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Cluster analysis : Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some specific sense defined by the analyst) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. Cluster analysis refers to a family of algorithms and tasks rather than one specific algorithm. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small distances between cluster members, dense areas of the data space, intervals or particular statistical distributions. Clustering can therefore be formulated as a multi-objective optimization problem. The appropriate clustering algorithm and parameter settings (including parameters such as the distance function to use, a density threshold or the number of expected clusters) depend on the individual data set and intended use of the results. Cluster analysis as such is not an automatic task, but an iterative process of knowledge discovery or interactive multi-objective optimization that involves trial and failure. It is often necessary to modify data preprocessing and model parameters until the result achieves the desired properties. Besides the term clustering, there are a number of terms with similar meanings, including automatic classification, numerical taxonomy, botryology (from Greek: βότρυς 'grape'), typological analysis, and community detection. The subtle differences are often in the use of the results: while in data mining, the resulting groups are the matter of interest, in automatic classification the resulting discriminative power is of interest. Cluster analysis originated in anthropology by Driver and Kroeber in 1932 and introduced to psychology by Joseph Zubin in 1938 and Robert Tryon in 1939 and famously used by Cattell beginning in 1943 for trait theory classification in personality psychology.
Cluster analysis : The notion of a "cluster" cannot be precisely defined, which is one of the reasons why there are so many clustering algorithms. There is a common denominator: a group of data objects. However, different researchers employ different cluster models, and for each of these cluster models again different algorithms can be given. The notion of a cluster, as found by different algorithms, varies significantly in its properties. Understanding these "cluster models" is key to understanding the differences between the various algorithms. Typical cluster models include: Connectivity models: for example, hierarchical clustering builds models based on distance connectivity. Centroid models: for example, the k-means algorithm represents each cluster by a single mean vector. Distribution models: clusters are modeled using statistical distributions, such as multivariate normal distributions used by the expectation-maximization algorithm. Density models: for example, DBSCAN and OPTICS defines clusters as connected dense regions in the data space. Subspace models: in biclustering (also known as co-clustering or two-mode-clustering), clusters are modeled with both cluster members and relevant attributes. Group models: some algorithms do not provide a refined model for their results and just provide the grouping information. Graph-based models: a clique, that is, a subset of nodes in a graph such that every two nodes in the subset are connected by an edge can be considered as a prototypical form of cluster. Relaxations of the complete connectivity requirement (a fraction of the edges can be missing) are known as quasi-cliques, as in the HCS clustering algorithm. Signed graph models: Every path in a signed graph has a sign from the product of the signs on the edges. Under the assumptions of balance theory, edges may change sign and result in a bifurcated graph. The weaker "clusterability axiom" (no cycle has exactly one negative edge) yields results with more than two clusters, or subgraphs with only positive edges. Neural models: the most well-known unsupervised neural network is the self-organizing map and these models can usually be characterized as similar to one or more of the above models, and including subspace models when neural networks implement a form of Principal Component Analysis or Independent Component Analysis. A "clustering" is essentially a set of such clusters, usually containing all objects in the data set. Additionally, it may specify the relationship of the clusters to each other, for example, a hierarchy of clusters embedded in each other. Clusterings can be roughly distinguished as: Hard clustering: each object belongs to a cluster or not Soft clustering (also: fuzzy clustering): each object belongs to each cluster to a certain degree (for example, a likelihood of belonging to the cluster) There are also finer distinctions possible, for example: Strict partitioning clustering: each object belongs to exactly one cluster Strict partitioning clustering with outliers: objects can also belong to no cluster; in which case they are considered outliers Overlapping clustering (also: alternative clustering, multi-view clustering): objects may belong to more than one cluster; usually involving hard clusters Hierarchical clustering: objects that belong to a child cluster also belong to the parent cluster Subspace clustering: while an overlapping clustering, within a uniquely defined subspace, clusters are not expected to overlap
Cluster analysis : As listed above, clustering algorithms can be categorized based on their cluster model. The following overview will only list the most prominent examples of clustering algorithms, as there are possibly over 100 published clustering algorithms. Not all provide models for their clusters and can thus not easily be categorized. An overview of algorithms explained in Wikipedia can be found in the list of statistics algorithms. There is no objectively "correct" clustering algorithm, but as it was noted, "clustering is in the eye of the beholder." In fact, an axiomatic approach to clustering demonstrates that it is impossible for any clustering method to meet three fundamental properties simultaneously: scale invariance (results remain unchanged under proportional scaling of distances), richness (all possible partitions of the data can be achieved), and consistency between distances and the clustering structure. The most appropriate clustering algorithm for a particular problem often needs to be chosen experimentally, unless there is a mathematical reason to prefer one cluster model over another. An algorithm that is designed for one kind of model will generally fail on a data set that contains a radically different kind of model. For example, k-means cannot find non-convex clusters. Most traditional clustering methods assume the clusters exhibit a spherical, elliptical or convex shape.
