Data Mining Cluster Analysis Basic Concepts and Algorithms
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
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 Inter-cluster distances are maximized Intra-cluster distances are minimized © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 2
Applications of Cluster Analysis l Understanding – Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations l Summarization – Reduce the size of large data sets Clustering precipitation in Australia © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 3
What is not Cluster Analysis? l Supervised classification – Have class label information l Simple segmentation – Dividing students into different registration groups alphabetically, by last name l Results of a query – Groupings are a result of an external specification l Graph partitioning – Some mutual relevance and synergy, but areas are not identical © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 4
Notion of a Cluster can be Ambiguous How many clusters? Six Clusters Two Clusters Four Clusters © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 5
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 6
Partitional Clustering Original Points © Tan, Steinbach, Kumar A Partitional Clustering Introduction to Data Mining 4/18/2004 7
Hierarchical Clustering Traditional Dendrogram Non-traditional Hierarchical Clustering Non-traditional Dendrogram © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 8
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 9
Types of Clusters l Well-separated clusters l Center-based clusters l Contiguous clusters l Density-based clusters l Property or Conceptual © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 10
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 11
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 12
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 13
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 14
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 15
Clustering Algorithms l K-means and its variants l Hierarchical clustering l Density-based clustering © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 16
K-means Clustering l Partitional clustering approach l Each cluster is associated with a centroid (center point) l Each point is assigned to the cluster with the closest centroid l Number of clusters, K, must be specified l The basic algorithm is very simple © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 17
K-means Clustering – Details l Initial centroids are often chosen randomly. – Clusters produced vary from one run to another. l The centroid is (typically) the mean of the points in the cluster. l ‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc. l K-means will converge for common similarity measures mentioned above. l Most of the convergence happens in the first few iterations. – l 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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 18
Two different K-means Clusterings Original Points Optimal Clustering © Tan, Steinbach, Kumar Introduction to Data Mining Sub-optimal Clustering 4/18/2004 19
Importance of Choosing Initial Centroids © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 20
Importance of Choosing Initial Centroids © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 21
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 22
Importance of Choosing Initial Centroids … © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 23
Importance of Choosing Initial Centroids … © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 24
Problems with Selecting Initial Points l 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 – Consider an example of five pairs of clusters Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 25
10 Clusters Example Starting with two initial centroids in one cluster of each pair of clusters © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 26
10 Clusters Example Starting with two initial centroids in one cluster of each pair of clusters © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 27
10 Clusters Example Starting with some pairs of clusters having three initial centroids, while other have only one. © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 28
10 Clusters Example Starting with some pairs of clusters having three initial centroids, while other have only one. © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 29
Solutions to Initial Centroids Problem l Multiple runs – Helps, but probability is not on your side Select most widely separated centroids l Postprocessing l Bisecting K-means l – Not as susceptible to initialization issues © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 30
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 31
Updating Centers Incrementally l In the basic K-means algorithm, centroids are updated after all points are assigned to a centroid l An alternative is to update the centroids after each assignment (incremental approach) – – Each assignment updates zero or two centroids More expensive Introduces an order dependency Never get an empty cluster © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 32
Pre-processing and Post-processing l Pre-processing – Normalize the data – Eliminate outliers l 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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 33
Bisecting K-means l Bisecting K-means algorithm – Variant of K-means that can produce a partitional or a hierarchical clustering © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 34
Bisecting K-means Example © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 35
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 36
Limitations of K-means: Differing Sizes K-means (3 Clusters) Original Points © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 37
Limitations of K-means: Differing Density K-means (3 Clusters) Original Points © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 38
Limitations of K-means: Non-globular Shapes Original Points © Tan, Steinbach, Kumar K-means (2 Clusters) Introduction to Data Mining 4/18/2004 39
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. © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 40
Overcoming K-means Limitations Original Points © Tan, Steinbach, Kumar K-means Clusters Introduction to Data Mining 4/18/2004 41
Overcoming K-means Limitations Original Points © Tan, Steinbach, Kumar K-means Clusters Introduction to Data Mining 4/18/2004 42
