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K-means properties

Pasi Fränti. K-means properties. 11.10.2017. K-means properties on six clustering benchmark datasets Pasi Fränti and Sami Sieranoja Algorithms , 2017. Goal of k-means. Input N points:. X={ x 1 ,  x 2 , …,  x N }. Output partition and k centroids:. P={ p 1 , p 2 , …,  p k }.

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K-means properties

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  1. Pasi Fränti K-means properties 11.10.2017 K-means properties on six clustering benchmark datasets Pasi Fränti and Sami Sieranoja Algorithms, 2017.

  2. Goal of k-means Input N points: X={x1, x2, …, xN} Output partition and k centroids: P={p1, p2, …, pk} C={c1, c2, …, ck} Objective function: SSE = sum-of-squared errors

  3. Goal of k-means Input N points: X={x1, x2, …, xN} Output partition and k centroids: P={p1, p2, …, pk} C={c1, c2, …, ck} Objective function: Assumptions: • SSE is suitable • k is known

  4. K-means algorithm X = Data set C = Cluster centroids P = Partition K-Means(X, C) → (C, P) REPEAT Cprev← C; FOR i=1 TO N DOpi ← FindNearest(xi, C); FOR j=1 TO k DO cj ← Average of xi pi = j; UNTIL C = Cprev Assignment step Centroid step

  5. K-means optimization steps Assignment step: Centroid step:

  6. Problems of k-meansDistance of clusters Cannot move centroids between clusters far away

  7. Problems of k-meansDependency of initial solution Initial solution: After k-means:

  8. K-means performanceHow affected by? 1. Overlap 3. Dimensionality 4. Unbalance of cluster sizes 2. Number of clusters

  9. Basic Benchmark

  10. Data sets statistics

  11. A1 A2 A3 A sets K=20 K=35 K=50 • Spherical clusters • Number of clusters changing from k=20 to 50 • Subsets of each other: A1  A2  A3. • Other parameters fixed: • Cluster size = 150 • Deviation = 1402 • Overlap = 0.30 • - Dimensionality = 2

  12. overlap increases S sets K=15 Gaussian clusters (few truncated) S S S S 1 2 3 4 9% 41% 22% 44% Least overlap Strong overlap but the clusters still recognizable

  13. Unbalance K=8 Dense clusters Sparse clusters st.dev=2043 st.dev=6637 100 100 Areas well-separated 2000 100 2000 2000 100 100 *Correct clustering can be obtained by minimizing SSE

  14. DIM sets K=16 DIM32 • Well-separated clusters in high-dimensional spaces • Dimensions vary: 32, 64, 128, 256, 512, 1024

  15. G2 Datasets K=2 G2-2-30 G2-2-50 G2-2-70 600,600 500,500 Dataset name: G2-dim-sd Centroid 1: [500,500, ...] Centroid 2: [600,600, ...] Dimensions: 2,4,8,16, ... 1024 St.dev. 10,20,30, ... 100

  16. Birch Birch1 Birch2 Regular 10x10 grid Constant variance offset = 43659 amplitude = -37819 phaseshift = 20.8388 frequency = 0.000004205 y(x) = amplitude * sin(2**frequency*x + phaseshift) + offset

  17. Birch2 subsets B2-random B2-sub N=100 000 k=100 N=99 000 k=99 k=98 N=98 000 N=97 000 k=97 k=96 k=95 …...………….. Random subsampling ….… Cutting off last cluster k=3 N=2 000 k=2 N=1 000 k=1

  18. Properties

  19. Measured properties • Overlap • Contrast • Intrinsic dimensionality • H-index • Distance profiles

  20. Misclassification probability Points from blue cluster that are closer to red centroid. Points from red cluster that are closer to blue centroid. Points = 2048 Incorrect = 20 Overlap = 20 / 2048 0.9 %

  21. Overlap Points in blue cluster whose red neigbor is closer than its centroids. Points in red cluster whose blue neighbor is closer than its centroids. Points = 2048 Evidence = 332 Overlap = 332 / 2048 16 % d1 = distance to nearest centroid d2 = distance to 2nd nearest

  22. Contrast

  23. Intrinsic dimensionality Average of distances Variance of distances Unbalance DIM 0.4 Birch2 Birch1 6.7 - 7.5 2.6 S sets A1 A2 A3 8.3 2.0 2.5 1.5 2.2

  24. H-index Hubness values: 2 0 4 2 Hub 2 2 2 Rank: 1 2 3 4 5 6 7 Hub: 4 2 2 2 2 2 0

  25. Distance profiles • Data that contains clusters tends to have two peaks: • Local distances: distances inside the clusters • Global distances: distances across different clusters A2 A1 A3 S1 S2 S3 S4 Birch1 Birch2 DIM32 Unbalance

  26. Distance profilesG2 datasets G2: dimension increases D=2 D=4 D=8 D=16 D=32 D=1024 G2: overlap increases sd=10 sd=20 sd=30 sd=40 sd=50 sd=60 D=2 sd=10 sd=20 sd=30 sd=40 sd=50 sd=60 D=128

  27. Summary of the properties

  28. G2 overlap Overlap decreases Overlap increases

  29. G2 contrast Contrast decreases Contrast decreases

  30. G2 Intrinsic dimensionality ID increases (if overlap) ID increases Most significant

  31. G2 H-index H-index increases No change Most significant

  32. Evaluation

  33. Internal measures Sum of squared distances (SSE) Normalized mean square error (nMSE) Approximation ratio ()

  34. External measuresCentroid index P. Fränti, M. Rezaei, Q. Zhao Centroid index: cluster level similarity measure Pattern Recognition 2014 CI=4 Missing centroids Too many centroids

  35. External measuresSuccess rate 17% CI=1 CI=2 CI=1 CI=0 CI=2 CI=2

  36. Results

  37. Summary of results

  38. overlap increases Dependency on overlapS datasets Success rates and CI-values: S S S S 1 2 3 4 3% 11% 12% 26% CI=1.3 CI=0.9 CI=1.8 CI=1.4

  39. Dependency on overlapG2 datasets

  40. Why overlap helps? Overlap = 22% 90 iterations Overlap = 7% 13 iterations

  41. Main observation 1. Overlap Overlap is good!

  42. A1 A2 A3 Clusters increases Dependency on clusters (k)A datasets K=20 K=35 K=50 0% 1% 0% Success: 4.5 2.5 6.6 CI: 13% 13% 13% Relative CI:

  43. Dependency on clusters (k)

  44. Dependency on data size (N)

  45. Main observation 2. Number of clusters Linear increase with k!

  46. Dimensions increases Dependency on dimensionsDIM datasets 32 64 128 512 1024 256 CI: 3.6 3.5 3.8 3.8 3.9 3.7 0% Success rate:

  47. Dependency on dimensionsG2 datasets Success degrades Success improves

  48. Lack of overlap is the cause! Overlap Correlation: 0.91 Success

  49. Main observation 3. Dimensionality No direct effect!

  50. Effect of unbalanceDIM datasets Success: Average CI: Problem originates from the random initialization. 3.9 0%

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