Cluster analysis : Evaluation (or "validation") of clustering results is as difficult as the clustering itself. Popular approaches involve "internal" evaluation, where the clustering is summarized to a single quality score, "external" evaluation, where the clustering is compared to an existing "ground truth" classification, "manual" evaluation by a human expert, and "indirect" evaluation by evaluating the utility of the clustering in its intended application. Internal evaluation measures suffer from the problem that they represent functions that themselves can be seen as a clustering objective. For example, one could cluster the data set by the Silhouette coefficient; except that there is no known efficient algorithm for this. By using such an internal measure for evaluation, one rather compares the similarity of the optimization problems, and not necessarily how useful the clustering is. External evaluation has similar problems: if we have such "ground truth" labels, then we would not need to cluster; and in practical applications we usually do not have such labels. On the other hand, the labels only reflect one possible partitioning of the data set, which does not imply that there does not exist a different, and maybe even better, clustering. Neither of these approaches can therefore ultimately judge the actual quality of a clustering, but this needs human evaluation, which is highly subjective. Nevertheless, such statistics can be quite informative in identifying bad clusterings, but one should not dismiss subjective human evaluation.
Archetypal analysis : Archetypal analysis in statistics is an unsupervised learning method similar to cluster analysis and introduced by Adele Cutler and Leo Breiman in 1994. Rather than "typical" observations (cluster centers), it seeks extremal points in the multidimensional data, the "archetypes". The archetypes are convex combinations of observations chosen so that observations can be approximated by convex combinations of the archetypes.
Archetypal analysis : Adele Cutler and Leo Breiman. Archetypal analysis. Technometrics, 36(4):338–347, November 1994. Manuel J. A. Eugster: Archetypal Analysis, Mining the Extreme. HIIT seminar, Helsinki Institute for Information Technology, 2012 Anil Damle, Yuekai Sun: A geometric approach to archetypal analysis and non-negative matrix factorization. arXiv preprint: arXiv : 1405.4275
Behavioral clustering : Behavioral clustering is a statistical analysis method used in retailing to identify consumer purchase trends and group stores based on consumer buying behaviors.
Behavioral clustering : Erickson, D., and Weber, W. (2009). "Five Pitfalls To Avoid When Clustering". Chain Store Age.: CS1 maint: multiple names: authors list (link)
Behavioral clustering : Behavioural Clustering 101: Here's What You Need To Know Personalize Content on Your Website with Behavioral Clustering A two-step segmentation algorithm for behavioral clustering of naturalistic driving styles
Biclustering : Biclustering, block clustering, Co-clustering or two-mode clustering is a data mining technique which allows simultaneous clustering of the rows and columns of a matrix. The term was first introduced by Boris Mirkin to name a technique introduced many years earlier, in 1972, by John A. Hartigan. Given a set of m samples represented by an n -dimensional feature vector, the entire dataset can be represented as m rows in n columns (i.e., an m × n matrix). The Biclustering algorithm generates Biclusters. A Bicluster is a subset of rows which exhibit similar behavior across a subset of columns, or vice versa.
Biclustering : Biclustering was originally introduced by John A. Hartigan in 1972. The term "Biclustering" was then later used and refined by Boris G. Mirkin. This algorithm was not generalized until 2000, when Y. Cheng and George M. Church proposed a biclustering algorithm based on the mean squared residue score (MSR) and applied it to biological gene expression data. In 2001 and 2003, I. S. Dhillon published two algorithms applying biclustering to files and words. One version was based on bipartite spectral graph partitioning. The other was based on information theory. Dhillon assumed the loss of mutual information during biclustering was equal to the Kullback–Leibler-distance (KL-distance) between P and Q. P represents the distribution of files and feature words before Biclustering, while Q is the distribution after Biclustering. KL-distance is for measuring the difference between two random distributions. KL = 0 when the two distributions are the same and KL increases as the difference increases. Thus, the aim of the algorithm was to find the minimum KL-distance between P and Q. In 2004, Arindam Banerjee used a weighted-Bregman distance instead of KL-distance to design a Biclustering algorithm that was suitable for any kind of matrix, unlike the KL-distance algorithm. To cluster more than two types of objects, in 2005, Bekkerman expanded the mutual information in Dhillon's theorem from a single pair into multiple pairs.
