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Clustering

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NLPseminar
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This document provides a short review of clustering techniques for students. It defines clustering and different types of grouping methods such as hard vs soft clustering. It discusses popular clustering algorithms like hierarchical clustering, k-means clustering, and density-based clustering. It also covers cluster validity, usability, preprocessing techniques, meta methods, and visual clustering. Open problems in clustering mentioned include how to identify outlier objects and accelerate classification.

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TECHNIQUES OF CLUSTERING (a short review for students) Mikhail Alexandrov 1,2 , Pavel Makagonov 3 1 Autonomous University of Barcelona, Spain 2 Social Network Research Center with UCE, Slovakia 3 Mixtec Technological University, Mexico dyner1950@mail.ru, mpp@mixteco.utm.mx Petersburg 2008
Introduction Definitions Clustering Discussion Open Problems CONTENTS 
Prof. Dr. Benno Stein Dr. Sven Meyer zu Eissen Weimar University, Germany Ideas, Materials, Collaboration - Structuring and Indexing - AI search in IR - Semi-Supervised Learning Dr. Xiaojin Zhu University of Wisconsin, Madison , USA
TEXTUAL DATA Subject of Grouping NON TEXTUAL DATA Local Terminology It is not important what is the source of data: textual or non textual. Data : work in the space of numerical parameters Texts : work in the space of words Example: typical dialog between passenger ( US ) and railway directory inquires ( DI )
TEXTS (‘indexed’) Presentation of Textual Data Vector model (‘parameterized’) Local Terminology Indexed texts are only parameterized texts in the space of words
TEXTS (‘indexed’) Presentation of Textual Data TEXTS (‘parameterized’) Local Terminology Indexed texts are only parameterized texts in the space of themes = > Category /Context Vector Models... Example : manually parameterized dialogs in the space of parameters (transport service and passenger needs)
Introduction Definitions Clustering Discussion Open Problems CONTENTS 
Unsupervised Learning Types of Grouping Supervised Learning Semi-Supervised Learning We know nothing about data sructure We know well data sructure We know something about data sructure
Clustering Classification Characteristics: Characteristics : Absence of patterns or descriptions Presence of patterns or descriptiones of classes, so the results are of classes, so the results are defined by the nature of defined by the user ( N>=1 ) the data themselves ( N >1 ) Synonyms: Synonyms : Classification without teacher Classificatio n with teacher Unsupervised learning Supervised learning Number of clusters Specials terms : [ ] is known exactly Categorization (of documents) [x] is known approximately Diagnostics (technics, medicine) [ ] is not known => searching Recognition (technics, science) Types of Grouping
“ Semi Clustering/Classification” Classification Characteristics: Characteristics : Presence of limited number Presence of patterns or of patterns, so the results are descriptiones of classes, so the results defined both by the user or are defined by the user ( N>=1 ) by the data themselves ( N >1 ) Synonyms: Synonyms : Semi-Classification Classificatio n with teacher Semi Supervised learning Supervised learning Number of clusters/categories Specials terms : [ ] is known exactly Categorization (of documents) [x] is known approximately Diagnostics (technics, medicine) [ ] is not known => searching Recognition (technics, science) Types of Grouping
Objectives of Grouping 1. Organization (structuring) of an object set Process is named data structuring 2. Searching interesting patterns Process is named navigation 3. Grouping for other applications: - Knowledge discovery (clustering) - Summarization of documents Note: Do not mix the type of grouping and its objective
Classification of methods Based on belonging to cluster/category Exclusive methods Every object belongs only to one cluster/category. Methods are named hard grouping methods Non-exclusive methods Every object can belong to several clusters/categories. Methods are named soft grouping methods. Based on data presentation Methods oriented on free metric space Every object is presented as a point in a free space Methods oriented on graphs Every object is presented as an element on graph
Hard grouping Hard clustering Hard categorization Soft grouping Soft clustering Soft categorization Example The distribution of letters of Moscovites to the Government is soft categorization (numbers in the table reflect the relative weight of each theme) Fuzzy Grouping
Preprocessing <= Processing General Scheme of Clustering Process - I Principal idea : To transform texts to num erical form in order to use matematical tools Remember : our problem is grouping textual documents but not undestanding Here: Both rude and good matrixes are matrix Object/Attributes
General Scheme of Clustering Process - II Preprocessing Processing <= Here: matrix Attribute/Attribute can be used instead of matrix Object/Object
Matrixes to be Considered
Clustering for Categorization Colour matrix “words-words” before clustering Matrix contains the value of word co-occurrences in texts. Red : if value more than some threshold. White : if less.
