ADVANCED MACHINE LEARNING MACHINE LEARNING. Kernel for Clustering kernel K-Means
|
|
|
- Andrew Spencer Rich
- 5 years ago
- Views:
Transcription
1 1 MACHINE LEARNING Kernel for Clustering ernel K-Means
2 Outline of Today s Lecture 1. Review principle and steps of K-Means algorithm. Derive ernel version of K-means 3. Exercise: Discuss the geometrical division of the space generated by RBF versus Polynomial ernels
3 3 K-means Clustering: Iterative Technique C x 1 C 1 x 1 1. Initialization: for C, 1... K clusters, pic K arbitrary centroids and set their geometric mean to random values.
4 4 K-means Clustering: E-step (expectation-step) C x i i arg min d x, 1 C 1 x 1 i x i th data point geometric centroid Assignment Step: Calculate the distance from each data point to each centroid. Assign each data point to its closest centroid. If a tie happens (i.e. two centroids are equidistant to a data point, one assigns the data point to the smallest winning centroid).
5 5 Update step (M-Step): Recompute the position of centroid based on the assignment of the points K-means Clustering: M-step (maximization-step) x 1 x number of datapoints in cluster : 1 i i i x C x C x m C m 1 C 1 C
6 6 K-means Clustering: E-step (expectation-step) x 1 C 1 C i i arg min d x, x 1 i x i th data point geometric centroid Go bac to the assignment step and repeat the update step.
7 7 K-means Clustering x C 1 C 1 x 1 Stopping Criterion: stop the process when the centers are stable.
8 8 K-means Clustering Intersection points x x 1 K-means creates a hard partitioning of the dataset
9 9 K-means Clustering Intersection points x x 1 K-Means clustering generates K disjoint clusters by minimizing the following quadratic cost function: K 1 K,...,, with, J d x d x x i i i i 1 x c
10 10 MACHINE LEARNING 01 K-means Clustering: Advantages The algorithm is guaranteed to converge in a finite number of iterations (but it converges to a local optimum!) It is computationally cheap and faster than other clustering techniques - update step is ~O(N).
11 11 MACHINE LEARNING 01 K-means Clustering: Sensitivity Very sensitive to the choice of the number of clusters K (hyperparameter) and the initialization.
12 MACHINE LEARNING 01 K-means Clustering: Hyperparameters K-means with usual norm- distance perform linear separation. Changing the power p of the metric allows to generate non-linear boundaries i i p d x, ; p L x : Lp norm P=1 P= P=3 P=4 1
13 13 Kernel K-means Idea: The objective function of K-means is composed of an inner product across datapoints. One can replace the inner product with a ernel to perform inner product in feature space. Exploit the principle of the ernel to perform classical K-means clustering with norm- in feature space: This yields non-linear boundaries. This retains simplicity of computation of linear K-means.
14 Kernel K-means K Means algorithm minimizes the objective function : J m K 1 K j j x C,..., x with 1 x C : number of datapoints in cluster j C m x j M i x i 1 Project into a feature space J K 1,..., K x j 1 x C j We cannot observe the mean in feature space. Construct the mean in feature space using images of points in same cluster. x j C m x j 14
15 15 Kernel K-means J K 1,..., x K j l x C = j m 1 x C K j j x, x T x x l x j j l x C T l T j x x x t x l t l x, x C 1 x C K = x, x j j m, t m j j x C l j x, x j l x, x m i t t l x x x, x t l x, x C 1 x C Objective function in feature space m
16 16 Kernel K-means Kernel K-means algorithm is also an iterative procedure: 1. Initialization: pic K clusters. Assignment Step: Assign each data point to its closest centroid (E-step). j i j x, x x, x j j l x x arg min, arg min, l i i i x C x, x C d x C m m 3. Update Step: Update the list of points belonging to each centroid (M-step) 4. Go bac to step and repeat the process until the clusters are stable.
17 17 Kernel K-means: Exercise I Metric used in ernel K-means to determine cluster assigment: arg min d x, C min, x x, i j j l x x x, x j l i i i j x C x, x C m m a) Draw the partitioning of the space when using two datapoints in dimensions with a rbf ernel and with K=. Discuss the effect of the initialization. b) Do a with a homogeneous polynomial ernel with p=. Is the result affected by the placement of the datapoints?
