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Data Mining Cluster Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 Introduction to Data Mining by Tan, Steinbach, Kumar © Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 Hierarchical Clustering Produces a set of nested clusters organized as a hierarchical tree z Can be visualized as a dendrogram z – A tree like diagram that records the sequences of merges or splits 5 6 0.2 4 3 4 2 0.15 5 2 0.1 1 0.05 3 0 1 3 2 5 4 6 1 Strengths of Hierarchical Clustering z 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 z They may correspond to meaningful taxonomies – Example in biological sciences (e (e.g., g animal kingdom kingdom, phylogeny reconstruction, …) Hierarchical Clustering z Hierarchical clustering is most frequently performed in an agglomerative manner – Start with the points as individual clusters – At each step, merge the closest pair of clusters until only one cluster (or k cclusters) (o uste s) left et z Traditional hierarchical algorithms use a similarity or distance matrix – Merge or split one cluster at a time Agglomerative Clustering Algorithm z Most popular hierarchical clustering technique z Basic algorithm is straightforward 1. Compute the proximity (distance) matrix 2. Let each data point be a cluster 3. Repeat 4. Merge the two closest clusters 5 5. Update the proximity matrix 6. z 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 l t di distinguish ti i h th the diff differentt algorithms l ith Starting Situation z Start with clusters of individual points and a proximity matrix p1 p2 p3 p4 p5 p1 p2 p3 p4 p5 . . . Proximity Matrix ... Intermediate Situation z After some merging steps, we have some clusters C1 C2 C3 C4 C1 C2 C3 C3 C4 C4 C5 Proximity Matrix C1 C2 C5 C5 Intermediate Situation z We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C1 C2 C3 C4 C5 C1 C2 C3 C3 C4 C4 C5 Proximity Matrix C1 C2 C5 After Merging z The question is “How do we update the proximity matrix?” C1 C1 C2 U C5 C3 C4 C2 U C5 C3 C4 ? ? ? ? ? C3 ? C4 ? Proximity Matrix C1 C2 U C5 How to Define Inter-Cluster Similarity p1 Similarity? p2 p3 p4 p5 p1 p2 p3 p4 z z z z z p5 MIN . MAX . Group Average . Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward Ward’s s Method uses squared error ... How to Define Inter-Cluster Similarity p1 p2 p3 p4 p5 p1 p2 p3 p4 z z z z z p5 MIN . MAX . Group Average . Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward Ward’s s Method uses squared error ... How to Define Inter-Cluster Similarity p1 p2 p3 p4 p5 p1 p2 p3 p4 z z z z z p5 MIN . MAX . Group Average . Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward Ward’s s Method uses squared error ... How to Define Inter-Cluster Similarity p1 p2 p3 p4 p5 p1 p2 p3 p4 z z z z z p5 MIN . MAX . Group Average . Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward Ward’s s Method uses squared error ... How to Define Inter-Cluster Similarity p1 p2 p3 p4 p5 p1 × × p2 p3 p4 z z z z z p5 MIN . MAX . Group Average . Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward Ward’s s method uses squared error ... Cluster Similarity: MIN or Single Link z Similarity of two clusters is based on the two most similar (closest) points in the different clusters – Determined by one pair of points, i.e., by one link in the proximity graph. I1 I2 I3 I4 I5 I1 1.00 0.90 0.10 0.65 0 20 0.20 I2 0.90 1.00 0.70 0.60 0 50 0.50 I3 0.10 0.70 1.00 0.40 0 30 0.30 I4 0.65 0.60 0.40 1.00 0 80 0.80 I5 0.20 0.50 0.30 0.80 1 00 1.00 1 2 3 4 5 Hierarchical Clustering: MIN 1 