Clustering & Dimensionality Reduction. 273A Intro Machine Learning

Size: px
Start display at page:

Download "Clustering & Dimensionality Reduction. 273A Intro Machine Learning"

Transcription

1 Clustering & Dimensionality Reduction 273A Intro Machine Learning

2 What is Unsupervised Learning? In supervised learning we were given attributes & targets (e.g. class labels). In unsupervised learning we are only given attributes. Our task is to discover structure in the data. Example I: the data may be structured in clusters: Example II: the data may live on a lower dimensional manifold:

3 Why Discover Structure? Data compression: If you have a good model you can encode the data more cheaply. Example PCA: To encode the data I have to encode the x and y position of each data-case. However, I could also encode the offset and angle of the line plus the deviations from the line. Small numbers can be encoded more cheaply than large numbers with the same precision. This idea is the basis for model selection: The complexity of your model (e.g. the number of parameters) should be such that you can encode the data-set with the fewest number of bits (up to a certain precision). Clustering: represent every data-case by a cluster representative plus deviations. ML is often trying to find semantically meaningful representations (abstractions). These are good as a basis for making new predictions.

4 Clustering: K-means We iterate two operations: 1. Update the assignment of data-cases to clusters 2. Update the location of the cluster. Denote Denote Denote the assignment of data-case i to cluster c. the position of cluster c in a d-dimensional space. the location of data-case i Then iterate until convergence: 1. For each data-case, compute distances to each cluster and the closest one: 2. For each cluster location, compute the mean location of all data-cases assigned to it: Nr. of data-cases in cluster c Set of data-cases assigned to cluster c

5 K-means Cost function: Each step in k-means decreases this cost function. Often initialization is very important since there are very many local minima in C. Relatively good initialization: place cluster locations on K randomly chosen data-cases. How to choose K? Add complexity term: Or X-validation Or Bayesian methods and minimize also over K

6 Vector Quantization K-means divides the space up in a Voronoi tesselation. Every point on a tile is summarized by the code-book vector +. This clearly allows for data compression!

7 Mixtures of Gaussians K-means assigns each data-case to exactly 1 cluster. But what if clusters are overlapping? Maybe we are uncertain as to which cluster it really belongs. The mixtures of Gaussians algorithm assigns data-cases to cluster with a certain probability.

8 MoG Clustering Covariance determines the shape of these contours Idea: fit these Gaussian densities to the data, one per cluster.

9 EM Algorithm: E-step r is the probability that data-case i belongs to cluster c. is the a priori probability of being assigned to cluster c. Note that if the Gaussian has high probability on data-case i (i.e. the bell-shape is on top of the data-case) then it claims high responsibility for this data-case. The denominator is just to normalize all responsibilities to 1:

10 EM Algorithm: M-Step total responsibility claimed by cluster c expected fraction of data-cases assigned to this cluster weighted sample mean where every data-case is weighted according to the probability that it belongs to that cluster. weighted sample covariance

11 EM-MoG EM comes from expectation maximization. We won t go through the derivation. If we are forced to decide, we should assign a data-case to the cluster which claims highest responsibility. For a new data-case, we should compute responsibilities as in the E-step and pick the cluster with the largest responsibility. E and M steps should be iterated until convergence (which is guaranteed). Every step increases the following objective function (which is the total log-probability of the data under the model we are learning):

Machine Learning and Data Mining. Clustering (1): Basics. Kalev Kask

Machine Learning and Data Mining. Clustering (1): Basics. Kalev Kask Machine Learning and Data Mining Clustering (1): Basics Kalev Kask Unsupervised learning Supervised learning Predict target value ( y ) given features ( x ) Unsupervised learning Understand patterns of

More information

Clustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin

Clustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin Clustering K-means Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin 2005-2014 1 Clustering images Set of Images [Goldberger et al.] Carlos Guestrin 2005-2014

More information

Clustering and Visualisation of Data

Clustering and Visualisation of Data Clustering and Visualisation of Data Hiroshi Shimodaira January-March 28 Cluster analysis aims to partition a data set into meaningful or useful groups, based on distances between data points. In some

More information

K-Means Clustering 3/3/17

K-Means Clustering 3/3/17 K-Means Clustering 3/3/17 Unsupervised Learning We have a collection of unlabeled data points. We want to find underlying structure in the data. Examples: Identify groups of similar data points. Clustering

More information

Note Set 4: Finite Mixture Models and the EM Algorithm

Note Set 4: Finite Mixture Models and the EM Algorithm Note Set 4: Finite Mixture Models and the EM Algorithm Padhraic Smyth, Department of Computer Science University of California, Irvine Finite Mixture Models A finite mixture model with K components, for

