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Ijcai 2007 Pedersen

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University of Minnesota, Duluth
University of Minnesota, Duluth
Technologie
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Language Independent Methods of Clustering Similar Contexts (with applications) Ted Pedersen University of Minnesota, Duluth [email_address] http://www.d.umn.edu/~tpederse/SCTutorial.html
Language Independent Methods ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
A Note on Tokenization ,[object Object],[object Object],[object Object],[object Object]
Clustering Similar Contexts ,[object Object],[object Object],[object Object],[object Object],[object Object]
Applications ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Tutorial Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
SenseClusters ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Many thanks… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Background and Motivations
Headed and Headless Contexts ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Headed Contexts (input) ,[object Object],[object Object],[object Object],[object Object],[object Object]
Headed Contexts (output) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Headless Contexts (input) ,[object Object],[object Object],[object Object],[object Object],[object Object]
Headless Contexts (output) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Web Search as Application ,[object Object],[object Object],[object Object],[object Object]
Name Discrimination
George Millers!
 
 
 
 
Email Foldering as Application ,[object Object],[object Object],[object Object],[object Object],[object Object]
Clustering News as Application ,[object Object],[object Object],[object Object],[object Object]
What is it to be “similar”? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
General Methodology ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Identifying Lexical Features Measures of Association and Tests of Significance
What are features? ,[object Object],[object Object],[object Object]
Feature Selection Data ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Feature Selection Data ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Lexical Features ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Lexical Features ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Bigrams ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Co-occurrences ,[object Object],[object Object],[object Object],[object Object],[object Object]
Bigrams and Co-occurrences ,[object Object],[object Object],[object Object],[object Object],[object Object]
“ occur together more often than expected by chance…” ,[object Object],[object Object],[object Object],[object Object]
2x2 Contingency Table 100,000 300 not Artificial 400 100 Artificial not Intelligence Intelligence
2x2 Contingency Table 100,000 99,700 300 99,600 99,400 200 not Artificial 400 300 100 Artificial not Intelligence Intelligence
2x2 Contingency Table 100,000 99,700 300 99,600 99,400.0 99,301.2 200.0 298.8 not Artificial 400 300.0 398.8 100.0 000.12 Artificial not Intelligence Intelligence
Measures of Association
Measures of Association
Interpreting the Scores… ,[object Object],[object Object]
 