Hierarchical Clustering Produces a set of nested clusters organized as a hierarchical tree l Can be visualized as a dendrogram l – A tree like diagram that records the sequences of merges or splits © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 43
Strengths of Hierarchical Clustering l Do not have to assume any particular number of clusters – Any desired number of clusters can be obtained by ‘cutting’ the dendrogram at the proper level l They may correspond to meaningful taxonomies – Example in biological sciences (e. g. , animal kingdom) © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 44
Hierarchical Clustering l Two main types of hierarchical clustering – Agglomerative: u Start with the points as individual clusters At each step, merge the closest pair of clusters until only one cluster (or k clusters) left u – Divisive: u Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point (or there are k clusters) u l Traditional hierarchical algorithms use a similarity or distance matrix – Merge or split one cluster at a time © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 45
Agglomerative Clustering Algorithm l More popular hierarchical clustering technique l Basic algorithm is straightforward 1. 2. 3. 4. 5. 6. l Compute the proximity matrix Let each data point be a cluster Repeat Merge the two closest clusters Update the proximity matrix 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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 46
Starting Situation l Start with clusters of individual points and a proximity matrix p 1 p 2 p 3 p 4 p 5 . . . p 1 p 2 p 3 p 4 p 5. . . © Tan, Steinbach, Kumar Introduction to Data Mining Proximity Matrix 4/18/2004 47
Intermediate Situation l After some merging steps, we have some clusters C 1 C 2 C 3 C 4 C 5 Proximity Matrix C 1 C 2 © Tan, Steinbach, Kumar C 5 Introduction to Data Mining 4/18/2004 48
Intermediate Situation l We want to merge the two closest clusters (C 2 and C 5) and update the proximity matrix. C 1 C 2 C 3 C 4 C 5 Proximity Matrix C 1 C 2 © Tan, Steinbach, Kumar C 5 Introduction to Data Mining 4/18/2004 49
After Merging l The question is “How do we update the proximity matrix? ” C 1 C 2 U C 5 C 3 C 4 ? ? ? C 3 ? C 4 ? Proximity Matrix C 1 C 2 U C 5 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 50
How to Define Inter-Cluster Similarity p 1 Similarity? p 2 p 3 p 4 p 5 . . . p 1 p 2 p 3 p 4 l l l p 5 MIN. MAX. Group Average. Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward’s Method uses squared error © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 51
How to Define Inter-Cluster Similarity p 1 p 2 p 3 p 4 p 5 . . . p 1 p 2 p 3 p 4 l l l p 5 MIN. MAX. Group Average. Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward’s Method uses squared error © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 52
How to Define Inter-Cluster Similarity p 1 p 2 p 3 p 4 p 5 . . . p 1 p 2 p 3 p 4 l l l p 5 MIN. MAX. Group Average. Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward’s Method uses squared error © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 53
How to Define Inter-Cluster Similarity p 1 p 2 p 3 p 4 p 5 . . . p 1 p 2 p 3 p 4 l l l p 5 MIN. MAX. Group Average. Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward’s Method uses squared error © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 54
How to Define Inter-Cluster Similarity p 1 p 2 p 3 p 4 p 5 . . . p 1 p 2 p 3 p 4 l l l p 5 MIN. MAX. Group Average. Proximity Matrix Distance Between Centroids Other methods driven by an objective function – Ward’s Method uses squared error © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 55
Cluster Similarity: MIN or Single Link l Similarity of two clusters is based on the two most similar (closest) points in the different clusters – Add links between points one at a time, shortest links first – Contiguity-based clustering 1 © Tan, Steinbach, Kumar Introduction to Data Mining 2 3 4 4/18/2004 5 56
Hierarchical Clustering: MIN 1 3 5 2 1 2 3 4 5 6 4 Nested Clusters © Tan, Steinbach, Kumar Dendrogram Introduction to Data Mining 4/18/2004 57
Strength of MIN Original Points Two Clusters • Can handle non-elliptical shapes © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 58
Limitations of MIN Original Points Two Clusters • Sensitive to noise and outliers © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 59
Cluster Similarity: MAX or Complete Link l Similarity of two clusters is based on the two least similar (most distant) points in the different clusters – Add links between points one at a time, shortest links first, a cluster is formed when a clique forms 1 © Tan, Steinbach, Kumar Introduction to Data Mining 2 3 4 4/18/2004 5 60
Hierarchical Clustering: MAX 4 1 2 5 5 2 3 3 6 1 4 Nested Clusters © Tan, Steinbach, Kumar Dendrogram Introduction to Data Mining 4/18/2004 61
Strength of MAX Original Points Two Clusters • Less susceptible to noise and outliers © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 62
Limitations of MAX Original Points Two Clusters • Tends to break large clusters • Biased towards globular clusters © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 63
Cluster Similarity: Group Average l Proximity of two clusters is the average of pairwise proximity between points in the two clusters. l Need to use average connectivity for scalability since total proximity favors large clusters 1 © Tan, Steinbach, Kumar Introduction to Data Mining 2 3 4 4/18/2004 5 64
Hierarchical Clustering: Group Average 5 4 1 2 5 2 3 6 1 4 3 Nested Clusters © Tan, Steinbach, Kumar Dendrogram Introduction to Data Mining 4/18/2004 65
Hierarchical Clustering: Group Average l Compromise between Single and Complete Link l Strengths – Less susceptible to noise and outliers l Limitations – Biased towards globular clusters © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 66
Cluster Similarity: Ward’s Method l 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 l Less susceptible to noise and outliers l Biased towards globular clusters l Hierarchical analogue of K-means – Can be used to initialize K-means © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 67
Hierarchical Clustering: Comparison 1 3 5 5 1 2 3 6 MIN MAX 5 2 3 3 5 1 5 Ward’s Method 2 3 3 6 4 1 2 5 2 Group Average 3 1 6 1 4 4 © Tan, Steinbach, Kumar 6 4 2 4 5 4 1 5 1 2 2 4 4 Introduction to Data Mining 3 4/18/2004 68
Hierarchical Clustering: Problems and Limitations l Once a decision is made to combine two clusters, it cannot be undone – Moving branches of the tree around – Use a partitional clustering technique to create many small clusters first No objective function is directly minimized l Different schemes have problems with one or more of the following: l – Sensitivity to noise and outliers – Difficulty handling different sized clusters and convex shapes – Breaking large clusters © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 69