More information

Unsupervised Learning

Unsupervised Learning Networks for Pattern Recognition, 2014 Networks for Single Linkage K-Means Soft DBSCAN PCA Networks for Kohonen Maps Linear Vector Quantization Networks for Problems/Approaches in Machine Learning Supervised

More information

COMS 4771 Clustering. Nakul Verma

COMS 4771 Clustering. Nakul Verma COMS 4771 Clustering Nakul Verma Supervised Learning Data: Supervised learning Assumption: there is a (relatively simple) function such that for most i Learning task: given n examples from the data, find

More information

Clustering web search results

Clustering web search results Clustering K-means Machine Learning CSE546 Emily Fox University of Washington November 4, 2013 1 Clustering images Set of Images [Goldberger et al.] 2 1 Clustering web search results 3 Some Data 4 2 K-means

More information

Grundlagen der Künstlichen Intelligenz

Grundlagen der Künstlichen Intelligenz Grundlagen der Künstlichen Intelligenz Unsupervised learning Daniel Hennes 29.01.2018 (WS 2017/18) University Stuttgart - IPVS - Machine Learning & Robotics 1 Today Supervised learning Regression (linear

More information

Unsupervised Learning

Unsupervised Learning Unsupervised Learning Unsupervised learning Until now, we have assumed our training samples are labeled by their category membership. Methods that use labeled samples are said to be supervised. However,

More information

Clustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin

Clustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin Clustering K-means Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin 2005-2014 1 Clustering images Set of Images [Goldberger et al.] Carlos Guestrin 2005-2014

More information

Network Traffic Measurements and Analysis

Network Traffic Measurements and Analysis DEIB - Politecnico di Milano Fall, 2017 Introduction Often, we have only a set of features x = x 1, x 2,, x n, but no associated response y. Therefore we are not interested in prediction nor classification,

More information

10/14/2017. Dejan Sarka. Anomaly Detection. Sponsors

10/14/2017. Dejan Sarka. Anomaly Detection. Sponsors Dejan Sarka Anomaly Detection Sponsors About me SQL Server MVP (17 years) and MCT (20 years) 25 years working with SQL Server Authoring 16 th book Authoring many courses, articles Agenda Introduction Simple

More information

CSC 411: Lecture 12: Clustering

CSC 411: Lecture 12: Clustering CSC 411: Lecture 12: Clustering Raquel Urtasun & Rich Zemel University of Toronto Oct 22, 2015 Urtasun & Zemel (UofT) CSC 411: 12-Clustering Oct 22, 2015 1 / 18 Today Unsupervised learning Clustering -means

More information

Machine Learning. Unsupervised Learning. Manfred Huber

Machine Learning. Unsupervised Learning. Manfred Huber Machine Learning Unsupervised Learning Manfred Huber 2015 1 Unsupervised Learning In supervised learning the training data provides desired target output for learning In unsupervised learning the training

More information

Clustering and The Expectation-Maximization Algorithm

Clustering and The Expectation-Maximization Algorithm Clustering and The Expectation-Maximization Algorithm Unsupervised Learning Marek Petrik 3/7 Some of the figures in this presentation are taken from An Introduction to Statistical Learning, with applications

More information

ECE 5424: Introduction to Machine Learning

ECE 5424: Introduction to Machine Learning ECE 5424: Introduction to Machine Learning Topics: Unsupervised Learning: Kmeans, GMM, EM Readings: Barber 20.1-20.3 Stefan Lee Virginia Tech Tasks Supervised Learning x Classification y Discrete x Regression

More information

Clustering Lecture 5: Mixture Model

Clustering Lecture 5: Mixture Model Clustering Lecture 5: Mixture Model Jing Gao SUNY Buffalo 1 Outline Basics Motivation, definition, evaluation Methods Partitional Hierarchical Density-based Mixture model Spectral methods Advanced topics

More information

Unsupervised Learning : Clustering

Unsupervised Learning : Clustering Unsupervised Learning : Clustering Things to be Addressed Traditional Learning Models. Cluster Analysis K-means Clustering Algorithm Drawbacks of traditional clustering algorithms. Clustering as a complex

More information

Clustering in R d. Clustering. Widely-used clustering methods. The k-means optimization problem CSE 250B

Clustering in R d. Clustering. Widely-used clustering methods. The k-means optimization problem CSE 250B Clustering in R d Clustering CSE 250B Two common uses of clustering: Vector quantization Find a finite set of representatives that provides good coverage of a complex, possibly infinite, high-dimensional

