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In data mining, k-means++ [1] [2] is an algorithm for choosing the initial values (or "seeds") for the k-means clustering algorithm. It was proposed in 2007 by David Arthur and Sergei Vassilvitskii, as an approximation algorithm for the NP-hard k-means problem—a way of avoiding the sometimes poor clusterings found by the standard k-means algorithm.
k-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean (cluster centers or cluster centroid), serving as a prototype of the cluster.
The average silhouette of the data is another useful criterion for assessing the natural number of clusters. The silhouette of a data instance is a measure of how closely it is matched to data within its cluster and how loosely it is matched to data of the neighboring cluster, i.e., the cluster whose average distance from the datum is lowest. [8]
Variations of k-means often include such optimizations as choosing the best of multiple runs, but also restricting the centroids to members of the data set (k-medoids), choosing medians (k-medians clustering), choosing the initial centers less randomly (k-means++) or allowing a fuzzy cluster assignment (fuzzy c-means). Most k-means-type ...
This relates directly to the k-median problem which is the problem of finding k centers such that the clusters formed by them are the most compact with respect to the 2-norm. Formally, given a set of data points x, the k centers c i are to be chosen so as to minimize the sum of the distances from each x to the nearest c i. The criterion ...
The equality ((+)) = (()) can also be understood as an equivalence of different counting problems: the number of k-tuples of non-negative integers whose sum is n equals the number of (n + 1)-tuples of non-negative integers whose sum is k − 1, which follows by interchanging the roles of bars and stars in the diagrams representing configurations.
Image credits: Ok-File-6997 Animal shelters in the US had a busy 2023, with over 6.5 million pets entering shelters and rescue organizations. That’s 3.3 million cats and 3.2 million dogs. It was ...
Lloyd's algorithm starts by an initial placement of some number k of point sites in the input domain. In mesh-smoothing applications, these would be the vertices of the mesh to be smoothed; in other applications they may be placed at random or by intersecting a uniform triangular mesh of the appropriate size with the input domain.
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