GapEvaluation

Gap criterion clustering evaluation object

Description

GapEvaluation is an object consisting of sample data (X), clustering data (OptimalY), and gap criterion values (CriterionValues) used to evaluate the optimal number of clusters (OptimalK). The gap criterion values correspond to the difference ExpectedLogW – LogW, where W is the within-cluster dispersion, ExpectedLogW is determined by Monte Carlo sampling from a reference distribution, and LogW is computed from the sample data. The optimal number of clusters corresponds to the solution with the largest local or global gap value within a tolerance range (SearchMethod). For more information, see Gap Value.

Creation

Create a gap criterion clustering evaluation object by using the evalclusters function and specifying the criterion as "gap".

You can then use compact to create a compact version of the gap 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.

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: char | function_handle

CriterionName — Name of criterion

'Gap'

This property is read-only.

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

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

Distance — Distance metric

'sqEuclidean' | 'Euclidean' | 'cityblock' | 'cosine' | 'correlation' | function handle

This property is read-only.

Distance metric used for clustering data and computing the criterion values, returned as one of the values in this table or a function handle.

Value	Description
'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)

Data Types: char | function_handle

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

SearchMethod — Method for selecting optimal number of clusters

'globalMaxSE' | 'firstMaxSE'

This property is read-only.

Method for selecting the optimal number of clusters, returned as 'globalMaxSE' or 'firstMaxSE'.

Value	Description
'globalMaxSE'	Evaluate each proposed number of clusters in InspectedK and select the smallest number of clusters satisfying $$\text{Gap}\left(K\right)\ge GAPMAX-\text{SE}(GAPMAX),$$ where K is the number of clusters, Gap(K) is the gap value for the clustering solution with K clusters, GAPMAX is the largest gap value, and SE(GAPMAX) is the standard error corresponding to the largest gap value.
'firstMaxSE'	Evaluate each proposed number of clusters in InspectedK and select the smallest number of clusters satisfying $$\text{Gap}(K)\ge \text{Gap}(K+1)-\text{SE}(K+1),$$ where K is the number of clusters, Gap(K) is the gap value for the clustering solution with K clusters, and SE(K + 1) is the standard error of the clustering solution with K + 1 clusters.

Sample Data Properties

LogW — Natural logarithm of within-cluster dispersion

numeric vector

This property is read-only.

Natural logarithm of the within-cluster dispersion W based on the sample data X, returned as a numeric vector. W is the within-cluster dispersion computed using the distance metric Distance. Each element of LogW corresponds to a specific number of proposed clusters (an element of InspectedK).

Data Types: double

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

Reference Data Properties

B — Number of reference data sets

positive integer scalar

This property is read-only.

Number of reference data sets generated from the reference distribution ReferenceDistribution, returned as a positive integer scalar.

Data Types: double

ExpectedLogW — Expectation of natural logarithm of within-cluster dispersion

numeric vector

This property is read-only.

Expectation of the natural logarithm of the within-cluster dispersion W based on the generated reference data, returned as a numeric vector. W is the within-cluster dispersion computed using the distance metric Distance. Each element of ExpectedLogW corresponds to a specific number of proposed clusters (an element of InspectedK).

Data Types: double

ReferenceDistribution — Reference data generation method

'PCA' | 'uniform'

This property is read-only.

Reference data generation method, returned as 'PCA' or 'uniform'.

Value	Description
'PCA'	Generate reference data from a uniform distribution over a box aligned with the principal components of the data matrix X.
'uniform'	Generate reference data uniformly over the range of each feature in the data matrix X.

SE — Standard error of natural logarithm of within-cluster dispersion

numeric vector

This property is read-only.

Standard error of the natural logarithm of the within-cluster dispersion W with respect to the reference data, returned as a numeric vector. W is the within-cluster dispersion computed using the distance metric Distance. Each element of SE corresponds to a specific number of proposed clusters (an element of InspectedK).

Data Types: double

StdLogW — Standard deviation of natural logarithm of within-cluster dispersion

numeric vector

This property is read-only.

Standard deviation of the natural logarithm of the within-cluster dispersion W with respect to the reference data, returned as a numeric vector. W is the within-cluster dispersion computed using the distance metric Distance. Each element of StdLogW corresponds to a specific number of proposed clusters (an element of InspectedK).

Data Types: double

Object Functions

addK	Evaluate additional numbers of clusters
compact	Compact clustering evaluation object
increaseB	Increase reference data sets
plot	Plot clustering evaluation object criterion values

Examples

Evaluate Clustering Solution Using Gap Criterion

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

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

load fisheriris

Evaluate the optimal number of clusters based on the gap criterion values. Cluster the data using kmeans.

rng("default") % For reproducibility
evaluation = evalclusters(meas,"kmeans","gap","KList",1:6)

evaluation = 
  GapEvaluation with properties:

    NumObservations: 150
         InspectedK: [1 2 3 4 5 6]
    CriterionValues: [0.0720 0.5928 0.8762 1.0114 1.0534 1.0720]
           OptimalK: 5

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

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

plot(evaluation)

Based on the plot, the maximum value of the gap criterion occurs at six clusters. However, the value at five clusters is within one standard error of the maximum, so the suggested optimal number of clusters is five.

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

PetalLength = meas(:,3);
PetalWidth = meas(:,4);
clusters = evaluation.OptimalY;
gscatter(PetalLength,PetalWidth,clusters,[],"xod^*");

The plot shows cluster 4 in the lower-left corner, completely separated from the other four clusters. Cluster 4 contains flowers with the smallest petal widths and lengths. Cluster 2 is in the upper-right corner, and contains flowers with the largest petal widths and lengths. Cluster 5 is next to cluster 2, and contains flowers with similar petal widths but smaller petal lengths compared to the flowers in cluster 2. Clusters 1 and 3 are near the center of the plot, and contain flowers with measurements between the extremes.

More About

Gap Value

A common graphical approach to clustering evaluation involves plotting an error measurement versus several proposed numbers of clusters, and locating the “elbow” of this plot. The “elbow” occurs at the most dramatic decrease in error measurement. The gap criterion formalizes this approach by estimating the “elbow” location as the number of clusters with the largest gap value. Therefore, under the gap criterion, the optimal number of clusters corresponds to the solution with the largest local or global gap value within a tolerance range.

The gap value is defined as

$$Ga{p}_n\left(k\right)=E_n^{*}\left\{\log \left(W_k\right)\right\}-\log \left(W_k\right),$$

where n is the sample size, k is the number of clusters being evaluated, and W_k is the pooled within-cluster dispersion measurement

$$W_k={\displaystyle \sum _{r=1}^k\frac{1}{2n_r}{D}_r,}$$

where n_r is the number of data points in cluster r, and D_r is the sum of the pairwise distances for all points in cluster r.

The expected value $$E_n^{*}\left\{\log \left(W_k\right)\right\}$$ is determined by Monte Carlo sampling from a reference distribution, and log(Wk) is computed from the sample data.

The gap value is defined even for clustering solutions that contain only one cluster, and can be used with any distance metric. However, the gap criterion is more computationally expensive than other clustering evaluation criteria, because the clustering algorithm must be applied to the reference data for each proposed clustering solution.

References

[1] Tibshirani, R., G. Walther, and T. Hastie. “Estimating the number of clusters in a data set via the gap statistic.” Journal of the Royal Statistical Society: Series B. Vol. 63, Part 2, 2001, pp. 411–423.

Version History

Introduced in R2013b