More information

Introduction to Machine Learning CMU-10701

Introduction to Machine Learning CMU-10701 Introduction to Machine Learning CMU-10701 Clustering and EM Barnabás Póczos & Aarti Singh Contents Clustering K-means Mixture of Gaussians Expectation Maximization Variational Methods 2 Clustering 3 K-

More information

Machine Learning A W 1sst KU. b) [1 P] Give an example for a probability distributions P (A, B, C) that disproves

Machine Learning A W 1sst KU. b) [1 P] Give an example for a probability distributions P (A, B, C) that disproves Machine Learning A 708.064 11W 1sst KU Exercises Problems marked with * are optional. 1 Conditional Independence I [2 P] a) [1 P] Give an example for a probability distribution P (A, B, C) that disproves

More information

Mixture Models and the EM Algorithm

Mixture Models and the EM Algorithm Mixture Models and the EM Algorithm Padhraic Smyth, Department of Computer Science University of California, Irvine c 2017 1 Finite Mixture Models Say we have a data set D = {x 1,..., x N } where x i is

More information

CPSC 340: Machine Learning and Data Mining. Principal Component Analysis Fall 2017

CPSC 340: Machine Learning and Data Mining. Principal Component Analysis Fall 2017 CPSC 340: Machine Learning and Data Mining Principal Component Analysis Fall 2017 Assignment 3: 2 late days to hand in tonight. Admin Assignment 4: Due Friday of next week. Last Time: MAP Estimation MAP

More information

Shape Classification and Cell Movement in 3D Matrix Tutorial (Part I)

Shape Classification and Cell Movement in 3D Matrix Tutorial (Part I) Shape Classification and Cell Movement in 3D Matrix Tutorial (Part I) Fred Park UCI icamp 2011 Outline 1. Motivation and Shape Definition 2. Shape Descriptors 3. Classification 4. Applications: Shape Matching,

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

Expectation Maximization (EM) and Gaussian Mixture Models

Expectation Maximization (EM) and Gaussian Mixture Models Expectation Maximization (EM) and Gaussian Mixture Models Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer 1 2 3 4 5 6 7 8 Unsupervised Learning Motivation

More information

CS839: Probabilistic Graphical Models. Lecture 10: Learning with Partially Observed Data. Theo Rekatsinas

CS839: Probabilistic Graphical Models. Lecture 10: Learning with Partially Observed Data. Theo Rekatsinas CS839: Probabilistic Graphical Models Lecture 10: Learning with Partially Observed Data Theo Rekatsinas 1 Partially Observed GMs Speech recognition 2 Partially Observed GMs Evolution 3 Partially Observed

More information

http://www.xkcd.com/233/ Text Clustering David Kauchak cs160 Fall 2009 adapted from: http://www.stanford.edu/class/cs276/handouts/lecture17-clustering.ppt Administrative 2 nd status reports Paper review

More information

Clustering. CE-717: Machine Learning Sharif University of Technology Spring Soleymani

Clustering. CE-717: Machine Learning Sharif University of Technology Spring Soleymani Clustering CE-717: Machine Learning Sharif University of Technology Spring 2016 Soleymani Outline Clustering Definition Clustering main approaches Partitional (flat) Hierarchical Clustering validation

More information

Unsupervised Learning: Clustering

Unsupervised Learning: Clustering Unsupervised Learning: Clustering Vibhav Gogate The University of Texas at Dallas Slides adapted from Carlos Guestrin, Dan Klein & Luke Zettlemoyer Machine Learning Supervised Learning Unsupervised Learning

More information

Machine Learning. B. Unsupervised Learning B.1 Cluster Analysis. Lars Schmidt-Thieme, Nicolas Schilling

Machine Learning. B. Unsupervised Learning B.1 Cluster Analysis. Lars Schmidt-Thieme, Nicolas Schilling Machine Learning B. Unsupervised Learning B.1 Cluster Analysis Lars Schmidt-Thieme, Nicolas Schilling Information Systems and Machine Learning Lab (ISMLL) Institute for Computer Science University of Hildesheim,

More information

Colorado School of Mines. Computer Vision. Professor William Hoff Dept of Electrical Engineering &Computer Science.

