evalclusters
Evaluate clustering solutions
Description
creates
a clustering evaluation object using additional options specified
by one or more name-value pair arguments.eva
= evalclusters(x
,clust
,criterion
,Name,Value
)
Examples
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: [NaN 513.9245 561.6278 530.4871 456.1279 469.5068] OptimalK: 3 Properties, Methods
The OptimalK
value indicates that, based on the Calinski-Harabasz criterion, the optimal number of clusters is three.
Use an input matrix of proposed clustering solutions to evaluate the optimal number of clusters.
Load the sample data.
load fisheriris;
The data contains length and width measurements from the sepals and petals of three species of iris flowers.
Use kmeans
to create an input matrix of proposed clustering solutions for the sepal length measurements, using 1, 2, 3, 4, 5, and 6 clusters.
clust = zeros(size(meas,1),6); for i=1:6 clust(:,i) = kmeans(meas,i,'emptyaction','singleton',... 'replicate',5); end
Each row of clust
corresponds to one sepal length measurement. Each of the six columns corresponds to a clustering solution containing 1 to 6 clusters.
Evaluate the optimal number of clusters using the Calinski-Harabasz criterion.
eva = evalclusters(meas,clust,'CalinskiHarabasz')
eva = CalinskiHarabaszEvaluation with properties: NumObservations: 150 InspectedK: [1 2 3 4 5 6] CriterionValues: [NaN 513.9245 561.6278 530.4871 456.1279 469.5068] OptimalK: 3 Properties, Methods
The OptimalK
value indicates that, based on the Calinski-Harabasz criterion, the optimal number of clusters is three.
Use a function handle to specify the clustering algorithm, then evaluate the optimal number of clusters.
Load the sample data.
load fisheriris;
The data contains length and width measurements from the sepals and petals of three species of iris flowers.
Use a function handle to specify the clustering algorithm.
myfunc = @(X,K)(kmeans(X,K,Emptyaction="singleton",Replicate=5));
Evaluate the optimal number of clusters for the sepal length data using the Calinski-Harabasz criterion.
eva = evalclusters(meas,myfunc,'CalinskiHarabasz',KList=1:6)
eva = CalinskiHarabaszEvaluation with properties: NumObservations: 150 InspectedK: [1 2 3 4 5 6] CriterionValues: [NaN 513.9245 561.6278 530.4871 456.1279 469.5068] OptimalK: 3 Properties, Methods
The OptimalK
value indicates that, based on the Calinski-Harabasz criterion, the optimal number of clusters is three.
Input Arguments
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. 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 beK
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
Clustering evaluation criterion, specified as one of the following.
'silhouette' | Create a SilhouetteEvaluation cluster evaluation object
containing silhouette values. For more information, see
Silhouette Value and Criterion. |
'DaviesBouldin' | Create a DaviesBouldinEvaluation cluster evaluation
object containing Davies-Bouldin index values. For more
information, see Davies-Bouldin Criterion. |
'CalinskiHarabasz' | Create a CalinskiHarabaszEvaluation clustering
evaluation object containing Calinski-Harabasz index values.
For more information, see Calinski-Harabasz Criterion. |
'gap' | Create a GapEvaluation
cluster evaluation object containing gap criterion values.
For more information, see Gap Value. |
The best choice of cluster evaluation method depends on the characteristics of your data set. Each method uses a different algorithm to derive an evaluation metric.
The silhouette method (Rouseeuw,1987) calculates a score ranging from +1 to –1 for each point in a cluster. The score measures how similar a point is to points in its own cluster, when compared to points in other clusters. The method can use any distance metric. You can use the mean silhouette scores as a numerical metric, or visualize the scores of points in each cluster by creating a
silhouette
plot. Because the method assumes that the clusters have convex shapes, the silhouette metric is not well suited for irregularly shaped clusters.The Davies-Bouldin method (Davies and Bouldin,1979) calculates a single index value that is based on a ratio of within-cluster and between-cluster Euclidean distances. Unlike the silhouette metric, this method makes no assumption about the shape of the clusters. A clustering solution generally has a lower (improved) Davies-Bouldin index value when there is a larger separation between the clusters and a smaller dispersion within the clusters.
The Calinski-Harabasz (CH) method (Calinski and Harabasz,1974) is similar to the Davies-Bouldin method, but instead uses squared Euclidean distances and variance statistics. The CH index is based on the ratio of between-cluster distance variance to within-cluster distance variance, normalized by degrees of freedom. The method makes no assumption about the shape of the clusters. A clustering solution generally has a higher (improved) CH index value when there is a larger separation between the clusters and a smaller dispersion within the clusters.
The gap value method (Tibshirani, Walther, and Hastie, 2001) calculates a single metric value by comparing a clustering solution to a simulated reference distribution that has the characteristics of the input points but lacks any clusters. Given a set of clustering solutions for the same data set, each with a different number of clusters k, the optimal solution has the highest gap value. The method can use any distance metric. The gap value method is more computationally expensive than other clustering evaluation methods, because the clustering algorithm must be applied to the reference data for each proposed clustering solution.
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: evalclusters(x,"kmeans","gap",KList=1:5,Distance="cityblock")
specifies to test 1, 2, 3, 4, and 5 clusters using the city block distance
metric.
For All Criteria
List of number of clusters to evaluate, specified as 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
For Silhouette and Gap
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. 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 specifyDistance
as the output vector created by the functionpdist
.When
clust
is'kmeans'
or'gmdistribution'
,evalclusters
uses the distance metric specified forDistance
to cluster the data.If
clust
is'linkage'
, andDistance
is either'sqEuclidean'
or'Euclidean'
, then the clustering algorithm uses the Euclidean distance and Ward linkage.If
clust
is'linkage'
andDistance
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
For Silhouette Only
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'
For Gap Only
Number of reference data sets generated from the reference distribution ReferenceDistribution
,
specified as the comma-separated pair consisting of 'B'
and
a positive integer value.
Example: 'B',150
Data Types: single
| double
Reference data generation method, specified as the comma-separated
pair consisting of 'ReferenceDistributions'
and
one of the following.
'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 . |
Example: 'ReferenceDistribution','uniform'
Method for selecting the optimal number of clusters, specified
as the comma-separated pair consisting of 'SearchMethod'
and
one of the following.
'globalMaxSE' |
Evaluate each proposed number of clusters in
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
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. |
Example: 'SearchMethod','globalMaxSE'
Output Arguments
Clustering evaluation data, returned as a clustering evaluation object.
References
[1] 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.
[2] 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.
[3] Calinski, T., and J. Harabasz. “A dendrite method for cluster analysis.” Communications in Statistics. Vol. 3, No. 1, 1974, pp. 1–27.
[4] 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
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