Interpreting the Scores… ,[object Object],[object Object],[object Object]
Measures of Association ,[object Object],[object Object],[object Object],[object Object]
Measures Supported in NSP ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
References ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Context Representations First and Second Order Methods
Once features selected… ,[object Object],[object Object],[object Object]
Possible Representations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
First Order Representation Native SenseClusters ,[object Object],[object Object],[object Object],[object Object]
Contexts ,[object Object],[object Object],[object Object],[object Object]
Unigram Features ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
First Order Vectors of Unigrams 1 0 1 0 1 x4 0 0 0 0 0 x3 1 1 0 1 0 x2 1 1 1 1 1 x1 child magic curse black island
Bigram Feature Set ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
First Order Vectors of Bigrams 1 0 1 1 0 x4 0 1 1 0 0 x3 1 0 0 0 1 x2 1 0 0 1 1 x1 voodoo child serious error military might island curse black magic
First Order Vectors ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Second Order Features ,[object Object],[object Object],[object Object],[object Object],[object Object]
Second Order Representation Native SenseClusters ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Word by Word Matrix 120.0 0 69.4 0 0 voodoo 0 89.2 0 21.2 0 serious 0 54.9 100.3 0 0 military 73.2 0 0 189.2 0 island 43.2 0 0 0 123.5 black child error might curse magic
Word by Word Matrix ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
There was an island curse of black magic cast by that voodoo child. 120.0 0 69.4 0 0 voodoo 73.2 0 0 189.2 0 island 43.2 0 0 0 123.5 black child error might curse magic
Second Order Co-Occurrences ,[object Object],[object Object]
Second Order Representation ,[object Object],[object Object],[object Object]
There was an island curse of black magic cast by that voodoo child. 78.8 0 24.4 63.1 41.2 x1 child error might curse magic
Second Order Representation Native SenseClusters ,[object Object],[object Object]
Second Order Representation Latent Semantic Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
First Order Vectors of Unigrams 1 0 1 0 1 x4 0 0 0 0 0 x3 1 1 0 1 0 x2 1 1 1 1 1 x1 child magic curse black island
Transposed 1 0 1 1 child 0 0 1 1 magic 1 0 0 1 curse 0 0 1 1 black 1 0 0 1 island x4 x3 x2 x1
harold a known voodoo child was gifted in the arts of black magic 1 0 1 1 child 0 0 1 1 magic 0 0 1 1 black x4 x3 x2 x1
Second Order Representation ,[object Object],[object Object],[object Object]
x2: harold a known voodoo child was gifted in the arts of black magic .3 0 1 1 x2 x4 x3 x2 x1
Second Order Representation Latent Semantic Analysis ,[object Object],[object Object],[object Object]
Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
References ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Dimensionality Reduction Singular Value Decomposition
Motivation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Many Methods ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Effect of SVD ,[object Object],[object Object]
Effect of SVD ,[object Object],[object Object],[object Object]
How can SVD be used? ,[object Object],[object Object],[object Object],[object Object]
Word by Word Matrix native SenseClusters 2 nd order 4 2 0 0 0 3 0 1 box 0 1 2 2 1 2 0 0 memory 0 0 0 1 0 0 2 0 organ 0 2 0 3 2 0 0 0 debt 0 1 0 3 1 0 0 2 linux 0 1 0 3 2 0 0 0 sales 3 0 2 2 0 3 0 0 lab 1 0 2 0 0 1 2 0 petri 0 1 0 0 2 0 0 1 disk 1 0 2 0 0 0 3 0 body 0 0 0 3 1 0 0 2 pc plasma graphics tissue data ibm cells blood apple
Singular Value Decomposition A=UDV’
U -.52 .39 -.48 .02 .09 .41 -.09 .40 -.30 .08 .31 .43 -.26 -.39 -.6 .20 .00 -.00 -.00 -.02 -.01 .00 -.02 -.00 -.07 -.3 .14 -.49 -.07 .30 .25 .56 -.01 .08 .05 -.01 .24 -.08 .11 .46 .08 .03 -.04 .72 .09 -.31 -.01 .37 -.07 .01 -.21 -.31 -.34 -.45 -.68 .29 .00 .05 .83 .17 -.02 .25 -.45 .08 .03 .20 -.22 .31 -.60 .39 .13 .35 -.01 -.04 -.44 .08 .44 .59 -.49 .05 -.02 .63 .02 -.09 .52 -.2 .09 .35
D 0.00 0.00 0.00 0.66 1.26 2.30 2.52 3.25 3.99 6.36 9.19
V -.20 .22 -.07 -.10 -.87 -.07 -.06 .17 .19 -.26 .04 .03 .17 -.32 .02 .13 -.26 -.17 .06 -.04 .86 .50 -.58 .12 .09 -.18 -.27 -.18 -.12 -.47 .11 -.03 .12 .31 -.32 -.04 .64 -.45 -.14 -.23 .28 .07 -.23 -.62 -.59 .05 .02 -.12 .15 .11 .25 -.71 -.31 -.04 .08 .29 -.05 .05 .20 -.51 .09 -.03 .12 .31 -.01 .02 -.45 -.32 .50 .27 .49 -.02 .08 .21 -.06 .08 -.09 .52 -.45 -.01 .63 .03 -.12 -.31 .71 -.13 .39 -.12 .12 .15 .37 .07 .58 -.41 .15 .17 -.30 -.32 -.27 -.39 .11 .44 .25 .03 -.02 .26 .23 .39 .57 -.37 .04 .03 -.12 -.31 -.05 -.05 .04 .28 -.04 .08 .21
