K-means algorithm

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The k-means algorithm is an algorithm to cluster n objects based on attributes into k partitions, k < n. It is similar to the expectation-maximization algorithm for mixtures of Gaussians in that they both attempt to find the centers of natural clusters in the data. It assumes that the object attributes form a vector space. The objective it tries to achieve is to minimize total intra-cluster variance, or, the squared error function

V = \sum_{i=1}^{k} \sum_{x_j \in S_i} (x_j - \mu_i)^2

where there are k clusters Si, i = 1, 2, ..., k, and µi is the centroid or mean point of all the points xj ∈ Si.

The most common form of the algorithm uses an iterative refinement heuristic known as Lloyd's algorithm. Lloyd's algorithm starts by partitioning the input points into k initial sets, either at random or using some heuristic data. It then calculates the mean point, or centroid, of each set. It constructs a new partition by associating each point with the closest centroid. Then the centroids are recalculated for the new clusters, and algorithm repeated by alternate application of these two steps until convergence, which is obtained when the points no longer switch clusters (or alternatively centroids are no longer changed).

Lloyd's algorithm and k-means are often used synonymously, but in reality Lloyd's algorithm is a heuristic for solving the k-means problem[1], but with certain combinations of starting points and centroids, Lloyd's algorithm can in fact converge to the wrong answer (ie a different and optimal answer to the minimization function above exists.)

Other variations exist[2], but Lloyd's algorithm has remained popular because it converges extremely quickly in practice. In fact, many have observed that the number of iterations is typically much less than the number of points. Recently, however, David Arthur and Sergei Vassilvitskii showed that there exist certain point sets on which k-means takes superpolynomial time: 2Ω(√n) to converge.[3]

Approximate k-means algorithms have been designed that make use of coresets: small subsets of the original data.

In terms of performance the algorithm is not guaranteed to return a global optimum. The quality of the final solution depends largely on the initial set of clusters, and may, in practice, be much poorer than the global optimum.[citation needed] Since the algorithm is extremely fast, a common method is to run the algorithm several times and return the best clustering found.

A drawback of the k-means algorithm is that the number of clusters k is an input parameter. An inappropriate choice of k may yield poor results. The algorithm also assumes that the variance is an appropriate measure of cluster scatter.

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[edit] Demonstration of the algorithm

The following images demonstrate the k-means clustering algorithm in action, for the two-dimensional case. The initial centres are generated randomly to demonstrate the stages in more detail.

Shows the initial randomized centroids and a number of points.

Points are associated with the nearest centroid.

Now the centroids are moved to the center of their respective clusters.

Steps 2 & 3 are repeated until a suitable level of convergence has been reached.

[edit] Applications of the algorithm

[edit] Image Segmentation

The k-means clustering algorithm is commonly used in computer vision as a form of image segmentation. The results of the segmentation are used to aid border detection and object recognition. In this context, the standard euclidean distance is usually insufficient in forming the clusters. Instead, a weighted distance measure utilizing pixel coordinates, RGB pixel color and/or intensity, and image texture is commonly used.[4]

[edit] Relation to PCA

It has been shown recently[5][6] that the relaxed solution of k-means clustering, specified by the cluster indicators, are given by the PCA (principal component analysis) principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace specified by the between-class scatter matrix.

[edit] Enhancements

In 2006 a new way of choosing the initial centers was proposed [1], dubbed "k-means++". The idea is to select centers in a way that they are already initially close to large quantities of points. The authors use L2 norm in selecting the centers, but general Ln may be used to tune the aggressiveness of the seeding.

This seeding method gives out considerable improvements in the final error of k-means. Although the initial selection in the algorithm takes considerable time, the k-means itself converges very fast after this seeding and thus the seeding actually lowers the computation time too. The authors tested their method with real and synthetic datasets and obtained typically 2-fold to 10-fold improvements in speed, and for certain datasets close to 1000-fold improvements in error. Their tests almost always showed the new method to be at least as good as vanilla k-means in both speed and error.

Additionally, the authors calculate an approximation ratio for their algorithm. This is something that has not been done with vanilla k-means (although with several variations of it). The k-means++ guarantees to have approximation ratio O(log(k)) where k is the number of clusters used.

[edit] Variations

The set of squared error minimizing cluster functions also includes the K-medoids algorithm, an approach which forces the center point of each cluster to be one of the actual points.

[edit] References

  • J. B. MacQueen (1967): "Some Methods for classification and Analysis of Multivariate Observations", Proceedings of 5-th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, University of California Press, 1:281-297
  • J. A. Hartigan (1975) "Clustering Algorithms". Wiley.
  1. ^ a b D. Arthur, S. Vassilvitskii: "k-means++ The Advantages of Careful Seeding" 2007 Symposium on Discrete Algorithms (SODA).
  2. ^ An efficient k-means clustering algorithm: Analysis and implementation, T. Kanungo, D. M. Mount, N. Netanyahu, C. Piatko, R. Silverman, and A. Y. Wu, IEEE Trans. Pattern Analysis and Machine Intelligence, 24 (2002), 881-892.
  3. ^ David Arthur & Sergei Vassilvitskii (2006). "How Slow is the k-means Method?". Proceedings of the 2006 Symposium on Computational Geometry (SoCG). 
  4. ^ Shapiro, Linda G. & Stockman, George C. (2001). Computer Vision. Upper Saddle River, NJ: Prentice Hall.
  5. ^ H. Zha, C. Ding, M. Gu, X. He and H.D. Simon. "Spectral Relaxation for K-means Clustering", Neural Information Processing Systems vol.14 (NIPS 2001). pp. 1057-1064, Vancouver, Canada. Dec. 2001.
  6. ^ Chris Ding and Xiaofeng He. "K-means Clustering via Principal Component Analysis". Proc. of Int'l Conf. Machine Learning (ICML 2004), pp 225-232. July 2004.

[edit] External links

[edit] See also