Gaussian mixture models (GMM) can be seen as the probabilistic counterparts of the k-means clustering algorithm. Weighted k-means takes a set of weighted samples and arranges the centroids according to weighted means of the data clusters, where the weights are the weights of the samples. I wonder if there is a GMM-like probabilistic counterpart of weighted k-means.
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3 Answers
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pKNN+AL (Jain and Kapoor, 2009) is a probabilistic modification of the KNN classifier. Given a set of points $\{x_1, \ldots, x_n\}$ from $\mathbb{R}^d$, labels $\{y_1, \ldots, y_n\}$ from $[1,C]$, and a Mercer kernel $K$, the probability of $x$ belonging to class $c$ is
$$\frac{\frac{1}{n_c} \sum_{\{i : y_i = c\}} K(x, x_i)}{\sum_{t=1}^C \frac{1}{n_t} \sum_{\{i : y_i = c\}} K(x, x_i)}$$
where $n_c$ is the number of $x_i$ that belong to class $c$. It is also an active learning algorithm and comes with a MATLAB implementation.
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$\begingroup$ I think our notion of the term weight is different. The weight $w_i$ of sample $x_i$ in the sense of weighted k-means is a measure for the importance of $x_i$ for the entire clustering. The weights are fixed and given with the data set a priori. Thus the k-means centriods are centered above clusters the samples of which have high weights. If there are clusters with low weights, they are ignored. $\endgroup$– chrivoCommented Mar 26, 2010 at 7:56
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How about Clara, or even PAM? I think Clara is an interesting mix between k-means and PAM.