Colorado School of Mines. Computer Vision. Professor William Hoff Dept of Electrical Engineering &Computer Science. Professor William Hoff Dept of Electrical Engineering &Computer Science http://inside.mines.edu/~whoff/ 1 Image Segmentation Some material for these slides comes from https://www.csd.uwo.ca/courses/cs4487a/

More information

CSE 158 Lecture 6. Web Mining and Recommender Systems. Community Detection

CSE 158 Lecture 6. Web Mining and Recommender Systems. Community Detection CSE 158 Lecture 6 Web Mining and Recommender Systems Community Detection Dimensionality reduction Goal: take high-dimensional data, and describe it compactly using a small number of dimensions Assumption:

More information

10. MLSP intro. (Clustering: K-means, EM, GMM, etc.)

10. MLSP intro. (Clustering: K-means, EM, GMM, etc.) 10. MLSP intro. (Clustering: K-means, EM, GMM, etc.) Rahil Mahdian 01.04.2016 LSV Lab, Saarland University, Germany What is clustering? Clustering is the classification of objects into different groups,

More information

CS325 Artificial Intelligence Ch. 20 Unsupervised Machine Learning

CS325 Artificial Intelligence Ch. 20 Unsupervised Machine Learning CS325 Artificial Intelligence Cengiz Spring 2013 Unsupervised Learning Missing teacher No labels, y Just input data, x What can you learn with it? Unsupervised Learning Missing teacher No labels, y Just

More information

CSE 255 Lecture 6. Data Mining and Predictive Analytics. Community Detection

CSE 255 Lecture 6. Data Mining and Predictive Analytics. Community Detection CSE 255 Lecture 6 Data Mining and Predictive Analytics Community Detection Dimensionality reduction Goal: take high-dimensional data, and describe it compactly using a small number of dimensions Assumption:

More information

University of Florida CISE department Gator Engineering. Clustering Part 2

University of Florida CISE department Gator Engineering. Clustering Part 2 Clustering Part 2 Dr. Sanjay Ranka Professor Computer and Information Science and Engineering University of Florida, Gainesville Partitional Clustering Original Points A Partitional Clustering Hierarchical

More information

The EM Algorithm Lecture What's the Point? Maximum likelihood parameter estimates: One denition of the \best" knob settings. Often impossible to nd di

The EM Algorithm Lecture What's the Point? Maximum likelihood parameter estimates: One denition of the \best knob settings. Often impossible to nd di The EM Algorithm This lecture introduces an important statistical estimation algorithm known as the EM or \expectation-maximization" algorithm. It reviews the situations in which EM works well and its

More information

Classification. Vladimir Curic. Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University

Classification. Vladimir Curic. Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University Classification Vladimir Curic Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University Outline An overview on classification Basics of classification How to choose appropriate

More information

Unsupervised: no target value to predict

Unsupervised: no target value to predict Clustering Unsupervised: no target value to predict Differences between models/algorithms: Exclusive vs. overlapping Deterministic vs. probabilistic Hierarchical vs. flat Incremental vs. batch learning

More information

22 October, 2012 MVA ENS Cachan. Lecture 5: Introduction to generative models Iasonas Kokkinos

22 October, 2012 MVA ENS Cachan. Lecture 5: Introduction to generative models Iasonas Kokkinos Machine Learning for Computer Vision 1 22 October, 2012 MVA ENS Cachan Lecture 5: Introduction to generative models Iasonas Kokkinos Iasonas.kokkinos@ecp.fr Center for Visual Computing Ecole Centrale Paris

More information

Introduction to Machine Learning. Xiaojin Zhu

Introduction to Machine Learning. Xiaojin Zhu Introduction to Machine Learning Xiaojin Zhu jerryzhu@cs.wisc.edu Read Chapter 1 of this book: Xiaojin Zhu and Andrew B. Goldberg. Introduction to Semi- Supervised Learning. http://www.morganclaypool.com/doi/abs/10.2200/s00196ed1v01y200906aim006

More information

Supervised vs. Unsupervised Learning

Supervised vs. Unsupervised Learning Clustering Supervised vs. Unsupervised Learning So far we have assumed that the training samples used to design the classifier were labeled by their class membership (supervised learning) We assume now

More information

Random projection for non-gaussian mixture models

Random projection for non-gaussian mixture models Random projection for non-gaussian mixture models Győző Gidófalvi Department of Computer Science and Engineering University of California, San Diego La Jolla, CA 92037 gyozo@cs.ucsd.edu Abstract Recently,

More information

Introduction to Mobile Robotics

Introduction to Mobile Robotics Introduction to Mobile Robotics Clustering Wolfram Burgard Cyrill Stachniss Giorgio Grisetti Maren Bennewitz Christian Plagemann Clustering (1) Common technique for statistical data analysis (machine learning,

More information

Classification. Vladimir Curic. Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University

Classification. Vladimir Curic. Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University Classification Vladimir Curic Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University Outline An overview on classification Basics of classification How to choose appropriate

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

CSE 258 Lecture 6. Web Mining and Recommender Systems. Community Detection

CSE 258 Lecture 6. Web Mining and Recommender Systems. Community Detection CSE 258 Lecture 6 Web Mining and Recommender Systems Community Detection Dimensionality reduction Goal: take high-dimensional data, and describe it compactly using a small number of dimensions Assumption:

More information

Statistics 202: Statistical Aspects of Data Mining

Statistics 202: Statistical Aspects of Data Mining Statistics 202: Statistical Aspects of Data Mining Professor Rajan Patel Lecture 11 = Chapter 8 Agenda: 1)Reminder about final exam 2)Finish Chapter 5 3)Chapter 8 1 Class Project The class project is due

More information

K-Means Clustering. Sargur Srihari

K-Means Clustering. Sargur Srihari K-Means Clustering Sargur srihari@cedar.buffalo.edu 1 Topics in Mixture Models and EM Mixture models K-means Clustering Mixtures of Gaussians Maximum Likelihood EM for Gaussian mistures EM Algorithm Gaussian

More information

MACHINE LEARNING: CLUSTERING, AND CLASSIFICATION. Steve Tjoa June 25, 2014

MACHINE LEARNING: CLUSTERING, AND CLASSIFICATION. Steve Tjoa June 25, 2014 MACHINE LEARNING: CLUSTERING, AND CLASSIFICATION Steve Tjoa kiemyang@gmail.com June 25, 2014 Review from Day 2 Supervised vs. Unsupervised Unsupervised - clustering Supervised binary classifiers (2 classes)

More information

CS 2750 Machine Learning. Lecture 19. Clustering. CS 2750 Machine Learning. Clustering. Groups together similar instances in the data sample

CS 2750 Machine Learning. Lecture 19. Clustering. CS 2750 Machine Learning. Clustering. Groups together similar instances in the data sample Lecture 9 Clustering Milos Hauskrecht milos@cs.pitt.edu 539 Sennott Square Clustering Groups together similar instances in the data sample Basic clustering problem: distribute data into k different groups

More information

Clustering. CS294 Practical Machine Learning Junming Yin 10/09/06

Clustering. CS294 Practical Machine Learning Junming Yin 10/09/06 Clustering CS294 Practical Machine Learning Junming Yin 10/09/06 Outline Introduction Unsupervised learning What is clustering? Application Dissimilarity (similarity) of objects Clustering algorithm K-means,

More information

Clustering k-mean clustering

Clustering k-mean clustering Clustering k-mean clustering Genome 373 Genomic Informatics Elhanan Borenstein The clustering problem: partition genes into distinct sets with high homogeneity and high separation Clustering (unsupervised)

More information

( ) =cov X Y = W PRINCIPAL COMPONENT ANALYSIS. Eigenvectors of the covariance matrix are the principal components

( ) =cov X Y = W PRINCIPAL COMPONENT ANALYSIS. Eigenvectors of the covariance matrix are the principal components Review Lecture 14 ! PRINCIPAL COMPONENT ANALYSIS Eigenvectors of the covariance matrix are the principal components 1. =cov X Top K principal components are the eigenvectors with K largest eigenvalues

More information

Unsupervised Data Mining: Clustering. Izabela Moise, Evangelos Pournaras, Dirk Helbing

Unsupervised Data Mining: Clustering. Izabela Moise, Evangelos Pournaras, Dirk Helbing Unsupervised Data Mining: Clustering Izabela Moise, Evangelos Pournaras, Dirk Helbing Izabela Moise, Evangelos Pournaras, Dirk Helbing 1 1. Supervised Data Mining Classification Regression Outlier detection

More information

CSE 255 Lecture 5. Data Mining and Predictive Analytics. Dimensionality Reduction

CSE 255 Lecture 5. Data Mining and Predictive Analytics. Dimensionality Reduction CSE 255 Lecture 5 Data Mining and Predictive Analytics Dimensionality Reduction Course outline Week 4: I ll cover homework 1, and get started on Recommender Systems Week 5: I ll cover homework 2 (at the

More information

CHAPTER 4: CLUSTER ANALYSIS

CHAPTER 4: CLUSTER ANALYSIS CHAPTER 4: CLUSTER ANALYSIS WHAT IS CLUSTER ANALYSIS? A cluster is a collection of data-objects similar to one another within the same group & dissimilar to the objects in other groups. Cluster analysis

More information

CS 1675 Introduction to Machine Learning Lecture 18. Clustering. Clustering. Groups together similar instances in the data sample

CS 1675 Introduction to Machine Learning Lecture 18. Clustering. Clustering. Groups together similar instances in the data sample CS 1675 Introduction to Machine Learning Lecture 18 Clustering Milos Hauskrecht milos@cs.pitt.edu 539 Sennott Square Clustering Groups together similar instances in the data sample Basic clustering problem:

More information

Computer vision: models, learning and inference. Chapter 10 Graphical Models

Computer vision: models, learning and inference. Chapter 10 Graphical Models Computer vision: models, learning and inference Chapter 10 Graphical Models Independence Two variables x 1 and x 2 are independent if their joint probability distribution factorizes as Pr(x 1, x 2 )=Pr(x

More information

Announcements. Image Segmentation. From images to objects. Extracting objects. Status reports next Thursday ~5min presentations in class

Announcements. Image Segmentation. From images to objects. Extracting objects. Status reports next Thursday ~5min presentations in class Image Segmentation Announcements Status reports next Thursday ~5min presentations in class Project voting From Sandlot Science Today s Readings Forsyth & Ponce, Chapter 1 (plus lots of optional references

More information

Statistics 202: Data Mining. c Jonathan Taylor. Week 8 Based in part on slides from textbook, slides of Susan Holmes. December 2, / 1

Statistics 202: Data Mining. c Jonathan Taylor. Week 8 Based in part on slides from textbook, slides of Susan Holmes. December 2, / 1 Week 8 Based in part on slides from textbook, slides of Susan Holmes December 2, 2012 1 / 1 Part I Clustering 2 / 1 Clustering Clustering Goal: Finding groups of objects such that the objects in a group

More information

Discrete geometry. Lecture 2. Alexander & Michael Bronstein tosca.cs.technion.ac.il/book

Discrete geometry. Lecture 2. Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Discrete geometry Lecture 2 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 The world is continuous, but the mind is discrete

More information

Gaussian Mixture Models For Clustering Data. Soft Clustering and the EM Algorithm

Gaussian Mixture Models For Clustering Data. Soft Clustering and the EM Algorithm Gaussian Mixture Models For Clustering Data Soft Clustering and the EM Algorithm K-Means Clustering Input: Observations: xx ii R dd ii {1,., NN} Number of Clusters: kk Output: Cluster Assignments. Cluster

More information

Clustering. k-mean clustering. Genome 559: Introduction to Statistical and Computational Genomics Elhanan Borenstein

Clustering. k-mean clustering. Genome 559: Introduction to Statistical and Computational Genomics Elhanan Borenstein Clustering k-mean clustering Genome 559: Introduction to Statistical and Computational Genomics Elhanan Borenstein A quick review The clustering problem: homogeneity vs. separation Different representations

More information

Lecture 11: Clustering Introduction and Projects Machine Learning

Lecture 11: Clustering Introduction and Projects Machine Learning Lecture 11: Clustering Introduction and Projects Machine Learning Andrew Rosenberg March 12, 2010 1/1 Last Time Junction Tree Algorithm Efficient Marginals in Graphical Models 2/1 Today Clustering Project

More information

Clustering. Supervised vs. Unsupervised Learning

Clustering. Supervised vs. Unsupervised Learning Clustering Supervised vs. Unsupervised Learning So far we have assumed that the training samples used to design the classifier were labeled by their class membership (supervised learning) We assume now

More information

Clustering. Image segmentation, document clustering, protein class discovery, compression

Clustering. Image segmentation, document clustering, protein class discovery, compression Clustering CS 444 Some material on these is slides borrowed from Andrew Moore's machine learning tutorials located at: Clustering The problem of grouping unlabeled data on the basis of similarity. A key

More information

Clustering CS 550: Machine Learning

Clustering CS 550: Machine Learning Clustering CS 550: Machine Learning This slide set mainly uses the slides given in the following links: http://www-users.cs.umn.edu/~kumar/dmbook/ch8.pdf http://www-users.cs.umn.edu/~kumar/dmbook/dmslides/chap8_basic_cluster_analysis.pdf

More information

What to come. There will be a few more topics we will cover on supervised learning

What to come. There will be a few more topics we will cover on supervised learning Summary so far Supervised learning learn to predict Continuous target regression; Categorical target classification Linear Regression Classification Discriminative models Perceptron (linear) Logistic regression

More information

A Course in Machine Learning

A Course in Machine Learning A Course in Machine Learning Hal Daumé III 13 UNSUPERVISED LEARNING If you have access to labeled training data, you know what to do. This is the supervised setting, in which you have a teacher telling

More information

Machine Learning. B. Unsupervised Learning B.1 Cluster Analysis. Lars Schmidt-Thieme

Machine Learning. B. Unsupervised Learning B.1 Cluster Analysis. Lars Schmidt-Thieme Machine Learning B. Unsupervised Learning B.1 Cluster Analysis Lars Schmidt-Thieme Information Systems and Machine Learning Lab (ISMLL) Institute for Computer Science University of Hildesheim, Germany

