Statistics 202: Data Mining. c Jonathan Taylor. Clustering Based in part on slides from textbook, slides of Susan Holmes.
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1 Clustering Based in part on slides from textbook, slides of Susan Holmes December 2, / 1
2 Clustering Clustering Goal: 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. An unsupervised problem that tries to produce labelled data from unlabelled data. Many different techniques, but most revolve around forming groups with small within cluster distances relative to between cluster distances. 2 / 1
3 What is Cluster Analysis? 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 Intra-cluster distances are minimized Inter-cluster distances are maximized Tan,Steinbach, Kumar Introduction to 4/18/ / 1
4 Clustering Applications Understanding of some structure in a dataset. Group related documents for browsing; Group genes and proteins that have similar functionality; Group stocks with similar price fluctuations Summarization: reduce the size of large data sets (sometimes known as vector quantization... ) 4 / 1
5 Cluster analysis 14.3 Cluster Analysis 527 FIGURE DNA microarray data: average linkage hierarchical clustering has been applied independently to the rows (genes) and columns (samples), determining the ordering of the rows and columns (see text). The colors range from 5 / 1
6 Cluster analysis Original image... 6 / 1
7 Cluster analysis Some compression... 7 / 1
8 Cluster analysis Too much compression? 8 / 1
9 Cluster analysis Notion of a Cluster can be Ambiguous How many clusters? Six Clusters Two Clusters Four Clusters Clusters can be ambiguous. 9 / 1 Tan,Steinbach, Kumar Introduction to 4/18/2004 5
10 Clustering Types of clustering Partitional A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset. Hierarchical A set of nested clusters organized as a hierarchical tree. Each data object is in exactly one subset for any horizontal cut of the tree / 1
11 Cluster analysis Unsupervised Learning X2 X 1 FIGURE Simulated data in the plane, clustered into three classes (represented A partitional by orange, example blue and green) by the K-means clustering algorithm 11 / 1
12 Cluster analysis Unsupervised Learning LEUKEMIA K562B-repro K562A-repro CNS CNS UNKNOWN OVARIAN MCF7A-repro MCF7D-repro LEUKEMIA LEUKEMIA LEUKEMIA LEUKEMIA OVARIAN OVARIAN LEUKEMIA OVARIAN OVARIAN PROSTATE OVARIAN PROSTATE CNS CNS CNS FIGURE Dendrogram from agglomerative hierarchical clustering with average linkage to the human tumor microarray data. A hierarchical example chical structure produced by the algorithm. Hierarchical methods impose 12 / 1
13 Clustering Other distinctions Exclusivity Are points in only one cluster? Soft vs. hard Can we give a score for each case and each cluster? Partial vs. complete Do we cluster all points, or only some? Heterogeneity Are the clusters similar in size, shape, etc. 13 / 1
14 Types of clusters: well-separated! Well-Separated Clusters: A cluster is a set of points such that 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. 3 well-separated clusters This type of cluster is a set of points such that 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. Tan,Steinbach, Kumar Introduction to 4/18/ / 1
15 ! Center-based A cluster is a set of objects such that an object in a cluster is closer (more similar) to the center of a cluster, than to the center of any other cluster Types of clusters: center-based 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 4 center-based clusters This type of cluster is a set of objects such that an object in a cluster is closer (more similar) to the center of a cluster, than to the center of any other cluster. Tan,Steinbach, Kumar Introduction to 4/18/ / 1
16 ! Center-based A cluster is a set of objects such that an object in a cluster is closer (more similar) to the center of a cluster, than to the center of any other cluster Types of clusters: center-based 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 4 center-based clusters The center of these clusters is usually, the average of all the points in the cluster, or a medoid, the most representative point of a cluster. Tan,Steinbach, Kumar Introduction to 4/18/ / 1
17 ! Contiguous Cluster (Nearest neighbor o Types of Transitive) clusters: contiguity-based A cluster is a set of points such that a point in a clus closer (or more similar) to one or more other points i cluster than to any point not in the cluster. This type of cluster is a set of points such that a point in a cluster is closer (or more similar) 8 contiguous to one or clusters more other points in the cluster than to any point not in the cluster. Tan,Steinbach, Kumar Introduction to 4/18 17 / 1
18 Types! Density-based of clusters: contiguity-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. These types of clusters 6 density-based are made upclusters of dense region of points, often described by one of the other cluster types, but separated by low-density regions, often in the form of noise. Tan,Steinbach, Kumar Introduction to 4/18/ / 1
19 Clustering Mathematical characterizations Most clustering algorithms are based on a dissimilarity measure d. Data may be of mixed type so some of the similarities we saw earlier may be used. Most clustering algorithms do not insist that the dissimilarity is truly a distance. (Local) Given two elements of the partition C r, C s, we might consider d SL (C r, C s ) = min d(x, y) x C r,y C s These types of measures are often used in hierarchical clustering algorithms. 19 / 1
20 Clustering Mathematical characterizations (Global) A clustering is a partition C = {C 1,..., C k } of the cases. It writes T = 1 2 n d 2 (x i, x j ) i,j=1 and tries to minimize W (C). This problem is NP Hard... = W (C) + B(C) These types of measures are often used in partitional clustering algorithms. 20 / 1
21 Clustering Most common models Partitional K-means / medoid ; mixture models Hierarchical agglomerative (bottom-up) hierarchical clustering. We ll spend a some time on each of these / 1
22 Cluster analysis Unsupervised Learning X2 X 1 FIGURE Simulated data in the plane, clustered into three classes (represented K-means by orange, clustering blue and green) by the K-means clustering algorithm 22 / 1
23 Cluster analysis Unsupervised Learning LEUKEMIA K562B-repro K562A-repro CNS CNS UNKNOWN OVARIAN MCF7A-repro MCF7D-repro LEUKEMIA LEUKEMIA LEUKEMIA LEUKEMIA OVARIAN OVARIAN LEUKEMIA OVARIAN OVARIAN PROSTATE OVARIAN PROSTATE CNS CNS CNS FIGURE Dendrogram from agglomerative hierarchical clustering with average linkage to the human tumor microarray data. Agglomerative hierarchical clustering chical structure produced by the algorithm. Hierarchical methods impose 23 / 1
24 24 / 1
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