# Documentation

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## Adjust Fuzzy Overlap in Fuzzy C-Means Clustering

This example shows how to adjust the amount of fuzzy overlap when performing fuzzy c-means clustering.

Create a random data set. For reproducibility, initialize the random number generator to its default value.

```rng('default') data = rand(100,2);```

Specify fuzzy partition matrix exponents.

`M = [1.1 2.0 3.0 4.0];`

The exponent values in `M` must be greater than `1`, with smaller values specifying a lower degree of fuzzy overlap. In other words, as `M` approaches `1`, the boundaries between the clusters become more crisp.

For each overlap exponent:

• Cluster the data.

• Classify each data point into the cluster for which it has the highest degree of membership.

• Find the data points with maximum membership values below `0.6`. These points have a more fuzzy classification.

• Calculate the average maximum membership value across all data points to quantify the degree of fuzzy overlap. A higher average maximum membership value indicates that there is less fuzzy overlap.

• Plot the clustering results.

```for i = 1:4 % Cluster the data. options = [M(i) NaN NaN 0]; [centers,U] = fcm(data,2,options); % Classify the data points. maxU = max(U); index1 = find(U(1,:) == maxU); index2 = find(U(2,:) == maxU); % Find data points with lower maximum membership values. index3 = find(maxU < 0.6); % Calculate the average maximum membership value. averageMax = mean(maxU); % Plot the results. subplot(2,2,i) plot(data(index1,1),data(index1,2),'ob') hold on plot(data(index2,1),data(index2,2),'or') plot(data(index3,1),data(index3,2),'xk','LineWidth',2) plot(centers(1,1),centers(1,2),'xb','MarkerSize',15,'LineWidth',3) plot(centers(2,1),centers(2,2),'xr','MarkerSize',15,'LineWidth',3) hold off title(['M = ' num2str(M(i)) ', Ave. Max. = ' num2str(averageMax,3)]) end```

A given data point is classified into the cluster for which it has the highest membership value, as indicated by `maxU`. A maximum membership value of `0.5` indicates that the point belongs to both clusters equally. The data points marked with a black x have maximum membership values below `0.6`. These points have a greater degree of uncertainty in their cluster membership.

More data points with low maximum membership values indicates a greater degree of fuzzy overlap in the clustering result. The average maximum membership value, `averageMax`, provides a quantitative description of the overlap. An `averageMax` value of `1` indicates completely crisp clusters, with smaller values indicating more overlap.