18 3 Kernel K-means With a RBF ernel Cst of value 1 If xi is close to all points in cluster, this is close to 1. If the points are well grouped in cluster, this sum is close to 1. arg min, min, x x, i j j l x x x, x j l i i i j x C x, x C d x C m m
19 4 Kernel K-means: Exercise II Metric used in ernel K-means to determine cluster assigment: arg min d x, C min, x x, i j j l x x x, x j l i i i j x C x, x C m m i) Draw the partitioning of the space for the datapoints on the right figure when using a rbf ernel and K. ii) Discuss the effect of. iii) Discuss the effect of the initialization.
20 6 Kernel K-means: Exercise III Metric used in ernel K-means to determine cluster assigment: arg min, min, x x, i j j l x x x, x j l i i i j x C x, x C d x C m m a) Draw the partitioning of the space for these two clusters with a rbf ernel. b) Discuss the effect of the number of datapoints in each cluster.
21 8 Kernel K-means With a polynomial ernel Positive value A: Some of the terms change sign depending on the angle between the pair of datapoints. The relative effect of the terms depends on the position from the origin (norm). B: If the points are aligned in the same Quadran, the sum is maximal arg min, min, x x, i j j l x x x, x j l i i i j x C x, x C d x C m m. A datapoint will be assigned to the cluster in the closest partition.
22 9 Kernel K-means: Exercise IV Metric used in ernel K-means to determine cluster assigment: arg min d x, C min, x x, i j j l x x x, x j l i i i j x C x, x C m m Draw the partitioning of the space when using 4 datapoints (K=4) with a polynomial ernel. Is the result affected by the placement of the datapoints? What is the effect of the power of the polynomial p?
23 35 Kernel K-means: examples Rbf Kernel, Clusters
24 36 Kernel K-means: examples Rbf Kernel, Clusters
25 37 Kernel K-means: examples Rbf Kernel, Clusters Kernel width: 0.5 Kernel width: 0.05
26 38 Kernel K-means: Limitations Choice of number of Clusters in Kernel K-means is important
27 39 Kernel K-means: Limitations Choice of number of Clusters in Kernel K-means is important
28 40 Kernel K-means: Limitations Choice of number of Clusters in Kernel K-means is important
29 41 MACHINE LEARNING 01 Limitations of ernel K-means Raw Data
30 4 MACHINE LEARNING 01 Limitations of ernel K-means ernel K-means with K=, RBF ernel
31 43 Summary 1. Kernel K-means follows the same principle as K-means. It is an iterative procedure, ain to Expectation-Maximization.. As K-means, it depends on initialization of the center which is random. 3. As K-means, the solution depends on choosing well the number of clusters K.
4. Ad-hoc I: Hierarchical clustering
4. Ad-hoc I: Hierarchical clustering Hierarchical versus Flat Flat methods generate a single partition into k clusters. The number k of clusters has to be determined by the user ahead of time. Hierarchical
Machine Learning for Signal Processing Clustering. Bhiksha Raj Class Oct 2016
Machine Learning for Signal Processing Clustering Bhiksha Raj Class 11. 13 Oct 2016 1 Statistical Modelling and Latent Structure Much of statistical modelling attempts to identify latent structure in the
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
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
k-means Clustering David S. Rosenberg April 24, 2018 New York University
k-means Clustering David S. Rosenberg New York University April 24, 2018 David S. Rosenberg (New York University) DS-GA 1003 / CSCI-GA 2567 April 24, 2018 1 / 19 Contents 1 k-means Clustering 2 k-means:
CSSE463: Image Recognition Day 21
CSSE463: Image Recognition Day 21 Sunset detector due. Foundations of Image Recognition completed This wee: K-means: a method of Image segmentation Questions? An image to segment Segmentation The process
CSE 5243 INTRO. TO DATA MINING
CSE 5243 INTRO. TO DATA MINING Cluster Analysis: Basic Concepts and Methods Huan Sun, CSE@The Ohio State University 09/25/2017 Slides adapted from UIUC CS412, Fall 2017, by Prof. Jiawei Han 2 Chapter 10.
