1. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 1 CSci 8980: Data Mining (Fall 2002) Vipin Kumar Army High Performance Computing Research Center Department of Computer Science University of Minnesota http://www.cs.umn.edu/~kumar

2. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 2 l For supervised classification we have a variety of measures to evaluate how good our model is –Accuracy, precision, recall l For cluster analysis, the analogous question is how to evaluate the “goodness” of a the resulting clusters? l However, if “clusters are in the eye of the beholder” then why should we want to evaluate them? –To avoid finding patterns in noise –To compare clustering algorithms –To compare two sets of clusters –To compare two clusters Cluster Validity

3. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 3 Clusters found in Random Data Random Points K-means DBSCAN Complete Link

4. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 4 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. Different Aspects of Cluster Validation

5. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 5 l The 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 of 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 l 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. Measures of Cluster Validity

6. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 6 l Two matrices –Proximity Matrix –“Incidence” Matrix  One row and one column for each data point  An entry is 1 if the associated pair of points belong to the same cluster  An entry is 0 if the associated pair of points belongs to different clusters l Compute the correlation between the two matrices –Since the matrices are symmetric, only the correlation between n(n-1) / 2 entries needs to be calculated. l High correlation indicates that points that belong to the same cluster are close to each other. l Not a good measure for some density or contiguity based clusters. Measuring Cluster Validity Via Correlation

7. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 7 l Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. Measuring Cluster Validity Via Correlation Corr = -0.9235Corr = -0.5810

8. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 8 l Order the similarity matrix with respect to cluster labels and inspect visually. Using the Similarity Matrix for Cluster Validation

9. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 9 l Clusters in random data are not so crisp Using the Similarity Matrix for Cluster Validation DBSCAN

10. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 10 l Clusters in random data are not so crisp Using the Similarity Matrix for Cluster Validation K-means

11. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 11 l Clusters in random data are not so crisp Using the Similarity Matrix for Cluster Validation Complete Link

12. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 12 Using the Similarity Matrix for Cluster Validation DBSCAN

13. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 13 l Clusters in more complicated figures aren’t well separated l Internal Index: Used to measure the goodness of a of clustering structure without respect to external information. –SSE l SSE is good for comparing two clusterings or two clusters (average SSE). l Can also be used to estimate the number of clusters Internal Measures of Cluster Validity: SSE

14. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 14 Internal Measures of Cluster Validity: SSE SSE of clusters found using K-means l SSE curve for a more complicated data set

15. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 15 l Need a framework to interpret any measure. –For example, if our measure of evaluation has the value, 10, is does that good, fair, or poor? l Statistics can provide a framework –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 hard to understand. l 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. Framework for Cluster Validity

16. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 16 l If you want a more absolute measure –Compare SSE of 0.005 against three clusters in random data –Histogram shows SSE of three clusters in 500 sets of random data points of size 100 distributed over the range 0.2 – 0.8 for x and y values l In general, it can be hard to have representative data to generate statistics. Statistical Framework for SSE

17. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 17 l Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. Statistical Framework for Correlation Corr = -0.9235Corr = -0.5810

18. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 18 l Cluster Cohesion: Measure of how closely related objects in a cluster are. –Example: SSE l Cluster Separation: Measure of how distinct or well-separated a cluster is from other clusters. l Example: Squared Error –Cohesion is measured by the within cluster sum of squares (SSE) –Separation is measured by the between cluster sum of squares. – –Between sum of squares + within sum of squares = constant Internal Measures of Cluster Validity: Cohesion and Separation

19. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 19 l A proximity graph based approach can also be used for cohesion and separation. –Cluster cohesion is the sum of the weight of all links within a cluster. –Cluster separation is the sum of the weights between nodes in the cluster and nodes outside the cluster. Internal Measures of Cluster Validity: Cohesion and Separation cohesionseparation

20. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 20 l Silhouette Coefficient combine ideas of both cohesion and separation, but for individual points, as well as clusters and clusterings. l For an individual point, i –Calculate a = average distance of i to the points in its cluster –Calculate b = min (average distance of i to points in another cluster) –The silhouette coefficient for a point is then given by s = 1 – a/b if a < b, (or s = b/a - 1 if a  b, not the usual case) –Typically between 0 and 1. –The closer to 1 the better. l Can calculate the Average Silhouette width for a cluster or a clustering. Internal Measures of Cluster Validity: Silhouette Coefficient

21. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 21 External Measures of Cluster Validity: Entropy and Purity

22. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 22 “The validation of clustering structures is the most difficult and frustrating part of cluster analysis. Without a strong effort in this direction, cluster analysis will remain a black art accessible only to those true believers who have experience and great courage.” Algorithms for Clustering Data, Jain and Dubes Final Comment on Cluster Validity

23. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 23 l Scalability l Independence of the order of input l Effective means of detecting and dealing with noise or outlying points l Effective means of evaluating the validity of clusters that are produced. l Easy interpretability of results l The ability to find clusters in subspaces of the original space. l The ability to handle distances in high dimensional spaces properly. l Robustness in the presence of different underlying data and cluster characteristics l An ability to estimate any parameters l An ability to function in an incremental manner Issues

24. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 24 l Robustness in the presence of different underlying data and cluster characteristics –Dimensionality –Noise and Outliers –Statistical Distribution. –Cluster Shape. –Cluster Size –Cluster Density –Cluster Separation –Type of data space, e.g., Euclidean or non-Euclidean. –Many and Mixed Attribute Types Issues - Handling Different Types of Data

25. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 25 Clustering Scalability for Large Data Sets l One very common solution is sampling, but the sampling could miss small clusters. –Data is sometimes not organized to make valid sampling easy or efficient. l Another approach is to compress the data or portions of the data. –Any such approach must ensure that not too much information is lost. (Scaling Clustering Algorithms to Large Databases, Bradley, Fayyad and Reina.)

26. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 26 Clustering Scalability for Large Data Sets: Birch l BIRCH (Balanced and Iterative Reducing and Clustering using Hierarchies) –BIRCH can efficiently cluster data with a single pass and can improve that clustering in additional passes. –Can work with a number of different distance metrics. –BIRCH can also also deal effectively with outliers.

27. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 27 Clustering Scalability for Large Data Sets: Birch l BIRCH is based on the notion of a clustering feature (CF) and a CF tree. l –A cluster of data points (vectors) can be represented by a triple of numbers (N, LS, SS)  N is the number of points in the cluster  LS is the linear sum of the points  SS is the sum of squares of the points. –Points are processed incrementally.  Each point is placed in the leaf node corresponding to the “closest” cluster (CF).  Clusters (CFs) are updated.

28. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 28 Clustering Scalability for Large Data Sets: Birch l Basic steps of BIRCH –Load the data into memory by creating a CF tree that “summarizes” the data. –Perform global clustering.  Produces a better clustering than the initial step.  An agglomerative, hierarchical technique was selected. –Redistribute the data points using the centroids of clusters discovered in the global clustering phase, and thus, discover a new (and hopefully better) set of clusters. ( See Zhang, Ramakrishnan and Livney or Ganti, Ramakrishnan, and Gehrke)

29. © University of Minnesota Data Mining CSCI 8980 (Fall 2002) 29 Other Clustering Approaches l Modeling clusters as a “mixture” of Multivariate Normal Distributions. (Raftery and Fraley) l Bayesian Approaches (AutoClass, Cheeseman ) l Neural Network Approaches (SOM, Kohonen) l Subspace Clustering (CLIQUE, Agrawal) l Many, many other variations and combinations of approaches.