Clustering for Categorizatión Colour matriz “words-words” after clustering Words are groupped. Cluster = > Subdictionary Absence of blocks means absence of Subthemes
Importance of Preprocessing ( it takes 60%-90% of efforts)
Introduction Definitions Clustering Discussion Open Problems CONTENTS 
Definitions Def. 1 “Let us V be the set of objects. Clustering C = { Ci | Ci є V } of V is division of V on subsets, for which we have : U i Ci = V and Ci ∩ Cj = 0 i ≠j“ Def. 2 “Let us V be the set of nodes, E be arcs, φ is weight function that reflects the distance between objects, so we have a weighted graph G = { V,E, φ }. In this case C is named as clustering of G .” In the framework of the second definition every Ci produced subgraph G(Ci). Both subsets Ci and subgraphs G(Ci) are named clusters . Graph Set Clique
Definitions Principal note Both definitions SAYS NOTHING : - about quality of clusters - about numbers of clusters Reason of difficulties Nowadays there is no any general agreement about any universal defintion of the term ‘ cluster ’ What means that clustering is good ? 1. Closeness between objets inside clusters is essentially more than the closeness between clusters themselves 2. Constructed clusters correspond to intuitive presentations of users ( they are natural clusters)
Classification of methods 1. Hierarchy based methods Any neighbors N =? N is not given 2. Exemplar based methods K-means N = ? N is given 3. Density based methods MajorClust N = ? N is calculated automatically Based on the way of grouping
Hierarchy based methods Neighbors. Every object is cluster General algorithm Initially every object is one cluster The series of steps are performed. On every step the pair of cluster being the closest ones are merged. At the end we have one cluster.
Hierarchy based methods Nearest neighbor method (NN)
Exemplar based methods K - means, centroid
Method K-means General algorithm Initially K centers are selected by any random way Series of steps are performed. On every step the objects are distributed between centers according the criterion of the nearest center . Then all centers are recalculated. The end is fixed when the centers are not changed . Exemplar based methods
Method X-means (Dan Pelleg, Andrew Moor) Approach Using evaluation of object distribution Selection of the most likely points Advantage - More rapid - Number of cluster is not fixed (in all cases it tends to be less) Exemplar based methods
Density based methods MajorClust method Principal idea Total closeness to the objects of his own cluster exceeds the closeness to any other cluster Suboptimal solution Only part of neighbors are considered on every step (to save time, to avoid mergence) .