Biclustering : The complexity of the Biclustering problem depends on the exact problem formulation, and particularly on the merit function used to evaluate the quality of a given Bicluster. However, the most interesting variants of this problem are NP-complete. NP-complete has two conditions. In the simple case that there is an only element a(i,j) either 0 or 1 in the binary matrix A, a Bicluster is equal to a biclique in the corresponding bipartite graph. The maximum size Bicluster is equivalent to the maximum edge biclique in the bipartite graph. In the complex case, the element in matrix A is used to compute the quality of a given Bicluster and solve the more restricted version of the problem. It requires either large computational effort or the use of lossy heuristics to short-circuit the calculation.
Biclustering : Bicluster with constant values (a) When a Biclustering algorithm tries to find a constant-value Bicluster, it reorders the rows and columns of the matrix to group together similar rows and columns, eventually grouping Biclusters with similar values. This method is sufficient when the data is normalized. A perfect constant Bicluster is a matrix(I,J) in which all values a(i,j) are equal to a given constant μ. In tangible data, these entries a(i,j) may be represented with the form n(i,j) + μ where n(i,j) denotes the noise. According to Hartigan's algorithm, by splitting the original data matrix into a set of Biclusters, variance is used to compute constant Biclusters. Hence, a perfect Bicluster may be equivalently defined as a matrix with a variance of zero. In order to prevent the partitioning of the data matrix into Biclusters with the only one row and one column; Hartigan assumes that there are, for example, K Biclusters within the data matrix. When the data matrix is partitioned into K Biclusters, the algorithm ends. Bicluster with constant values on rows (b) or columns (c) Unlike the constant-value Biclusters, these types of Biclusters cannot be evaluated solely based on the variance of their values. To finish the identification, the columns and the rows should be normalized first. There are, however, other algorithms, without the normalization step, that can find Biclusters which have rows and columns with different approaches. Bicluster with coherent values (d, e) For Biclusters with coherent values on rows and columns, an overall improvement over the algorithms for Biclusters with constant values on rows or on columns should be considered. This algorithm may contain analysis of variance between groups, using co-variance between both rows and columns. In Cheng and Church's theorem, a Bicluster is defined as a subset of rows and columns with almost the same score. The similarity score is used to measure the coherence of rows and columns. The relationship between these cluster models and other types of clustering such as correlation clustering is discussed in.
Biclustering : There are many Biclustering algorithms developed for bioinformatics, including: block clustering, CTWC (Coupled Two-Way Clustering), ITWC (Interrelated Two-Way Clustering), δ-bicluster, δ-pCluster, δ-pattern, FLOC, OPC, Plaid Model, OPSMs (Order-preserving submatrixes), Gibbs, SAMBA (Statistical-Algorithmic Method for Bicluster Analysis), Robust Biclustering Algorithm (RoBA), Crossing Minimization, cMonkey, PRMs, DCC, LEB (Localize and Extract Biclusters), QUBIC (QUalitative BIClustering), BCCA (Bi-Correlation Clustering Algorithm) BIMAX, ISA and FABIA (Factor analysis for Bicluster Acquisition), runibic, and recently proposed hybrid method EBIC (evolutionary-based Biclustering), which was shown to detect multiple patterns with very high accuracy. More recently, IMMD-CC is proposed that is developed based on the iterative complexity reduction concept. IMMD-CC is able to identify co-cluster centroids from highly sparse transformation obtained by iterative multi-mode discretization. Biclustering algorithms have also been proposed and used in other application fields under the names co-clustering, bi-dimensional clustering, and subspace clustering. Given the known importance of discovering local patterns in time-series data. Recent proposals have addressed the Biclustering problem in the specific case of time-series gene expression data. In this case, the interesting Biclusters can be restricted to those with contiguous columns. This restriction leads to a tractable problem and enables the development of efficient exhaustive enumeration algorithms such as CCC-Biclustering and e-CCC-Biclustering. The approximate patterns in CCC-Biclustering algorithms allow a given number of errors, per gene, relatively to an expression profile representing the expression pattern in the Bicluster. The e-CCC-Biclustering algorithm uses approximate expressions to find and report all maximal CCC-Bicluster's by a discretized matrix A and efficient string processing techniques. These algorithms find and report all maximal Biclusters with coherent and contiguous columns with perfect/approximate expression patterns, in time linear/polynomial which is obtained by manipulating a discretized version of original expression matrix in the size of the time-series gene expression matrix using efficient string