Tan Steinbach Kumar Introduction to Data Mining 4182004

© Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Significantly modified and extended by Ch. Eick Lecture Notes for Chapter 8 Critical Issues with Respect to Clustering Introduction to Data Mining by Tan, Steinbach, Kumar All covered in Part 1 of Clustering in 2013!

What is Cluster Analysis? l 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

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

Types of Clusterings l A clustering is a set of clusters l Important distinction between hierarchical and partitional sets of clusters l Partitional Clustering – A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset l Hierarchical clustering – A set of nested clusters organized as a hierarchical tree

Partitional Clustering Original Points A Partitional Clustering

Hierarchical Clustering Traditional Dendrogram Non-traditional Hierarchical Clustering Non-traditional Dendrogram

Other Distinctions Between Sets of Clusters l Exclusive versus non-exclusive – In non-exclusive clusterings, points may belong to multiple clusters. – Can represent multiple classes or ‘border’ points l 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 – Probabilistic clustering has similar characteristics l Partial versus complete – In some cases, we only want to cluster some of the data l Heterogeneous versus homogeneous – Cluster of widely different sizes, shapes, and densities

Types of Clusters l Well-separated clusters l Center-based clusters l Contiguous clusters l Density-based clusters l Property or Conceptual l Described by an Objective Function Skip/Part 2

Types of Clusters: Well-Separated l 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

Types of Clusters: Center-Based l 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 – 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

Types of Clusters: Contiguity-Based l Contiguous Cluster (Nearest neighbor or Transitive) – A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster. 8 contiguous clusters

Types of Clusters: Density-Based l 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. 6 density-based clusters

Types of Clusters: Conceptual Clusters l Shared Property or Conceptual Clusters – Finds clusters that share some common property or represent a particular concept. . 2 Overlapping Circles

Types of Clusters: Objective Function l Clusters Defined by an Objective Function – Finds clusters that minimize or maximize an objective function. – Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard) – Can have global or local objectives. u Hierarchical clustering algorithms typically have local objectives u Partitional algorithms typically have global objectives – A variation of the global objective function approach is to fit the data to a parameterized model. u Parameters for the model are determined from the data. Mixture models assume that the data is a ‘mixture' of a number of statistical distributions. u

Types of Clusters: Objective Function … l Map the clustering problem to a different domain and solve a related problem in that domain – Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points – Clustering is equivalent to breaking the graph into connected components, one for each cluster. – Want to minimize the edge weight between clusters and maximize the edge weight within clusters

Two different K-means Clusterings Resume! Original Points Optimal Clustering Sub-optimal Clustering

Importance of Choosing Initial Centroids

Importance of Choosing Initial Centroids

Evaluating K-means Clusters l Most common measure is Sum of Squared Error (SSE) – For each point, the error is the distance to the nearest cluster – To get SSE, we square these errors and sum them. – x is a data point in cluster Ci and mi is the representative point for cluster Ci u can show that mi corresponds to the center (mean) of the cluster – Given two clusters, we can choose the one with the smallest error – One easy way to reduce SSE is to increase K, the number of clusters A good clustering with smaller K can have a lower SSE than a poor clustering with higher K u

Importance of Choosing Initial Centroids …

Importance of Choosing Initial Centroids …

Problems with Selecting Initial Points l skip If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small. – – Chance is relatively small when K is large – – For example, if K = 10, then probability = 10!/1010 = 0. 00036 – Consider an example of five pairs of clusters If clusters are the same size, n, then Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t

Solutions to Initial Centroids Problem l Multiple runs – Helps, but probability is not on your side Sample and use hierarchical clustering to determine initial centroids l Select more than k initial centroids and then select among these initial centroids l – Select most widely separated Postprocessing l Bisecting K-means l – Not as susceptible to initialization issues

Handling Empty Clusters l Basic K-means algorithm can yield empty clusters l Several strategies – Choose the point that contributes most to SSE – Choose a point from the cluster with the highest SSE – If there are several empty clusters, the above can be repeated several times.

Limitations of K-means l K-means has problems when clusters are of differing – Sizes – Densities – Non-globular shapes l K-means has problems when the data contains outliers.

Limitations of K-means: Differing Sizes Original Points K-means (3 Clusters)

Limitations of K-means: Differing Density Original Points K-means (3 Clusters)

Limitations of K-means: Non-globular Shapes Original Points K-means (2 Clusters)

Overcoming K-means Limitations Original Points K-means Clusters One solution is to use many clusters. Find parts of clusters, but need to put together.

Overcoming K-means Limitations Original Points K-means Clusters Example: Achieved 97% purity for the Iris Dataset by running K-means with k=11!

Overcoming K-means Limitations Original Points K-means Clusters

Cluster Validity l Part 1 Clustering 2013 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 the resulting clusters? l But “clusters are in the eye of the beholder”! l Then why do 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

Clusters found in Random Data Random Points K-means DBSCAN Complete Link

Different Aspects of Cluster Validation 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.