Density based methods MajorClust method General algorithm Initially every object is one cluster and it joins to the nearest neighbor Every object evaluates the total closeness to his own cluster and separately to all other clusters. After such evaluation the objects change its belonging and go off to the closest one The end of searching is fixed when clusters do not change . Preprocessing for MajorClust Many weak links can be stronger than the several strongest ones that disfigures results. So: weak links should be eliminated before clustering
Cluster Validity Definition It reflects cluster separability and formally depends on : - Scatters inside clusters - Separation between clusters Indexes It is formal characteristics of structure Dunn index Davies Bouldin index Hypervolume criterion ( Andre Hardy ) Density expected measure DEM ( Benno Stein ) Dunn index (to be max )
Cluster Validity Number of clusters Geometrical approach, two variants: Optimum (min, max) of curve Jump of curve Dunn index (to be max ) is too sensible to extremal cases
Cluster Usability Definition It reflects user’s opinion and formally expresses the difference between : - Classes selected manually by a user - Clusters constructed by a given method Cluster F -measure ( Benno Stein ) Data Expert Method Here: i, j are indexes of clusses and clusters C * i , C j are classes and clusters prec(i,j), rec(i,j) are precision and recall
Validity and Usability Conclusion Density expected measure corresponds to F -measure reflecting expert ’s opinion. So, DEM can be an indicator of expert opinion
Tecnologies of Clustering Meta methods They construct separated data sets using criteria of optimization and limitations : Neither much nor small number of clusters Neither large nor small size of clusters etc. Visual methods They present visual images to a user in order to select manually the clusters Using different methods Comparing results
Meta Methods Algorithm (example) Notations: N is the number of objects in a given cluster D is the diagonal of a given cluster Initially N 0 and their centers Ci are given Steps 1. Method K - medoid (or any other one) is performed 2. If N > N max or D > D max (in any cluster), then this cluster is divided on 2 parts. Go to p.1 3. If N < N min or D < D min (in any cluster), then this and the closest clusters are joined. Go to p.1 4. When the number of iteration I > I max , Stop Otherwise go to p.1
Visual Clustering Clustering on dendrite Clustering in space of factors
Problem Authorship of Molier dramatic works (comedies, dramas,...). Corneille and/or Molier ? Approach Style based indexing ( NooJ can be used ) Clustering all dramatic works Well-known dramatic works should be marked Style - Formal style estimations Informal style estimations Formal style indicators - Text Complexity - Text Harmonicity Authorship References : Labbe C., Labbe D. Inter-textual distance and authorship attribution Corneille and Molier. Journ. of Quantitative Linguistics. 2001. Vol.8, N_3, pp.213-331
Clustering Authorship Results 1) 18 comedies of Molier should be belonged to Corneille 2) 15 comedies of Mollier are weak connected with all his other works. So, they can be written by two authors 3) 2 comedies of Corneille now are considered as works of Molier . etc. Note : During a certain time Molier and Corneille were friends
Special and Universal packages with algorithms of С lustering 1. ClustAn ( Scotland ) www. clustan.com Clustan Graphics-7 (2006) 2. MatLab Descriptions are in Internet 3. Statistica Descriptions are in Internet Learning Journals and С ongresses about Clustering 1. Journal “Journal of Classification”, Springer 2. IFCS - International Federation of Classification Societies, Conferences 3. CSNA - Classification Society of North America, Seminars, Workshops
Introduction Definitions Clustering Discussion Open Problems CONTENTS 
Certain Observations The numbers of methods for grouping data is a little bit more than the numbers of researchers working in this area. Problem does not consist in searching the best method for all cases. Problem consists in searching the method being relevant for your data. Only you know what methods are the best for you own data . Principal problems consist in choice of indexes (parameters) and measure of closeness to be adecuate to a given problem and given data Frecuently the results are bad because of the bad indexes and bad measure but not the bad method !
Certain Observations Antipodal methods To be sure that results are really good and do not depend on the method used one should test these results using any antipodal methods Solomon G, 1977: “The most antipodes are: NN-method and K-means ” Sensibility To be sure that results do not depend essentially on the method’s parameters one should perform the analysis of sensibility by changing parameters of adjustment.