Word by Word Matrix After SVD 1.1 1.0 .98 1.7 .86 .72 .85 .77 memory .00 .00 .17 1.2 .77 .00 .84 .00 organ .00 1.5 .00 3.2 2.1 .00 .00 1.2 debt .13 1.1 .03 2.7 1.7 .16 .00 .96 linux .41 .85 .35 2.2 1.3 .39 .15 .73 sales 2.3 .18 2.5 1.7 .35 2.0 1.7 .21 lab 1.4 .00 1.5 .49 .00 1.2 1.1 .00 germ .00 .91 .00 2.1 1.3 .01 .00 .76 disk 1.5 .00 1.6 .33 .00 1.3 1.2 .00 body .09 .86 .01 2.0 1.3 .11 .00 .73 pc plasma graphics tissue data ibm cells blood apple
Second Order Co-Occurrences ,[object Object],[object Object],[object Object],[object Object],1.0 .72 memory .00 .00 organ .13 1.1 .03 2.7 1.7 .16 .00 .96 linux .00 .91 .00 2.1 1.3 .01 .00 .76 disk Plasma graphics tissue data ibm cells blood apple
References ,[object Object],[object Object],[object Object],[object Object]
Clustering Partitional Methods Cluster Stopping Cluster Labeling
Many many methods… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
General Methodology ,[object Object],[object Object],[object Object],[object Object],[object Object]
Agglomerative Clustering ,[object Object],[object Object],[object Object]
Measuring Similarity ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Agglomerative Clustering ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Average Link Clustering 1 2 4 S3 1 2 4 S3 0 2 S4 0 3 S2 2 3 S1 S4 S2 S1 0 S4 0 S2 S1S3 S4 S2 S1S3 S4 S1S3S2 S4 S1S3S2
Partitional Methods ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Partitional Methods ,[object Object],[object Object],[object Object],[object Object],[object Object]
Vectors to be clustered
Random Initial Centroids (k=2)
Assignment of Clusters
Recalculation of Centroids
Reassignment of Clusters
Recalculation of Centroid
Reassignment of Clusters
Partitional Criterion Functions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Intra Cluster Similarity ,[object Object],[object Object],[object Object],[object Object]
Contexts to be Clustered
Ball of String (I1 Internal Criterion Function)
Flower (I2 Internal Criterion Function)
Inter Cluster Similarity ,[object Object],[object Object],[object Object]
The Fan (E1 External Criterion Function)
Hybrid Criterion Functions ,[object Object],[object Object],[object Object],[object Object],[object Object]
Cluster Stopping
Cluster Stopping ,[object Object],[object Object]
Criterion Functions Can Help ,[object Object],[object Object],[object Object],[object Object]
H2 versus k T. Blair – V. Putin – S. Hussein
PK2 ,[object Object],[object Object],[object Object]
PK2 predicts 3 senses T. Blair – V. Putin – S. Hussein
PK3 ,[object Object],[object Object],[object Object],[object Object]
PK3 predicts 3 senses T. Blair – V. Putin – S. Hussein
Adapted Gap Statistic ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Adapted Gap Statistic
Gap predicts 3 senses T. Blair – V. Putin – S. Hussein
References ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Cluster Labeling
Cluster Labeling ,[object Object],[object Object]
Results of Clustering ,[object Object],[object Object],[object Object],[object Object]
Label Types ,[object Object],[object Object]
George Miller Labels ,[object Object],[object Object],[object Object]
Evaluation Techniques Comparison to gold standard data
Evaluation ,[object Object],[object Object],[object Object],[object Object]
Evaluation ,[object Object],[object Object],[object Object],[object Object]
Evaluation ,[object Object],[object Object],[object Object]
Baseline Algorithm ,[object Object],[object Object]
Baseline Performance ,[object Object],170 55 35 80 Totals 170 55 35 80 C3 0 0 0 0 C2 0 0 0 0 C1 Totals S3 S2 S1 170 80 35 55 Totals 170 80 35 55 C3 0 0 0 0 C2 0 0 0 0 C1 Totals S1 S2 S3
Evaluation ,[object Object],[object Object],[object Object],[object Object],[object Object],170 55 35 80 Totals 65 10 5 50 C3 60 40 0 20 C2 45 5 30 10 C1 Totals S3 S2 S1
Evaluation ,[object Object],[object Object],[object Object],170 80 55 35 Totals 65 50 10 5 C3 60 20 40 0 C2 45 10 5 30 C1 Totals S1 S3 S2
Analysis ,[object Object],[object Object],[object Object],[object Object]
Hands on Experience Experiments with SenseClusters
Things to Try ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Experimental Data ,[object Object],[object Object],[object Object],[object Object]
Creating Experimental Data ,[object Object],[object Object],[object Object],[object Object],[object Object]
 
