B Classification k-nearest neighbor classifier Naïve Bayes Logistic Regression Support Vector Machines NEAREST NEIGHBOR CLASSIFICATION 10 10 Illustrating Classification Task Tid Attrib1

More information

Classification: Feature Vectors

Classification: Feature Vectors Classification: Feature Vectors Hello, Do you want free printr cartriges? Why pay more when you can get them ABSOLUTELY FREE! Just # free YOUR_NAME MISSPELLED FROM_FRIEND... : : : : 2 0 2 0 PIXEL 7,12

More information

Lecture 9: Support Vector Machines

Lecture 9: Support Vector Machines Lecture 9: Support Vector Machines William Webber (william@williamwebber.com) COMP90042, 2014, Semester 1, Lecture 8 What we ll learn in this lecture Support Vector Machines (SVMs) a highly robust and

More information

Constrained optimization

Constrained optimization Constrained optimization A general constrained optimization problem has the form where The Lagrangian function is given by Primal and dual optimization problems Primal: Dual: Weak duality: Strong duality:

More information

Voronoi Region. K-means method for Signal Compression: Vector Quantization. Compression Formula 11/20/2013

Voronoi Region. K-means method for Signal Compression: Vector Quantization. Compression Formula 11/20/2013 Voronoi Region K-means method for Signal Compression: Vector Quantization Blocks of signals: A sequence of audio. A block of image pixels. Formally: vector example: (0.2, 0.3, 0.5, 0.1) A vector quantizer

More information

Lecture 7: Decision Trees

Lecture 7: Decision Trees Lecture 7: Decision Trees Instructor: Outline 1 Geometric Perspective of Classification 2 Decision Trees Geometric Perspective of Classification Perspective of Classification Algorithmic Geometric Probabilistic...

More information

Artificial Intelligence. Programming Styles

Artificial Intelligence. Programming Styles Artificial Intelligence Intro to Machine Learning Programming Styles Standard CS: Explicitly program computer to do something Early AI: Derive a problem description (state) and use general algorithms to

More information

Lecture Linear Support Vector Machines

Lecture Linear Support Vector Machines Lecture 8 In this lecture we return to the task of classification. As seen earlier, examples include spam filters, letter recognition, or text classification. In this lecture we introduce a popular method

More information

Pattern Recognition for Neuroimaging Data

Pattern Recognition for Neuroimaging Data Pattern Recognition for Neuroimaging Data Edinburgh, SPM course April 2013 C. Phillips, Cyclotron Research Centre, ULg, Belgium http://www.cyclotron.ulg.ac.be Overview Introduction Univariate & multivariate

More information

CS229 Lecture notes. Raphael John Lamarre Townshend

CS229 Lecture notes. Raphael John Lamarre Townshend CS229 Lecture notes Raphael John Lamarre Townshend Decision Trees We now turn our attention to decision trees, a simple yet flexible class of algorithms. We will first consider the non-linear, region-based

More information

Mathematics of Data. INFO-4604, Applied Machine Learning University of Colorado Boulder. September 5, 2017 Prof. Michael Paul

Mathematics of Data. INFO-4604, Applied Machine Learning University of Colorado Boulder. September 5, 2017 Prof. Michael Paul Mathematics of Data INFO-4604, Applied Machine Learning University of Colorado Boulder September 5, 2017 Prof. Michael Paul Goals In the intro lecture, every visualization was in 2D What happens when we

More information

CSE 158. Web Mining and Recommender Systems. Midterm recap

CSE 158. Web Mining and Recommender Systems. Midterm recap CSE 158 Web Mining and Recommender Systems Midterm recap Midterm on Wednesday! 5:10 pm 6:10 pm Closed book but I ll provide a similar level of basic info as in the last page of previous midterms CSE 158

More information

Cross-validation. Cross-validation is a resampling method.

Cross-validation. Cross-validation is a resampling method. Cross-validation Cross-validation is a resampling method. It refits a model of interest to samples formed from the training set, in order to obtain additional information about the fitted model. For example,

More information

Classification and Regression Trees

Classification and Regression Trees Classification and Regression Trees David S. Rosenberg New York University April 3, 2018 David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 3, 2018 1 / 51 Contents 1 Trees 2 Regression

More information

Support Vector Machines.

Support Vector Machines. Support Vector Machines srihari@buffalo.edu SVM Discussion Overview 1. Overview of SVMs 2. Margin Geometry 3. SVM Optimization 4. Overlapping Distributions 5. Relationship to Logistic Regression 6. Dealing

More information

CPSC 340: Machine Learning and Data Mining. Principal Component Analysis Fall 2016

CPSC 340: Machine Learning and Data Mining. Principal Component Analysis Fall 2016 CPSC 340: Machine Learning and Data Mining Principal Component Analysis Fall 2016 A2/Midterm: Admin Grades/solutions will be posted after class. Assignment 4: Posted, due November 14. Extra office hours:

More information

Lecture 7: Linear Regression (continued)

Lecture 7: Linear Regression (continued) Lecture 7: Linear Regression (continued) Reading: Chapter 3 STATS 2: Data mining and analysis Jonathan Taylor, 10/8 Slide credits: Sergio Bacallado 1 / 14 Potential issues in linear regression 1. Interactions

More information

Topic 4: Support Vector Machines

Topic 4: Support Vector Machines CS 4850/6850: Introduction to achine Learning Fall 2018 Topic 4: Support Vector achines Instructor: Daniel L Pimentel-Alarcón c Copyright 2018 41 Introduction Support vector machines (SVs) are considered

More information

LARGE MARGIN CLASSIFIERS

LARGE MARGIN CLASSIFIERS Admin Assignment 5 LARGE MARGIN CLASSIFIERS David Kauchak CS 451 Fall 2013 Midterm Download from course web page when you re ready to take it 2 hours to complete Must hand-in (or e-mail in) by 11:59pm

More information

Announcements. CS 188: Artificial Intelligence Spring Generative vs. Discriminative. Classification: Feature Vectors. Project 4: due Friday.

Announcements. CS 188: Artificial Intelligence Spring Generative vs. Discriminative. Classification: Feature Vectors. Project 4: due Friday. CS 188: Artificial Intelligence Spring 2011 Lecture 21: Perceptrons 4/13/2010 Announcements Project 4: due Friday. Final Contest: up and running! Project 5 out! Pieter Abbeel UC Berkeley Many slides adapted

More information

An Introduction to Machine Learning

An Introduction to Machine Learning TRIPODS Summer Boorcamp: Topology and Machine Learning August 6, 2018 General Set-up Introduction Set-up and Goal Suppose we have X 1,X 2,...,X n data samples. Can we predict properites about any given

More information