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Possibilistic C-means (PCM)

FCM forces the memberships of each sample to sum to one, so an outlier still belongs to some cluster with high membership and drags its center. PCM1 drops that constraint: \(u_{ij}\) is the typicality of sample \(j\) to cluster \(i\), computed independently for each cluster.

\[ u_{ij} = \frac{1}{1 + \left(d_{ij}^2 / \eta_i\right)^{1/(m-1)}} \qquad \eta_i = \frac{\sum_j u_{ij}^m d_{ij}^2}{\sum_j u_{ij}^m} \]

Centers are updated as in FCM. The \(\eta_i\) and the starting centers come from an initial FCM run and stay fixed while PCM iterates, because PCM has no term that keeps the clusters apart and can otherwise collapse onto the same center.

Usage

import numpy as np
from fcmeans import PCM

X = np.random.normal(size=(100, 2))
pcm = PCM(n_clusters=3, random_state=42)
pcm.fit(X)
pcm.centers  # cluster centers
pcm.soft_predict(X)  # typicalities, rows do not sum to one
pcm.predict(X)  # cluster of highest typicality

PCM takes the same parameters as FCM. partition_coefficient and partition_entropy_coefficient raise NotImplementedError, since they assume rows of u sum to one.

  1. Krishnapuram, R., and J. M. Keller. "A possibilistic approach to clustering." IEEE Transactions on Fuzzy Systems 1.2 (1993): 98-110. ↩