Introduction Definitions Clustering Conclusions Open Problems CONTENTS 
Some Problems Question 1 How to reveal alien objects? Solution (idea) Revealing a stable structure on different sets of objects. They are subsets of a given set. Object distribution reflects: real structure ( nature ) + noise ( alien objects )
Some Problems Question 2 How to accelerate classification? Solution (idea) Filtering objects, which give a minimum contribution to decisive function Representative objects of each cluster
CONTACT INFORMATION Mikhail Alexandrov 1,2 , Pavel Makagonov 3 1 Autonomous University of Barcelona, Spain 2 Social Network Research Center with UCE, Slovakia 3 Mixtec Technological University, Mexico dyner1950@mail.ru, mpp@mixteco.utm.mx Petersburg 2008

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Clustering

  • 1. TECHNIQUES OF CLUSTERING (a short review for students) Mikhail Alexandrov 1,2 , Pavel Makagonov 3 1 Autonomous University of Barcelona, Spain 2 Social Network Research Center with UCE, Slovakia 3 Mixtec Technological University, Mexico dyner1950@mail.ru, mpp@mixteco.utm.mx Petersburg 2008
  • 2. Introduction Definitions Clustering Discussion Open Problems CONTENTS 
  • 3. Prof. Dr. Benno Stein Dr. Sven Meyer zu Eissen Weimar University, Germany Ideas, Materials, Collaboration - Structuring and Indexing - AI search in IR - Semi-Supervised Learning Dr. Xiaojin Zhu University of Wisconsin, Madison , USA
  • 4. TEXTUAL DATA Subject of Grouping NON TEXTUAL DATA Local Terminology It is not important what is the source of data: textual or non textual. Data : work in the space of numerical parameters Texts : work in the space of words Example: typical dialog between passenger ( US ) and railway directory inquires ( DI )
  • 5. TEXTS (‘indexed’) Presentation of Textual Data Vector model (‘parameterized’) Local Terminology Indexed texts are only parameterized texts in the space of words
  • 6. TEXTS (‘indexed’) Presentation of Textual Data TEXTS (‘parameterized’) Local Terminology Indexed texts are only parameterized texts in the space of themes = > Category /Context Vector Models... Example : manually parameterized dialogs in the space of parameters (transport service and passenger needs)
  • 7. Introduction Definitions Clustering Discussion Open Problems CONTENTS 
  • 8. Unsupervised Learning Types of Grouping Supervised Learning Semi-Supervised Learning We know nothing about data sructure We know well data sructure We know something about data sructure
  • 9. Clustering Classification Characteristics: Characteristics : Absence of patterns or descriptions Presence of patterns or descriptiones of classes, so the results are of classes, so the results are defined by the nature of defined by the user ( N>=1 ) the data themselves ( N >1 ) Synonyms: Synonyms : Classification without teacher Classificatio n with teacher Unsupervised learning Supervised learning Number of clusters Specials terms : [ ] is known exactly Categorization (of documents) [x] is known approximately Diagnostics (technics, medicine) [ ] is not known => searching Recognition (technics, science) Types of Grouping
  • 10. “ Semi Clustering/Classification” Classification Characteristics: Characteristics : Presence of limited number Presence of patterns or of patterns, so the results are descriptiones of classes, so the results defined both by the user or are defined by the user ( N>=1 ) by the data themselves ( N >1 ) Synonyms: Synonyms : Semi-Classification Classificatio n with teacher Semi Supervised learning Supervised learning Number of clusters/categories Specials terms : [ ] is known exactly Categorization (of documents) [x] is known approximately Diagnostics (technics, medicine) [ ] is not known => searching Recognition (technics, science) Types of Grouping
  • 11. Objectives of Grouping 1. Organization (structuring) of an object set Process is named data structuring 2. Searching interesting patterns Process is named navigation 3. Grouping for other applications: - Knowledge discovery (clustering) - Summarization of documents Note: Do not mix the type of grouping and its objective
  • 12. Classification of methods Based on belonging to cluster/category Exclusive methods Every object belongs only to one cluster/category. Methods are named hard grouping methods Non-exclusive methods Every object can belong to several clusters/categories. Methods are named soft grouping methods. Based on data presentation Methods oriented on free metric space Every object is presented as a point in a free space Methods oriented on graphs Every object is presented as an element on graph
  • 13. Hard grouping Hard clustering Hard categorization Soft grouping Soft clustering Soft categorization Example The distribution of letters of Moscovites to the Government is soft categorization (numbers in the table reflect the relative weight of each theme) Fuzzy Grouping
  • 14. Preprocessing <= Processing General Scheme of Clustering Process - I Principal idea : To transform texts to num erical form in order to use matematical tools Remember : our problem is grouping textual documents but not undestanding Here: Both rude and good matrixes are matrix Object/Attributes
  • 15. General Scheme of Clustering Process - II Preprocessing Processing <= Here: matrix Attribute/Attribute can be used instead of matrix Object/Object
  • 16. Matrixes to be Considered
  • 17. Clustering for Categorization Colour matrix “words-words” before clustering Matrix contains the value of word co-occurrences in texts. Red : if value more than some threshold. White : if less.