processing techniques based on suffix trees. These algorithms are also applied to solve problems and sketch the analysis of computational complexity. Some recent algorithms have attempted to include additional support for Biclustering rectangular matrices in the form of other datatypes, including cMonkey. There is an ongoing debate about how to judge the results of these methods, as Biclustering allows overlap between clusters and some algorithms allow the exclusion of hard-to-reconcile columns/conditions. Not all of the available algorithms are deterministic and the analyst must pay attention to the degree to which results represent stable minima. Because this is an unsupervised classification problem, the lack of a gold standard makes it difficult to spot errors in the results. One approach is to utilize multiple Biclustering algorithms, with the majority or super-majority voting amongst them to decide the best result. Another way is to analyze the quality of shifting and scaling patterns in Biclusters. Biclustering has been used in the domain of text mining (or classification) which is popularly known as co-clustering. Text corpora are represented in a vectoral form as a matrix D whose rows denote the documents and whose columns denote the words in the dictionary. Matrix elements Dij denote occurrence of word j in document i. Co-clustering algorithms are then applied to discover blocks in D that correspond to a group of documents (rows) characterized by a group of words(columns). Text clustering can solve the high-dimensional sparse problem, which means clustering text and words at the same time. When clustering text, we need to think about not only the words information, but also the information of words clusters that was composed by words. Then, according to similarity of feature words in the text, will eventually cluster the feature words. This is called co-clustering. There are two advantages of co-clustering: one is clustering the test based on words clusters can extremely decrease the dimension of clustering, it can also appropriate to measure the distance between the tests. Second is mining more useful information and can get the corresponding information in test clusters and words clusters. This corresponding information can be used to describe the type of texts and words, at the same time, the result of words clustering can be also used to text mining and information retrieval. Several approaches have been proposed based on the information contents of the resulting blocks: matrix-based approaches such as SVD and BVD, and graph-based approaches. Information-theoretic algorithms iteratively assign each row to a cluster of documents and each column to a cluster of words such that the mutual information is maximized. Matrix-based methods focus on the decomposition of matrices into blocks such that the error between the original matrix and the regenerated matrices from the decomposition is minimized. Graph-based methods tend to minimize the cuts between the clusters. Given two groups of documents d1 and d2, the number of cuts can be measured as the number of words that occur in documents of groups d1 and d2. More recently (Bisson and Hussain) have proposed a new approach of using the similarity between words and the similarity between documents to co-cluster the matrix. Their method (known as χ-Sim, for cross similarity) is based on finding document-document similarity and word-word similarity, and then using classical clustering methods such as hierarchical clustering. Instead of explicitly clustering rows and columns alternately, they consider higher-order occurrences of words, inherently taking into account the documents in which they occur. Thus, the similarity between two words is calculated based on the documents in which they occur and also the documents in which "similar" words occur. The idea here is that two documents about the same topic do not necessarily use the same set of words to describe it, but a subset of the words and other similar words that are characteristic of that topic. This approach of taking higher-order similarities takes the latent semantic structure of the whole corpus into consideration with the result of generating a better clustering of the documents and words. In text databases, for a document collection defined by a document by term D matrix (of size m by n, m: number of documents, n: number of terms) the cover-coefficient based clustering methodology yields the same number of clusters both for documents and terms (words) using a double-stage probability experiment. According to the cover coefficient concept number of clusters can also be roughly estimated by the following formula ( m × n ) / t where t is the number of non-zero entries in D. Note that in D each row and each column must contain at least one non-zero element. In contrast to other approaches, FABIA is a multiplicative model that assumes realistic non-Gaussian signal distributions with heavy tails. FABIA utilizes well understood model selection techniques like variational approaches and applies the Bayesian framework. The generative framework allows FABIA to determine the information content of each Bicluster to separate spurious Biclusters from true Biclusters.