 
 
 
Thank you! ,[object Object],[object Object],[object Object],[object Object],[object Object]
Target Word Clustering SenseClusters Native Mode ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Target Word Clustering Latent Semantic Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Feature Clustering Latent Semantic Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
 
 
 
 
 
 
 
 
 

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Ijcai 2007 Pedersen

  • 1. Language Independent Methods of Clustering Similar Contexts (with applications) Ted Pedersen University of Minnesota, Duluth [email_address] http://www.d.umn.edu/~tpederse/SCTutorial.html
  • 2.
  • 3.
  • 4.
  • 5.
  • 6.
  • 7.
  • 8.
  • 10.
  • 11.
  • 12.
  • 13.
  • 14.
  • 15.
  • 18.  
  • 19.  
  • 20.  
  • 21.  
  • 22.
  • 23.
  • 24.
  • 25.
  • 26. Identifying Lexical Features Measures of Association and Tests of Significance
  • 27.
  • 28.
  • 29.
  • 30.
  • 31.
  • 32.
  • 33.
  • 34.
  • 35.
  • 36. 2x2 Contingency Table 100,000 300 not Artificial 400 100 Artificial not Intelligence Intelligence
  • 37. 2x2 Contingency Table 100,000 99,700 300 99,600 99,400 200 not Artificial 400 300 100 Artificial not Intelligence Intelligence
  • 38. 2x2 Contingency Table 100,000 99,700 300 99,600 99,400.0 99,301.2 200.0 298.8 not Artificial 400 300.0 398.8 100.0 000.12 Artificial not Intelligence Intelligence
  • 41.
  • 42.  
  • 43.
  • 44.
  • 45.
  • 46.
  • 47.
  • 48. Context Representations First and Second Order Methods
  • 49.
  • 50.
  • 51.
  • 52.
  • 53.
  • 54. First Order Vectors of Unigrams 1 0 1 0 1 x4 0 0 0 0 0 x3 1 1 0 1 0 x2 1 1 1 1 1 x1 child magic curse black island
  • 55.
  • 56. First Order Vectors of Bigrams 1 0 1 1 0 x4 0 1 1 0 0 x3 1 0 0 0 1 x2 1 0 0 1 1 x1 voodoo child serious error military might island curse black magic
  • 57.
  • 58.
  • 59.
  • 60. Word by Word Matrix 120.0 0 69.4 0 0 voodoo 0 89.2 0 21.2 0 serious 0 54.9 100.3 0 0 military 73.2 0 0 189.2 0 island 43.2 0 0 0 123.5 black child error might curse magic
  • 61.
  • 62. There was an island curse of black magic cast by that voodoo child. 120.0 0 69.4 0 0 voodoo 73.2 0 0 189.2 0 island 43.2 0 0 0 123.5 black child error might curse magic
  • 63.
  • 64.
  • 65. There was an island curse of black magic cast by that voodoo child. 78.8 0 24.4 63.1 41.2 x1 child error might curse magic
  • 66.
  • 67.
  • 68. First Order Vectors of Unigrams 1 0 1 0 1 x4 0 0 0 0 0 x3 1 1 0 1 0 x2 1 1 1 1 1 x1 child magic curse black island
  • 69. Transposed 1 0 1 1 child 0 0 1 1 magic 1 0 0 1 curse 0 0 1 1 black 1 0 0 1 island x4 x3 x2 x1
  • 70. harold a known voodoo child was gifted in the arts of black magic 1 0 1 1 child 0 0 1 1 magic 0 0 1 1 black x4 x3 x2 x1
  • 71.
  • 72. x2: harold a known voodoo child was gifted in the arts of black magic .3 0 1 1 x2 x4 x3 x2 x1
  • 73.
  • 74.
  • 75.
  • 76. Dimensionality Reduction Singular Value Decomposition
  • 77.
  • 78.
  • 79.
  • 80.
  • 81.
