What is Cluster Analysis? Cluster: a collection of data objects Similar to one another within the same cluster Dissimilar to the objects in other clusters Cluster analysis Grouping a set of data objects into clusters Clustering is unsupervised classification: no predefined classes Typical applications As a stand-alone tool to get insight into data distribution As a preprocessing step for other algorithms Examples of Clustering Applications Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs Land use: Identification of areas of similar land use in an earth observation database Insurance: Identifying groups of motor insurance policy holders with a high average claim cost City-planning: Identifying groups of houses according to their house type, value, and geographical location Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults What Is Good Clustering?

A good clustering method will produce high quality clusters with high intra-class similarity low inter-class similarity The quality of a clustering result depends on both the similarity measure used by the method and its implementation. The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns . Requirements of Clustering in Data Mining Scalability Ability to deal with different types of attributes Discovery of clusters with arbitrary shape Minimal requirements for domain knowledge to determine input parameters Able to deal with noise and outliers Insensitive to order of input records

High dimensionality Incorporation of user-specified constraints Interpretability and usability Data Structures Data matrix (two modes) Dissimilarity matrix (one mode) x11 ... x i1 ... x n1 ... x1f ... ... ... ... ... xif ... ...

... ... xnf ... ... 0 d(2,1) 0 d(3,1) d ( 3,2) 0 : : : d ( n,1) d ( n,2) ... x1p ... xip ... xnp ... 0 Measure the Quality of Clustering Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically metric: d(i, j) There is a separate quality function that measures the goodness of a cluster. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. Weights should be associated with different variables

based on applications and data semantics. It is hard to define similar enough or good enough the answer is typically highly subjective. Major Clustering Approaches Partitioning algorithms: Construct various partitions and then evaluate them by some criterion Hierarchy algorithms: Create a hierarchical decomposition of the set of data (or objects) using some criterion Density-based: based on connectivity and density functions Grid-based: based on a multiple-level granularity structure Model-based: A model is hypothesized for each of the clusters and the idea is to find the best fit of that model to each other Partitioning Algorithms: Basic Concept Partitioning method: Construct a partition of a database D of n objects into a set of k clusters Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion Global optimal: exhaustively enumerate all partitions Heuristic methods: k-means and k-medoids algorithms k-means (MacQueen67): Each cluster is represented by the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw87): Each cluster is represented by one of the objects in the cluster The K-Means Clustering Method Given k, the k-means algorithm is implemented in four steps: Partition objects into k nonempty subsets Compute seed points as the centroids of the clusters of the current partition (the centroid is the center, i.e., mean point, of the cluster) Assign each object to the cluster with the nearest seed point Go back to Step 2, stop when no more new assignment The K-Means Clustering Method Example 10

9 8 7 6 5 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 2 1 0 0 1 2 3

4 5 6 7 8 K=2 Arbitrarily choose K object as initial cluster center 9 10 Assign each objects to most similar center 3 2 1 0 0 1 2 3 4 5 6

7 8 9 10 Update the cluster means 4 3 2 1 0 0 1 2 3 4 5 6 reassign 10 9 9 8 8 7

7 6 6 5 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 8 9 10 reassign 10 0

7 10 Update the cluster means 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Comments on the K-Means Method Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k)) Weakness Applicable only when mean is defined, then what about

categorical data? Need to specify k, the number of clusters, in advance Unable to handle noisy data and outliers Not suitable to discover clusters with non-convex shapes Variations of the K-Means Method A few variants of the k-means which differ in Selection of the initial k means Dissimilarity calculations Strategies to calculate cluster means Handling categorical data: k-modes (Huang98) Replacing means of clusters with modes Using new dissimilarity measures to deal with categorical objects Using a frequency-based method to update modes of clusters A mixture of categorical and numerical data: k-prototype method What is the problem of k-Means Method? The k-means algorithm is sensitive to outliers ! Since an object with an extremely large value may substantially distort the distribution of the data. K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster. 10 10 9 9 8 8 7

7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 5 6 7

8 9 10 0 1 2 3 4 5 6 7 8 9 10 The K-Medoids Clustering Method Find representative objects, called medoids, in clusters PAM (Partitioning Around Medoids, 1987) starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering PAM works effectively for small data sets, but does not scale well for large data sets CLARA (Kaufmann & Rousseeuw, 1990)

CLARANS (Ng & Han, 1994): Randomized sampling Focusing + spatial data structure (Ester et al., 1995) Typical k-medoids algorithm (PAM) Total Cost = 20 10 10 10 9 9 9 8 8 8 7 7 6 5 4 3 2 1 0 0 1 2

3 4 5 K=2 6 7 8 9 10 Arbitrar y choose k object as initial medoid s 6 5 4 3 2 1 0 0 1 2 3 4

5 6 7 8 9 Total Cost = 26 10 Do loop Until no change 10 Assign each remaini ng object to nearest medoid s 7 6 5 4 3 2 1 0 0 10 9 Compute total cost

of swapping Swapping O and Oramdom If quality is improved. 3 3 2 2 1 1 6 5 4 0 2 3 4 5 6 7 8 9 Randomly select a nonmedoid

object,Oramdom 9 8 7 1 8 7 6 5 4 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3