Biclustering : Formal concept analysis Biclique Galois connection
Biclustering : FABIA: Factor Analysis for Bicluster Acquisition, an R package —software
Calinski–Harabasz index : The Calinski–Harabasz index (CHI), also known as the Variance Ratio Criterion (VRC), is a metric for evaluating clustering algorithms, introduced by Tadeusz Caliński and Jerzy Harabasz in 1974. It is an internal evaluation metric, where the assessment of the clustering quality is based solely on the dataset and the clustering results, and not on external, ground-truth labels.
Calinski–Harabasz index : Given a data set of n points: , and the assignment of these points to k clusters: , the Calinski–Harabasz (CH) Index is defined as the ratio of the between-cluster separation (BCSS) to the within-cluster dispersion (WCSS), normalized by their number of degrees of freedom: C H = B C S S / ( k − 1 ) W C S S / ( n − k ) BCSS (Between-Cluster Sum of Squares) is the weighted sum of squared Euclidean distances between each cluster centroid (mean) and the overall data centroid (mean): B C S S = ∑ i = 1 k n i \| \| c i − c \| \| 2 ^n_\|\|\mathbf _-\mathbf \|\|^ where ni is the number of points in cluster Ci, ci is the centroid of Ci, and c is the overall centroid of the data. BCSS measures how well the clusters are separated from each other (the higher the better). WCSS (Within-Cluster Sum of Squares) is the sum of squared Euclidean distances between the data points and their respective cluster centroids: W C S S = ∑ i = 1 k ∑ x ∈ C i \| \| x − c i \| \| 2 ^\sum _ \in C_\|\|\mathbf -\mathbf _\|\|^ WCSS measures the compactness or cohesiveness of the clusters (the smaller the better). Minimizing the WCSS is the objective of centroid-based clustering algorithms such as k-means.
Calinski–Harabasz index : The numerator of the CH index is the between-cluster separation (BCSS) divided by its degrees of freedom. The number of degrees of freedom of BCSS is k - 1, since fixing the centroids of k - 1 clusters also determines the kth centroid, as its value makes the weighted sum of all centroids match the overall data centroid. The denominator of the CH index is the within-cluster dispersion (WCSS) divided by its degrees of freedom. The number of degrees of freedom of WCSS is n - k, since fixing the centroid of each cluster reduces the degrees of freedom by one. This is because given a centroid ci of cluster Ci, the assignment of ni - 1 points to that cluster also determines the assignment of the nith point, since the overall mean of the points assigned to the cluster should be equal to ci. Dividing both the BCSS and WCSS by their degrees of freedom helps to normalize the values, making them comparable across different numbers of clusters. Without this normalization, the CH index could be artificially inflated for higher values of k, making it hard to determine whether an increase in the index value is due to genuinely better clustering or just due to the increased number of clusters. A higher value of CH indicates a better clustering, because it means that the data points are more spread out between clusters than they are within clusters. Although there is no satisfactory probabilistic foundation to support the use of CH index, the criterion has some desirable mathematical properties as shown in. For example, in the special case of equal distances between all pairs of points, the CH index is equal to 1. In addition, it is analogous to the F-test statistic in univariate analysis. Liu et al. discuss the effectiveness of using CH index for cluster evaluation relative to other internal clustering evaluation metrics. Maulik and Bandyopadhyay evaluate the performance of three clustering algorithms using four cluster validity indices, including Davies–Bouldin index, Dunn index, Calinski–Harabasz index and a newly developed index. Wang et al. have suggested an improved index for clustering validation based on Silhouette indexing and Calinski–Harabasz index.
Calinski–Harabasz index : Similar to other clustering evaluation metrics such as Silhouette score, the CH index can be used to find the optimal number of clusters k in algorithms like k-means, where the value of k is not known a priori. This can be done by following these steps: Perform clustering for different values of k. Compute the CH index for each clustering result. The value of k that yields the maximum CH index is chosen as the optimal number of clusters.
Calinski–Harabasz index : The scikit-learn Python library provides an implementation of this metric in the sklearn.metrics module. R provides a similar implementation in its fpc package.
