fcm

Fuzzy c-means clustering

Syntax

[centers,U] = fcm(data)

[centers,U] = fcm(data,options)

[centers,U,objFunc] = fcm(___)

Description

[centers,U] = fcm(data) performs fuzzy c-means clustering on the data using default options.

[centers,U] = fcm(data,options) specifies clustering options, including the number of clusters and the clustering exponent.

[centers,U,objFunc] = fcm(___) returns the objective function values at each optimization iteration for all of the previous syntaxes.

Examples

Cluster Data Using Fuzzy C-Means Clustering

Load the data to cluster. Each row of fcmdata contains one data point. The two columns of fcmdata contain the feature values for each data point.

load fcmdata.dat

Specify clustering options using an fcmOptions object. For this example, set the number of clusters to 2 and use default values for the other options.

options = fcmOptions(NumClusters=2);

Find the cluster centers using fuzzy c-means clustering.

[centers,U] = fcm(fcmdata,options);

Iteration count = 1, obj. fcn = 8.970479
Iteration count = 2, obj. fcn = 7.197402
Iteration count = 3, obj. fcn = 6.325579
Iteration count = 4, obj. fcn = 4.586142
Iteration count = 5, obj. fcn = 3.893114
Iteration count = 6, obj. fcn = 3.810804
Iteration count = 7, obj. fcn = 3.799801
Iteration count = 8, obj. fcn = 3.797862
Iteration count = 9, obj. fcn = 3.797508
Iteration count = 10, obj. fcn = 3.797444
Iteration count = 11, obj. fcn = 3.797432
Iteration count = 12, obj. fcn = 3.797430
Minimum improvement reached.

Classify each data point into the cluster with the largest membership value.

maxU = max(U);
index1 = find(U(1,:) == maxU);
index2 = find(U(2,:) == maxU);

Plot the clustered data and cluster centers.

plot(fcmdata(index1,1),fcmdata(index1,2),"ob")
hold on
plot(fcmdata(index2,1),fcmdata(index2,2),"or")
plot(centers(1,1),centers(1,2),"xb",MarkerSize=15,LineWidth=3)
plot(centers(2,1),centers(2,2),"xr",MarkerSize=15,LineWidth=3)
xlabel("Feature 1")
ylabel("Feature 2")
hold off

Specify Fuzzy Overlap Between Clusters

Create a random data set.

data = rand(100,2);

Specify the following FCM clustering options.

• Compute two clusters.

• To increase the amount of fuzzy overlap between the clusters, specify a large fuzzy partition matrix exponent.

• Suppress the command-window display of the objective function values for each iteration.

options = fcmOptions(...
    NumClusters=2,...
    Exponent=3.0,...
    Verbose=false);

Cluster the data.

[centers,U] = fcm(data,options);

Configure Clustering Termination Conditions

Load the clustering data.

load clusterDemo.dat

Configure an options object for computing three clusters and suppress the command-window output of the objective function values. Also, set the clustering termination conditions such that the optimization stops when either of the following occurs:

• The number of iterations reaches a maximum of 50.

• The objective function improves by less than 0.001 between two consecutive iterations.

options = fcmOptions(...
    NumClusters=3,...
    MaxNumIteration=50,...
    MinImprovement=0.001,...
    Verbose=false);

Cluster the data.

[centers,U,objFun] = fcm(clusterDemo,options);

The length of the objective function vector is less than 50; therefore the clustering did not reach the maximum number of iterations.

View the final three values of the objective function vector.

objFun(end-2:end)

ans = 3×1

   15.4353
   15.4306
   15.4305

The optimization stopped because the objective function improved by less than 0.001 between the final two iterations.

Input Arguments

data — Data set to be clustered

matrix

Data set to be clustered, specified as a matrix with N_d rows, where N_d is the number of data points. The number of columns in data is equal to the data dimensionality, that is, the number of features in each data point.

options — Clustering options

fcmOptions object

Clustering options, specified as an fcmOptions object.