4 5 6 7 8 9 10 10 PAM (Partitioning Around Medoids) (1987) PAM (Kaufman and Rousseeuw, 1987), built in Splus Use real object to represent the cluster Select k representative objects arbitrarily For each pair of non-selected object h and selected object i, calculate the total swapping cost TCih For each pair of i and h, If TCih < 0, i is replaced by h Then assign each non-selected object to the most similar representative object repeat steps 2-3 until there is no change PAM Clustering: Total swapping cost TCih=jCjih 10 10 9 9 t 8 7 j

t 8 7 j 6 5 i 4 3 6 5 h h i 4 3 2 2 1 1 0 0 0 1 2 3

4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 Cjih = 0 Cjih = d(j, h) - d(j, i) 10 10 9 9 h 8

8 j 7 6 5 6 i 5 i 4 7 t 3 h 4 3 t 2 2 1 1 j 0 0

0 1 2 3 4 5 6 7 8 9 Cjih = d(j, t) - d(j, i) 10 0 1 2 3 4 5 6 7 8 9 Cjih = d(j, h) - d(j, t)

10 8 9 10 What is the problem with PAM? Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean Pam works efficiently for small data sets but does not scale well for large data sets. O(k(n-k)2 ) for each iteration where n is # of data,k is # of clusters Sampling based method, CLARA(Clustering LARge Applications) CLARA (Clustering Large Applications) (1990) CLARA (Kaufmann and Rousseeuw in 1990) Built in statistical analysis packages, such as S+ It draws multiple samples of the data set, applies PAM on each sample, and gives the best clustering as the output Strength: deals with larger data sets than PAM Weakness: Efficiency depends on the sample size A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased K-Means Example Given: {2,4,10,12,3,20,30,11,25}, k=2 Randomly assign means: m1=3,m2=4 Solve for the rest . Similarly try for k-medoids Clustering Approaches

Clustering Hierarchical Agglomerative Partitional Divisive Categorical Sampling Large DB Compression Cluster Summary Parameters Distance Between Clusters Single Link: smallest distance between points Complete Link: largest distance between points Average Link: average distance between points Centroid: distance between centroids Hierarchical Clustering Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step 0 a b Step 1 Step 2 Step 3 Step 4

ab abcde c cde d de e Step 4 agglomerative (AGNES) Step 3 Step 2 Step 1 Step 0 divisive (DIANA) Hierarchical Clustering Clusters are created in levels actually creating sets of clusters at each level. Agglomerative Initially each item in its own cluster Iteratively clusters are merged together Bottom Up Divisive Initially all items in one cluster Large clusters are successively divided Top Down Hierarchical Algorithms Single Link

MST Single Link Complete Link Average Link Dendrogram Dendrogram: a tree data structure which illustrates hierarchical clustering techniques. Each level shows clusters for that level. Leaf individual clusters Root one cluster A cluster at level i is the union of its children clusters at level i+1. Levels of Clustering Agglomerative Example A B C D E A 0 1 2 2 3 B 1 0 2 4 3

C 2 2 0 1 5 D 2 4 1 0 3 E 3 3 5 3 0 A B E C

D Threshold of 1 2 34 5 A B C D E MST Example A A B C D E A 0 1 2 2 3 B 1 0 2 4 3 C 2 2 0 1 5

D 2 4 1 0 3 E 3 3 5 3 0 B E C D Agglomerative Algorithm Single Link View all items with links (distances) between them. Finds maximal connected components in this graph. Two clusters are merged if there is at least one edge which connects them. Uses threshold distances at each level. Could be agglomerative or divisive. MST Single Link Algorithm

Single Link Clustering AGNES (Agglomerative Nesting) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Use the Single-Link method and the dissimilarity matrix. Merge nodes that have the least dissimilarity Go on in a non-descending fashion Eventually all nodes belong to the same cluster 10 10 10 9 9 9 8 8 8

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0 0 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10 DIANA (Divisive Analysis) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Inverse order of AGNES Eventually each node forms a cluster on its own 10 10 10 9 9 9 8 8

8 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0

0 1 2 3 4 5 6 7 8 9 10 0 0 0 1 2 3 4 5 6 7 8 9

10 0 1 2 3 4 5 6 7 8 9 10 Readings CHAMELEON: A Hierarchical Clustering Algorithm Using Dynamic Modeling. George Karypis, Eui-Hong Han, Vipin Kumar, IEEE Computer 32(8): 68-75, 1999 (http://glaros.dtc.umn.edu/gkhome/node/152) A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise. Martin Ester, Hans-Peter Kriegel, Jrg Sander, Xiaowei Xu. Proceedings of 2nd International Conference on Knowledge Discovery and Data Mining (KDD-96) BIRCH: A New Data Clustering Algorithm and Its Applications. Data Mining and Knowledge Discovery Volume 1 , Issue 2 (1997)