. Learn the number of classes and the structure of each class using similarity between unlabeled training patterns

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Unsupervised Learning -- Ana Fred

Outline

Part 1: Basic Concepts of data clustering

 Non-Supervised Learning and Clustering

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Problem formulation – cluster analysis

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Taxonomies of Clustering Techniques

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Data types and Proximity Measures

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Difficulties and open problems

Part 2: Clustering Algorithms

 Hierarchical methods

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Single-link

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Complete-link

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Clustering Based on Dissimilarity Increments Criteria

From Single Clustering to Ensemble Methods - April 2009

Unsupervised Learning Basic Concepts

Pattern Recognition – Decision Making

 Supervised Learning

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training samples, labeled by their category membership, are used to design a classifier

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^Labeledtraining patterns

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Labels represent true categories of patterns

 Unsupervised Learning

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Based on a collection of samples without being told their categories

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Learn the number of classes and the structure of each class using similarity between unlabeledtraining patterns

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^Datamining

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Unsupervised Learning / Clustering

 Unsupervised Learning

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Learn the structure of multidimensional patterns

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Mixture Densities

– Gaussian Mixture Decomposition

» The probability structure is known with the exception of the values of the parameters

 Clustering Procedures

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Find subclasses

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Data description in terms of clusters or groups of data points that possess strong internal similarities

 Typical applications:

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As a stand-alone toolto get insight into data distribution

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As a preprocessing stepfor other algorithms

Organize data into sensible groupings (either as a grouping of patterns or a hierarchy of groups)

 Clustering

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The process of grouping a set of objects into classes of similar objects (extracting hidden structure from data)

 Cluster

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A collection of objects that are similar to one another within the same cluster and are dissimilar to the objects in other clusters

Cluster Analysis

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Shape Clustering

Right Ventricle from MR brain images

Duta, Jain and Jolly, “Automatic Construction of 2-D Shape Models”, IEEE PAMI, May 2001 Cistern from MR brain images

The main cluster is drawn using multicolor dots, secondary clusters are drawn in red, green and magenta.

Shape Clustering

Right Ventricle from MR brain images

Duta, Jain and Jolly, “Automatic Construction of 2-D Shape Models”, IEEE PAMI, May 2001 Cistern from MR brain images

The main cluster is drawn using multicolor dots, secondary clusters are drawn in red, green and magenta.

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Identification of Writing Styles

 122,000 “online” characters written by 100 writers

 Lexemes are identified by clustering data within each character class into subclasses:

a string matching measure used to calculate distance between 2 characters

Connell and Jain, “Writer Adaptation for Online Handwriting Recognition”, IEEE PAMI, Mar 2002

Segmentation of Natural Scenes

Hermes, Zoller, Bumannn, “Parametric Distributional Clustering for Image Segmentation”, ECCV 2002

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What is a Cluster?

 A set of entities which are alike; entities from different clusters are not alike

 An aggregation of points such that the distance between any two points in a cluster is less than the distance between any point in the cluster and any point not in it.

 A relatively high density of points, surrounded by a relatively low density of points

Ideal cluster: Compact and Isolated

Taxonomy of Clustering Approaches

Two main strategies:

 Hierarchical Methods

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Propose a sequence of nested data partitions in a hierarchical structure

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Single-Link

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Complete Link

 Partitional Methods

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Organize patterns into a small number of clusters

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^K-means

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Spectral clustering

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Taxonomy of Clustering Approaches

Clustering Principles:

 Compactness

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^K-means

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Complete-link

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Histogram clustering

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Pairwise data clustering

 Connectedness

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Single-linkage

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Dissimilarity Increments

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Mean Shift clustering

 Separation

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Normalized Cut

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Spectral clustering

Taxonomy of Clustering Approaches

 Compactness

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^K-means

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Complete-link

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 Connectedness

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Single-linkage

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 Separation

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Normalized Cut

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Spectral clustering

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1 2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

Taxonomy of Clustering Approaches

 Compactness

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^K-means

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Complete-link

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 Connectedness

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Single-linkage

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 Separation

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Normalized Cut

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Spectral clustering

Taxonomy of Clustering Approaches

Approaches:

 Model-based

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Patterns can be given a simple and compact description in terms of

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Parametrical distribution --Parametric density approaches(Mixture models)

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A representative element, such as a centroid, median (central clustering, square-error clustering, k-means, k-medoids) – or multiple prototypes per cluster (CURE) --Prototype-based methods

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Some geometrical primitives (lines, planes, circles, curves, surfaces) –Shape fitting approaches

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These approaches assume particular cluster shapes, partitions being in general obtained as a result of an optimization process using a global criterion

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Taxonomy of Clustering Approaches

 Graph-theoretical

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Mostly explored in hierarchical methods that can be represented graphically as a tree or dendrogram

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Agglomerative methods (Single-link, complete-link)

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Divisive approaches (ex. Based on Minimum Spanning Tree)

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View clustering as a graph partitioning problem

 Non parametric density-based

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Attempt to identify high density clusters separated by low density regions (local cluster criterion, such as density-connected points) (valley seeking clustering algorithms)

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DBSCAN, OPTICS, DENCLUE, CLIQUE

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Discover clusters of arbitrary shape

Data Types in Clustering Problems

Data representations:

 Vector data: n vectors in R

^d

 Proximity data: n x n pairwise proximity matrix

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All types of data may be addressed by choosing adequate proximity measures

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Similarity and Dissimilarity Between Objects

 Distances are normally used to measure the similarity or dissimilarity between two data objects

 Metrics:

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Positivity: d(a, b) >0 and d(a, b)=0 , a=b

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Symmetry property: d(a,b)=d(b,a).