Unsupervised Learning Partitioning Methods
Unsupervised Learning Partitioning Methods Road Map 1. Basic Concepts 2. K-Means 3. K-Medoids 4. CLARA & CLARANS Cluster Analysis Unsupervised learning (i.e., Class label is unknown) Group data to form
Introduction to Artificial Intelligence
Introduction to Artificial Intelligence COMP307 Machine Learning 2: 3-K Techniques Yi Mei yi.mei@ecs.vuw.ac.nz 1 Outline K-Nearest Neighbour method Classification (Supervised learning) Basic NN (1-NN)
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:
k-means, k-means++ Barna Saha March 8, 2016
k-means, k-means++ Barna Saha March 8, 2016 K-Means: The Most Popular Clustering Algorithm k-means clustering problem is one of the oldest and most important problem. K-Means: The Most Popular Clustering
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
Clustering: Centroid-Based Partitioning
Clustering: Centroid-Based Partitioning Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong 1 / 29 Y Tao Clustering: Centroid-Based Partitioning In this lecture, we
Clustering. Chapter 10 in Introduction to statistical learning
Clustering Chapter 10 in Introduction to statistical learning 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 1 Clustering ² Clustering is the art of finding groups in data (Kaufman and Rousseeuw, 1990). ² What
k-means A classical clustering algorithm
k-means A classical clustering algorithm Devert Alexandre School of Software Engineering of USTC 30 November 2012 Slide 1/65 Table of Contents 1 Introduction 2 Visual demo Step by step Voronoi diagrams
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
Applications. Foreground / background segmentation Finding skin-colored regions. Finding the moving objects. Intelligent scissors
Segmentation I Goal Separate image into coherent regions Berkeley segmentation database: http://www.eecs.berkeley.edu/research/projects/cs/vision/grouping/segbench/ Slide by L. Lazebnik Applications Intelligent
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
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
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,
Clustering. (Part 2)
Clustering (Part 2) 1 k-means clustering 2 General Observations on k-means clustering In essence, k-means clustering aims at minimizing cluster variance. It is typically used in Euclidean spaces and works
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:
CLUSTERING. CSE 634 Data Mining Prof. Anita Wasilewska TEAM 16
CLUSTERING CSE 634 Data Mining Prof. Anita Wasilewska TEAM 16 1. K-medoids: REFERENCES https://www.coursera.org/learn/cluster-analysis/lecture/nj0sb/3-4-the-k-medoids-clustering-method https://anuradhasrinivas.files.wordpress.com/2013/04/lesson8-clustering.pdf
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
Exploratory Data Analysis using Self-Organizing Maps. Madhumanti Ray
Exploratory Data Analysis using Self-Organizing Maps Madhumanti Ray Content Introduction Data Analysis methods Self-Organizing Maps Conclusion Visualization of high-dimensional data items Exploratory data
Clustering: K-means and Kernel K-means
Clustering: K-means and Kernel K-means Piyush Rai Machine Learning (CS771A) Aug 31, 2016 Machine Learning (CS771A) Clustering: K-means and Kernel K-means 1 Clustering Usually an unsupervised learning problem
1 Case study of SVM (Rob)
DRAFT a final version will be posted shortly COS 424: Interacting with Data Lecturer: Rob Schapire and David Blei Lecture # 8 Scribe: Indraneel Mukherjee March 1, 2007 In the previous lecture we saw how
INF4820, Algorithms for AI and NLP: Evaluating Classifiers Clustering
INF4820, Algorithms for AI and NLP: Evaluating Classifiers Clustering Erik Velldal University of Oslo Sept. 18, 2012 Topics for today 2 Classification Recap Evaluating classifiers Accuracy, precision,
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
Cluster Analysis. Ying Shen, SSE, Tongji University
Cluster Analysis Ying Shen, SSE, Tongji University Cluster analysis Cluster analysis groups data objects based only on the attributes in the data. The main objective is that The objects within a group
Applications of Linear Programming
Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 1 Why LP? Linear programming (LP, also called linear
INF4820. Clustering. Erik Velldal. Nov. 17, University of Oslo. Erik Velldal INF / 22
INF4820 Clustering Erik Velldal University of Oslo Nov. 17, 2009 Erik Velldal INF4820 1 / 22 Topics for Today More on unsupervised machine learning for data-driven categorization: clustering. The task
Introduction to Information Retrieval
Introduction to Information Retrieval http://informationretrieval.org IIR 6: Flat Clustering Wiltrud Kessler & Hinrich Schütze Institute for Natural Language Processing, University of Stuttgart 0-- / 83
