Clustering. Adrian Groza. Department of Computer Science Technical University of Cluj-Napoca

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1 Clustering Adrian Groza Department of Computer Science Technical University of Cluj-Napoca

2 Outline 1 Cluster Analysis What is Datamining? Cluster Analysis 2 K-means 3 Hierarchical Clustering

3 What is Datamining? Why mine data? 1 Commercial viewpoint: lots of data is being collected and warehoused: web data, e-commerce purchases, credit card transactions 2 Scientific viewpoint: data collected and stored at enormous speeds (GB/hour): CERN experiment remote sensors on a satellite, telescopes scanning the skies, microarrays generating gene expression data Definition (What is data mining?) Exploration and analysis, by automatic or semi-automatic means, of large quantities of data in order to discover meaningful patterns.

4 What is Datamining? Data Mining Tasks Prediction methods : use some variables to predict unknown or future values of other variables Description methods : find human-interpretable patterns that describe the data Tasks Classification [Predictive] Clustering [Descriptive] Association Rule Discovery [Descriptive] Sequential Pattern Discovery [Descriptive] Regression [Predictive] Dataset: collection of data objects and their attributes Data arrangement : difference data understanding

5 Cluster Analysis What is Cluster Analysis? Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups The greater the simmilarity within the group, and the greater the difference between groups, the better the clustering

6 Cluster Analysis Applications of Clustes Analysis Clustering for understanding Biology: creating taxonomies, finding groups of genes that have similar functions.

7 Cluster Analysis Applications of Cluster Analysis Clustering for understanding Information retrieval: group the search results into a small number of clusters: Query: movie - clusters: reviews, stars, trailers

8 Cluster Analysis Applications of Clustes Analysis Clustering for understanding Business: segment customers in small groups

9 Cluster Analysis Applications of Clustes Analysis Clustering for understanding Climate: finding patterns in the atmosphere and ocean (atmosphere pressure in the polar regions) Psycology and medicine: identify variations of illness (used to identify types of depression), spatial and temporal distribution of a disease Clustering for utility : finding the most representative cluster prototype Summarisation: many data analysis techniques are impractical for large data sets apply algorithms on the reduced data set conssisting only of cluster prototypes. Compression: applied to image, sound where: 1 Many of data objects are highly similar 2 Some loss of information is acceptaple 3 But you obtain considerable reduction in the datasize

10 Cluster Analysis Notion of a Cluster can be Ambiguous How many? Six clusters Two clusters Four clusters

11 Cluster Analysis Types of Clusterings 1 Partitional Clustering : a division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset 2 Hierarchical clustering: a set of nested clusters organized as a hierarchical tree

12 Cluster Analysis Other Distinctions Between Sets of Clusters 1 Exclusive versus non-exclusive In non-exclusive clustering, points may belong to multiple clusters. Can represent multiple classes (a person at the university can be both an enrolled student and an employee) or border points 2 Fuzzy versus non-fuzzy In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1. Weights must sum to 1. 3 Partial versus complete In some cases, we only want to cluster some of the data (find important topic in last month s stories) 4 Heterogeneous versus homogeneous Cluster of widely different sizes, shapes, and densities

13 Cluster Analysis Types of Clusters 1 Well-Separated Clusters: any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster. 2 Center-based: an object in a cluster is closer (more similar) to the center of a cluster, than to the center of any other cluster- the center of a cluster is often a centroid (the average of all the points in the cluster), or a medoid (the most representative point of a cluster) when data has categorical attributes

14 Cluster Analysis Types of Clusters 1 Graph-based: a cluster can be defined as a connected component (a group of objects that are connected to each other) 2 Contiguous Cluster (Nearest neighbor or Transitive): two objects are connected only if they are within a specific distance (each object in a contiguous cluster is closer to some other object, that to any point in different cluster 8 contiguous clusters 3 Density-based: a cluster is a dense region of points, which is separated by low-density regions, from other regions of high density. (used when the clusters are irregular or intertwined, and when noise and outliers are present.

