INFO 4300 / CS4300 Information Retrieval slides adapted from - - PowerPoint PPT Presentation

INFO 4300 / CS4300 Information Retrieval slides adapted from Hinrich Sch¨ utze’s, linked from http://informationretrieval.org/

IR 22/25: Hierarchical Clustering

Paul Ginsparg

Cornell University, Ithaca, NY

17 Nov 2011

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Overview

1

Recap

2

Feature selection

3

Introduction to Hierarchical clustering

4

Single-link/Complete-link

5

Centroid/GAAC

6

Variants

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Administrativa

Assignment 4 due 2 Dec (extended til 4 Dec). Sample of minHash duplicate detection . . .

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Worked example, assignment 4 first part

doc1: “This is a six word sentence” (0 1 2 3) doc2: “six word sentence this is a” (3 4 5 0) doc3: “sentence this is a six word” (5 0 1 2) doc4: “this this this this this this” (6 6 6 6) 3-grams: 0 “this is a” 1 “is a six” 2 “a six word” 3 “six word sentence” 4 “word sentence this” 5 “sentence this is” 6 “this this this”

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Documents contain the following 3-grams: doc1: {0, 1, 2, 3} doc2: {0, 3, 4, 5} doc3: {0, 1, 2, 5} doc4: {6} J(S1, S2) = |S1 ∩ S2|/|S1 ∪ S2| J(d1, d2) = 2/6, J(d1, d3) = 3/5, J(d2, d3) = 2/6, J(di, d4) = 0 To estimate, pick three random functions fi(x) = (ax + b) mod7: f1(x) = 2x + 3 mod7 : (0,1,2,3,4,5,6) → (3,5,0,2,4,6,1) f2(x) = x + 5 mod7 : (0,1,2,3,4,5,6) → (5,6,0,1,2,3,4) f3(x) = 4x + 1 mod7 : (0,1,2,3,4,5,6) → (1,5,2,6,3,0,4)

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doc1 f1 : (0,1,2,3) → (3,5,0,2) f2 : (0,1,2,3) → (5,6,0,1) f3 : (0,1,2,3) → (1,5,2,6) ⇒ sketch for doc1 is (0,0,1) doc2 f1 : (0,3,4,5) → (3,2,4,6) f2 : (0,3,4,5) → (5,1,2,3) f3 : (0,3,4,5) → (1,6,3,0) ⇒ sketch for doc2 is (2,1,0) doc3 f1 : (0,1,2,5) → (3,5,0,6) f2 : (0,1,2,5) → (5,6,0,3) f3 : (0,1,2,5) → (1,5,2,0) ⇒ sketch for doc3 is (0,0,0) doc4 f1 : (6) → (1) f2 : (6) → (4) f3 : (6) → (4) ⇒ sketch for doc4 is (1,4,4)

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doc1=(0,0,1), doc2=(2,1,0) : estimate k/n = 0/3, exact J(d1, d2) = 1/3 doc1=(0,0,1), doc3=(0,0,0) : estimate k/n = 2/3, exact J(d1, d3) = 3/5 doc2=(2,1,0), doc3=(0,0,0) : estimate k/n = 1/3, exact J(d2, d3) = 1/3 doc1=(0,0,1), doc4=(1,4,4) : estimate k/n = 0/3, exact J(d1, d4) = 0 doc2=(2,1,0), doc4=(1,4,4) : estimate k/n = 0/3, exact J(d2, d4) = 0 doc3=(0,0,0), doc4=(1,4,4) : estimate k/n = 0/3, exact J(d3, d4) = 0

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Outline

1

Recap

2

Feature selection

3

Introduction to Hierarchical clustering

4

Single-link/Complete-link

5

Centroid/GAAC

6

Variants

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Major issue in clustering – labeling

After a clustering algorithm finds a set of clusters: how can they be useful to the end user? We need a pithy label for each cluster. For example, in search result clustering for “jaguar”, The labels of the three clusters could be “animal”, “car”, and “operating system”. Topic of this section: How can we automatically find good labels for clusters?