5 3 5 0.2 2 1 2 3 0.15 6 0.1 0.05 4 4 Nested Clusters 0 3 6 2 5 Dendrogram 4 1 Strength of MIN Original Points • Can handle non-elliptical p shapes p Two Clusters Limitations of MIN Original Points • Sensitive to noise and outliers Two Clusters Cluster Similarity: MAX or Complete Linkage z Similarity of two clusters is based on the two least similar (most distant) points in the different clusters – Determined by all pairs of points in the two clusters I1 I2 I3 I4 I1 1.00 0.90 0.10 0.65 I2 0.90 1.00 0.70 0.60 I3 0.10 0 10 0.70 0 0 1.00 1 00 0.40 0 40 I4 0.65 0.60 0.40 1.00 I5 0.20 0 20 0.50 0 50 0.30 0 30 0.80 0 80 I5 0.20 0.50 0 30 0.30 0.80 1 00 1.00 1 2 3 4 5 Hierarchical Clustering: MAX 4 1 2 5 5 0.4 0.35 03 0.3 2 0.25 3 3 6 1 4 0.2 0.15 0.1 0.05 0 Nested Clusters 3 6 4 Dendrogram 1 2 5 Strength of MAX Original Points • Less susceptible p to noise and outliers Two Clusters Limitations of MAX Original Points • Tends to break large clusters • Biased towards globular clusters Two Clusters Cluster Similarity: Group Average z Proximity of two clusters is the average of pairwise proximity between points in the two clusters. ∑ proximity(p , p ) i proximity(Clusteri , Clusterj ) = z j pi∈Clusteri p j∈Clusterj |Clusteri |∗|Clusterj | Need to use average connectivity for scalability since total proximity favors large clusters I1 I2 I3 I4 I5 I1 1.00 0 90 0.90 0.10 0.65 0.20 I2 0.90 1 00 1.00 0.70 0.60 0.50 I3 0.10 0 70 0.70 1.00 0.40 0.30 I4 0.65 0 60 0.60 0.40 1.00 0.80 I5 0.20 0 50 0.50 0.30 0.80 1.00 1 2 3 4 5 Hierarchical Clustering: Group Average 5 4 1 0.25 2 5 0.2 2 0.15 3 6 1 4 3 Nested Clusters 0.1 0.05 0 3 6 4 1 Dendrogram g 2 5 Hierarchical Clustering: Group Average z Compromise between Single and Complete Link z St Strengths th – Less susceptible to noise and outliers z Limitations – Biased towards globular clusters Cluster Similarity: Ward’s Method z 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 z Less susceptible to noise and outliers z Biased towards globular clusters z Hierarchical analogue of K-means – Can be used to initialize K-means K means Hierarchical Clustering: Comparison 1 5 4 3 5 5 2 2 5 1 2 1 MIN 3 2 MAX 6 3 3 4 1 4 4 1 5 5 2 5 6 4 1 2 Ward’s Method 2 3 3 6 5 2 Group Average 3 1 4 4 6 1 4 3 Hierarchical Clustering: Problems and Limitations z Once a decision is made to combine two clusters, it cannot be undone z No objective function is directly minimized z Different schemes have problems with one or more off the th following: f ll i – Sensitivity to noise and outliers – Difficulty Diffi lt handling h dli diff differentt sized i d clusters l t and d convex shapes – Breaking large clusters Cluster Validity z For supervised classification we have a variety of measures to evaluate how good our model is – Accuracy, Accuracy sensitivity, sensitivity specificity specificity... z For cluster analysis, the analogous question is how to evaluate the “goodness” goodness of the resulting clusters? z But “clusters are in the eye of the beholder”! z Then why do we want to evaluate them? – – – – To avoid finding gp patterns in noise To compare clustering algorithms To compare two sets of clusters To compare p two clusters 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 y Random Points y Clusters found in Random Data 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.2 0.4 0.6 0.8 0 1 DBSCAN (densitybased) 0 0.2 0.4 x 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 y y K-means 0.4 0.4 0.3 0.3 0.2 0.2 01 0.1 01 0.1 0 0 0.2 0.4 0.6 x 0.6 0.8 1 x 0.8 1 0 Complete