  • 18. Clustering for Categorizatión Colour matriz “words-words” after clustering Words are groupped. Cluster = > Subdictionary Absence of blocks means absence of Subthemes
  • 19. Importance of Preprocessing ( it takes 60%-90% of efforts)
  • 20. Introduction Definitions Clustering Discussion Open Problems CONTENTS 
  • 21. Definitions Def. 1 “Let us V be the set of objects. Clustering C = { Ci | Ci є V } of V is division of V on subsets, for which we have : U i Ci = V and Ci ∩ Cj = 0 i ≠j“ Def. 2 “Let us V be the set of nodes, E be arcs, φ is weight function that reflects the distance between objects, so we have a weighted graph G = { V,E, φ }. In this case C is named as clustering of G .” In the framework of the second definition every Ci produced subgraph G(Ci). Both subsets Ci and subgraphs G(Ci) are named clusters . Graph Set Clique
  • 22. Definitions Principal note Both definitions SAYS NOTHING : - about quality of clusters - about numbers of clusters Reason of difficulties Nowadays there is no any general agreement about any universal defintion of the term ‘ cluster ’ What means that clustering is good ? 1. Closeness between objets inside clusters is essentially more than the closeness between clusters themselves 2. Constructed clusters correspond to intuitive presentations of users ( they are natural clusters)
  • 23. Classification of methods 1. Hierarchy based methods Any neighbors N =? N is not given 2. Exemplar based methods K-means N = ? N is given 3. Density based methods MajorClust N = ? N is calculated automatically Based on the way of grouping
  • 24. Hierarchy based methods Neighbors. Every object is cluster General algorithm Initially every object is one cluster The series of steps are performed. On every step the pair of cluster being the closest ones are merged. At the end we have one cluster.
  • 25. Hierarchy based methods Nearest neighbor method (NN)
  • 26. Exemplar based methods K - means, centroid
  • 27. Method K-means General algorithm Initially K centers are selected by any random way Series of steps are performed. On every step the objects are distributed between centers according the criterion of the nearest center . Then all centers are recalculated. The end is fixed when the centers are not changed . Exemplar based methods
  • 28. Method X-means (Dan Pelleg, Andrew Moor) Approach Using evaluation of object distribution Selection of the most likely points Advantage - More rapid - Number of cluster is not fixed (in all cases it tends to be less) Exemplar based methods
  • 29. Density based methods MajorClust method Principal idea Total closeness to the objects of his own cluster exceeds the closeness to any other cluster Suboptimal solution Only part of neighbors are considered on every step (to save time, to avoid mergence) .