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Exercise

Come up with an algorithm for labeling clusters Input: a set of documents, partitioned into K clusters (flat clustering) Output: A label for each cluster Part of the exercise: What types of labels should we consider? Words?

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Discriminative labeling

To label cluster ω, compare ω with all other clusters Find terms or phrases that distinguish ω from the other clusters We can use any of the feature selection criteria used in text classification to identify discriminating terms: (i) mutual information, (ii) χ2, (iii) frequency (but the latter is actually not discriminative)

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Non-discriminative labeling

Select terms or phrases based solely on information from the cluster itself Terms with high weights in the centroid (if we are using a vector space model) Non-discriminative methods sometimes select frequent terms that do not distinguish clusters. For example, Monday, Tuesday, . . . in newspaper text

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Using titles for labeling clusters

Terms and phrases are hard to scan and condense into a holistic idea of what the cluster is about. Alternative: titles For example, the titles of two or three documents that are closest to the centroid. Titles are easier to scan than a list of phrases.

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Cluster labeling: Example

labeling method # docs centroid mutual information title 4 622

• il plant mexico pro-

duction crude power 000 refinery gas bpd plant oil production barrels crude bpd mexico dolly capac- ity petroleum MEXICO: Hurricane Dolly heads for Mex- ico coast 9 1017 police security rus- sian people military peace killed told grozny court police killed military security peace told troops forces rebels people RUSSIA: Russia’s Lebed meets rebel chief in Chechnya 10 1259 00 000 tonnes traders futures wheat prices cents september tonne delivery traders fu- tures tonne tonnes desk wheat prices 000 00 USA: Export Business

• Grain/oilseeds com-

plex Three methods: most prominent terms in centroid, differential labeling using MI, title of doc closest to centroid All three methods do a pretty good job.

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Outline

1

Recap

2

Feature selection

3

Introduction to Hierarchical clustering

4

Single-link/Complete-link

5

Centroid/GAAC

6

Variants

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Feature selection

In text classification, we usually represent documents in a high-dimensional space, with each dimension corresponding to a term. In this lecture: axis = dimension = word = term = feature Many dimensions correspond to rare words. Rare words can mislead the classifier. Rare misleading features are called noise features. Eliminating noise features from the representation increases efficiency and effectiveness of text classification. Eliminating features is called feature selection.

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Example for a noise feature

Let’s say we’re doing text classification for the class China. Suppose a rare term, say arachnocentric, has no information about China . . . . . . but all instances of arachnocentric happen to occur in China documents in our training set. Then we may learn a classifier that incorrectly interprets arachnocentric as evidence for the China. Such an incorrect generalization from an accidental property

• f the training set is called overfitting.

Feature selection reduces overfitting and improves the accuracy of the classifier.

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Basic feature selection algorithm

SelectFeatures(D, c, k) 1 V ← ExtractVocabulary(D) 2 L ← [] 3 for each t ∈ V 4 do A(t, c) ← ComputeFeatureUtility(D, t, c) 5 Append(L, A(t, c), t) 6 return FeaturesWithLargestValues(L, k) How do we compute A, the feature utility?

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Different feature selection methods

A feature selection method is mainly defined by the feature utility measures it employs. Feature utility measures: Frequency – select the most frequent terms Mutual information – select the terms with the highest mutual information (mutual information is also called information gain in this context) χ2 (Chi-square)

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Information

H[p] =

i=1,n −pi log2 pi measures information uncertainty

(p.91 in book) has maximum H = log2 n for all pi = 1/n Consider two probability distributions: p(x) for x ∈ X and p(y) for y ∈ Y MI: I[X; Y ] = H[p(x)] + H[p(y)] − H[p(x, y)] measures how much information p(x) gives about p(y) (and vice versa) MI is zero iff p(x, y) = p(x)p(y), i.e., x and y are independent for all x ∈ X and y ∈ Y can be as large as H[p(x)] or H[p(y)] I[X; Y ] =

• x∈X,y∈Y

p(x, y) log2 p(x, y) p(x)p(y)

