DaviesBouldinEvaluation

Davies-Bouldin criterion clustering evaluation object

Description

DaviesBouldinEvaluation is an object consisting of sample data (X), clustering data (OptimalY), and Davies-Bouldin criterion values (CriterionValues) used to evaluate the optimal number of clusters (OptimalK). The Davies-Bouldin criterion is based on a ratio of within-cluster and between-cluster distances. The optimal clustering solution has the smallest Davies-Bouldin index value. For more information, see Davies-Bouldin Criterion.

Creation

Create a Davies-Bouldin criterion clustering evaluation object by using the evalclusters function and specifying the criterion as "DaviesBouldin".

You can then use compact to create a compact version of the Davies-Bouldin criterion clustering evaluation object. The function removes the contents of the properties X, OptimalY, and Missing.

Properties

Clustering Evaluation Properties

ClusteringFunction — Clustering algorithm

'kmeans' | 'linkage' | 'gmdistribution' | function handle | []

This property is read-only.

Clustering algorithm used to cluster the sample data, returned as 'kmeans', 'linkage', 'gmdistribution', or a function handle. If you specify the clustering solutions as an input argument to evalclusters when you create the clustering evaluation object, then ClusteringFunction is empty.

Value	Description
'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.

Data Types: double | char | function_handle

CriterionName — Name of criterion

'DaviesBouldin'

This property is read-only.

Name of the criterion used for clustering evaluation, returned as 'DaviesBouldin'.

CriterionValues — Criterion values

numeric vector

This property is read-only.

Criterion values, returned as a numeric vector. Each value corresponds to a proposed number of clusters in InspectedK.

Data Types: double

InspectedK — List of number of proposed clusters

positive integer vector

This property is read-only.

List of the number of proposed clusters for which to compute criterion values, returned as a positive integer vector.

Data Types: double

OptimalK — Optimal number of clusters

positive integer scalar

This property is read-only.

Optimal number of clusters, returned as a positive integer scalar.

Data Types: double

OptimalY — Optimal clustering solution

positive integer column vector | []

This property is read-only.

Optimal clustering solution corresponding to OptimalK, returned as a positive integer column vector. Each row of OptimalY represents the cluster index of the corresponding observation (or row) in X. If you specify the clustering solutions as an input argument to evalclusters when you create the clustering evaluation object, or if the clustering evaluation object is compact (see compact), then OptimalY is empty.

Data Types: double

Sample Data Properties

Missing — Excluded data

logical column vector | []

This property is read-only.

Excluded data, returned as a logical column vector. If an element of Missing is true, then the corresponding observation (or row) in the data matrix X is not used in the clustering solutions. If the clustering evaluation object is compact (see compact), then Missing is empty.

Data Types: double | logical

NumObservations — Number of observations

positive integer scalar

This property is read-only.

Number of observations in the data matrix X, ignoring observations with missing (NaN) values, returned as a positive integer scalar.

Data Types: double

X — Data used for clustering

numeric matrix | []

This property is read-only.

Data used for clustering, returned as a numeric matrix. Rows correspond to observations, and columns correspond to variables. If the clustering evaluation object is compact (see compact), then X is empty.

Data Types: single | double

Object Functions

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

Examples

Evaluate Clustering Solution Using Davies-Bouldin Criterion

Evaluate the optimal number of clusters using the Davies-Bouldin clustering evaluation criterion.

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

rng("default") % For reproducibility
n = 200;

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];

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

Evaluate the optimal number of clusters using the Davies-Bouldin criterion. Cluster the data using kmeans.

evaluation = evalclusters(X,"kmeans","DaviesBouldin","KList",1:6)

evaluation = 
  DaviesBouldinEvaluation with properties:

    NumObservations: 600
         InspectedK: [1 2 3 4 5 6]
    CriterionValues: [NaN 0.4663 0.4454 0.8316 1.0444 0.9236]
           OptimalK: 3

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

Plot the Davies-Bouldin criterion values for each number of clusters tested.

plot(evaluation)

The plot shows that the lowest Davies-Bouldin 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.

clusters = evaluation.OptimalY;
gscatter(X(:,1),X(:,2),clusters,[],"xod")

The plot shows three distinct clusters within the data: cluster 1 in the lower-left corner, cluster 2 in the upper-right corner, and cluster 3 near the center of the plot.

More About

Davies-Bouldin Criterion

The Davies-Bouldin criterion is based on a ratio of within-cluster and between-cluster distances. The Davies-Bouldin index is defined as

$$DB=\frac{1}{k}{\displaystyle \sum _{i=1}^k{\max }_{j\ne i}\left\{D_{i,j}\right\},}$$

where D_i,j is the within-to-between cluster distance ratio for the ith and jth clusters.

In mathematical terms,

$$D_{i,j}=\frac{\left({\overline{d}}_i+{\overline{d}}_j\right)}{d_{i,j}}.$$

$${\overline{d}}_i$$ is the average distance between each point in the ith cluster and the centroid of the ith cluster. $${\overline{d}}_j$$ is the average distance between each point in the jth cluster and the centroid of the jth cluster. $$d_{i,j}$$ is the Euclidean distance between the centroids of the ith and jth clusters.

The maximum value of D_i,j represents the worst-case within-to-between cluster ratio for cluster i. The optimal clustering solution has the smallest Davies-Bouldin index value.

References

[1] Davies, D. L., and D. W. Bouldin. “A Cluster Separation Measure.” IEEE Transactions on Pattern Analysis and Machine Intelligence. Vol. PAMI-1, No. 2, 1979, pp. 224–227.

Version History

Introduced in R2013b