INF 4300 Classification III Anne Solberg The agenda today:
INF 4300 Classification III Anne Solberg 28.10.15 The agenda today: More on estimating classifier accuracy Curse of dimensionality and simple feature selection knn-classification K-means clustering 28.10.15
K Nearest Neighbor Wrap Up K- Means Clustering. Slides adapted from Prof. Carpuat
K Nearest Neighbor Wrap Up K- Means Clustering Slides adapted from Prof. Carpuat K Nearest Neighbor classification Classification is based on Test instance with Training Data K: number of neighbors that
Clustering. SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic
Clustering SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic Clustering is one of the fundamental and ubiquitous tasks in exploratory data analysis a first intuition about the
CPSC 425: Computer Vision
1 / 31 CPSC 425: Computer Vision Instructor: Jim Little little@cs.ubc.ca Department of Computer Science University of British Columbia Lecture Notes 2016/2017 Term 2 2 / 31 Menu March 16, 2017 Topics:
The Simplex Algorithm for LP, and an Open Problem
The Simplex Algorithm for LP, and an Open Problem Linear Programming: General Formulation Inputs: real-valued m x n matrix A, and vectors c in R n and b in R m Output: n-dimensional vector x There is one
K-means and Hierarchical Clustering
K-means and Hierarchical Clustering Xiaohui Xie University of California, Irvine K-means and Hierarchical Clustering p.1/18 Clustering Given n data points X = {x 1, x 2,, x n }. Clustering is the partitioning
Flat Clustering. Slides are mostly from Hinrich Schütze. March 27, 2017
Flat Clustering Slides are mostly from Hinrich Schütze March 7, 07 / 79 Overview Recap Clustering: Introduction 3 Clustering in IR 4 K-means 5 Evaluation 6 How many clusters? / 79 Outline Recap Clustering:
Machine Learning Department School of Computer Science Carnegie Mellon University. K- Means + GMMs
10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University K- Means + GMMs Clustering Readings: Murphy 25.5 Bishop 12.1, 12.3 HTF 14.3.0 Mitchell
Search Engines. Information Retrieval in Practice
Search Engines Information Retrieval in Practice All slides Addison Wesley, 2008 Classification and Clustering Classification and clustering are classical pattern recognition / machine learning problems
Data Mining Cluster Analysis: Basic Concepts and Algorithms. Slides From Lecture Notes for Chapter 8. Introduction to Data Mining
Data Mining Cluster Analysis: Basic Concepts and Algorithms Slides From Lecture Notes for Chapter 8 Introduction to Data Mining by Tan, Steinbach, Kumar Tan,Steinbach, Kumar Introduction to Data Mining
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
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
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
CLASSIFICATION WITH RADIAL BASIS AND PROBABILISTIC NEURAL NETWORKS
CLASSIFICATION WITH RADIAL BASIS AND PROBABILISTIC NEURAL NETWORKS CHAPTER 4 CLASSIFICATION WITH RADIAL BASIS AND PROBABILISTIC NEURAL NETWORKS 4.1 Introduction Optical character recognition is one of
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-
Parallel Algorithms K means Clustering
CSE 633: Parallel Algorithms Spring 2014 Parallel Algorithms K means Clustering Final Results By: Andreina Uzcategui Outline The problem Algorithm Description Parallel Algorithm Implementation(MPI) Test
COMP 551 Applied Machine Learning Lecture 13: Unsupervised learning
COMP 551 Applied Machine Learning Lecture 13: Unsupervised learning Associate Instructor: Herke van Hoof (herke.vanhoof@mail.mcgill.ca) Slides mostly by: (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/comp551
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
Lecture 2 Convex Sets
Optimization Theory and Applications Lecture 2 Convex Sets Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall 2016 2016/9/29 Lecture 2: Convex Sets 1 Outline
Lecture 12 Recognition
Institute of Informatics Institute of Neuroinformatics Lecture 12 Recognition Davide Scaramuzza http://rpg.ifi.uzh.ch/ 1 Lab exercise today replaced by Deep Learning Tutorial by Antonio Loquercio Room
ENG 8801/ Special Topics in Computer Engineering: Pattern Recognition. Memorial University of Newfoundland Pattern Recognition
Memorial University of Newfoundland Pattern Recognition Lecture 15, June 29, 2006 http://www.engr.mun.ca/~charlesr Office Hours: Tuesdays & Thursdays 8:30-9:30 PM EN-3026 July 2006 Sunday Monday Tuesday
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
Introduction to Information Retrieval
Introduction to Information Retrieval http://informationretrieval.org IIR 16: Flat Clustering Hinrich Schütze Institute for Natural Language Processing, Universität Stuttgart 2009.06.16 1/ 64 Overview
