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Cluster Validation

Cluster Validation. Cluster validation Assess the quality and reliability of clustering results. Why validation? To avoid finding clusters formed by chance To compare clustering algorithms To choose clustering parameters e.g., the number of clusters in the K-means algorithm. DBSCAN.

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Cluster Validation

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  1. Cluster Validation • Cluster validation • Assess the quality and reliability of clustering results. • Why validation? • To avoid finding clusters formed by chance • To compare clustering algorithms • To choose clustering parameters • e.g., the number of clusters in the K-means algorithm

  2. DBSCAN K-means Complete Link Clusters found in Random Data Random Points

  3. Aspects of Cluster Validation • Comparing the clustering results to ground truth (externally known results). • External Index • Evaluating the quality of clusters without reference to external information. • Use only the data • Internal Index • Determining thereliabilityof clusters. • To what confidence level, the clusters are not formed by chance • Statistical framework

  4. Comparing to Ground Truth • Notation • N: number of objects in the data set; • P={P1,…,Pm}: the set of “ground truth” clusters; • C={C1,…,Cn}: the set of clusters reported by a clustering algorithm. • The “incidence matrix” • N  N (both rows and columns correspond to objects). • Pij = 1 if Oi and Oj belong to the same “ground truth” cluster in P; Pij=0 otherwise. • Cij = 1 if Oi and Oj belong to the same cluster in C; Cij=0 otherwise.

  5. External Index • A pair of data object (Oi,Oj) falls into one of the following categories • SS: Cij=1 and Pij=1; (agree) • DD: Cij=0 and Pij=0; (agree) • SD: Cij=1 and Pij=0; (disagree) • DS: Cij=0 and Pij=1; (disagree) • Rand index • may be dominated by DD • Jaccard Coefficient

  6. Internal Index • “Ground truth” may be unavailable • Use only the data to measure cluster quality • Measure the “homogeneity” and “separation” of clusters. • SSE: Sum of squared errors. • Calculate the correlation between clustering results and distance matrix.

  7. Sum of Squared Error • Homogeneity is measured by the within cluster sum of squares Exactly the objective function of K-means. • Separation is measured by the between cluster sum of squares Where |Ci| is the size of cluster i, m is the centroid of the whole data set. • BSS + WSS = constant • A larger number of clusters tend to result in smaller WSS.

  8. m    1 m1 2 3 4 m2 5 Sum of Squared Error K=1 : K=2 : K=4:

  9. Sum of Squared Error • Can also be used to estimate the number of clusters.

  10. Internal Measures: SSE • SSE curve for a more complicated data set SSE of clusters found using K-means

  11. Correlation with Distance Matrix • Distance Matrix • Dij is the similarity between object Oi and Oj. • Incidence Matrix • Cij=1 if Oi and Oj belong to the same cluster, Cij=0 otherwise • Compute the correlation between the two matrices • Only n(n-1)/2 entries needs to be calculated. • High correlation indicates good clustering.

  12. Correlation with Distance Matrix • Given Distance Matrix D = {d11,d12, …, dnn } and Incidence Matrix C= { c11, c12,…, cnn } . • Correlation r between D and C is given by

  13. Measuring Cluster Validity Via Correlation • Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. Corr = -0.9235 Corr = -0.5810

  14. DBSCAN K-means Complete Link Clusters found in Random Data Random Points

  15. Using Similarity Matrix for Cluster Validation • Order the similarity matrix with respect to cluster labels and inspect visually.

  16. Using Similarity Matrix for Cluster Validation • Clusters in random data are not so crisp K-means

  17. Using Similarity Matrix for Cluster Validation • Clusters in random data are not so crisp Complete Link

  18. Using Similarity Matrix for Cluster Validation • Clusters in random data are not so crisp DBSCAN

  19. Reliability of Clusters • Need a framework to interpret any measure. • For example, if our measure of evaluation has the value, 10, is that good, fair, or poor? • Statistics provide a framework for cluster validity • The more “atypical” a clustering result is, the more likely it represents valid structure in the data.

  20. Statistical Framework for SSE • Example • Compare SSE of 0.005 against three clusters in random data • SSE Histogram of 500 sets of random data points of size 100 distributed over the range 0.2 – 0.8 for x and y values SSE = 0.005

  21. Statistical Framework for Correlation • Correlation of incidence and distance matrices for the K-means of the following two data sets. Correlation histogram of random data Corr = -0.5810 Corr = -0.9235

  22. Hyper-geometric Distribution • Given the total number of genes in the data set associated with term T is M, if randomly draw n genes from the data set N, what is the probability that m of the selected n genes will be associated with T?

  23. P-Value Based on Hyper-geometric distribution, the probability of having m genes or fewer associated to T in N can be calculated by summing the probabilities of a random list of N genes having 1, 2, …, m genes associated to T. So the p-value of over-representation is as follows:

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