More information

CSE 258 Lecture 5. Web Mining and Recommender Systems. Dimensionality Reduction

CSE 258 Lecture 5. Web Mining and Recommender Systems. Dimensionality Reduction CSE 258 Lecture 5 Web Mining and Recommender Systems Dimensionality Reduction This week How can we build low dimensional representations of high dimensional data? e.g. how might we (compactly!) represent

More information

Unsupervised Learning. CS 3793/5233 Artificial Intelligence Unsupervised Learning 1

Unsupervised Learning. CS 3793/5233 Artificial Intelligence Unsupervised Learning 1 Unsupervised CS 3793/5233 Artificial Intelligence Unsupervised 1 EM k-means Procedure Data Random Assignment Assign 1 Assign 2 Soft k-means In clustering, the target feature is not given. Goal: Construct

More information

Clustering: Classic Methods and Modern Views

Clustering: Classic Methods and Modern Views Clustering: Classic Methods and Modern Views Marina Meilă University of Washington mmp@stat.washington.edu June 22, 2015 Lorentz Center Workshop on Clusters, Games and Axioms Outline Paradigms for clustering

More information

Content-based image and video analysis. Machine learning

Content-based image and video analysis. Machine learning Content-based image and video analysis Machine learning for multimedia retrieval 04.05.2009 What is machine learning? Some problems are very hard to solve by writing a computer program by hand Almost all

More information

Clustering. Mihaela van der Schaar. January 27, Department of Engineering Science University of Oxford

Clustering. Mihaela van der Schaar. January 27, Department of Engineering Science University of Oxford Department of Engineering Science University of Oxford January 27, 2017 Many datasets consist of multiple heterogeneous subsets. Cluster analysis: Given an unlabelled data, want algorithms that automatically

More information

Data Mining: Data. Lecture Notes for Chapter 2. Introduction to Data Mining

Data Mining: Data. Lecture Notes for Chapter 2. Introduction to Data Mining Data Mining: Data Lecture Notes for Chapter 2 Introduction to Data Mining by Tan, Steinbach, Kumar Data Preprocessing Aggregation Sampling Dimensionality Reduction Feature subset selection Feature creation

More information

Overview Citation. ML Introduction. Overview Schedule. ML Intro Dataset. Introduction to Semi-Supervised Learning Review 10/4/2010

Overview Citation. ML Introduction. Overview Schedule. ML Intro Dataset. Introduction to Semi-Supervised Learning Review 10/4/2010 INFORMATICS SEMINAR SEPT. 27 & OCT. 4, 2010 Introduction to Semi-Supervised Learning Review 2 Overview Citation X. Zhu and A.B. Goldberg, Introduction to Semi- Supervised Learning, Morgan & Claypool Publishers,

More information

Segmentation Computer Vision Spring 2018, Lecture 27

Segmentation Computer Vision Spring 2018, Lecture 27 Segmentation http://www.cs.cmu.edu/~16385/ 16-385 Computer Vision Spring 218, Lecture 27 Course announcements Homework 7 is due on Sunday 6 th. - Any questions about homework 7? - How many of you have

More information

Clustering in Data Mining

Clustering in Data Mining Clustering in Data Mining Classification Vs Clustering When the distribution is based on a single parameter and that parameter is known for each object, it is called classification. E.g. Children, young,

More information

CSC 411: Lecture 14: Principal Components Analysis & Autoencoders

CSC 411: Lecture 14: Principal Components Analysis & Autoencoders CSC 411: Lecture 14: Principal Components Analysis & Autoencoders Raquel Urtasun & Rich Zemel University of Toronto Nov 4, 2015 Urtasun & Zemel (UofT) CSC 411: 14-PCA & Autoencoders Nov 4, 2015 1 / 18

More information

Parameter Selection for EM Clustering Using Information Criterion and PDDP

Parameter Selection for EM Clustering Using Information Criterion and PDDP Parameter Selection for EM Clustering Using Information Criterion and PDDP Ujjwal Das Gupta,Vinay Menon and Uday Babbar Abstract This paper presents an algorithm to automatically determine the number of

More information

Solution Sketches Midterm Exam COSC 6342 Machine Learning March 20, 2013

Solution Sketches Midterm Exam COSC 6342 Machine Learning March 20, 2013 Your Name: Your student id: Solution Sketches Midterm Exam COSC 6342 Machine Learning March 20, 2013 Problem 1 [5+?]: Hypothesis Classes Problem 2 [8]: Losses and Risks Problem 3 [11]: Model Generation

More information

Slide07 Haykin Chapter 9: Self-Organizing Maps

Slide07 Haykin Chapter 9: Self-Organizing Maps Slide07 Haykin Chapter 9: Self-Organizing Maps CPSC 636-600 Instructor: Yoonsuck Choe Spring 2012 Introduction Self-organizing maps (SOM) is based on competitive learning, where output neurons compete

