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Clustering Basic Concepts and Algorithms Bamshad Mobasher DePaul University What is Clustering in Data Mining? Clustering is a process of partitioning a set of data (or objects) in a set of meaningful sub-classes, called clusters Helps users understand the natural grouping or structure in a data set Cluster: a collection of data objects that are “similar” to one another and thus can be treated collectively as one group but as a collection, they are sufficiently different from other groups Clustering unsupervised classification no predefined classes 2 Applications of Cluster Analysis Data reduction Summarization: Preprocessing for regression, PCA, classification, and association analysis Compression: Image processing: vector quantization Hypothesis generation and testing Prediction based on groups Cluster & find characteristics/patterns for each group Finding K-nearest Neighbors Localizing search to one or a small number of clusters Outlier detection: Outliers are often viewed as those “far away” from any cluster 3 Basic Steps to Develop a Clustering Task Feature selection / Preprocessing Select info concerning the task of interest Minimal information redundancy May need to do normalization/standardization Distance/Similarity measure Similarity of two feature vectors Clustering criterion Expressed via a cost function or some rules Clustering algorithms Choice of algorithms Validation of the results Interpretation of the results with applications 4 Distance or Similarity Measures Common Distance Measures: Manhattan distance: Euclidean distance: Cosine similarity: dist ( X , Y ) 1 sim ( X , Y ) sim ( X , Y ) ( xi y i ) i i xi y i 2 2 i 5 More Similarity Measures In vector-space model many similarity measures can be used in clustering Simple Matching Cosine Coefficient Dice’s Coefficient Jaccard’s Coefficient 6 Quality: What Is Good Clustering? A good clustering method will produce high quality clusters high intra-class similarity: cohesive within clusters low inter-class similarity: distinctive between clusters The quality of a clustering method depends on the similarity measure used its implementation, and Its ability to discover some or all of the hidden patterns 7 Major Clustering Approaches Partitioning approach: Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum of square errors Typical methods: k-means, k-medoids, CLARANS Hierarchical approach: Create a hierarchical decomposition of the set of data (or objects) using some criterion Typical methods: Diana, Agnes, BIRCH, CAMELEON Density-based approach: Based on connectivity and density functions Typical methods: DBSCAN, OPTICS, DenClue Model-based: A model is hypothesized for each of the clusters and tries to find the best fit of that model to each other Typical methods: EM, SOM, COBWEB 8 Partitioning Approaches The notion of comparing item similarities can be extended to clusters themselves, by focusing on a representative vector for each cluster cluster representatives can be actual items in the cluster or other “virtual” representatives such as the centroid this methodology reduces the number of similarity computations in clustering clusters are revised successively until a stopping condition is satisfied, or until no more changes to clusters can be made Reallocation-Based Partitioning Methods Start with an initial assignment of items to clusters and then move items from cluster to cluster to obtain an improved partitioning Most common algorithm: k-means 9 The K-Means Clustering Method Given the number of desired clusters k, the k-means algorithm follows four steps: 1. Randomly assign objects to create k nonempty initial partitions (clusters) 2. Compute the centroids of the clusters of the current partitioning (the centroid is the center, i.e., mean point, of the cluster) 