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Mutual information

Compute the feature utility A(t, c) as the expected mutual information (MI) of term t and class c. MI tells us “how much information” the term contains about the class and vice versa. For example, if a term’s occurrence is independent of the class (same proportion of docs within/without class contain the term), then MI is 0. Definition:

I(U; C)=

• et∈{1,0}
• ec∈{1,0}

P(U =et, C =ec) log2 P(U =et, C =ec) P(U =et)P(C =ec) = p(t, c) log2 p(t, c) p(t)p(c) + p(t, c) log2 p(t, c) p(t)p(c) + p(t, c) log2 p(t, c) p(t)p(c) + p(t, c) log2 p(t, c) p(t)p(c)

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How to compute MI values

Based on maximum likelihood estimates, the formula we actually use is: I(U; C) = N11 N log2 NN11 N1.N.1 + N10 N log2 NN10 N1.N.0 (1) +N01 N log2 NN01 N0.N.1 + N00 N log2 NN00 N0.N.0

N11: # of documents that contain t (et = 1) and are in c (ec = 1) N10: # of documents that contain t (et = 1) and not in c (ec = 0) N01: # of documents that don’t contain t (et = 0) and in c (ec = 1) N00: # of documents that don’t contain t (et = 0) and not in c (ec = 0) N = N00 + N01 + N10 + N11 p(t, c) ≈ N11/N, p(t, c) ≈ N01/N, p(t, c) ≈ N10/N, p(t, c) ≈ N00/N

• N1. = N10 + N11: # documents that contain t, p(t) ≈ N1./N

N.1 = N01 + N11: # documents in c, p(c) ≈ N.1/N

• N0. = N00 + N01: # documents that don’t contain t, p(t) ≈ N0./N

N.0 = N00 + N10: # documents not in c, p(c) ≈ N.0/N

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MI example for poultry/export in Reuters

ec = epoultry = 1 ec = epoultry = 0 et = eexport = 1 N11 = 49 N10 = 141 et = eexport = 0 N01 = 27,652 N00 = 774,106 Plug these values into formula: I(U; C) = 49 801,948 log2 801,948 · 49 (49+27,652)(49+141) + 141 801,948 log2 801,948 · 141 (141+774,106)(49+141) + 27,652 801,948 log2 801,948 · 27,652 (49+27,652)(27,652+774,106) +774,106 801,948 log2 801,948 · 774,106 (141+774,106)(27,652+774,106) ≈ 0.000105

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MI feature selection on Reuters

Terms with highest mutual information for three classes: coffee coffee 0.0111 bags 0.0042 growers 0.0025 kg 0.0019 colombia 0.0018 brazil 0.0016 export 0.0014 exporters 0.0013 exports 0.0013 crop 0.0012 sports soccer 0.0681 cup 0.0515 match 0.0441 matches 0.0408 played 0.0388 league 0.0386 beat 0.0301 game 0.0299 games 0.0284 team 0.0264 poultry poultry 0.0013 meat 0.0008 chicken 0.0006 agriculture 0.0005 avian 0.0004 broiler 0.0003 veterinary 0.0003 birds 0.0003 inspection 0.0003 pathogenic 0.0003 I(export,poultry) ≈ .000105 not among the ten highest for class poultry, but still potentially significant.

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Outline

1

Recap

2

Feature selection

3

Introduction to Hierarchical clustering

4

Single-link/Complete-link

5

Centroid/GAAC

6

Variants

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Hierarchical clustering

Our goal in hierarchical clustering is to create a hierarchy like the one we saw earlier in Reuters:

coffee poultry

• il & gas

France UK China Kenya industries regions TOP

We want to create this hierarchy automatically. We can do this either top-down or bottom-up. The best known bottom-up method is hierarchical agglomerative clustering.

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Hierarchical agglomerative clustering (HAC)

HAC creates a hierachy in the form of a binary tree. Assumes a similarity measure for determining the similarity of two clusters. Up to now, our similarity measures were for documents. We will look at four different cluster similarity measures.

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Hierarchical agglomerative clustering (HAC)

Start with each document in a separate cluster Then repeatedly merge the two clusters that are most similar Until there is only one cluster The history of merging is a hierarchy in the form of a binary tree. The standard way of depicting this history is a dendrogram.

