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Data clustering is a common technique for statistical data analysis, which is used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics. Clustering is the classification of similar objects into different groups, or more precisely, the partitioning of a data set into subsets (clusters), so that the data in each subset (ideally) share some common trait - often proximity according to some defined distance measure. Machine learning typically regards data clustering as a form of unsupervised learning. Besides the term data clustering (or just clustering), there are a number of terms with similar meanings, including cluster analysis, automatic classification, numerical taxonomy, botryology and typological analysis. Types of clustering Data clustering algorithms can be hierarchical or partitional. Hierarchical algorithms find successive clusters using previously established clusters, whereas partitional algorithms determine all clusters at once. Hierarchical algorithms can be agglomerative (bottom-up) or divisive (top-down). Agglomerative algorithms begin with each element as a separate cluster and merge them in successively larger clusters. Divisive algorithms begin with the whole set and proceed to divide it into successively smaller clusters. Two-way clustering, co-clustering or bi-clustering are the names for clusterings where not only the objects are clustered but also the features of the objects, i.e., if the data is represented in a data matrix the row and columns are clustered simultaneously. Distance measure A key step in a hierarchical clustering is to select a distance measure. A simple measure is manhattan distance, equal to the sum of absolute distances for each variable. The name comes from the fact that in a two-variable case, the variables can be plotted on a grid that can be compared to city streets, and the distance between two points is the number of blocks a person would walk. A more common measure is Euclidean distance, computed by finding the square of the distance between each variable, summing the squares, and finding the square root of that sum. In the two-variable case, the distance is analogous to finding the length of the hypotenuse in a triangle; that is, it is the distance "as the crow flies." A review of cluster analysis in health psychology research found that the most common distance measure in published studies in that research area is the Euclidean distance or the squared Euclidean distance. Creating clusters Given a distance measure, elements can be combined. Hierarchical clustering builds (agglomerative), or breaks up (divisive), a hierarchy of clusters. The traditional representation of this hierarchy is a tree data structure (called a dendrogram), with individual elements at one end and a single cluster with every element at the other. Agglomerative algorithms begin at the top of the tree, whereas divisive algorithms begin at the bottom. (In the figure, the arrows indicate an agglomerative clustering.) Cutting the tree at a given height will give a clustering at a selected precision. In the following example, cutting after the second row will yield clusters . Cutting after the third row will yield clusters , which is a coarser clustering, with a fewer number of larger clusters. Agglomerative hierarchical clustering
K-means clustering The K-means algorithm assigns each point to the cluster whose center (also called centroid) is nearest. The center is the average of all the points in the cluster — that is, its coordinates are the arithmetic mean for each dimension separately over all the points in the cluster. Example: The data set has three dimensions and the cluster has two points: X = (x1, x2, x3) and Y = (y1, y2, y3). Then the centroid Z becomes Z = (z1, z2, z3), where z1 = (x1 + y1)/2 and z2 = (x2 + y2)/2 and z3 = (x3 + y3)/2. The algorithm is roughly (J. MacQueen, 1967): The main advantages of this algorithm are its simplicity and speed which allows it to run on large datasets. Its disadvantage is that it does not yield the same result with each run, since the resulting clusters depend on the initial random assignments. It maximizes inter-cluster (or minimizes intra-cluster) variance, but does not ensure that the result has a global minimum of variance. QT Clust algorithm QT (Quality Threshold) Clustering (Heyer et al, 1999) is an alternative method of partitioning data, invented for gene clustering. It requires more computing power than k-means, but does not require specifying the number of clusters a priori, and always returns the same result when run several times. The algorithm is: The distance between a point and a group of points is computed using complete linkage, i.e. as the maximum distance from the point to any member of the group (see the "Agglomerative hierarchical clustering" section about distance between clusters). Fuzzy c-means clustering In fuzzy clustering, each point has a degree of belonging to clusters, as in fuzzy logic, rather than belonging completely to just one cluster. Thus, points on the edge of a cluster, may be in the cluster to a lesser degree than points in the center of cluster. For each point x we have a coefficient giving the degree of being in the kth cluster . Usually, the sum of those coefficients is defined to be 1, so that denotes a probability of belonging to a certain cluster: With fuzzy c-means, the centroid of a cluster is the mean of all points, weighted by their degree of belonging to the cluster: The degree of belonging is related to the inverse of the distance to the cluster then the coefficients are normalized and fuzzyfied with a real parameter so that their sum is 1. So For m equal to 2, this is equivalent to normalising the coefficient linearly to make their sum 1. When m is close to 1, then cluster center closest to the point is given much more weight than the others, and the algorithm is similar to k-means. The fuzzy c-means algorithm is very similar to the k-means algorithm: The algorithm minimizes intra-cluster variance as well, but has the same problems as k-means, the minimum is a local minimum, and the results depend on the initial choice of weights. Graph-theoretic methods Formal concept analysis is a technique for generating clusters of objects and attributes, given a bipartite graph representing the relations between the objects and attributes. Other methods for generating overlapping clusters (a cover rather than a partition) are discussed by Jardine and Sibson (1968) and Cole and Wishart (1970). Elbow criterion
Spectral clustering Given a set of data points A, the similarity matrix may be defined as a matrix where represents a measure of the similarity between points . Spectral clustering techniques make use of the spectrum of the similarity matrix of the data to cluster the points. Sometimes such techniques are also used to perform dimensionality reduction for clustering in fewer dimensions. One such technique is the Shi-Malik algorithm, commonly used for image segmentation. It partitions points into two sets based on the eigenvector corresponding to the second-smallest eigenvalue of the Laplacian matrix of , where is the diagonal matrix This partitioning may be done in various ways, such as by taking the median of the components in , and placing all points whose component in is greater than in , and the rest in . The algorithm can be used for hierarchical clustering, by repeatedly partitioning the subsets in this fashion. A related algorithm is the Meila-Shi algorithm, which takes the eigenvectors corresponding to the k largest eigenvalues of the matrix for some k, and then invokes another (e.g. k-means) to cluster points by their respective k components in these eigenvectors. Biology In biology clustering has many applications Market research Cluster analysis is widely used in market research when working with multivariate data from surveys and test panels. Market researchers use cluster analysis to partition the general population of consumers into market segments and to better understand the relationships between different groups of consumers/potential customers. experimental techniques) Other applications Social network analysis: In the study of social networks, clustering may be used to recognize communities within large groups of people. Image segmentation: Clustering can be used to divide a digital image into distinct regions for border detection or object recognition. Data mining: Many data mining applications involve partitioning data items into related subsets; the marketing applications discussed above represent some examples. Another common application is the division of documents, such as World Wide Web pages, into genres. Slippy map optimisation: Flickr's map of photos and other map sites use clustering to reduce the number of markers on a map. This makes it both faster and reduces the amount of visual clutter. Comparisons between data clusterings There have been several suggestions for a measure of similarity between two clusterings. Such a measure can be used to compare how well different data clustering algorithms perform on a set of data. Many of these measures are derived from the matching matrix (aka confusion matrix), e.g., the Rand measure and the Fowlkes-Mallows Bk measures. See also Bibliography For spectral clustering For estimating number of clusters: For discussion of the elbow criterion: Free Model Based Cluster And Discriminant Analysis. Code in C++, interface with Matlab and Scilab Weka contains tools for data pre-processing, classification, regression, clustering, association rules, and visualization. It is also well-suited for developing new machine learning schemes. a free data mining software including several clustering algorithms such as K-MEANS, SOM, Clustering Tree, HAC and more. Non-free | |||||||||||
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