Presentation on theme: "What is Cluster Analysis?"— Presentation transcript:
1 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 groupsInter-cluster distances are maximizedIntra-cluster distances are minimized
2 Applications of Cluster Analysis UnderstandingGroup related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuationsSummarizationReduce the size of large data setsClustering precipitation in Australia
3 What is not Cluster Analysis? Supervised classificationHave class label informationSimple segmentationDividing students into different registration groups alphabetically, by last nameResults of a queryGroupings are a result of an external specification
4 Notion of a cluster can be ambiguous How many clusters?Six ClustersTwo ClustersFour Clusters
5 Types of Clusterings A clustering is a set of clusters Important distinction between hierarchical and partitional sets of clustersPartitional ClusteringA division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subsetHierarchical clusteringA set of nested clusters organized as a hierarchical tree
6 Partitional Clustering A Partitional ClusteringOriginal Points
8 Other Distinctions Between Sets of Clusters Exclusive versus non-exclusiveIn non-exclusive clusterings, points may belong to multiple clusters.Can represent multiple classes or ‘border’ pointsFuzzy versus non-fuzzyIn fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1Weights must sum to 1Probabilistic clustering has similar characteristicsPartial versus completeIn some cases, we only want to cluster some of the dataHeterogeneous versus homogeneousCluster of widely different sizes, shapes, and densities
9 Types of Clusters Well-separated clusters Center-based clusters Contiguous clustersDensity-based clustersDescribed by an Objective Function
10 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
11 Types of Clusters: 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 clusterThe 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 cluster4 center-based clusters
12 Types of Clusters: Contiguity-Based 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
13 Types of Clusters: 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
14 Types of Clusters: Objective Function Clusters Defined by an Objective FunctionFinds clusters that minimize or maximize an objective function.
15 Characteristics of the Input Data Are Important Type of proximity or density measureThis is a derived measure, but central to clusteringSparsenessDictates type of similarityAdds to efficiencyAttribute typeType of DataDimensionalityNoise and OutliersType of Distribution
16 Clustering Algorithms K-means and its variantsHierarchical clusteringDensity-based clustering
17 K-means Clustering Partitional clustering approach Each cluster is associated with a centroid (center point)Each point is assigned to the cluster with the closest centroidNumber of clusters, K, must be specifiedThe basic algorithm is very simple
18 K-means Clustering – Details Initial centroids are often chosen randomly.Clusters produced vary from one run to another.The centroid is (typically) 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
24 Evaluating K-means Clusters Most common measure is Sum of Squared Error (SSE)For each point, the error is the distance to the nearest clusterTo 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 Cican show that mi corresponds to the center (mean) of the clusterGiven two clusterings, we can choose the one with the smallest errorOne easy way to reduce SSE is to increase K, the number of clusters
25 Evaluating K-means Clusters Need to incorporate measure of separation between clusters, not only similarity inside each clusterDissimilarity ratio = (inter-cluster distance / intra-cluster distance)Computing dissimilarity ratio from JMP output
26 Computing dissimilarity ratio from JMP output: Centroid coordinates are given in the output. However each value has to be normalized.Obtain mean and standard deviation of each dimension to compute normalized coordinates of centroids. These can be obtained from Histograms.Distances between centroids X and Y are given byThe inter-cluster distance will be the smallest of the distances between centroids.The intra-cluster distance is the mean distance (obtained from JMP histogram)Compute ratioDissimilarity ratio = (inter-cluster distance / intra-cluster distance)
27 Two different K-means Clusterings Original PointsOptimal ClusteringSub-optimal Clustering
32 Problems with Selecting Initial Points 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 largeIf clusters are the same size, n, thenFor example, if K = 10, then probability = 10!/1010 =Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’tConsider an example of five pairs of clusters
33 10 Clusters ExampleStarting with two initial centroids in one cluster of each pair of clusters
34 10 Clusters ExampleStarting with two initial centroids in one cluster of each pair of clusters
35 10 Clusters ExampleStarting with some pairs of clusters having three initial centroids, while other have only one.
36 10 Clusters ExampleStarting with some pairs of clusters having three initial centroids, while other have only one.
37 Solutions to Initial Centroids Problem Multiple runsHelps, but probability is not on your sideSample and use hierarchical clustering to determine initial centroidsSelect more than k initial centroids and then select among these initial centroidsSelect most widely separatedPostprocessing
38 Pre-processing and Post-processing Normalize the dataEliminate outliersPost-processingEliminate small clusters that may represent outliersSplit ‘loose’ clusters, i.e., clusters with relatively high SSEMerge clusters that are ‘close’ and that have relatively low SSE
39 Limitations of K-means K-means has problems when clusters are of differingSizesDensitiesNon-globular shapesK-means has problems when the data contains outliers.
40 Limitations of K-means: Differing Sizes Original PointsK-means (3 Clusters)
41 Limitations of K-means: Differing Density Original PointsK-means (3 Clusters)
42 Limitations of K-means: Non-globular Shapes Original PointsK-means (2 Clusters)
43 Overcoming K-means Limitations Original Points K-means ClustersOne solution is to use many clusters.Find parts of clusters, but need to put together.
44 Overcoming K-means Limitations Original Points K-means Clusters
45 Overcoming K-means Limitations Original Points K-means Clusters