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A dendrogram

1.0 0.8 0.6 0.4 0.2 0.0 Ag trade reform. Back−to−school spending is up Lloyd’s CEO questioned Lloyd’s chief / U.S. grilling Viag stays positive Chrysler / Latin America Ohio Blue Cross Japanese prime minister / Mexico CompuServe reports loss Sprint / Internet access service Planet Hollywood Trocadero: tripling of revenues German unions split War hero Colin Powell War hero Colin Powell Oil prices slip Chains may raise prices Clinton signs law Lawsuit against tobacco companies suits against tobacco firms Indiana tobacco lawsuit Most active stocks Mexican markets Hog prices tumble NYSE closing averages British FTSE index Fed holds interest rates steady Fed to keep interest rates steady Fed keeps interest rates steady Fed keeps interest rates steady

The history of mergers can be read off from left to right. The vertical line of each merger tells us what the similarity of the merger was. We can cut the dendrogram at a particular point (e.g., at 0.1 or 0.4) to get a flat clustering.

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Divisive clustering

Divisive clustering is top-down. Alternative to HAC (which is bottom up). Divisive clustering:

Start with all docs in one big cluster Then recursively split clusters Eventually each node forms a cluster on its own.

→ Bisecting K-means at the end For now: HAC (= bottom-up)

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Naive HAC algorithm

SimpleHAC(d1, . . . , dN) 1 for n ← 1 to N 2 do for i ← 1 to N 3 do C[n][i] ← Sim(dn, di) 4 I[n] ← 1 (keeps track of active clusters) 5 A ← [] (collects clustering as a sequence of merges) 6 for k ← 1 to N − 1 7 do i, m ← arg max{i,m:i=m∧I[i]=1∧I[m]=1} C[i][m] 8 A.Append(i, m) (store merge) 9 for j ← 1 to N 10 do (use i as representative for < i, m >) 11 C[i][j] ← Sim(< i, m >, j) 12 C[j][i] ← Sim(< i, m >, j) 13 I[m] ← 0 (deactivate cluster) 14 return A

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Computational complexity of the naive algorithm

First, we compute the similarity of all N × N pairs of documents. Then, in each of N iterations:

We scan the O(N × N) similarities to find the maximum similarity. We merge the two clusters with maximum similarity. We compute the similarity of the new cluster with all other (surviving) clusters.

There are O(N) iterations, each performing a O(N × N) “scan” operation. Overall complexity is O(N3). We’ll look at more efficient algorithms later.

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Key question: How to define cluster similarity

Single-link: Maximum similarity

Maximum similarity of any two documents

Complete-link: Minimum similarity

Minimum similarity of any two documents

Centroid: Average “intersimilarity”

Average similarity of all document pairs (but excluding pairs of docs in the same cluster) This is equivalent to the similarity of the centroids.

Group-average: Average “intrasimilarity”

Average similary of all document pairs, including pairs of docs in the same cluster

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Cluster similarity: Example

1 2 3 4 5 6 7 1 2 3 4

b b b b

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Single-link: Maximum similarity

1 2 3 4 5 6 7 1 2 3 4

b b b b

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Complete-link: Minimum similarity

1 2 3 4 5 6 7 1 2 3 4

b b b b

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Centroid: Average intersimilarity

intersimilarity = similarity of two documents in different clusters 1 2 3 4 5 6 7 1 2 3 4

b b b b

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Group average: Average intrasimilarity

intrasimilarity = similarity of any pair, including cases where the two documents are in the same cluster 1 2 3 4 5 6 7 1 2 3 4

b b b b

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Cluster similarity: Larger Example

1 2 3 4 5 6 7 1 2 3 4

b b b b b b b b b b b b b b b b b b b b

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Single-link: Maximum similarity

1 2 3 4 5 6 7 1 2 3 4

b b b b b b b b b b b b b b b b b b b b

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Complete-link: Minimum similarity

1 2 3 4 5 6 7 1 2 3 4

b b b b b b b b b b b b b b b b b b b b

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Centroid: Average intersimilarity