  • 30. Density based methods MajorClust method General algorithm Initially every object is one cluster and it joins to the nearest neighbor Every object evaluates the total closeness to his own cluster and separately to all other clusters. After such evaluation the objects change its belonging and go off to the closest one The end of searching is fixed when clusters do not change . Preprocessing for MajorClust Many weak links can be stronger than the several strongest ones that disfigures results. So: weak links should be eliminated before clustering
  • 31. Cluster Validity Definition It reflects cluster separability and formally depends on : - Scatters inside clusters - Separation between clusters Indexes It is formal characteristics of structure Dunn index Davies Bouldin index Hypervolume criterion ( Andre Hardy ) Density expected measure DEM ( Benno Stein ) Dunn index (to be max )
  • 32. Cluster Validity Number of clusters Geometrical approach, two variants: Optimum (min, max) of curve Jump of curve Dunn index (to be max ) is too sensible to extremal cases
  • 33. Cluster Usability Definition It reflects user’s opinion and formally expresses the difference between : - Classes selected manually by a user - Clusters constructed by a given method Cluster F -measure ( Benno Stein ) Data Expert Method Here: i, j are indexes of clusses and clusters C * i , C j are classes and clusters prec(i,j), rec(i,j) are precision and recall
  • 34. Validity and Usability Conclusion Density expected measure corresponds to F -measure reflecting expert ’s opinion. So, DEM can be an indicator of expert opinion
  • 35. Tecnologies of Clustering Meta methods They construct separated data sets using criteria of optimization and limitations : Neither much nor small number of clusters Neither large nor small size of clusters etc. Visual methods They present visual images to a user in order to select manually the clusters Using different methods Comparing results
  • 36. Meta Methods Algorithm (example) Notations: N is the number of objects in a given cluster D is the diagonal of a given cluster Initially N 0 and their centers Ci are given Steps 1. Method K - medoid (or any other one) is performed 2. If N > N max or D > D max (in any cluster), then this cluster is divided on 2 parts. Go to p.1 3. If N < N min or D < D min (in any cluster), then this and the closest clusters are joined. Go to p.1 4. When the number of iteration I > I max , Stop Otherwise go to p.1
  • 37. Visual Clustering Clustering on dendrite Clustering in space of factors
  • 38. Problem Authorship of Molier dramatic works (comedies, dramas,...). Corneille and/or Molier ? Approach Style based indexing ( NooJ can be used ) Clustering all dramatic works Well-known dramatic works should be marked Style - Formal style estimations Informal style estimations Formal style indicators - Text Complexity - Text Harmonicity Authorship References : Labbe C., Labbe D. Inter-textual distance and authorship attribution Corneille and Molier. Journ. of Quantitative Linguistics. 2001. Vol.8, N_3, pp.213-331
  • 39. Clustering Authorship Results 1) 18 comedies of Molier should be belonged to Corneille 2) 15 comedies of Mollier are weak connected with all his other works. So, they can be written by two authors 3) 2 comedies of Corneille now are considered as works of Molier . etc. Note : During a certain time Molier and Corneille were friends
  • 40. Special and Universal packages with algorithms of С lustering 1. ClustAn ( Scotland ) www. clustan.com Clustan Graphics-7 (2006) 2. MatLab Descriptions are in Internet 3. Statistica Descriptions are in Internet Learning Journals and С ongresses about Clustering 1. Journal “Journal of Classification”, Springer 2. IFCS - International Federation of Classification Societies, Conferences 3. CSNA - Classification Society of North America, Seminars, Workshops
  • 41. Introduction Definitions Clustering Discussion Open Problems CONTENTS 
  • 42. Certain Observations The numbers of methods for grouping data is a little bit more than the numbers of researchers working in this area. Problem does not consist in searching the best method for all cases. Problem consists in searching the method being relevant for your data. Only you know what methods are the best for you own data . Principal problems consist in choice of indexes (parameters) and measure of closeness to be adecuate to a given problem and given data Frecuently the results are bad because of the bad indexes and bad measure but not the bad method !
  • 43. Certain Observations Antipodal methods To be sure that results are really good and do not depend on the method used one should test these results using any antipodal methods Solomon G, 1977: “The most antipodes are: NN-method and K-means ” Sensibility To be sure that results do not depend essentially on the method’s parameters one should perform the analysis of sensibility by changing parameters of adjustment.
  • 44. Introduction Definitions Clustering Conclusions Open Problems CONTENTS 
  • 45. Some Problems Question 1 How to reveal alien objects? Solution (idea) Revealing a stable structure on different sets of objects. They are subsets of a given set. Object distribution reflects: real structure ( nature ) + noise ( alien objects )
  • 46. Some Problems Question 2 How to accelerate classification? Solution (idea) Filtering objects, which give a minimum contribution to decisive function Representative objects of each cluster
  • 47. CONTACT INFORMATION Mikhail Alexandrov 1,2 , Pavel Makagonov 3 1 Autonomous University of Barcelona, Spain 2 Social Network Research Center with UCE, Slovakia 3 Mixtec Technological University, Mexico dyner1950@mail.ru, mpp@mixteco.utm.mx Petersburg 2008