Cluster validity 1.ppt on cluster validity
- 2. WHY TO EVALUATE THE “GOODNESS” OF THE RESULTING CLUSTERS? To avoid finding patterns in noise To compare clustering algorithms To compare two sets of clusters To compare two clusters
- 3. DIFFERENT ASPECTS OF CLUSTER VALIDATION 1. Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data. 2. Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels. 3. Evaluating how well the results of a cluster analysis fit the data without reference to external information. - Use only the data 4. Comparing the results of two different sets of cluster analyses to determine which is better. 5. Determining the ‘correct’ number of clusters. For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.
- 4. FRAMEWORK FOR CLUSTER VALIDITY 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 Can compare the values of an index that result from random data or clusterings to those of a clustering result. If the value of the index is unlikely, then the cluster results are valid These approaches are more complicated and harder to understand. For comparing the results of two different sets of cluster analyses, a framework is less necessary. However, there is the question of whether the difference between two index values is significant
- 5. MEASURES OF CLUSTER VALIDITY Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types. External Index: Used to measure the extent to which cluster labels match externally supplied class labels. Entropy Internal Index: Used to measure the goodness of a clustering structure without respect to external information. Sum of Squared Error (SSE) Relative Index: Used to compare two different clusterings or clusters. Often an external or internal index is used for this function, e.g., SSE or entropy Sometimes these are referred to as criteria instead of indices However, sometimes criterion is the general strategy and index is the numerical measure that implements the criterion.
- 6. EXTERNAL MEASURES The correct or ground truth clustering is known priori. Given a clustering partition C and ground truth partitioning T, we redefine TP, TN, FP, FN in the context of clustering. Given the number of pairs N N=TP+FP+FN+TN
- 7. EXTERNAL MEASURES … True Positives (TP): Xi and Xj are a true positive pair if they belong to the same partition in T, and they are also in the same cluster in C. TP is defined as the number of true positive pairs. False Negatives (FN): Xi and Xj are a false negative pair if they belong to the same partition in T, but they do not belong to the same cluster in C. FN is defined as the number of false negative pairs. • False Positives (FP): Xi and Xj are a false positive pair if the do not belong to the same partition in T, but belong to the same cluster in C. FP is the number of false positive pairs. True Negatives (TN): Xi and Xj are a false negative pair if they do not belong to the same partition in T, nor to the same cluster in C. TN is the number of true negative pairs.
- 8. JACCARD COEFFICIENT Measures the fraction of true positive point pairs, but after ignoring the true negatives as, Jaccard = TP/ (TP+FP+FN) For a perfect clustering C, the coefficient is one, that is, there are no false positives nor false negatives. Note that the Jaccard coefficient is asymmetric in that it ignores the true negatives
- 9. RAND STATISTIC Measures the fraction of true positives and true negatives over all pairs as Rand = (TP + TN)/ N The Rand statistic measures the fraction of point pairs where both the clustering C and the ground truth T agree. A perfect clustering has a value of 1 for the statistic. The adjusted rand index is the extension of the rand statistic corrected for chance.
- 10. FOWLKES-MALLOWS MEASURE Define precision and recall analogously to what done for classification, Prec = TP/ (TP+FP) and Recall = TP / (TP+FN) • The Fowlkes–Mallows (FM) measure is defined as the geometric mean of the pairwise precision and recall FM = (precision recall) √ ∙ FM is also asymmetric in terms of the true positives and negatives because it ignores the true negatives. Its highest value is also 1, achieved when there are no false positives or negatives.
- 11. INTERNAL MEASURES: COHESION AND SEPARATION Cluster Cohesion (Compactness): Measures how closely related are objects in a cluster. Cluster Separation (Separation): Measure how distinct or well- separated a cluster is from other clusters. Example: Squared Error Cohesion is measured by the within cluster sum of squares (SSE) Separation is measured by the between cluster sum of squares Where |Ci| is the size of cluster i i i i m m C BSS 2 ) (
- 12. INTERNAL MEASURES: COHESION AND SEPARATION 10 9 1 9 ) 3 5 . 4 ( 2 ) 5 . 1 3 ( 2 1 ) 5 . 4 5 ( ) 5 . 4 4 ( ) 5 . 1 2 ( ) 5 . 1 1 ( 2 2 2 2 2 2 Total BSS WSS 1 2 3 4 5 m1 m2 m K=2 clusters: 10 0 10 0 ) 3 3 ( 4 10 ) 3 5 ( ) 3 4 ( ) 3 2 ( ) 3 1 ( 2 2 2 2 2 Total BSS WSS K=1 cluster:
- 13. INTERNAL MEASURES: COHESION AND SEPARATION
- 16. DUNN’S INDEX:
- 18. XIE-BENI INDEX: In the definition of XB-index, the numerator indicates the compactness of the obtained cluster while the denominator indicates the strength of the separation between clusters. The objective is to minimize the XB-index for achieving proper clustering.
- 19. PS INDEX:
- 20. I-INDEX:
- 21. CS-INDEX: Used for tackling clusters of different densities and/or sizes. where mi , i = 1, . . . K are the cluster centers.