  • 82. Word by Word Matrix native SenseClusters 2 nd order 4 2 0 0 0 3 0 1 box 0 1 2 2 1 2 0 0 memory 0 0 0 1 0 0 2 0 organ 0 2 0 3 2 0 0 0 debt 0 1 0 3 1 0 0 2 linux 0 1 0 3 2 0 0 0 sales 3 0 2 2 0 3 0 0 lab 1 0 2 0 0 1 2 0 petri 0 1 0 0 2 0 0 1 disk 1 0 2 0 0 0 3 0 body 0 0 0 3 1 0 0 2 pc plasma graphics tissue data ibm cells blood apple
  • 84. U -.52 .39 -.48 .02 .09 .41 -.09 .40 -.30 .08 .31 .43 -.26 -.39 -.6 .20 .00 -.00 -.00 -.02 -.01 .00 -.02 -.00 -.07 -.3 .14 -.49 -.07 .30 .25 .56 -.01 .08 .05 -.01 .24 -.08 .11 .46 .08 .03 -.04 .72 .09 -.31 -.01 .37 -.07 .01 -.21 -.31 -.34 -.45 -.68 .29 .00 .05 .83 .17 -.02 .25 -.45 .08 .03 .20 -.22 .31 -.60 .39 .13 .35 -.01 -.04 -.44 .08 .44 .59 -.49 .05 -.02 .63 .02 -.09 .52 -.2 .09 .35
  • 85. D 0.00 0.00 0.00 0.66 1.26 2.30 2.52 3.25 3.99 6.36 9.19
  • 86. V -.20 .22 -.07 -.10 -.87 -.07 -.06 .17 .19 -.26 .04 .03 .17 -.32 .02 .13 -.26 -.17 .06 -.04 .86 .50 -.58 .12 .09 -.18 -.27 -.18 -.12 -.47 .11 -.03 .12 .31 -.32 -.04 .64 -.45 -.14 -.23 .28 .07 -.23 -.62 -.59 .05 .02 -.12 .15 .11 .25 -.71 -.31 -.04 .08 .29 -.05 .05 .20 -.51 .09 -.03 .12 .31 -.01 .02 -.45 -.32 .50 .27 .49 -.02 .08 .21 -.06 .08 -.09 .52 -.45 -.01 .63 .03 -.12 -.31 .71 -.13 .39 -.12 .12 .15 .37 .07 .58 -.41 .15 .17 -.30 -.32 -.27 -.39 .11 .44 .25 .03 -.02 .26 .23 .39 .57 -.37 .04 .03 -.12 -.31 -.05 -.05 .04 .28 -.04 .08 .21
  • 87. Word by Word Matrix After SVD 1.1 1.0 .98 1.7 .86 .72 .85 .77 memory .00 .00 .17 1.2 .77 .00 .84 .00 organ .00 1.5 .00 3.2 2.1 .00 .00 1.2 debt .13 1.1 .03 2.7 1.7 .16 .00 .96 linux .41 .85 .35 2.2 1.3 .39 .15 .73 sales 2.3 .18 2.5 1.7 .35 2.0 1.7 .21 lab 1.4 .00 1.5 .49 .00 1.2 1.1 .00 germ .00 .91 .00 2.1 1.3 .01 .00 .76 disk 1.5 .00 1.6 .33 .00 1.3 1.2 .00 body .09 .86 .01 2.0 1.3 .11 .00 .73 pc plasma graphics tissue data ibm cells blood apple
  • 88.
  • 89.
  • 90. Clustering Partitional Methods Cluster Stopping Cluster Labeling
  • 91.
  • 92.
  • 93.
  • 94.
  • 95.
  • 96. Average Link Clustering 1 2 4 S3 1 2 4 S3 0 2 S4 0 3 S2 2 3 S1 S4 S2 S1 0 S4 0 S2 S1S3 S4 S2 S1S3 S4 S1S3S2 S4 S1S3S2
  • 97.
  • 98.
  • 99. Vectors to be clustered
  • 106.
  • 107.
  • 108. Contexts to be Clustered
  • 109. Ball of String (I1 Internal Criterion Function)
  • 110. Flower (I2 Internal Criterion Function)
  • 111.
  • 112. The Fan (E1 External Criterion Function)
  • 113.
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  • 117. H2 versus k T. Blair – V. Putin – S. Hussein
  • 118.
  • 119. PK2 predicts 3 senses T. Blair – V. Putin – S. Hussein
  • 120.
  • 121. PK3 predicts 3 senses T. Blair – V. Putin – S. Hussein
  • 122.
  • 124. Gap predicts 3 senses T. Blair – V. Putin – S. Hussein
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  • 131. Evaluation Techniques Comparison to gold standard data
  • 132.
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  • 139.
  • 140. Hands on Experience Experiments with SenseClusters
  • 141.
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  • 200.  

Hinweis der Redaktion

  1. The main idea is to assume a null hypothesis of single cluster and to see if the alternative hypothesis of k>1 clusters is able to refute the null hypothesis.