More information

Today. Lecture 4: Last time. The EM algorithm. We examine clustering in a little more detail; we went over it a somewhat quickly last time

Today. Lecture 4: Last time. The EM algorithm. We examine clustering in a little more detail; we went over it a somewhat quickly last time Today Lecture 4: We examine clustering in a little more detail; we went over it a somewhat quickly last time The CAD data will return and give us an opportunity to work with curves (!) We then examine

More information

ALTERNATIVE METHODS FOR CLUSTERING

ALTERNATIVE METHODS FOR CLUSTERING ALTERNATIVE METHODS FOR CLUSTERING K-Means Algorithm Termination conditions Several possibilities, e.g., A fixed number of iterations Objects partition unchanged Centroid positions don t change Convergence

More information

Bayes Classifiers and Generative Methods

Bayes Classifiers and Generative Methods Bayes Classifiers and Generative Methods CSE 4309 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1 The Stages of Supervised Learning To

More information

Hierarchical Clustering 4/5/17

Hierarchical Clustering 4/5/17 Hierarchical Clustering 4/5/17 Hypothesis Space Continuous inputs Output is a binary tree with data points as leaves. Useful for explaining the training data. Not useful for making new predictions. Direction

More information

IBL and clustering. Relationship of IBL with CBR

IBL and clustering. Relationship of IBL with CBR IBL and clustering Distance based methods IBL and knn Clustering Distance based and hierarchical Probability-based Expectation Maximization (EM) Relationship of IBL with CBR + uses previously processed

More information

Latent Variable Models and Expectation Maximization

Latent Variable Models and Expectation Maximization Latent Variable Models and Expectation Maximization Oliver Schulte - CMPT 726 Bishop PRML Ch. 9 2 4 6 8 1 12 14 16 18 2 4 6 8 1 12 14 16 18 5 1 15 2 25 5 1 15 2 25 2 4 6 8 1 12 14 2 4 6 8 1 12 14 5 1 15

More information

A Comparative study of Clustering Algorithms using MapReduce in Hadoop

A Comparative study of Clustering Algorithms using MapReduce in Hadoop A Comparative study of Clustering Algorithms using MapReduce in Hadoop Dweepna Garg 1, Khushboo Trivedi 2, B.B.Panchal 3 1 Department of Computer Science and Engineering, Parul Institute of Engineering

More information

Spectral Classification

Spectral Classification Spectral Classification Spectral Classification Supervised versus Unsupervised Classification n Unsupervised Classes are determined by the computer. Also referred to as clustering n Supervised Classes

More information

Unsupervised Learning

Unsupervised Learning Unsupervised Learning Unsupervised learning Until now, we have assumed our training samples are labeled by their category membership. Methods that use labeled samples are said to be supervised. However,

More information

Chapter 7: Competitive learning, clustering, and self-organizing maps

Chapter 7: Competitive learning, clustering, and self-organizing maps Chapter 7: Competitive learning, clustering, and self-organizing maps António R. C. Paiva EEL 6814 Spring 2008 Outline Competitive learning Clustering Self-Organizing Maps What is competition in neural

More information

Further Applications of a Particle Visualization Framework

Further Applications of a Particle Visualization Framework Further Applications of a Particle Visualization Framework Ke Yin, Ian Davidson Department of Computer Science SUNY-Albany 1400 Washington Ave. Albany, NY, USA, 12222. Abstract. Our previous work introduced

More information

10-701/15-781, Fall 2006, Final

10-701/15-781, Fall 2006, Final -7/-78, Fall 6, Final Dec, :pm-8:pm There are 9 questions in this exam ( pages including this cover sheet). If you need more room to work out your answer to a question, use the back of the page and clearly

More information

Cluster Analysis. Jia Li Department of Statistics Penn State University. Summer School in Statistics for Astronomers IV June 9-14, 2008

Cluster Analysis. Jia Li Department of Statistics Penn State University. Summer School in Statistics for Astronomers IV June 9-14, 2008 Cluster Analysis Jia Li Department of Statistics Penn State University Summer School in Statistics for Astronomers IV June 9-1, 8 1 Clustering A basic tool in data mining/pattern recognition: Divide a

More information

Lecture-17: Clustering with K-Means (Contd: DT + Random Forest)

Lecture-17: Clustering with K-Means (Contd: DT + Random Forest) Lecture-17: Clustering with K-Means (Contd: DT + Random Forest) Medha Vidyotma April 24, 2018 1 Contd. Random Forest For Example, if there are 50 scholars who take the measurement of the length of the

More information