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clustering.evaluation.SilhouetteEvaluation class

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

Silhouette criterion clustering evaluation object

Description

clustering.evaluation.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.

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x — Input datamatrix

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

clust — Clustering algorithm'kmeans' | 'linkage' | 'gmdistribution' | matrix of clustering solutions | function handle

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 'CalinskHarabasz', 'DaviesBouldin', or 'silhouette', you can specify a clustering algorithm using the function_handle (@) operator. 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 'CalinskHarabasz', '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.

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 single 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 sum of absolute differences distance measure.

'ClusterPriors' — Prior probabilities for each cluster'empirical' (default) | 'equal'

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' — Distance metric'sqEuclidean' (default) | 'Euclidean' | 'cityblock' | vector | function | ...

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 '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 'Jaccard' Percentage of nonzero coordinates that differ

For detailed information about each distance metric, see pdist.

You can also specify a function for the distance metric by using the function_handle (@) operator. 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,:).

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

When Clust a string representing a built-in clustering algorithm, evalclusters uses the distance metric specified for Distance to cluster the data, except for the following:

• If Clust is 'linkage', and Distance is either 'sqEuclidean' or 'Euclidean', then the clustering algorithm uses 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'

'KList' — List of number of clusters to evaluatevector

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 string or a function handle. When criterion is 'gap', clust must be a string or a function handle, and you must specify KList.

Example: 'KList',[1:6]

Properties

 ClusteringFunction Clustering algorithm used to cluster the input data, stored as a valid clustering algorithm name string 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 string. 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 string. CriterionValues Criterion values corresponding to each proposed number of clusters in InspectedK, stored as a vector of numerical values. Distance Distance measure used for clustering data, stored as a valid distance measure name string. 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

Definitions

Silhouette Value

The silhouette value for each point is a measure of how similar that point is to points in its own cluster, when compared to points in other clusters. The silhouette value for the ith point, Si, is defined as

`Si = (bi-ai)/ max(ai,bi)`

where ai is the average distance from the ith point to the other points in the same cluster as i, and bi is the minimum average distance from the ith point to points in a different cluster, minimized over clusters.

The silhouette value ranges from -1 to +1. A high silhouette value indicates that i is well-matched to its own cluster, and poorly-matched to neighboring clusters. If most points have a high silhouette value, then the clustering solution is appropriate. If many points have a low or negative silhouette value, then the clustering solution may have either too many or too few clusters. The silhouette clustering evaluation criterion can be used with any distance metric.

Examples

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Evaluate the Clustering Solution Using Silhouette Criterion

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 0.7170 0.7376 0.6239]
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 near the center of the plot, and cluster 3 is in the upper-right corner.

References

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

[2] 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.