1 2 3 4 5 6 7 1 2 3 4

b b b b b b b b b b b b b b b b b b b b

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Group average: Average intrasimilarity

1 2 3 4 5 6 7 1 2 3 4

b b b b b b b b b b b b b b b b b b b b

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Outline

1

Recap

2

Feature selection

3

Introduction to Hierarchical clustering

4

Single-link/Complete-link

5

Centroid/GAAC

6

Variants

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Single link HAC

The similarity of two clusters is the maximum intersimilarity – the maximum similarity of a document from the first cluster and a document from the second cluster. Once we have merged two clusters, how do we update the similarity matrix? This is simple for single link: sim(ωi, (ωk1 ∪ ωk2)) = max(sim(ωi, ωk1), sim(ωi, ωk2))

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This dendrogram was produced by single-link

1.0 0.8 0.6 0.4 0.2 0.0 Ag trade reform. Back−to−school spending is up Lloyd’s CEO questioned Lloyd’s chief / U.S. grilling Viag stays positive Chrysler / Latin America Ohio Blue Cross Japanese prime minister / Mexico CompuServe reports loss Sprint / Internet access service Planet Hollywood Trocadero: tripling of revenues German unions split War hero Colin Powell War hero Colin Powell Oil prices slip Chains may raise prices Clinton signs law Lawsuit against tobacco companies suits against tobacco firms Indiana tobacco lawsuit Most active stocks Mexican markets Hog prices tumble NYSE closing averages British FTSE index Fed holds interest rates steady Fed to keep interest rates steady Fed keeps interest rates steady Fed keeps interest rates steady

Notice: many small clusters (1 or 2 members) being added to the main cluster There is no balanced 2-cluster or 3-cluster clustering that can be derived by cutting the dendrogram.

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Complete link HAC

The similarity of two clusters is the minimum intersimilarity – the minimum similarity of a document from the first cluster and a document from the second cluster. Once we have merged two clusters, how do we update the similarity matrix? Again, this is simple: sim(ωi, (ωk1 ∪ ωk2)) = min(sim(ωi, ωk1), sim(ωi, ωk2)) We measure the similarity of two clusters by computing the diameter of the cluster that we would get if we merged them.

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Complete-link dendrogram

1.0 0.8 0.6 0.4 0.2 0.0 NYSE closing averages Hog prices tumble Oil prices slip Ag trade reform. Chrysler / Latin America Japanese prime minister / Mexico Fed holds interest rates steady Fed to keep interest rates steady Fed keeps interest rates steady Fed keeps interest rates steady Mexican markets British FTSE index War hero Colin Powell War hero Colin Powell Lloyd’s CEO questioned Lloyd’s chief / U.S. grilling Ohio Blue Cross Lawsuit against tobacco companies suits against tobacco firms Indiana tobacco lawsuit Viag stays positive Most active stocks CompuServe reports loss Sprint / Internet access service Planet Hollywood Trocadero: tripling of revenues Back−to−school spending is up German unions split Chains may raise prices Clinton signs law

Notice that this dendrogram is much more balanced than the single-link one. We can create a 2-cluster clustering with two clusters of about the same size.

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Exercise: Compute single and complete link clusterings

1 2 3 4 1 2 3

×

d5

×

d6

×

d7

×

d8

×

d1

×

d2

×

d3

×

d4

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Single-link clustering

1 2 3 4 1 2 3

×

d5

×

d6

×

d7

×

d8

×

d1

×

d2

×

d3

×

d4

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Complete link clustering

1 2 3 4 1 2 3

×

d5

×

d6

×

d7

×

d8

×

d1

×

d2

×

d3

×

d4

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Single-link vs. Complete link clustering

1 2 3 4 1 2 3

×

d5

×

d6

×

d7

×

d8

×

d1

×

d2

×

d3

×

d4 1 2 3 4 1 2 3

×

d5

×

d6

×

d7

×

d8

×

d1

×

d2

×

d3

×

d4

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Single-link: Chaining

0 1 2 3 4 5 6 1 2

× × × × × × × × × × × ×

Single-link clustering often produces long, straggly clusters. For most applications, these are undesirable.