CSE 5243 INTRO. TO DATA MINING
CSE 5243 INTRO. TO DATA MINING Cluster Analysis: Basic Concepts and Methods Huan Sun, CSE@The Ohio State University Slides adapted from UIUC CS412, Fall 2017, by Prof. Jiawei Han 2 Chapter 10. Cluster
Clustering Results. Result List Example. Clustering Results. Information Retrieval
Information Retrieval INFO 4300 / CS 4300! Presenting Results Clustering Clustering Results! Result lists often contain documents related to different aspects of the query topic! Clustering is used to
Introduction to Computer Science
DM534 Introduction to Computer Science Clustering and Feature Spaces Richard Roettger: About Me Computer Science (Technical University of Munich and thesis at the ICSI at the University of California at
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
Clustering. Robert M. Haralick. Computer Science, Graduate Center City University of New York
Clustering Robert M. Haralick Computer Science, Graduate Center City University of New York Outline K-means 1 K-means 2 3 4 5 Clustering K-means The purpose of clustering is to determine the similarity
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
Hierarchical Clustering
Hierarchical Clustering Hierarchical Clustering Produces a set of nested clusters organized as a hierarchical tree Can be visualized as a dendrogram A tree-like diagram that records the sequences of merges
Based on Raymond J. Mooney s slides
Instance Based Learning Based on Raymond J. Mooney s slides University of Texas at Austin 1 Example 2 Instance-Based Learning Unlike other learning algorithms, does not involve construction of an explicit
Document Clustering: Comparison of Similarity Measures
Document Clustering: Comparison of Similarity Measures Shouvik Sachdeva Bhupendra Kastore Indian Institute of Technology, Kanpur CS365 Project, 2014 Outline 1 Introduction The Problem and the Motivation
Olmo S. Zavala Romero. Clustering Hierarchical Distance Group Dist. K-means. Center of Atmospheric Sciences, UNAM.
Center of Atmospheric Sciences, UNAM November 16, 2016 Cluster Analisis Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster)
ECG782: Multidimensional Digital Signal Processing
ECG782: Multidimensional Digital Signal Processing Object Recognition http://www.ee.unlv.edu/~b1morris/ecg782/ 2 Outline Knowledge Representation Statistical Pattern Recognition Neural Networks Boosting
Automatic Cluster Number Selection using a Split and Merge K-Means Approach
Automatic Cluster Number Selection using a Split and Merge K-Means Approach Markus Muhr and Michael Granitzer 31st August 2009 The Know-Center is partner of Austria's Competence Center Program COMET. Agenda
Robust PDF Table Locator
Robust PDF Table Locator December 17, 2016 1 Introduction Data scientists rely on an abundance of tabular data stored in easy-to-machine-read formats like.csv files. Unfortunately, most government records
Clustering. Lecture 6, 1/24/03 ECS289A
Clustering Lecture 6, 1/24/03 What is Clustering? Given n objects, assign them to groups (clusters) based on their similarity Unsupervised Machine Learning Class Discovery Difficult, and maybe ill-posed
Function approximation using RBF network. 10 basis functions and 25 data points.
1 Function approximation using RBF network F (x j ) = m 1 w i ϕ( x j t i ) i=1 j = 1... N, m 1 = 10, N = 25 10 basis functions and 25 data points. Basis function centers are plotted with circles and data
CSE 7/5337: Information Retrieval and Web Search Document clustering I (IIR 16)
CSE 7/5337: Information Retrieval and Web Search Document clustering I (IIR 16) Michael Hahsler Southern Methodist University These slides are largely based on the slides by Hinrich Schütze Institute for
Behavioral Data Mining. Lecture 18 Clustering
Behavioral Data Mining Lecture 18 Clustering Outline Why? Cluster quality K-means Spectral clustering Generative Models Rationale Given a set {X i } for i = 1,,n, a clustering is a partition of the X i
An Unsupervised Technique for Statistical Data Analysis Using Data Mining
International Journal of Information Sciences and Application. ISSN 0974-2255 Volume 5, Number 1 (2013), pp. 11-20 International Research Publication House http://www.irphouse.com An Unsupervised Technique
Clustering. Informal goal. General types of clustering. Applications: Clustering in information search and analysis. Example applications in search
Informal goal Clustering Given set of objects and measure of similarity between them, group similar objects together What mean by similar? What is good grouping? Computation time / quality tradeoff 1 2
Contents. I The Basic Framework for Stationary Problems 1
page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other
Cluster Analysis. Summer School on Geocomputation. 27 June July 2011 Vysoké Pole
