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Clustering Algorithms Hierarchical k-Means Distance Measures

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Clustering Distance Measures Hierarchical Clustering k -Means Algorithms
The Problem of Clustering ,[object Object]
Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x
Problems With Clustering ,[object Object],[object Object],[object Object]
The Curse of Dimensionality ,[object Object],[object Object],[object Object]
Example : SkyCat ,[object Object],[object Object],[object Object]
Example : Clustering CD’s (Collaborative Filtering) ,[object Object],[object Object],[object Object],[object Object]
The Space of CD’s ,[object Object],[object Object],[object Object],[object Object]
Example : Clustering Documents ,[object Object],[object Object],[object Object]
Example : Protein Sequences ,[object Object],[object Object],[object Object]
Distance Measures ,[object Object],[object Object],[object Object],[object Object]
Euclidean Vs. Non-Euclidean ,[object Object],[object Object],[object Object],[object Object]
Axioms of a Distance Measure ,[object Object],[object Object],[object Object],[object Object],[object Object]
Some Euclidean Distances ,[object Object],[object Object],[object Object],[object Object]
Examples of Euclidean Distances x = (5,5) y = (9,8) L 2 -norm: dist(x,y) =  (4 2 +3 2 ) = 5 L 1 -norm: dist(x,y) = 4+3 = 7 4 3 5
Another Euclidean Distance ,[object Object],[object Object]
Non-Euclidean Distances ,[object Object],[object Object],[object Object]
Jaccard Distance ,[object Object],[object Object],[object Object],[object Object]
Why J.D. Is a Distance Measure ,[object Object],[object Object],[object Object],[object Object]
Triangle Inequality for J.D. ,[object Object],[object Object],[object Object],[object Object]
Triangle Inequality --- (2) ,[object Object],[object Object]
Cosine Distance ,[object Object],[object Object],[object Object],[object Object],[object Object]
Cosine-Measure Diagram p 1 p 2 p 1 .p 2  |p 2 | dist(p 1 , p 2 ) =  = arccos(p 1 .p 2 /|p 2 ||p 1 |)
Why? p 1 = (x 1 ,y 1 ) p 2 = (x 2 ,0) x 1  Dot product is invariant under rotation, so pick convenient coordinate system. x 1 = p 1 .p 2 /|p 2 | p 1 .p 2 = x 1 *x 2 . |p 2 | = x 2 .
Why C.D. Is a Distance Measure ,[object Object],[object Object],[object Object],[object Object]
Edit Distance ,[object Object],[object Object],[object Object]
Example ,[object Object],[object Object],[object Object],[object Object],[object Object]
Why E.D. Is a Distance Measure ,[object Object],[object Object],[object Object],[object Object]
Variant Edit Distance ,[object Object],[object Object],[object Object]
Methods of Clustering ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Hierarchical Clustering ,[object Object],[object Object],[object Object]
Example (5,3) o (1,2) o o (2,1) o (4,1) o (0,0) o (5,0) x (1.5,1.5) x (4.5,0.5) x (1,1) x (4.7,1.3)
And in the Non-Euclidean Case? ,[object Object],[object Object],[object Object],[object Object]
“Closest”? ,[object Object],[object Object],[object Object],[object Object]
Example 1 2 3 4 5 6 intercluster distance clustroid clustroid
Other Approaches to Defining “Nearness” of Clusters ,[object Object],[object Object],[object Object]
k –Means Algorithm(s) ,[object Object],[object Object],[object Object],[object Object]
Populating Clusters ,[object Object],[object Object],[object Object],[object Object]
Example 1 2 3 4 5 6 7 8 x x Clusters after first round Reassigned points
Getting k Right ,[object Object],[object Object],k Average distance to centroid Best value of k
Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too few; many long distances to centroid.
Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Just right; distances rather short.
Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too many; little improvement in average distance.
BFR Algorithm ,[object Object],[object Object],[object Object]
BFR --- (2) ,[object Object],[object Object],[object Object]
Initialization: k -Means ,[object Object],[object Object],[object Object]
Three Classes of Points ,[object Object],[object Object],[object Object]
Representing Sets of Points ,[object Object],[object Object],[object Object],[object Object]
Comments ,[object Object],[object Object],[object Object],[object Object]
Comments --- (2) ,[object Object],[object Object],[object Object]
“Galaxies” Picture A cluster. Its points are in the DS. The centroid Compressed sets. Their points are in the CS. Points in the RS
Processing a “Memory-Load” of Points ,[object Object],[object Object],[object Object]
Processing --- (2) ,[object Object],[object Object],[object Object]
A Few Details . . . ,[object Object],[object Object]
How Close is Close Enough? ,[object Object],[object Object],[object Object],[object Object]
Mahalanobis Distance ,[object Object],[object Object],[object Object],[object Object],[object Object]
Mahalanobis Distance --- (2) ,[object Object],[object Object],[object Object]
Picture: Equal M.D. Regions  2 
Should Two CS Subclusters Be Combined? ,[object Object],[object Object],[object Object]

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Clustering Algorithms Hierarchical k-Means Distance Measures

  • 1. Clustering Distance Measures Hierarchical Clustering k -Means Algorithms
  • 2.
  • 3. Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x
  • 4.
  • 5.
  • 6.
  • 7.
  • 8.
  • 9.
  • 10.
  • 11.
  • 12.
  • 13.
  • 14.
  • 15. Examples of Euclidean Distances x = (5,5) y = (9,8) L 2 -norm: dist(x,y) =  (4 2 +3 2 ) = 5 L 1 -norm: dist(x,y) = 4+3 = 7 4 3 5
  • 16.
  • 17.
  • 18.
  • 19.
  • 20.
  • 21.
  • 22.
  • 23. Cosine-Measure Diagram p 1 p 2 p 1 .p 2  |p 2 | dist(p 1 , p 2 ) =  = arccos(p 1 .p 2 /|p 2 ||p 1 |)
  • 24. Why? p 1 = (x 1 ,y 1 ) p 2 = (x 2 ,0) x 1  Dot product is invariant under rotation, so pick convenient coordinate system. x 1 = p 1 .p 2 /|p 2 | p 1 .p 2 = x 1 *x 2 . |p 2 | = x 2 .
  • 25.
  • 26.
  • 27.
  • 28.
  • 29.
  • 30.
  • 31.
  • 32. Example (5,3) o (1,2) o o (2,1) o (4,1) o (0,0) o (5,0) x (1.5,1.5) x (4.5,0.5) x (1,1) x (4.7,1.3)
  • 33.
  • 34.
  • 35. Example 1 2 3 4 5 6 intercluster distance clustroid clustroid
  • 36.
  • 37.
  • 38.
  • 39. Example 1 2 3 4 5 6 7 8 x x Clusters after first round Reassigned points
  • 40.
  • 41. Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too few; many long distances to centroid.
  • 42. Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Just right; distances rather short.
  • 43. Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too many; little improvement in average distance.
  • 44.
  • 45.
  • 46.
  • 47.
  • 48.
  • 49.
  • 50.
  • 51. “Galaxies” Picture A cluster. Its points are in the DS. The centroid Compressed sets. Their points are in the CS. Points in the RS
  • 52.
  • 53.
  • 54.
  • 55.
  • 56.
  • 57.
  • 58. Picture: Equal M.D. Regions  2 
  • 59.