15 Outline 1 Cluster Analysis What is Datamining? Cluster Analysis 2 K-means 3 Hierarchical Clustering

16 K-Means Example

17 K-Means Algorithm Partitional clustering approach Each cluster is associated with a centroid (center point) Each point is assigned to the cluster with the closest centroid Number of clusters, K, must be specified The basic algorithm is very simple: 1: Select K points as initial centroids 2: repeat 3: Form K clusters by assigning all points to the closest centroid 4: Recompute the centroid of each cluster 5: until The centroids do not change

18 Remarks Initial centroids are often chosen randomly clusters produced vary from one run to another The centroid is the mean of the points in the cluster Closeness is measured by Euclidean distance, cosine similarity, correlation, etc. K-means will converge for common similarity measures mentioned above - most of the convergence happens in the first few iterations Often the stopping condition is changed to Until relatively few points change clusters Complexity is O(n K I d) n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

19 Two different K-means Clusterings

20 Evaluating K-means Clusters Most common measure is Sum of Squared Error (SSE) For each point, the error is the distance to the nearest centroid To get SSE, we square these errors and sum them SSE = K i=1 x C i dist 2 (m i, x) where x is a data point in cluster C i and m i is the representative point for cluster C i Given two clusters, choose the one with the smallest error One easy way to reduce SSE is to increase K

21 Importance of Choosing Initial Centroids

22 Importance of Choosing Initial Centroids

23 Exercise If there are 3 real (natural) clusters, which is the chance of selecting one centroid from each cluster?

24 Problems with Selecting Initial Points If there are K real (natural) clusters then the chance of selecting one centroid from each cluster is small. Chance is relatively small when K is large If clusters are the same size, n, then P = number of ways to select a centroid from each cluster number of ways to select K centroids For example, if K = 10, then probability = 10!/10 10 = = K!nK (Kn) = K! K K K Sometimes the initial centroids will readjust themselves in right way, and sometimes they don t Consider an example of five pairs of clusters

25 10 Clusters Example Starting with two initial centroids in one cluster of each pair of clusters Optimal clustering: two initial centroids fall anywhere in the pair of clusters

26 10 Clusters Example Starting with some pairs of clusters having three initial centroids, while other have only one

27 Solutions to Initial Centroids Problem 1 Multiple runs Helps, but probability is not on your side 2 Use hierarchical clustering extract the centroids of the resulted clusters and use them as initial centroids for k-means algorithm 3 Select more than k initial centroids and then select among these initial centroids Select most widely separated Disadvantage: you can select points that are not in dense regions (but you can apply it on data samples); 4 Postprocessing 5 Bisecting K-means

28 Handling Empty Clusters Basic K-means algorithm can yield empty clusters No points are allocated to a cluster during the assignment Strategies for centroid replacement 1 Choose the point that is farthest away from any current centroid (it contributes most to SSE) 2 Choose a point from the cluster with the highest SSE Split the cluster and reduce the overall SSE

29 Outliers When outliers are present the resulting centroids are less representative SSE will be higher Pre-processing - eliminate outliers Post processing - eliminate points with unusually high contributions to SSE over multiple runs For some applications this is not a good idea Compression: every point must be clustered Financial analysis: unusually profitable customers can be the most interesting points

30 Updating Centers Incrementally In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid An alternative is to update the centroids after each assignment (incremental approach) Each assignment updates zero or two centroids Can use weights to change the impact The weight of points are often decreased as clutering More expensive Introduces an order dependency - clustering depends on the order in which the points are processed

31 Pre-processing and Post-processing Pre-processing Normalize the data Eliminate outliers Post-processing Eliminate small clusters that may represent outliers Split loose clusters, i.e., clusters with relatively high SSE Merge clusters that are close and that have relatively low SSE

32 Bisecting K-means Example

33 Bisecting K-means Idea Split the set of all points into two clusters Select one cluster to split 1: Initialize the list of clusters to contain the cluster containing all po 2: repeat 3: Select a cluster from the list of clusters 4: for i = 1 to number of iterations do 5: Bisect the current cluster using basic K-means 6: endfor 7: Add the two clusters from the bisection with the lowest SSE to 8: until The list of clusters contains K clusters

