K-means clustering – wikipedia


k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. Futures tradingcharts market quotes k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. Decimal to binary This results in a partitioning of the data space into Voronoi cells.

The problem is computationally difficult ( NP-hard); however, there are efficient heuristic algorithms that are commonly employed and converge quickly to a local optimum. Binary calculator online These are usually similar to the expectation-maximization algorithm for mixtures of Gaussian distributions via an iterative refinement approach employed by both algorithms. Usd price Additionally, they both use cluster centers to model the data; however, k-means clustering tends to find clusters of comparable spatial extent, while the expectation-maximization mechanism allows clusters to have different shapes.

The algorithm has a loose relationship to the k-nearest neighbor classifier, a popular machine learning technique for classification that is often confused with k-means because of the k in the name. Usd to aed conversion One can apply the 1-nearest neighbor classifier on the cluster centers obtained by k-means to classify new data into the existing clusters. Euro to indian rupee exchange rate today This is known as nearest centroid classifier or Rocchio algorithm.

As it is a heuristic algorithm, there is no guarantee that it will converge to the global optimum, and the result may depend on the initial clusters. Futures market history As the algorithm is usually very fast, it is common to run it multiple times with different starting conditions. Cnn money markets futures However, in the worst case, k-means can be very slow to converge: in particular it has been shown that there exist certain point sets, even in 2 dimensions, on which k-means takes exponential time, that is 2 Ω( n), to converge. Cad to usd history [10] These point sets do not seem to arise in practice: this is corroborated by the fact that the smoothed running time of k-means is polynomial. Binary file [11]

The “assignment” step is also referred to as expectation step, the “update step” as maximization step, making this algorithm a variant of the generalized expectation-maximization algorithm. Python commands Complexity [ edit ]

• If k and d (the dimension) are fixed, the problem can be exactly solved in time O ( n d k + 1 ) {\displaystyle O(n^{dk+1})} , where n is the number of entities to be clustered [15]

The running time of Lloyd’s algorithm is often given as O ( n k d i ) {\displaystyle O(nkdi)} , where n is the number of d-dimensional vectors, k the number of clusters and i the number of iterations needed until convergence. Euro to dollar conversion rate today On data that does have a clustering structure, the number of iterations until convergence is often small, and results only improve slightly after the first dozen iterations. Exchange rate usd to aed Lloyd’s algorithm is therefore often considered to be of “linear” complexity in practice.

• Lloyd’s k-means algorithm has polynomial smoothed running time. Usd to euro graph It is shown that [11] for arbitrary set of n points in [ 0 , 1 ] d {\displaystyle [0,1]^{d}} , if each point is independently perturbed by a normal distribution with mean 0 and variance σ 2 {\displaystyle \sigma ^{2}} , then the expected running time of k-means algorithm is bounded by O ( n 34 k 34 d 8 log 4 ⁡ ( n ) / σ 6 ) {\displaystyle O(n^{34}k^{34}d^{8}\log ^{4}(n)/\sigma ^{6})} , which is a polynomial in n, k, d and 1 / σ {\displaystyle 1/\sigma } .

• Better bounds are proved for simple cases. Usd to inr exchange rate today For example, [16] showed that the running time of k-means algorithm is bounded by O ( d n 4 M 2 ) {\displaystyle O(dn^{4}M^{2})} for n points in an integer lattice { 1 , … , M } d {\displaystyle \{1,\dots ,M\}^{d}} .

Lloyd’s algorithm is the standard approach for this problem, However, it spends a lot of processing time computing the distances between each of the k cluster centers and the n data points. Exchange rate cny to usd Since points usually stay in the same clusters after a few iterations, much of this work is unnecessary, making the naive implementation very inefficient. Inr to usd conversion rate today Some implementations use the triangle inequality in order to create bounds and accelerate Lloyd’s algorithm. Convert usd to euro calculator [17] [18] [19] Variations [ edit ]

• k-medoids (also: Partitioning Around Medoids, PAM) uses the medoid instead of the mean, and this way minimizes the sum of distances for arbitrary distance functions.

• Gaussian mixture models trained with expectation-maximization algorithm (EM algorithm) maintains probabilistic assignments to clusters, instead of deterministic assignments, and multivariate Gaussian distributions instead of means.

• Hierarchical variants such as Bisecting k-means, [24] X-means clustering [25] and G-means clustering [26] repeatedly split clusters to build a hierarchy, and can also try to automatically determine the optimal number of clusters in a dataset.

• Minkowski weighted k-means automatically calculates cluster specific feature weights, supporting the intuitive idea that a feature may have different degrees of relevance at different features. Premarket stock futures cnn [27] These weights can also be used to re-scale a given data set, increasing the likelihood of a cluster validity index to be optimized at the expected number of clusters. Gold manufacturing [28]

However, the pure k-means algorithm is not very flexible, and as such is of limited use (except for when vector quantization as above is actually the desired use case!). 1 usd to sbd In particular, the parameter k is known to be hard to choose (as discussed above) when not given by external constraints. Non binary gender quiz Another limitation of the algorithm is that it cannot be used with arbitrary distance functions or on non-numerical data. Gender spectrum For these use cases, many other algorithms have been developed since. Joy news Feature learning [ edit ]

k-means clustering has been used as a feature learning (or dictionary learning) step, in either ( semi-) supervised learning or unsupervised learning. Gold price today [32] The basic approach is first to train a k-means clustering representation, using the input training data (which need not be labelled). What is a futures market Then, to project any input datum into the new feature space, we have a choice of “encoding” functions, but we can use for example the thresholded matrix-product of the datum with the centroid locations, the distance from the datum to each centroid, or simply an indicator function for the nearest centroid, [32] [33] or some smooth transformation of the distance. Math puzzles printable [34] Alternatively, by transforming the sample-cluster distance through a Gaussian RBF, one effectively obtains the hidden layer of a radial basis function network. Investing futures market [35]

