CalinskiHarabaszEvaluation

Package: clustering.evaluation
Superclasses: ClusterCriterion

Calinski-Harabasz criterion clustering evaluation object

Description

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

Construction

eva = evalclusters(x,clust,'CalinskiHarabasz') creates a Calinski-Harabasz criterion clustering evaluation object.

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

Input Arguments

x — Input data

matrix

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 'CalinskiHarabasz', 'DaviesBouldin', or 'silhouette', you can specify a clustering algorithm using a function handle. 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',[1:6] specifies to test 1, 2, 3, 4, 5, and 6 clusters to find the optimal number.

'KList' — List of number of clusters to evaluate

vector

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.
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.
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

Evaluate the Clustering Solution Using Calinski-Harabasz Criterion

Evaluate the optimal number of clusters using the Calinski-Harabasz clustering evaluation criterion.

Load the sample data.

load fisheriris;

The data contains length and width measurements from the sepals and petals of three species of iris flowers.

Evaluate the optimal number of clusters using the Calinski-Harabasz criterion. Cluster the data using kmeans.

rng('default');  % For reproducibility
eva = evalclusters(meas,'kmeans','CalinskiHarabasz','KList',[1:6])

eva = 
  CalinskiHarabaszEvaluation with properties:

    NumObservations: 150
         InspectedK: [1 2 3 4 5 6]
    CriterionValues: [Inf 513.9245 561.6278 530.4871 456.1279 469.5068]
           OptimalK: 1

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

Plot the Calinski-Harabasz criterion values for each number of clusters tested.

figure;
plot(eva);

The plot shows that the highest Calinski-Harabasz value occurs at three clusters, suggesting that the optimal number of clusters is three.

Create a grouped scatter plot to examine the relationship between petal length and width. Group the data by suggested clusters.

PetalLength = meas(:,3);
PetalWidth = meas(:,4);
ClusterGroup = eva.OptimalY;
figure;
gscatter(PetalLength,PetalWidth,ClusterGroup,'rbg','xod');

The plot shows cluster 3 in the lower-left corner, completely separated from the other two clusters. Cluster 3 contains flowers with the smallest petal widths and lengths. Cluster 1 is in the upper-right corner, and contains flowers with the largest petal widths and lengths. Cluster 2 is near the center of the plot, and contains flowers with measurements between these two extremes.

More About

Calinski-Harabasz Criterion

The Calinski-Harabasz criterion is sometimes called the variance ratio criterion (VRC). The Calinski-Harabasz index is defined as

$$VR{C}_k=\frac{S{S}_B}{S{S}_W}\times \frac{\left(N-k\right)}{\left(k-1\right)},$$

where SS_B is the overall between-cluster variance, SS_W is the overall within-cluster variance, k is the number of clusters, and N is the number of observations.

The overall between-cluster variance SS_B is defined as

$$S{S}_B={\displaystyle \sum _{i=1}^k{n}_i{\Vert m_i-m\Vert }^2},$$

where k is the number of clusters, n_i is the number of observations in cluster i, m_i is the centroid of cluster i, m is the overall mean of the sample data, and $$\Vert m_i-m\Vert$$ is the L² norm (Euclidean distance) between the two vectors.

The overall within-cluster variance SS_W is defined as

$$S{S}_W={\displaystyle \sum _{i=1}^k{{\displaystyle \sum _{x\in c_i}\Vert x-m_i\Vert }}^2,}$$

where k is the number of clusters, x is a data point, c_i is the ith cluster, m_i is the centroid of cluster i, and $$\Vert x-m_i\Vert$$ is the L² norm (Euclidean distance) between the two vectors.

Well-defined clusters have a large between-cluster variance (SS_B) and a small within-cluster variance (SS_W). The larger the VRC_k ratio, the better the data partition. To determine the optimal number of clusters, maximize VRC_k with respect to k. The optimal number of clusters is the solution with the highest Calinski-Harabasz index value.

The Calinski-Harabasz criterion is best suited for k-means clustering solutions with squared Euclidean distances.

References

[1] Calinski, T., and J. Harabasz. “A dendrite method for cluster analysis.” Communications in Statistics. Vol. 3, No. 1, 1974, pp. 1–27.