Fuzzy C-Means Clustering for Iris Data

This example shows how to use fuzzy c-means clustering for the iris data set. This dataset was collected by botanist Edgar Anderson and contains random samples of flowers belonging to three species of iris flowers: setosa, versicolor, and virginica. For each of the species, the data set contains 50 observations for sepal length, sepal width, petal length, and petal width.

Load Data

Load the data set from the iris.dat data file.

load iris.dat

Partition the data into three groups named setosa, versicolor, and virginica.

setosaIndex = iris(:,5)==1;
versicolorIndex = iris(:,5)==2;
virginicaIndex = iris(:,5)==3;

setosa = iris(setosaIndex,:);
versicolor = iris(versicolorIndex,:);
virginica = iris(virginicaIndex,:);

Plot Data in 2-D

The iris data contains four dimensions representing sepal length, sepal width, petal length, and petal width. Plot the data points for each combination of two dimensions.

characteristics = ["sepal length","sepal width",...
    "petal length","petal width"];
pairs = [1 2; 1 3; 1 4; 2 3; 2 4; 3 4];

for i = 1:6
    x = pairs(i,1); 
    y = pairs(i,2);   
    subplot(2,3,i)
    plot([setosa(:,x) versicolor(:,x) virginica(:,x)],...
         [setosa(:,y) versicolor(:,y) virginica(:,y)],".")
    xlabel(characteristics(x))
    ylabel(characteristics(y))
end

Setup Parameters

Specify the options for clustering the data using fuzzy c-means clustering. These options are:

• Nc — Number of clusters

• M — Fuzzy partition matrix exponent, which indicates the degree of fuzzy overlap between clusters. For more information, see Adjust Fuzzy Overlap in Fuzzy C-Means Clustering.

• maxIter — Maximum number of iterations. The clustering process stops after this number of iterations.

• minImprove — Minimum improvement. The clustering process stops when the objective function improvement between two consecutive iterations is less than this value.

options = fcmOptions(...
    NumClusters=3,...
    Exponent=2.0, ...
    MaxNumIteration=100, ...
    MinImprovement=1e-6);

For more information about these options and the fuzzy c-means algorithm, see fcm and fcmOptions.

Compute Clusters

Fuzzy c-means clustering is an iterative process. Initially, the fcm function generates a random fuzzy partition matrix. This matrix indicates the degree of membership of each data point in each cluster.

In each clustering iteration, fcm calculates the cluster centers and updates the fuzzy partition matrix using the calculated center locations. It then computes the objective function value.

Cluster the data, displaying the objective function value after each iteration.

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

Iteration count = 1, obj. fcn = 28838.424340
Iteration count = 2, obj. fcn = 21010.880067
Iteration count = 3, obj. fcn = 15272.280943
Iteration count = 4, obj. fcn = 11029.756194
Iteration count = 5, obj. fcn = 10550.015503
Iteration count = 6, obj. fcn = 10301.776800
Iteration count = 7, obj. fcn = 9283.793786
Iteration count = 8, obj. fcn = 7344.379868
Iteration count = 9, obj. fcn = 6575.117093
Iteration count = 10, obj. fcn = 6295.215539
Iteration count = 11, obj. fcn = 6167.772051
Iteration count = 12, obj. fcn = 6107.998500
Iteration count = 13, obj. fcn = 6080.461019
Iteration count = 14, obj. fcn = 6068.116247
Iteration count = 15, obj. fcn = 6062.713326
Iteration count = 16, obj. fcn = 6060.390433
Iteration count = 17, obj. fcn = 6059.403978
Iteration count = 18, obj. fcn = 6058.988494
Iteration count = 19, obj. fcn = 6058.814438
Iteration count = 20, obj. fcn = 6058.741777
Iteration count = 21, obj. fcn = 6058.711512
Iteration count = 22, obj. fcn = 6058.698925
Iteration count = 23, obj. fcn = 6058.693695
Iteration count = 24, obj. fcn = 6058.691523
Iteration count = 25, obj. fcn = 6058.690622
Iteration count = 26, obj. fcn = 6058.690247
Iteration count = 27, obj. fcn = 6058.690092
Iteration count = 28, obj. fcn = 6058.690028
Iteration count = 29, obj. fcn = 6058.690001
Iteration count = 30, obj. fcn = 6058.689990
Iteration count = 31, obj. fcn = 6058.689985
Iteration count = 32, obj. fcn = 6058.689983
Iteration count = 33, obj. fcn = 6058.689983
Minimum improvement reached.

The clustering stops when the objective function improvement is below the specified minimum threshold.

Plot the computed cluster centers as bold numbers.

for i = 1:6
    subplot(2,3,i)
    for j = 1:options.NumClusters
        x = pairs(i,1);
        y = pairs(i,2);
        text(centers(j,x),centers(j,y),int2str(j),...
            FontWeight="bold");
    end
end