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Triangle inequality: d(a,c)·d(a,b) + d(b,c).

Metric Models in Feature Spaces

 Minskowski distance:

(Euclidean distance corresponds to r = 2)

 Maximum Value Metric:

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Considers only most distant features

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Metric Models in Feature Spaces

 Absolute Value Metric, Manhattan Distance or City-block (r = 1)

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Reduced computational time; does not penalize much the features with higher dissimilarity

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^{In R}²^{: dist}1((x₁,y₁),(x₂,y₂))=|x₂-x₁|+|y₂-y₁| , city-block

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It is not possible to make short-cuts through corners: it counts the number of blocks that is necessary to pass in order to move from one corner to another

Constant Manhattan distance curves:

1

( , ) ( , )

d

M i i

i

d a b d a b b a



 





Metric Models in Feature Spaces

 Euclidean Distance:

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R2: dist2((x1,y1),(x2,y2))=((x2-x1)2+(y2-y1)2)1/2.

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Emphasizes more features with higher dissimilarity.

 Mahalanobis Distance

 

²

2

1

( , ) ( , )

d

e i i

i

d a b d a b b a



 





 

¹

 

( , ) ^T

Mahalanobis

d x y  xy ^ xy

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Dissimilarity for Sequences of Symbols

 Dissimilarity based on String Editing operations

. . . .

 The Levensthein distancebetween two strings s1, s2 2^*, DL(s1, s2), is defined as the minimum number of editing operations needed in order to transform s₁into s₂.

Dissimilarity for Sequences of Symbols

 The Weighted Levensthein distancebetween two strings s1, s2 2^*, is defined by

where

 Normalized Weighted Levensthein distance

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Dissimilarity for Sequences of Symbols

 String Editing operations and String Matching

String matching. In (b), diagonal path elements represent substitutions, vertical segments correspond to insertions, and horizontal

segments correspond to deletions.

(a) String matching using editing operations. (b) Editing path.

Dissimilarity for Sequences of Symbols

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Normalized Edit Distance

Marzal and Vidal, “Computation of normalized edit distance and applications”, IEEE PAMI, 1993

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Dissimilarity for Sequences of Symbols

 Dissimilarity based on Error-Correcting Parsing [Fu]

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distance between strings based on the modelling of string structure by means of grammars and on the concept of error-correcting parsing

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the distance between a string and a reference string is given by the error-correcting parser as the weighted Levensthein distance between the string and the nearest (in terms of edit operations) string generated by the grammar inferred from the reference string (thus exhibiting a similar structure):

ECP distance

1

2

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Dissimilarity for Sequences of Symbols

 Dissimilarity based on Error-Correcting Parsing [Fu]

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distance between strings based on the modelling of string structure by means of grammars and on the concept of error-correcting parsing

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the distance between a string and a reference string is given by the error-correcting parser as the weighted Levensthein distance between the string and the nearest (in terms of edit operations) string generated by the grammar inferred from the reference string (thus exhibiting a similar structure):

:

In order to preserve symmetry

Dissimilarity for Sequences of Symbols

 Grammar Complexity-based Similarity

The basic idea is that, if two sentences are structurally similar, then their joint description will be more compact than their isolated description due to sharing of rules of symbol composition; the compactness of the representation is quantified by the grammar complexity, and the similarity is measured by the ratio of decrease in grammar complexity

where C(Gsi) denotes grammar complexity.

Fred. “Similarity measures and clustering of string patterns”. In Dechang Chen and Xiuzhen Cheng, editors, Pattern Recognition and String Matching, Kluwer Academic, 2002,

Fred, “Clustering of Sequences using a Minimum Grammar Complexity Criterion”, ICGI 1996

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RDGC Similarity

Dissimilarity for Sequences of Symbols

 Grammar Complexity-based Similarity– RDGC

Let G=(V_N, , R, ) be a context-free grammar, where VN,are the sets of nonterminal and terminal symbols, respectively, is the grammar’s start symbol and R is the set of productions written in the form:

Let 2 (VN)^*, be a grammatical sentence of length n, in which the symbols a1, a2, … , amappear k1, k2, … , kmtimes, respectively. The complexity of the sentence, C(), is given by [Fu]

The complexity of the grammar G is defined as

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Dissimilarity for Sequences of Symbols

 Minimum Code Length-based Similarity

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Based on Solomonoff’s code: a string is represented by a triplet

where a coded string is obtained in an iterative procedure where, in each step, intermediate codes are produced by defining sequences of two symbols, which are represented by special symbols, and rewriting the sequences using them. Compact codes are produced when sequences exhibit local or distant inter-symbol interactions.

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Code length: sum of the lengths of the descriptions of the three part code above

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Extension to sets of strings

Fred. “Similarity measures and clustering of string patterns”. In Dechang Chen and Xiuzhen Cheng, editors, Pattern Recognition and String Matching, Kluwer Academic, 2002,

Fred and Leitão, “A Minimum Code Length Technique for Clustering of Syntactic Patterns”, ICPR 1996

Dissimilarity for Sequences of Symbols

 Minimum Code Length-based Similarity

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The basic idea is that global compact codes are produced by considering the inter-symbol dependencies on the ensemble of the strings. The quantification of this reduction in code length forms the basis of the similarity measure designated by Normalized Ratio of decrease in code length - NRDCL