Hierarchical Clustering: Problems and Limitations l Centroid-based methods (HW#4) – Inversions – two clusters that are merged may be more similar than the pair of clusters merged in a previous step (page 523) – A point in one cluster may be closer to the centroid of some other cluster than its centroid (page 526) © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 70
DBSCAN l DBSCAN is a density-based algorithm. – Density = number of points within a specified radius (Eps) – A point is a core point if it has more than a specified number of points (Min. Pts) within Eps u These are points that are at the interior of a cluster – A border point has fewer than Min. Pts within Eps, but is in the neighborhood of a core point – A noise point is any point that is not a core point or a border point. © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 71
DBSCAN: Core, Border, and Noise Points © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 72
DBSCAN Algorithm Label all points as core, border, or noise points l Eliminate noise points l Put an edge between all core points that are within Eps of each other l Make each group of connected core points into a separate cluster l Assign each border point to one of the clusters of its associated core points l © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 73
DBSCAN: Core, Border and Noise Points Original Points Point types: core, border and noise Eps = 10, Min. Pts = 4 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 74
When DBSCAN Works Well Original Points Clusters • Resistant to Noise • Can handle clusters of different shapes and sizes © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 75
DBSCAN: Determining EPS and Min. Pts l l l Idea is that for points in a cluster, their kth nearest neighbors are at roughly the same distance Noise points have the kth nearest neighbor at farther distance So, plot sorted distance of every point to its kth nearest neighbor © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 76
DBSCAN: Determining EPS and Min. Pts If the value of k is too small, then even a small number of closely spaced points that are noise will be labeled as clusters l If the value of k is too large, then small clusters of size lass than k are likely to be labeled as noise l © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 77
When DBSCAN Does NOT Work Well (Min. Pts=4, Eps=9. 75) Original Points • Varying densities • High-dimensional data © Tan, Steinbach, Kumar (Min. Pts=4, Eps=9. 92) Introduction to Data Mining 4/18/2004 78
Cluster Validity l 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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 79
Clusters found in Random Data Random Points DBSCAN K-means © Tan, Steinbach, Kumar Complete Link Introduction to Data Mining 4/18/2004 80
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 clusters 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. © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 81
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 Often an external or internal index is used for this function, e. g. , SSE or entropy © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 82
Measuring Cluster Validity Via Correlation l l Two matrices – Proximity Matrix – “Incidence” Matrix l One row and one column for each data point u An entry is 1 if the associated pair of points belong to the same cluster u An entry is 0 if the associated pair of points belongs to different clusters Compute the correlation between the two matrices – l u 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. © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 83
Measuring Cluster Validity Via Correlation l Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. Corr = -0. 9235 © Tan, Steinbach, Kumar Corr = -0. 5810 Introduction to Data Mining 4/18/2004 84
Using Similarity Matrix for Cluster Validation l Order the similarity matrix with respect to cluster labels and inspect visually. © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 85
Using Similarity Matrix for Cluster Validation l Clusters in random data are not so crisp DBSCAN © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 86
Using Similarity Matrix for Cluster Validation l Clusters in random data are not so crisp K-means © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 87
Using Similarity Matrix for Cluster Validation l Clusters in random data are not so crisp Complete Link © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 88
Using Similarity Matrix for Cluster Validation DBSCAN © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 89
Internal Measures: SSE l Clusters in more complicated figures aren’t well separated l Internal Index: Used to measure the goodness of a clustering structure without respect to external information – SSE l l SSE is good for comparing two clusterings or two clusters (average SSE). Can also be used to estimate the number of clusters © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 90
Internal Measures: SSE l SSE curve for a more complicated data set SSE of clusters found using K-means © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 91
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 wellseparated a cluster is from other clusters l Example: Squared Error – Cohesion is measured by the within cluster sum of squares (SSE) – Separation is measured by the between cluster sum of squares – Where |Ci| is the size of cluster i © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 92
Internal Measures: Cohesion and Separation l Example: SSE – BSS + WSS = constant (page 540) 1 m 2 3 4 m 2 5 K=1 cluster: K=2 clusters: © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 93
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 © Tan, Steinbach, Kumar separation Introduction to Data Mining 4/18/2004 94
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 -1 and 1. – The closer to 1 the better. l Can calculate the Average Silhouette coefficient for a cluster © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 95
External Measures of Cluster Validity: Entropy and Purity © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 96
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 © Tan, Steinbach, Kumar Introduction to Data Mining 4/18/2004 97
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