3. Assign each object to the cluster with the nearest centroid (reallocation step) 4. Go back to Step 2, stop when the assignment does not change 10 K-Means Example: Document Clustering T1 0 4 0 0 0 2 1 3 4/2 0/2 2/2 D1 D2 D3 D4 D5 D6 D7 D8 C1 C2 C3 Initial (arbitrary) assignment: C1 = {D1,D2}, C2 = {D3,D4}, C3 = {D5,D6} Cluster Centroids T2 3 1 4 3 1 2 0 1 4/2 7/2 3/2 T3 3 0 0 0 3 0 3 0 3/2 0/2 3/2 T4 0 1 0 3 0 0 2 0 1/2 3/2 0/2 T5 2 2 2 3 1 4 0 2 4/2 5/2 5/2 Now compute the similarity (or distance) of each item to each cluster, resulting a clusterdocument similarity matrix (here we use dot product as the similarity measure). C1 C2 C3 D1 29/2 31/2 28/2 D2 29/2 20/2 21/2 D3 24/2 38/2 22/2 D4 27/2 45/2 24/2 D5 17/2 12/2 17/2 D6 32/2 34/2 30/2 D7 15/2 6/2 11/2 D8 24/2 17/2 19/2 11 Example (Continued) C1 C2 C3 D1 29/2 31/2 28/2 D2 29/2 20/2 21/2 D3 24/2 38/2 22/2 D4 27/2 45/2 24/2 D5 17/2 12/2 17/2 D6 32/2 34/2 30/2 D7 15/2 6/2 11/2 D8 24/2 17/2 19/2 For each document, reallocate the document to the cluster to which it has the highest similarity (shown in red in the above table). After the reallocation we have the following new clusters. Note that the previously unassigned D7 and D8 have been assigned, and that D1 and D6 have been reallocated from their original assignment. C1 = {D2,D7,D8}, C2 = {D1,D3,D4,D6}, C3 = {D5} This is the end of first iteration (i.e., the first reallocation). Next, we repeat the process for another reallocation… 12 Example (Continued) Now compute new cluster centroids using the original documentterm matrix C1 = {D2,D7,D8}, C2 = {D1,D3,D4,D6}, C3 = {D5} D1 D2 D3 D4 D5 D6 D7 D8 C1 C2 C3 This will lead to a new cluster-doc similarity matrix similar to previous slide. Again, the items are reallocated to clusters with highest similarity. C1 C2 C3 D1 7.67 16.75 14.00 New assignment T1 0 4 0 0 0 2 1 3 8/3 2/4 0/1 D2 15.01 11.25 3.00 D3 5.34 17.50 6.00 T2 3 1 4 3 1 2 0 1 2/3 12/4 1/1 D4 9.00 19.50 6.00 D5 5.00 8.00 11.00 T3 3 0 0 0 3 0 3 0 3/3 3/4 3/1 D6 12.00 6.68 9.34 T4 0 1 0 3 0 0 2 0 3/3 3/4 0/1 D7 7.67 4.25 9.00 T5 2 2 2 3 1 4 0 2 4/3 11/4 1/1 D8 11.34 10.00 3.00 C1 = {D2,D6,D8}, C2 = {D1,D3,D4}, C3 = {D5,D7} Note: This process is now repeated with new clusters. However, the next iteration in this example Will show no change to the clusters, thus terminating the algorithm. 13 K-Means Algorithm Strength of the k-means: Relatively efficient: O(tkn), where n is # of objects, k is # of clusters, and t is # of iterations. Normally, k, t << n Often terminates at a local optimum Weakness of the k-means: Applicable only when mean is defined; what about categorical data? Need to specify k, the number of clusters, in advance Unable to handle noisy data and outliers Variations of K-Means usually differ in: Selection of the initial k means Distance or similarity measures used Strategies to calculate cluster means 14 A Disk Version of k-means k-means can be implemented with data on disk In each iteration, it scans the database once The centroids are computed incrementally It can be used to cluster large datasets that do not fit in main memory We need to control the number of iterations In practice, a limited is set (< 50) There are better algorithms that scale up for large data sets, e.g., BIRCH 15 BIRCH Designed for very large data sets Time and memory are limited Incremental and dynamic clustering of incoming objects Only one scan of data is necessary Does not need the whole data set in advance Two key phases: Scans the database to build an in-memory tree Applies clustering algorithm to cluster the leaf nodes 16 Hierarchical Clustering Algorithms • Two main types of hierarchical clustering – Agglomerative: • Start with the points as individual clusters • At each step, merge the closest pair of clusters until only one cluster (or k clusters) left – Divisive: • Start with