34 Remarks 1 What cluster to split? Larger cluster 2 Less susceptible to initializations problems Only two centroids at each step

35 Limitations of K-means - Differing Sizes Original points K-means (3 clusters) The largest cluster is broken

36 Limitations of K-means - Differing Density Original points K-means (3 clusters)

37 Limitations of K-means - Non Globular Shapes Original points K-means (2 clusters)

38 Overcoming K-means Limitations One solution is to use many clusters Original points K-means (2 clusters) Find parts of clusters, but need to put together

39 Overcoming K-means Limitations One solution is to use many clusters Original points K-means (2 clusters) Find parts of clusters, but need to put together

40 Overcoming K-means Limitations One solution is to use many clusters Original points K-means (2 clusters) Find parts of clusters, but need to put together

41 Strenghts and Weaknesses Advantages Simple Efficient: rapid convergerce Some techniques exist (increase the number of clusters, bisecting, eliminate outliers) to overcome to following... Disadvatanges Problems when clusters are of different sizes, densities, non-globular shapes Problems when the data contains outliers Initialization problems (number of clusters, initial centroids) Restricted to data for each there is a notion of centroid

42 Outline 1 Cluster Analysis What is Datamining? Cluster Analysis 2 K-means 3 Hierarchical Clustering

43 Hierarchical Clustering Produces a set of nested clusters organized as a hierarchical tree Can be visualized as a dendrogram: a tree like diagram that records the sequences of merges or splits Strengths of Hierarchical Clustering Do not have to assume any particular number of clusters (any desired number of clusters can be obtained by cutting the dendogram at the proper level) They may correspond to meaningful taxonomies

44 Types of Hierarchical Clustering 1 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 2 Divisive: Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point (or there are k clusters)

45 Agglomerative Clustering Algorithm the most popular hierarchical clustering technique 1: Compute the proximity matrix 2: Let each data point be a cluster 3: repeat 4: Merge the two closest clusters 4: Update the proximity matrix 5: until Only a single cluster remains Key operation is the computation of the proximity of two clusters Different approaches to defining the distance between clusters distinguish the different algorithms

46 Intermediate Situation After some merging steps, we have some clusters

47 Intermediate Situation We want to merge the two closest clusters (C2 and C5) and update the proximity matrix.

48 After Merging The question is How do we update the proximity matrix?

49 How to Define Inter-Cluster Similarity 1 Min (single link) 2 Max (complete ink) 3 Group average 4 Distance between centroids 5 Ward s method: proximity is given by the increase in SSE if the clusters would merge

50 Hierarchical Clustering - Min Similarity of two clusters is based on the two most similar (closest) points in the different clusters

51 Hierarchical Clustering - Min dist({3,6},{2,5}) = min(dist(3,2), dist(6,2), dist(3,5),dist(6,5) = min(0.15,0.25,0.28,0.39) = 0.15

52 Remarks 1 Advantage: can handle non-elliptical shapes 2 Limitation: sensitive to noise and outliers

53 Hierarchical Clustering - Max

54 Remarks 1 Advantage: less susceptible to noise and outliers 2 Limitation: tends to break large clusters, biased towards globular clusters

55 Cluster Similarity: Group Average Proximity of two clusters is the average of pairwise proximity between points in the two clusters. p i C i,p j C j proximity(p i, p j ) proximity(c i, C j ) = C i C j Need to use average connectivity for scalability since total proximity favors large clusters Compromise between Min and Max Strength: less susceptible to noise and outliers Limitation: biased towards globular clusters

56 Hierarchical Clustering - Group Average

57 Cluster Similarity: Ward s Method Similarity of two clusters is based on the increase in squared error when two clusters are merged Similar to group average if distance between points is distance squared Less susceptible to noise and outliers Biased towards globular clusters Can be used to initialize K-mean

58 Hierarchical Clustering: Comparison

59 Readings Acknowledgment Slides adapted from Tan, Steinbach, Kumar, Introduction to datamining Readings Tan, Steinbach, Kumar, Introduction to datamining - chapter 8, pages

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