Documentation

SilhouetteEvaluation

Package: clustering.evaluation
Superclasses: clustering.evaluation.ClusterCriterion

Silhouette criterion clustering evaluation object

Description

SilhouetteEvaluation is an object consisting of sample data, clustering data, and silhouette criterion values used to evaluate the optimal number of data clusters. Create a silhouette criterion clustering evaluation object using evalclusters.

Construction

eva = evalclusters(x,clust,'Silhouette') creates a silhouette criterion clustering evaluation object.

eva = evalclusters(x,clust,'Silhouette',Name,Value) creates a silhouette criterion clustering evaluation object using additional options specified by one or more name-value pair arguments.

Input Arguments

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Input data, specified as an N-by-P matrix. N is the number of observations, and P is the number of variables.

Data Types: single | double

Clustering algorithm, specified as one of the following.

 'kmeans' Cluster the data in x using the kmeans clustering algorithm, with 'EmptyAction' set to 'singleton' and 'Replicates' set to 5. 'linkage' Cluster the data in x using the clusterdata agglomerative clustering algorithm, with 'Linkage' set to 'ward'. 'gmdistribution' Cluster the data in x using the gmdistribution Gaussian mixture distribution algorithm, with 'SharedCov' set to true and 'Replicates' set to 5.

If criterion is 'CalinskiHarabasz', 'DaviesBouldin', or 'silhouette', you can specify a clustering algorithm using a function handle (MATLAB). The function must be of the form C = clustfun(DATA,K), where DATA is the data to be clustered, and K is the number of clusters. The output of clustfun must be one of the following:

• A vector of integers representing the cluster index for each observation in DATA. There must be K unique values in this vector.

• A numeric n-by-K matrix of score for n observations and K classes. In this case, the cluster index for each observation is determined by taking the largest score value in each row.

If criterion is 'CalinskiHarabasz', 'DaviesBouldin', or 'silhouette', you can also specify clust as a n-by-K matrix containing the proposed clustering solutions. n is the number of observations in the sample data, and K is the number of proposed clustering solutions. Column j contains the cluster indices for each of the N points in the jth clustering solution.

Data Types: single | double | char | string | function_handle

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'KList',[1:5],'Distance','cityblock' specifies to test 1, 2, 3, 4, and 5 clusters using the city block distance metric.

Prior probabilities for each cluster, specified as the comma-separated pair consisting of 'ClusterPriors' and one of the following.

 'empirical' Compute the overall silhouette value for the clustering solution by averaging the silhouette values for all points. Each cluster contributes to the overall silhouette value proportionally to its size. 'equal' Compute the overall silhouette value for the clustering solution by averaging the silhouette values for all points within each cluster, and then averaging those values across all clusters. Each cluster contributes equally to the overall silhouette value, regardless of its size.

Example: 'ClusterPriors','empirical'

Distance metric used for computing the criterion values, specified as the comma-separated pair consisting of 'Distance' and one of the following.

 'sqEuclidean' Squared Euclidean distance 'Euclidean' Euclidean distance. This option is not valid for the kmeans clustering algorithm. 'cityblock' Sum of absolute differences 'cosine' One minus the cosine of the included angle between points (treated as vectors) 'correlation' One minus the sample correlation between points (treated as sequences of values) 'Hamming' Percentage of coordinates that differ. This option is only valid for the Silhouette criterion. 'Jaccard' Percentage of nonzero coordinates that differ. This option is only valid for the Silhouette criterion.

For detailed information about each distance metric, see pdist.

You can also specify a function for the distance metric using a function handle (MATLAB). The distance function must be of the form d2 = distfun(XI,XJ), where XI is a 1-by-n vector corresponding to a single row of the input matrix X, and XJ is an m2-by-n matrix corresponding to multiple rows of X. distfun must return an m2-by-1 vector of distances d2, whose kth element is the distance between XI and XJ(k,:).

Distance only accepts a function handle if the clustering algorithm clust accepts a function handle as the distance metric. For example, the kmeans clustering algorithm does not accept a function handle as the distance metric. Therefore, if you use the kmeans algorithm and then specify a function handle for Distance, the software errors.

• If criterion is 'silhouette', you can also specify Distance as the output vector created by the function pdist.

• When clust is 'kmeans' or 'gmdistribution', evalclusters uses the distance metric specified for Distance to cluster the data.