Cluster Analysis Summer School on Geocomputation 27 June 2011 2 July 2011 Vysoké Pole Lecture delivered by: doc. Mgr. Radoslav Harman, PhD. Faculty of Mathematics, Physics and Informatics Comenius University,
Machine Learning (BSMC-GA 4439) Wenke Liu
Machine Learning (BSMC-GA 4439) Wenke Liu 01-31-017 Outline Background Defining proximity Clustering methods Determining number of clusters Comparing two solutions Cluster analysis as unsupervised 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
Clustering. K-means clustering
Clustering K-means clustering Clustering Motivation: Identify clusters of data points in a multidimensional space, i.e. partition the data set {x 1,...,x N } into K clusters. Intuition: A cluster is a
Dynamic Collision Detection
Distance Computation Between Non-Convex Polyhedra June 17, 2002 Applications Dynamic Collision Detection Applications Dynamic Collision Detection Evaluating Safety Tolerances Applications Dynamic Collision
Kernel Methods. Chapter 9 of A Course in Machine Learning by Hal Daumé III. Conversion to beamer by Fabrizio Riguzzi
Kernel Methods Chapter 9 of A Course in Machine Learning by Hal Daumé III http://ciml.info Conversion to beamer by Fabrizio Riguzzi Kernel Methods 1 / 66 Kernel Methods Linear models are great because
Introduction to Data Mining
Introduction to Data Mining Lecture #14: Clustering Seoul National University 1 In This Lecture Learn the motivation, applications, and goal of clustering Understand the basic methods of clustering (bottom-up
Machine Learning (BSMC-GA 4439) Wenke Liu
Machine Learning (BSMC-GA 4439) Wenke Liu 01-25-2018 Outline Background Defining proximity Clustering methods Determining number of clusters Other approaches Cluster analysis as unsupervised Learning Unsupervised
6. Dicretization methods 6.1 The purpose of discretization
6. Dicretization methods 6.1 The purpose of discretization Often data are given in the form of continuous values. If their number is huge, model building for such data can be difficult. Moreover, many
Unsupervised Learning
Outline Unsupervised Learning Basic concepts K-means algorithm Representation of clusters Hierarchical clustering Distance functions Which clustering algorithm to use? NN Supervised learning vs. unsupervised
Cluster Evaluation and Expectation Maximization! adapted from: Doug Downey and Bryan Pardo, Northwestern University
Cluster Evaluation and Expectation Maximization! adapted from: Doug Downey and Bryan Pardo, Northwestern University Kinds of Clustering Sequential Fast Cost Optimization Fixed number of clusters Hierarchical
In this lecture, we are going to talk about image segmentation, essentially defined as methods for grouping pixels together.
In this lecture, we are going to talk about image segmentation, essentially defined as methods for grouping pixels together. We will first define the segmentation problem, overview some basic ideas of
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:
K-Means. Oct Youn-Hee Han
K-Means Oct. 2015 Youn-Hee Han http://link.koreatech.ac.kr ²K-Means algorithm An unsupervised clustering algorithm K stands for number of clusters. It is typically a user input to the algorithm Some criteria
SUPPORT VECTOR MACHINES
SUPPORT VECTOR MACHINES Today Reading AIMA 18.9 Goals (Naïve Bayes classifiers) Support vector machines 1 Support Vector Machines (SVMs) SVMs are probably the most popular off-the-shelf classifier! Software
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
Open and Closed Sets
Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
Machine Learning : Clustering, Self-Organizing Maps
Machine Learning Clustering, Self-Organizing Maps 12/12/2013 Machine Learning : Clustering, Self-Organizing Maps Clustering The task: partition a set of objects into meaningful subsets (clusters). The
9/17/2009. Wenyan Li (Emily Li) Sep. 15, Introduction to Clustering Analysis
Introduction ti to K-means Algorithm Wenan Li (Emil Li) Sep. 5, 9 Outline Introduction to Clustering Analsis K-means Algorithm Description Eample of K-means Algorithm Other Issues of K-means Algorithm
Mathematical Programming and Research Methods (Part II)
Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types
ECLT 5810 Clustering
ECLT 5810 Clustering What is Cluster Analysis? Cluster: a collection of data objects Similar to one another within the same cluster Dissimilar to the objects in other clusters Cluster analysis Grouping
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
Copyright 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin Introduction to the Design & Analysis of Algorithms, 2 nd ed., Ch.
Iterative Improvement Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible
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