Link 0 0.2 0.4 0.6 x 0.8 1 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 3. Evaluating how well the results of a cluster analysis fit the data without reference to external information. - Use only the data 4 4. Comparing C i th the results lt off ttwo different diff t sets t off cluster l t analyses l to t 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 clusters. Measures of Cluster Validity z 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. Entropy – Internal Index: Used to measure the goodness of a clustering structure without respect to external information. Sum of Squared Error (SSE) – Relative Index: Used to compare two different clusterings or clusters. z 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. Measuring Cluster Validity Via Correlation z z Two matrices – Proximity Matrix – “Incidence” Incidence Matrix z One row and one column for each data point An entry is 1 if the associated pair of points belong to the same cluster An entry is 0 if the associated pair of points belongs to different clusters Compute the correlation between the two matrices – z Since the matrices are symmetric, only the correlation between n(n-1) / 2 entries needs to be calculated. High correlation indicates that points that belong to the same cluster are close to each other. Not a good measure for some density or contiguity based clusters. Measuring Cluster Validity Via Correlation Correlation of incidence and proximity matrices for the K-means K means clusterings of the following two data sets. 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 y y z 0.4 0.4 0.3 0.3 0.2 0.2 01 0.1 01 0.1 0 0 0.2 0.4 0.6 x Corr = -0.9235 0.8 1 0 0 0.2 0.4 0.6 x Corr = -0.5810 0.8 1 Using Similarity Matrix for Cluster Validation Order the similarity matrix with respect to cluster labels and inspect visually. 1 1 0.9 0.8 0.7 P Points 0.6 y z 0.5 0.4 0.3 0.2 0.1 0 10 0.9 20 08 0.8 30 0.7 40 0.6 50 0.5 60 0.4 70 0.3 80 0.2 90 0.1 100 0 0.2 0.4 0.6 x 0.8 1 20 40 60 Points 80 0 100Similarity Using Similarity Matrix for Cluster Validation z Clusters in random data are not so crisp 1 10 0.9 0.9 20 0.8 0.8 30 07 0.7 07 0.7 40 0.6 0.6 50 0.5 0.5 60 0.4 0.4 70 0.3 0.3 80 0.2 0.2 90 0.1 0.1 100 20 40 60 80 0 100Similarity Points y Points 1 0 0 0.2 0.4 0.6 x DBSCAN 0.8 1 Using Similarity Matrix for Cluster Validation z Clusters in random data are not so crisp 1 10 0.9 0.9 20 0.8 0.8 30 0.7 0.7 40 0.6 0.6 50 0.5 0.5 60 0.4 0.4 70 0.3 0.3 80 0.2 0.2 90 0.1 0.1 100 20 40 60 80 0 100Similarity y Points 1 0 0 0.2 0.4 0.6 x Points K-means 0.8 1 Using Similarity Matrix for Cluster Validation z Clusters in random data are not so crisp 1 10 0.9 0.9 20 0.8 0.8 30 07 0.7 07 0.7 40 0.6 0.6 50 0.5 0.5 60 0.4 0.4 70 0.3 0.3 80 0.2 0.2 90 0.1 0.1 100 20 40 60 80 0 100Similarity y Points 1 0 0 Points 0.2 0.4 0.6 x Complete Link 0.8 1 Using Similarity Matrix for Cluster Validation 1 0.9 500 1 2 0.8 6 0.7 1000 3 0.6 4 1500 0.5 0.4 2000 0.3 5 02 0.2 2500 0.1 7 3000 DBSCAN 500 1000 1500 2000 2500 3000 0 Internal Measures: SSE z Clusters in more complicated figures aren’t well separated z Internal Index: Used to measure the goodness of a clustering structure without respect to external information – SSE z SSE is g good for comparing p g two clusterings g or two clusters (average SSE). Can also be used to estimate the number of clusters 10 9 6 8 4 7 6 2 SSE E z 0 5 4 -2 3 2 -4 1 -6 0 5 10 15 2 5 10 15 K 20 25 30 Internal Measures: SSE z SSE curve for a more complicated data set 1 2 6 3 4 5 7 SSE of clusters found using K-means