• If clust is 'linkage', and Distance is either 'sqEuclidean' or 'Euclidean', then the clustering algorithm uses the Euclidean distance and Ward linkage.

• If clust is 'linkage' and Distance is any other metric, then the clustering algorithm uses the specified distance metric and average linkage.

• In all other cases, the distance metric specified for Distance must match the distance metric used in the clustering algorithm to obtain meaningful results.

Example: 'Distance','Euclidean'

Data Types: single | double | char | string | function_handle

List of number of clusters to evaluate, specified as the comma-separated pair consisting of 'KList' and a vector of positive integer values. You must specify KList when clust is a clustering algorithm name or a function handle. When criterion is 'gap', clust must be a character vector, a string scalar, or a function handle, and you must specify KList.

Example: 'KList',[1:6]

Data Types: single | double

Properties

 ClusteringFunction Clustering algorithm used to cluster the input data, stored as a valid clustering algorithm name or function handle. If the clustering solutions are provided in the input, ClusteringFunction is empty. ClusterPriors Prior probabilities for each cluster, stored as valid prior probability name. ClusterSilhouettes Silhouette values corresponding to each proposed number of clusters in InspectedK, stored as a cell array of vectors. CriterionName Name of the criterion used for clustering evaluation, stored as a valid criterion name. CriterionValues Criterion values corresponding to each proposed number of clusters in InspectedK, stored as a vector of numerical values. Distance Distance metric used for clustering data, stored as a valid distance metric name. InspectedK List of the number of proposed clusters for which to compute criterion values, stored as a vector of positive integer values. Missing Logical flag for excluded data, stored as a column vector of logical values. If Missing equals true, then the corresponding value in the data matrix x is not used in the clustering solution. NumObservations Number of observations in the data matrix X, minus the number of missing (NaN) values in X, stored as a positive integer value. OptimalK Optimal number of clusters, stored as a positive integer value. OptimalY Optimal clustering solution corresponding to OptimalK, stored as a column vector of positive integer values. If the clustering solutions are provided in the input, OptimalY is empty. X Data used for clustering, stored as a matrix of numerical values.

Methods

Inherited Methods

 addK Evaluate additional numbers of clusters compact Compact clustering evaluation object plot Plot clustering evaluation object criterion values

Examples

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Evaluate the optimal number of clusters using the silhouette clustering evaluation criterion.

Generate sample data containing random numbers from three multivariate distributions with different parameter values.

rng('default');  % For reproducibility
mu1 = [2 2];
sigma1 = [0.9 -0.0255; -0.0255 0.9];

mu2 = [5 5];
sigma2 = [0.5 0 ; 0 0.3];

mu3 = [-2, -2];
sigma3 = [1 0 ; 0 0.9];

N = 200;

X = [mvnrnd(mu1,sigma1,N);...
mvnrnd(mu2,sigma2,N);...
mvnrnd(mu3,sigma3,N)];

Evaluate the optimal number of clusters using the silhouette criterion. Cluster the data using kmeans.

E = evalclusters(X,'kmeans','silhouette','klist',[1:6])
E =
SilhouetteEvaluation with properties:

NumObservations: 600
InspectedK: [1 2 3 4 5 6]
CriterionValues: [NaN 0.8055 0.8551 0.7155 0.6071 0.6232]
OptimalK: 3

The OptimalK value indicates that, based on the silhouette criterion, the optimal number of clusters is three.

Plot the silhouette criterion values for each number of clusters tested.

figure;
plot(E) The plot shows that the highest silhouette value occurs at three clusters, suggesting that the optimal number of clusters is three.

Create a grouped scatter plot to visually examine the suggested clusters.

figure;
gscatter(X(:,1),X(:,2),E.OptimalY,'rbg','xod') The plot shows three distinct clusters within the data: Cluster 1 is in the lower-left corner, cluster 2 is in the upper-right corner, and cluster 3 is near the center of the plot.

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References

 Kaufman L. and P. J. Rouseeuw. Finding Groups in Data: An Introduction to Cluster Analysis. Hoboken, NJ: John Wiley & Sons, Inc., 1990.

 Rouseeuw, P. J. “Silhouettes: a graphical aid to the interpretation and validation of cluster analysis.” Journal of Computational and Applied Mathematics. Vol. 20, No. 1, 1987, pp. 53–65.