Calinski–Harabasz index : Cluster analysis Silhouette score Davies–Bouldin index Dunn index == References ==
Clustering high-dimensional data : Clustering high-dimensional data is the cluster analysis of data with anywhere from a few dozen to many thousands of dimensions. Such high-dimensional spaces of data are often encountered in areas such as medicine, where DNA microarray technology can produce many measurements at once, and the clustering of text documents, where, if a word-frequency vector is used, the number of dimensions equals the size of the vocabulary.
Clustering high-dimensional data : Four problems need to be overcome for clustering in high-dimensional data: Multiple dimensions are hard to think in, impossible to visualize, and, due to the exponential growth of the number of possible values with each dimension, complete enumeration of all subspaces becomes intractable with increasing dimensionality. This problem is known as the curse of dimensionality. The concept of distance becomes less precise as the number of dimensions grows, since the distance between any two points in a given dataset converges. The discrimination of the nearest and farthest point in particular becomes meaningless: lim d → ∞ d i s t max − d i s t min d i s t min = 0 _-__=0 A cluster is intended to group objects that are related, based on observations of their attribute's values. However, given a large number of attributes some of the attributes will usually not be meaningful for a given cluster. For example, in newborn screening a cluster of samples might identify newborns that share similar blood values, which might lead to insights about the relevance of certain blood values for a disease. But for different diseases, different blood values might form a cluster, and other values might be uncorrelated. This is known as the local feature relevance problem: different clusters might be found in different subspaces, so a global filtering of attributes is not sufficient. Given a large number of attributes, it is likely that some attributes are correlated. Hence, clusters might exist in arbitrarily oriented affine subspaces. Recent research indicates that the discrimination problems only occur when there is a high number of irrelevant dimensions, and that shared-nearest-neighbor approaches can improve results.
Clustering high-dimensional data : Approaches towards clustering in axis-parallel or arbitrarily oriented affine subspaces differ in how they interpret the overall goal, which is finding clusters in data with high dimensionality. An overall different approach is to find clusters based on pattern in the data matrix, often referred to as biclustering, which is a technique frequently utilized in bioinformatics.
Clustering high-dimensional data : ELKI includes various subspace and correlation clustering algorithms FCPS includes over fifty clustering algorithms == References ==
Clustering illusion : The clustering illusion is the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random. The illusion is caused by a human tendency to underpredict the amount of variability likely to appear in a small sample of random or pseudorandom data. Thomas Gilovich, an early author on the subject, argued that the effect occurs for different types of random dispersions. Some might perceive patterns in stock market price fluctuations over time, or clusters in two-dimensional data such as the locations of impact of World War II V-1 flying bombs on maps of London. Although Londoners developed specific theories about the pattern of impacts within London, a statistical analysis by R. D. Clarke originally published in 1946 showed that the impacts of V-2 rockets on London were a close fit to a random distribution.
Clustering illusion : Using this cognitive bias in causal reasoning may result in the Texas sharpshooter fallacy, in which differences in data are ignored and similarities are overemphasized. More general forms of erroneous pattern recognition are pareidolia and apophenia. Related biases are the illusion of control which the clustering illusion could contribute to, and insensitivity to sample size in which people don't expect greater variation in smaller samples. A different cognitive bias involving misunderstanding of chance streams is the gambler's fallacy.
Clustering illusion : Daniel Kahneman and Amos Tversky explained this kind of misprediction as being caused by the representativeness heuristic (which itself they also first proposed).
Clustering illusion : Apophenia Alignments of random points Complete spatial randomness Confirmation bias List of cognitive biases Numeracy bias Poisson distribution Statistical randomness
Clustering illusion : Skeptic's Dictionary: The clustering illusion Hot Hand website: Statistical analysis of sports streakiness
Consensus clustering : Consensus clustering is a method of aggregating (potentially conflicting) results from multiple clustering algorithms. Also called cluster ensembles or aggregation of clustering (or partitions), it refers to the situation in which a number of different (input) clusterings have been obtained for a particular dataset and it is desired to find a single (consensus) clustering which is a better fit in some sense than the existing clusterings. Consensus clustering is thus the problem of reconciling clustering information about the same data set coming from different sources or from different runs of the same algorithm. When cast as an optimization problem, consensus clustering is known as median partition, and has been shown to be NP-complete, even when the number of input clusterings is three. Consensus clustering for unsupervised learning is analogous to ensemble learning in supervised learning.