one, all-inclusive cluster • At each step, split a cluster until each cluster contains a point (or there are k clusters) • Traditional hierarchical algorithms use a similarity or distance matrix – Merge or split one cluster at a time Hierarchical Clustering Algorithms Use dist / sim matrix as clustering criteria does not require the no. of clusters as input, but needs a termination condition Step 0 Step 1 Step 2 Step 3 Step 4 Agglomerative a ab b abcde c cd d cde e Divisive Step 4 Step 3 Step 2 Step 1 Step 0 18 Hierarchical Agglomerative Clustering Basic procedure 1. Place each of N items into a cluster of its own. 2. Compute all pairwise item-item similarity coefficients Total of N(N-1)/2 coefficients 3. Form a new cluster by combining the most similar pair of current clusters i and j (methods for determining which clusters to merge: single-link, complete link, group average, etc.); update similarity matrix by deleting the rows and columns corresponding to i and j; calculate the entries in the row corresponding to the new cluster i+j. 4. Repeat step 3 if the number of clusters left is great than 1. 19 Hierarchical Agglomerative Clustering :: Example 4 1 2 5 5 0.4 0.35 2 0.3 0.25 3 3 6 1 4 0.2 0.15 0.1 0.05 0 Nested Clusters 3 6 4 1 Dendrogram 2 5 Distance Between Two Clusters The basic procedure varies based on the method used to determine inter-cluster distances or similarities Different methods results in different variants of the algorithm Single link Complete link Average link Ward’s method Etc. 21 Single Link Method The distance between two clusters is the distance between two closest data points in the two clusters, one data point from each cluster It can find arbitrarily shaped clusters, but It may cause the undesirable “chain effect” due to noisy points Two natural clusters are split into two 22 Distance between two clusters Single-link distance between clusters Ci and Cj is the minimum distance between any object in Ci and any object in Cj The distance is defined by the two most similar objects D sl C i , C j min I1 I2 I3 I4 I5 I1 1.00 0.90 0.10 0.65 0.20 I2 0.90 1.00 0.70 0.60 0.50 I3 0.10 0.70 1.00 0.40 0.30 x,y I4 0.65 0.60 0.40 1.00 0.80 d ( x , y ) x C , y C i I5 0.20 0.50 0.30 0.80 1.00 1 j 2 3 4 5 Complete Link Method The distance between two clusters is the distance of two furthest data points in the two clusters It is sensitive to outliers because they are far away 24 Distance between two clusters Complete-link distance between clusters Ci and Cj is the maximum distance between any object in Ci and any object in Cj The distance is defined by the two least similar objects D cl C i , C j max I1 I2 I3 I4 I5 I1 1.00 0.90 0.10 0.65 0.20 I2 0.90 1.00 0.70 0.60 0.50 I3 0.10 0.70 1.00 0.40 0.30 x,y I4 0.65 0.60 0.40 1.00 0.80 d ( x , y ) x C , y C I5 0.20 0.50 0.30 0.80 1.00 i 1 j 2 3 4 5 Average link and centroid methods Average link: A compromise between the sensitivity of complete-link clustering to outliers and the tendency of single-link clustering to form long chains that do not correspond to the intuitive notion of clusters as compact, spherical objects In this method, the distance between two clusters is the average distance of all pair-wise distances between the data points in two clusters. Centroid method: In this method, the distance between two clusters is the distance between their centroids 26 Distance between two clusters Group average distance between clusters Ci and Cj is the average distance between objects in Ci and objects in Cj The distance is defined by the average similarities D avg C i , C j I1 I2 I3 I4 I5 I1 1.00 0.90 0.10 0.65 0.20 I2 0.90 1.00 0.70 0.60 0.50 I3 0.10 0.70 1.00 0.40 0.30 1 Ci C j I4 0.65 0.60 0.40 1.00 0.80 I5 0.20 0.50 0.30 0.80 1.00 d ( x, y ) x C i , y C 1 j 2 3 4 5 Clustering Basic Concepts and Algorithms Bamshad Mobasher DePaul University