This use of k-means has been successfully combined with simple, linear classifiers for semi-supervised learning in NLP (specifically for named entity recognition) [36] and in computer vision. Binary solver On an object recognition task, it was found to exhibit comparable performance with more sophisticated feature learning approaches such as autoencoders and restricted Boltzmann machines. Dollar euro forecast [34] However, it generally requires more data than the sophisticated methods, for equivalent performance, because each data point only contributes to one “feature” rather than multiple. Us canadian dollar exchange rate [32] Relation to other statistical machine learning algorithms [ edit ] Gaussian Mixture Model [ edit ]

k-means clustering, and its associated expectation-maximization algorithm, is a special case of a Gaussian mixture model, specifically, the limit of taking all covariances as diagonal, equal, and small. What is futures and options in stock market with examples It is often easy to generalize a k-means problem into a Gaussian mixture model. Futures markets today [37] Another generalization of the k-means algorithm is the K-SVD algorithm, which estimates data points as a sparse linear combination of “codebook vectors”. Convert malawi kwacha to usd K-means corresponds to the special case of using a single codebook vector, with a weight of 1. Exchange rate uk to us dollar [38] Principal component analysis (PCA) [ edit ]

It was proved [39] [40] that the relaxed solution of k-means clustering, specified by the cluster indicators, is given by principal component analysis (PCA), and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. Dollar to euro chart The intuition is that k-means describe spherically shaped (ball-like) clusters. Binary joke If the data have 2 clusters, the line connecting the two centroids is the best 1-dimensional projection direction, which is also the 1st PCA direction. Usd graduate programs Cutting the line at the center of mass separate the clusters (this is the continuous relaxation of the discreet cluster indicator). Convert usd to hkd If the data have 3 clusters, the 2-dimensional plane spanned by 3 cluster centroids is the best 2-D projection. Stock meaning in tagalog This plane is also the first 2 PCA dimensions. How to read binary code Well-separated clusters are effectively modeled by ball-shape clusters and thus discovered by K-means. Commodity futures intraday market Non-ball-shaped clusters are hard to separate when they are close-by. Marriott travel agent rates For example, two half-moon shaped clusters intertwined in space does not separate well when projected to PCA subspace. Mortgage meaning in hindi But neither is k-means supposed to do well on this data. Usd to canadian However, that PCA is a useful relaxation of k-means clustering was not a new result, [41] and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions. Exchange rate inr to usd [42] Mean shift clustering [ edit ]

Basic mean shift clustering algorithms maintain a set of data points the same size as the input data set. Math jokes for teachers Initially, this set is copied from the input set. Exchange rate us dollar to pound sterling Then this set is iteratively replaced by the mean of those points in the set that are within a given distance of that point. Usd to rmb exchange rate history By contrast, k-means restricts this updated set to k points usually much less than the number of points in the input data set, and replaces each point in this set by the mean of all points in the input set that are closer to that point than any other (e.g. Exchange rate british pound to us dollar within the Voronoi partition of each updating point). Xauusd forecast A mean shift algorithm that is similar then to k-means, called likelihood mean shift, replaces the set of points undergoing replacement by the mean of all points in the input set that are within a given distance of the changing set. Usps shipping calculator [43] One of the advantages of mean shift over k-means is that there is no need to choose the number of clusters, because mean shift is likely to find only a few clusters if indeed only a small number exist. Binary table However, mean shift can be much slower than k-means, and still requires selection of a bandwidth parameter. Binary code chart Mean shift has soft variants much as k-means does. List of binary numbers Independent component analysis (ICA) [ edit ]

It has been shown in [44] that under sparsity assumptions and when input data is pre-processed with the whitening transformation k-means produces the solution to the linear Independent component analysis task. Binary sunset piano sheet music This aids in explaining the successful application of k-means to feature learning. Binary search tree java code Bilateral filtering [ edit ]

k-means implicitly assumes that the ordering of the input data set does not matter. Usa today games The bilateral filter is similar to K-means and mean shift in that it maintains a set of data points that are iteratively replaced by means. The box tops However, the bilateral filter restricts the calculation of the (kernel weighted) mean to include only points that are close in the ordering of the input data. Usd jpy chart live [43] This makes it applicable to problems such as image denoising, where the spatial arrangement of pixels in an image is of critical importance. Chf to usd chart Similar problems [ edit ]

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, i.e., it uses medoids in place of centroids. Usd myr Software implementations [ edit ]

Different implementations of the same algorithm were found to exhibit enormous performance differences, with the fastest on a test data set finishing in 10 seconds, the slowest taking 25988 seconds. Exchange rate usd [45] The differences can be attributed to implementation quality, language and compiler differences, and the use of indexes for acceleration. Ip address to binary Free Software/Open Source [ edit ]

• ELKI contains k-means (with Lloyd and MacQueen iteration, along with different initializations such as k-means++ initialization) and various more advanced clustering algorithms.