Consensus clustering : Current clustering techniques do not address all the requirements adequately. Dealing with large number of dimensions and large number of data items can be problematic because of time complexity; Effectiveness of the method depends on the definition of "distance" (for distance-based clustering) If an obvious distance measure doesn't exist, we must "define" it, which is not always easy, especially in multidimensional spaces. The result of the clustering algorithm (that, in many cases, can be arbitrary itself) can be interpreted in different ways.
Consensus clustering : There are potential shortcomings for all existing clustering techniques. This may cause interpretation of results to become difficult, especially when there is no knowledge about the number of clusters. Clustering methods are also very sensitive to the initial clustering settings, which can cause non-significant data to be amplified in non-reiterative methods. An extremely important issue in cluster analysis is the validation of the clustering results, that is, how to gain confidence about the significance of the clusters provided by the clustering technique (cluster numbers and cluster assignments). Lacking an external objective criterion (the equivalent of a known class label in supervised analysis), this validation becomes somewhat elusive. Iterative descent clustering methods, such as the SOM and k-means clustering circumvent some of the shortcomings of hierarchical clustering by providing for univocally defined clusters and cluster boundaries. Consensus clustering provides a method that represents the consensus across multiple runs of a clustering algorithm, to determine the number of clusters in the data, and to assess the stability of the discovered clusters. The method can also be used to represent the consensus over multiple runs of a clustering algorithm with random restart (such as K-means, model-based Bayesian clustering, SOM, etc.), so as to account for its sensitivity to the initial conditions. It can provide data for a visualization tool to inspect cluster number, membership, and boundaries. However, they lack the intuitive and visual appeal of hierarchical clustering dendrograms, and the number of clusters must be chosen a priori.
Consensus clustering : The Monti consensus clustering algorithm is one of the most popular consensus clustering algorithms and is used to determine the number of clusters, K . Given a dataset of N total number of points to cluster, this algorithm works by resampling and clustering the data, for each K and a N × N consensus matrix is calculated, where each element represents the fraction of times two samples clustered together. A perfectly stable matrix would consist entirely of zeros and ones, representing all sample pairs always clustering together or not together over all resampling iterations. The relative stability of the consensus matrices can be used to infer the optimal K . More specifically, given a set of points to cluster, D = ,e_,...e_\ , let D 1 , D 2 , . . . , D H ,D^,...,D^ be the list of H perturbed (resampled) datasets of the original dataset D , and let M h denote the N × N connectivity matrix resulting from applying a clustering algorithm to the dataset D h . The entries of M h are defined as follows: M h ( i , j ) = (i,j)=1,&\\0,&\end Let I h be the N × N identicator matrix where the ( i , j ) -th entry is equal to 1 if points i and j are in the same perturbed dataset D h , and 0 otherwise. The indicator matrix is used to keep track of which samples were selected during each resampling iteration for the normalisation step. The consensus matrix C is defined as the normalised sum of all connectivity matrices of all the perturbed datasets and a different one is calculated for every K . C ( i , j ) = ( ∑ h = 1 H M h ( i , j ) ∑ h = 1 H I h ( i , j ) ) ^M^(i,j)\displaystyle ^I^(i,j)\right) That is the entry ( i , j ) in the consensus matrix is the number of times points i and j were clustered together divided by the total number of times they were selected together. The matrix is symmetric and each element is defined within the range [ 0 , 1 ] . A consensus matrix is calculated for each K to be tested, and the stability of each matrix, that is how far the matrix is towards a matrix of perfect stability (just zeros and ones) is used to determine the optimal K . One way of quantifying the stability of the K th consensus matrix is examining its CDF curve (see below).
Consensus clustering : Monti consensus clustering can be a powerful tool for identifying clusters, but it needs to be applied with caution as shown by Şenbabaoğlu et al. It has been shown that the Monti consensus clustering algorithm is able to claim apparent stability of chance partitioning of null datasets drawn from a unimodal distribution, and thus has the potential to lead to over-interpretation of cluster stability in a real study. If clusters are not well separated, consensus clustering could lead one to conclude apparent structure when there is none, or declare cluster stability when it is subtle. Identifying false positive clusters is a common problem throughout cluster research, and has been addressed by methods such as SigClust and the GAP-statistic. However, these methods rely on certain assumptions for the null model that may not always be appropriate. Şenbabaoğlu et al demonstrated the original delta K metric to decide K in the Monti algorithm performed poorly, and proposed a new superior metric for measuring the stability of consensus matrices using their CDF curves. In the CDF curve of a consensus matrix, the lower left portion represents sample pairs rarely clustered together, the upper right portion represents those almost always clustered together, whereas the middle segment represent those with ambiguous assignments in different clustering runs. The proportion of ambiguous clustering (PAC) score measure quantifies this middle segment; and is defined as the fraction of sample pairs with consensus indices falling in the interval (u1, u2) ∈ [0, 1] where u1 is a value close to 0 and u2 is a value close to 1 (for instance u1=0.1 and u2=0.9). A low value of PAC indicates a flat middle segment, and a low rate of discordant assignments across permuted clustering runs. One can therefore infer the optimal number of clusters by the K value having the lowest PAC.