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What 2-cluster clustering will complete-link produce?

0 1 2 3 4 5 6 7 1

×

d1

×

d2

×

d3

×

d4

×

d5 Coordinates: 1 + 2ε, 4, 5 + 2ε, 6, 7 − ε, so that distance(d2, d1) = 3 − 2ε is less than distance(d2, d5) = 3 − ε and d2 joins d1 rather than d3, d4, d5.

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What 2-cluster clustering will complete-link produce?

0 1 2 3 4 5 6 7 1

×

d1

×

d2

×

d3

×

d4

×

d5 Coordinates: 1 + 2ε, 4, 5 + 2ε, 6, 7 − ε, so that distance(d2, d1) = 3 − 2ε is less than distance(d2, d5) = 3 − ε and d2 joins d1 rather than d3, d4, d5.

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Complete-link: Sensitivity to outliers

0 1 2 3 4 5 6 7 1

×

d1

×

d2

×

d3

×

d4

×

d5 The complete-link clustering of this set splits d2 from its right neighbors – clearly undesirable. The reason is the outlier d1. This shows that a single outlier can negatively affect the

• utcome of complete-link clustering.

Single-link clustering does better in this case.

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Outline

1

Recap

2

Feature selection

3

Introduction to Hierarchical clustering

4

Single-link/Complete-link

5

Centroid/GAAC

6

Variants

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Centroid HAC

The similarity of two clusters is the average intersimilarity – the average similarity of documents from the first cluster with documents from the second cluster. A naive implementation of this definition is inefficient (O(N2)), but the definition is equivalent to computing the similarity of the centroids: sim-cent(ωi, ωj) = µ(ωi) · µ(ωj) = 1 Ni

• dm∈ωi
• dm
• ·

1 Nj

• dm∈ωj
• dm
• =

1 NiNj

• dm∈ωi
• dn∈ωj
• dm ·

dn Hence the name: centroid HAC Note: this is the dot product, not cosine similarity!

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Exercise: Compute centroid clustering

1 2 3 4 5 6 7 1 2 3 4 5

× d1 × d2 × d3 × d4 ×

d5

× d6

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Centroid clustering

1 2 3 4 5 6 7 1 2 3 4 5

× d1 × d2 × d3 × d4 ×

d5

× d6

b c

µ1

b c µ2 b c

µ3

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Inversion in centroid clustering

In an inversion, the similarity increases during a merge

• sequence. Results in an “inverted” dendrogram.

Below: d1 = (1 + ε, 1), d2 = (5, 1), d3 = (3, 1 + 2 √ 3) Similarity of the first merger (d1 ∪ d2) is -4.0, similarity of second merger ((d1 ∪ d2) ∪ d3) is ≈ −3.5. 1 2 3 4 5 1 2 3 4 5

× × ×

b c

d1 d2 d3 −4 −3 −2 −1 d1 d2 d3

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Inversions

Hierarchical clustering algorithms that allow inversions are inferior. The rationale for hierarchical clustering is that at any given point, we’ve found the most coherent clustering of a given size. Intuitively: smaller clusterings should be more coherent than larger clusterings. An inversion contradicts this intuition: we have a large cluster that is more coherent than one of its subclusters.

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Group-average agglomerative clustering (GAAC)

GAAC also has an “average-similarity” criterion, but does not have inversions. idea is that next merge cluster ωk = ωi ∩ ωj should be coherent: look at all doc–doc similarities within ωk, including those within ωi and within ωj The similarity of two clusters is the average intrasimilarity – the average similarity of all document pairs (including those from the same cluster). But we exclude self-similarities.

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Group-average agglomerative clustering (GAAC)

Again, a naive implementation is inefficient (O(N2)) and there is an equivalent, more efficient, centroid-based definition: sim-ga(ωi, ωj) = 1 (Ni + Nj)(Ni + Nj − 1)

• dm∈ωi∪ωj
• dn∈ωi∪ωj

dn=dm

• dm·

dn = 1 (Ni + Nj)(Ni + Nj − 1)

• dm∈ωi∪ωj
• dm

2 − (Ni + Nj)

• Again, this is the dot product, not cosine similarity.