Measures of Cluster Validity l 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. u Entropy – Internal Index: Used to measure the goodness of a clustering structure without respect to external information. u Sum of Squared Error (SSE) – Relative Index: Used to compare two different clusterings or clusters. u l Often an external or internal index is used for this function, e. g. , SSE or entropy 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.

Framework for Cluster Validity l Need a framework to interpret any measure. – l For example, if our measure of evaluation has the value, 10, is that good, fair, or poor? Statistics provide a framework for cluster validity – 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. u – l If the value of the index is unlikely, then the cluster results are valid These approaches are more complicated and harder to understand. 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

Internal Measures: Cohesion and Separation l Cluster Cohesion: Measures how closely related are objects in a cluster – Example: SSE l Cluster Separation: Measure how distinct or well-separated a cluster is from other clusters

Internal Measures: Cohesion and Separation 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. cohesion separation

skip Example Cluster Evaluation Measures { Remark: k is the number of clusters and X={C 1, …, Ck} are the clusters found by the clustering algorithm; ci refers to the centroid if the i-th cluster.

Internal Measures: Silhouette Coefficient l l Silhouette Coefficient combine ideas of both cohesion and separation, but for individual points, as well as clusters and clusterings 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 = (b-a)/max(a, b) – Typically between 0 and 1. – The closer to 1 the better – Negative values are pathological l Can calculate the Average Silhouette width for a cluster or for the whole clustering

Silhouette in R require("cluster") setwd("C: \Users\C. Eick\Desktop") a<-read. csv("c 8. csv") d<-data. frame(a=a[, 1], b=a[, 2], c=a[, 3]) d<-data. frame(a=d$a, b=d$b, c=d$c+1) plot(d$a, d$b, col=c('grey', 'blue', 'green', 'brown', 'black', 'orange', 'red', 'purple')[d$c]) set. seed(2) y<-pam(d[1: 2], 10) plot(d$a, d$b, col=c('gray', 'blue', 'green', 'brown', 'black', 'orange', 'red', 'purple', 'aquamarine', 'coral', 'slateblue')[y$cluster]) s<-silhouette(y) ss<-summary(s) ss$si. summary ss s[1, ] s[, 3] #computing the Silhouette for DBSCAN and other clusterings is more complicated! #You need to create a distance matrix for the dataset and pass it to Silhouette #in general, cluster 0 needs to be removed before computing Silhouette, which #is not done in the code below! require('ggplot 2') require('lattice') require('fpc') y<-dbscan(d[1: 2], 22, 20, showplot=1) y dm<-dist(d[1: 2]) s<-silhouette(y$cluster, dist=dm) ss<-summary(s) # there are too many clusters with a negative Silhouette; there should be only one; something is wrong!

Silhouette in R require("cluster") setwd("C: \Users\C. Eick\Desktop") a<-read. csv("c 8. csv") d<-data. frame(a=a[, 1], b=a[, 2], c=a[, 3]) d<-data. frame(a=d$a, b=d$b, c=d$c+1) plot(d$a, d$b, col=c('grey', 'blue', 'green', 'brown', 'black', 'orange', 'red', 'purple')[d$c]) set. seed(2) y<-pam(d[1: 2], 10) plot(d$a, d$b, col=c('gray', 'blue', 'green', 'brown', 'black', 'orange', 'red', 'purple', 'aquamarine', 'coral', 'slateblue')[y$cluster]) s<-silhouette(y) ss<-summary(s) ss$si. summary ss s[1, ] s[, 3]

Silhouette in R #computing the Silhouette for DBSCAN and other clusterings is more complicated! #You need to create a distance matrix for the dataset and pass it to Silhouette #in general, cluster 0 needs to be removed before computing Silhouette, which #is not done in the code below! require('ggplot 2') require('lattice') require('fpc') y<-dbscan(d[1: 2], 22, 20, showplot=1) y dm<-dist(d[1: 2]) s<-silhouette(y$cluster, dist=dm) ss<-summary(s) ss y<-dbscan(d[1: 2], 22, 20, showplot=1) #Let us take a look at cluster 4 with the negative silouette c<-data. frame(a=d$a, b=d$b, c=y$cluster) dd<-subset(c, c==4) plot(dd$a, dd$b) #if you take a look you understand why the silhouette of cluster 4 is negative, although cluster is "really good“ ! y<-dbscan(d[1: 2], 22, 20, showplot=1) Summary: Silhouette is an reasonable evaluation measure for clustering algorithms that find globular shapes, such as K-means and PAM, but it is of limited use for DBSCAN which finds clusters of non-globular shape.

External Measures of Cluster Validity: Entropy and Purity

Final Comment on Cluster Validity “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

Summary Cluster Evaluation Clustering Tendency: Compare dataset distribution with a random distribution textbook l Comparing clustering results: l – Use internal measures; e. g. silhouette – Use external measure; e. g purity; domain expert; or pre-classified examples (last approach is called semisupervised clustering; expert assigns link/do not link to some examples that is used to evaluate clusterings and/or to guide the search for good clusters) l How many clusters? textbook