Output Arguments

centers — Cluster centers

matrix

Final cluster centers, returned as a matrix with N_c rows containing the coordinates of each cluster center, where N_c is the number of clusters specified using options.NumClusters. The number of columns in centers is equal to the dimensionality of the data being clustered.

U — Fuzzy partition matrix

matrix

Fuzzy partition matrix, returned as an N_c-by-N_d matrix. Element U(i,j) indicates the degree of membership μ_ij of the jth data point in the ith cluster. For a given data point, the sum of the membership values for all clusters is one.

objFunc — Objective function values

vector

Objective function values for each iteration, returned as a vector.

Tips

• To generate a fuzzy inference system using FCM clustering, use the genfis function. For example, suppose that you cluster your data using the following syntax.

  [centers,U] = fcm(data,fcmOpt);

  The first M columns of data correspond to input variables and the remaining columns correspond to output variables.

  You can generate a fuzzy system using the same training data and FCM clustering configuration. To do so:

  1. Configure the clustering options.

     opt = genfisOptions("FCMClustering");
     opt.NumClusters = fcmOpt.NumClusters;
     opt.Exponent = fcmOpt.Exponent;
     opt.MaxNumIteration = fcmOpt.MaxNumIteration;
     opt.MinImprovement = fcmOpt.MinImprovement;
     opt.DistanceMetric = fcmOpt.DistanceMetric;
     opt.Verbose = fcmOpt.Verbose;
     

  2. Extract the input and output variable data.

     inputData = data(:,1:M);
     outputData = data(:,M+1:end);
     

  3. Generate the FIS structure.

     fis = genfis(inputData,outputData,opt);

  The fuzzy system fis contains one fuzzy rule for each cluster, and each input and output variable has one membership function per cluster. For more information, see genfis and genfisOptions.

Algorithms

FCM is a clustering method that allows each data point to belong to multiple clusters with varying degrees of membership. To configure clustering options, create an fcmOptions object.

The FCM algorithm computes cluster centers and membership values to minimize the following objective function.

$$J_m={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^N{\mu }_{ij}^m{D}_{ij}^2}}$$

Here:

• N is the number of data points.

• C is the number of clusters. To specify this value, use the NumClusters option.

• m is fuzzy partition matrix exponent for controlling the degree of fuzzy overlap, with m > 1. Fuzzy overlap refers to how fuzzy the boundaries between clusters are, that is, the number of data points that have significant membership in more than one cluster. To specify the fuzzy partition matrix exponent, use the Exponent option.

• D_ij is the distance from the jth data point to the ith cluster.

• μ_ij is the degree of membership of the jth data point in the ith cluster. For a given data point, the sum of the membership values for all clusters is one.

The fcm function supports two types of FCM clustering: classical FCM and Gustafson-Kessel FCM. These methods differ in the distance metric used for computing D_ij. For more information, see Fuzzy Clustering.

References

[1] Bezdek, James C. Pattern Recognition with Fuzzy Objective Function Algorithms. Boston, MA: Springer US, 1981. https://doi.org/10.1007/978-1-4757-0450-1.

Version History

Introduced before R2006a

R2023a: Gustafson-Kessel FCM algorithm

You can now cluster data using the Gustafson-Kessel FCM algorithm, which allows you to detect clusters with different geometrical shapes within the same data set. This algorithm uses a Mahalanobis distance metric instead of the Euclidean distance metric used in classical FCM clustering.

To use this algorithm, create an fcmOptions object and set the DistanceMetric property to "mahalanobis".

R2023a: Specify options using fcmOptions object

To specify options for clustering data using FCM, you now use an fcmOptions object.

Previously, you specified the number of clusters using an input argument and specified other options in a vector format.

Nc = 3;
exp = 2.5;
maxIter = 200;
minImprove = 1e-4;
verbose = false;
options = [exp maxIter minImprove verbose];
[centers,U] = fcm(data,Nc,options);

Now, you specify these clustering options using an fcmOptions object.

options = fcmOptions(...
    NumClusters=Nc,...
    Exponent=exp,...
    MaxNumIteration=maxIter,...
    MinImprovement=minImprove,...
    Verbose=verbose);
[centers,U] = fcm(data,options);