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Which HAC clustering should I use?

Don’t use centroid HAC because of inversions. In most cases: GAAC is best since it isn’t subject to chaining and sensitivity to outliers. However, we can only use GAAC for vector representations. For other types of document representations (or if only pairwise similarities for document are available): use complete-link. There are also some applications for single-link (e.g., duplicate detection in web search).

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Flat or hierarchical clustering?

For high efficiency, use flat clustering (or perhaps bisecting k-means) For deterministic results: HAC When a hierarchical structure is desired: hierarchical algorithm HAC also can be applied if K cannot be predetermined (can start without knowing K)

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Outline

1

Recap

2

Feature selection

3

Introduction to Hierarchical clustering

4

Single-link/Complete-link

5

Centroid/GAAC

6

Variants

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Efficient single link clustering

SingleLinkClustering(d1, . . . , dN) 1 for n ← 1 to N 2 do for i ← 1 to N 3 do C[n][i].sim ← SIM(dn, di) 4 C[n][i].index ← i 5 I[n] ← n 6 NBM[n] ← arg maxX∈{C[n][i]:n=i} X.sim 7 A ← [] 8 for n ← 1 to N − 1 9 do i1 ← arg max{i:I[i]=i} NBM[i].sim 10 i2 ← I[NBM[i1].index] 11 A.Append(i1, i2) 12 for i ← 1 to N 13 do if I[i] = i ∧ i = i1 ∧ i = i2 14 then C[i1][i].sim ← C[i][i1].sim ← max(C[i1][i].sim, C[i2][i].sim) 15 if I[i] = i2 16 then I[i] ← i1 17 NBM[i1] ← arg maxX∈{C[i1][i]:I[i]=i∧i=i1} X.sim 18 return A

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Time complexity of HAC

The single-link algorithm we just saw is O(N2). Much more efficient than the O(N3) algorithm we looked at earlier! There is no known O(N2) algorithm for complete-link, centroid and GAAC. Best time complexity for these three is O(N2 log N): See book. In practice: little difference between O(N2 log N) and O(N2).

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Combination similarities of the four algorithms

clustering algorithm sim(ℓ, k1, k2) single-link max(sim(ℓ, k1), sim(ℓ, k2)) complete-link min(sim(ℓ, k1), sim(ℓ, k2)) centroid ( 1

Nm

vm) · ( 1

Nℓ

vℓ) group-average

1 (Nm+Nℓ)(Nm+Nℓ−1)[(

vm + vℓ)2 − (Nm + Nℓ)]

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Comparison of HAC algorithms

method combination similarity time compl.

• ptimal?

comment single-link max intersimilarity of any 2 docs Θ(N2) yes chaining effect complete-link min intersimilarity of any 2 docs Θ(N2 log N) no sensitive to outliers group-average average of all sims Θ(N2 log N) no best choice for most applications centroid average intersimilarity Θ(N2 log N) no inversions can occur

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What to do with the hierarchy?

Use as is (e.g., for browsing as in Yahoo hierarchy) Cut at a predetermined threshold Cut to get a predetermined number of clusters K

Ignores hierarchy below and above cutting line.

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Bisecting K-means: A top-down algorithm

Start with all documents in one cluster Split the cluster into 2 using K-means Of the clusters produced so far, select one to split (e.g. select the largest one) Repeat until we have produced the desired number of clusters

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Bisecting K-means

BisectingKMeans(d1, . . . , dN) 1 ω0 ← { d1, . . . , dN} 2 leaves ← {ω0} 3 for k ← 1 to K − 1 4 do ωk ← PickClusterFrom(leaves) 5 {ωi, ωj} ← KMeans(ωk, 2) 6 leaves ← leaves \ {ωk} ∪ {ωi, ωj} 7 return leaves

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Bisecting K-means

If we don’t generate a complete hierarchy, then a top-down algorithm like bisecting K-means is much more efficient than HAC algorithms. But bisecting K-means is not deterministic. There are deterministic versions of bisecting K-means but they are much less efficient.

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