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What to Do When K -Means Clustering Fails: A Simple yet Principled Alternative Algorithm

View Article: PubMed Central - PubMed

ABSTRACT

The K-means algorithm is one of the most popular clustering algorithms in current use as it is relatively fast yet simple to understand and deploy in practice. Nevertheless, its use entails certain restrictive assumptions about the data, the negative consequences of which are not always immediately apparent, as we demonstrate. While more flexible algorithms have been developed, their widespread use has been hindered by their computational and technical complexity. Motivated by these considerations, we present a flexible alternative to K-means that relaxes most of the assumptions, whilst remaining almost as fast and simple. This novel algorithm which we call MAP-DP (maximum a-posteriori Dirichlet process mixtures), is statistically rigorous as it is based on nonparametric Bayesian Dirichlet process mixture modeling. This approach allows us to overcome most of the limitations imposed by K-means. The number of clusters K is estimated from the data instead of being fixed a-priori as in K-means. In addition, while K-means is restricted to continuous data, the MAP-DP framework can be applied to many kinds of data, for example, binary, count or ordinal data. Also, it can efficiently separate outliers from the data. This additional flexibility does not incur a significant computational overhead compared to K-means with MAP-DP convergence typically achieved in the order of seconds for many practical problems. Finally, in contrast to K-means, since the algorithm is based on an underlying statistical model, the MAP-DP framework can deal with missing data and enables model testing such as cross validation in a principled way. We demonstrate the simplicity and effectiveness of this algorithm on the health informatics problem of clinical sub-typing in a cluster of diseases known as parkinsonism.

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Clustering solution obtained by K-means and MAP-DP for synthetic elliptical Gaussian data.All clusters share exactly the same volume and density, but one is rotated relative to the others. There is no appreciable overlap. K-means fails because the objective function which it attempts to minimize measures the true clustering solution as worse than the manifestly poor solution shown here.
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pone.0162259.g002: Clustering solution obtained by K-means and MAP-DP for synthetic elliptical Gaussian data.All clusters share exactly the same volume and density, but one is rotated relative to the others. There is no appreciable overlap. K-means fails because the objective function which it attempts to minimize measures the true clustering solution as worse than the manifestly poor solution shown here.

Mentions: By use of the Euclidean distance (algorithm line 9) K-means treats the data space as isotropic (distances unchanged by translations and rotations). This means that data points in each cluster are modeled as lying within a sphere around the cluster centroid. A sphere has the same radius in each dimension. Furthermore, as clusters are modeled only by the position of their centroids, K-means implicitly assumes all clusters have the same radius. When this implicit equal-radius, spherical assumption is violated, K-means can behave in a non-intuitive way, even when clusters are very clearly identifiable by eye (see Figs 1 and 2 and discussion in Sections 5.1, 5.4).


What to Do When K -Means Clustering Fails: A Simple yet Principled Alternative Algorithm
Clustering solution obtained by K-means and MAP-DP for synthetic elliptical Gaussian data.All clusters share exactly the same volume and density, but one is rotated relative to the others. There is no appreciable overlap. K-means fails because the objective function which it attempts to minimize measures the true clustering solution as worse than the manifestly poor solution shown here.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5036949&req=5

pone.0162259.g002: Clustering solution obtained by K-means and MAP-DP for synthetic elliptical Gaussian data.All clusters share exactly the same volume and density, but one is rotated relative to the others. There is no appreciable overlap. K-means fails because the objective function which it attempts to minimize measures the true clustering solution as worse than the manifestly poor solution shown here.
Mentions: By use of the Euclidean distance (algorithm line 9) K-means treats the data space as isotropic (distances unchanged by translations and rotations). This means that data points in each cluster are modeled as lying within a sphere around the cluster centroid. A sphere has the same radius in each dimension. Furthermore, as clusters are modeled only by the position of their centroids, K-means implicitly assumes all clusters have the same radius. When this implicit equal-radius, spherical assumption is violated, K-means can behave in a non-intuitive way, even when clusters are very clearly identifiable by eye (see Figs 1 and 2 and discussion in Sections 5.1, 5.4).

View Article: PubMed Central - PubMed

ABSTRACT

The K-means algorithm is one of the most popular clustering algorithms in current use as it is relatively fast yet simple to understand and deploy in practice. Nevertheless, its use entails certain restrictive assumptions about the data, the negative consequences of which are not always immediately apparent, as we demonstrate. While more flexible algorithms have been developed, their widespread use has been hindered by their computational and technical complexity. Motivated by these considerations, we present a flexible alternative to K-means that relaxes most of the assumptions, whilst remaining almost as fast and simple. This novel algorithm which we call MAP-DP (maximum a-posteriori Dirichlet process mixtures), is statistically rigorous as it is based on nonparametric Bayesian Dirichlet process mixture modeling. This approach allows us to overcome most of the limitations imposed by K-means. The number of clusters K is estimated from the data instead of being fixed a-priori as in K-means. In addition, while K-means is restricted to continuous data, the MAP-DP framework can be applied to many kinds of data, for example, binary, count or ordinal data. Also, it can efficiently separate outliers from the data. This additional flexibility does not incur a significant computational overhead compared to K-means with MAP-DP convergence typically achieved in the order of seconds for many practical problems. Finally, in contrast to K-means, since the algorithm is based on an underlying statistical model, the MAP-DP framework can deal with missing data and enables model testing such as cross validation in a principled way. We demonstrate the simplicity and effectiveness of this algorithm on the health informatics problem of clinical sub-typing in a cluster of diseases known as parkinsonism.

No MeSH data available.