Consensus clustering : Clustering ensemble (Strehl and Ghosh): They considered various formulations for the problem, most of which reduce the problem to a hyper-graph partitioning problem. In one of their formulations they considered the same graph as in the correlation clustering problem. The solution they proposed is to compute the best k-partition of the graph, which does not take into account the penalty for merging two nodes that are far apart. Clustering aggregation (Fern and Brodley): They applied the clustering aggregation idea to a collection of soft clusterings they obtained by random projections. They used an agglomerative algorithm and did not penalize for merging dissimilar nodes. Fred and Jain: They proposed to use a single linkage algorithm to combine multiple runs of the k-means algorithm. Dana Cristofor and Dan Simovici: They observed the connection between clustering aggregation and clustering of categorical data. They proposed information theoretic distance measures, and they propose genetic algorithms for finding the best aggregation solution. Topchy et al.: They defined clustering aggregation as a maximum likelihood estimation problem, and they proposed an EM algorithm for finding the consensus clustering.
Consensus clustering : This approach by Strehl and Ghosh introduces the problem of combining multiple partitionings of a set of objects into a single consolidated clustering without accessing the features or algorithms that determined these partitionings. They discuss three approaches towards solving this problem to obtain high quality consensus functions. Their techniques have low computational costs and this makes it feasible to evaluate each of the techniques discussed below and arrive at the best solution by comparing the results against the objective function.
Consensus clustering : Punera and Ghosh extended the idea of hard clustering ensembles to the soft clustering scenario. Each instance in a soft ensemble is represented by a concatenation of r posterior membership probability distributions obtained from the constituent clustering algorithms. We can define a distance measure between two instances using the Kullback–Leibler (KL) divergence, which calculates the "distance" between two probability distributions. sCSPA: extends CSPA by calculating a similarity matrix. Each object is visualized as a point in dimensional space, with each dimension corresponding to probability of its belonging to a cluster. This technique first transforms the objects into a label-space and then interprets the dot product between the vectors representing the objects as their similarity. sMCLA:extends MCLA by accepting soft clusterings as input. sMCLA's working can be divided into the following steps: Construct Soft Meta-Graph of Clusters Group the Clusters into Meta-Clusters Collapse Meta-Clusters using Weighting Compete for Objects sHBGF:represents the ensemble as a bipartite graph with clusters and instances as nodes, and edges between the instances and the clusters they belong to. This approach can be trivially adapted to consider soft ensembles since the graph partitioning algorithm METIS accepts weights on the edges of the graph to be partitioned. In sHBGF, the graph has n + t vertices, where t is the total number of underlying clusters. Bayesian consensus clustering (BCC): defines a fully Bayesian model for soft consensus clustering in which multiple source clusterings, defined by different input data or different probability models, are assumed to adhere loosely to a consensus clustering. The full posterior for the separate clusterings, and the consensus clustering, are inferred simultaneously via Gibbs sampling. Ensemble Clustering Fuzzification Means (ECF-Means): ECF-means is a clustering algorithm, which combines different clustering results in ensemble, achieved by different runs of a chosen algorithm (k-means), into a single final clustering configuration.
Consensus clustering : Aristides Gionis, Heikki Mannila, Panayiotis Tsaparas. Clustering Aggregation. 21st International Conference on Data Engineering (ICDE 2005) Hongjun Wang, Hanhuai Shan, Arindam Banerjee. Bayesian Cluster Ensembles, SIAM International Conference on Data Mining, SDM 09 Nguyen, Nam; Caruana, Rich (2007). "Consensus Clusterings". Seventh IEEE International Conference on Data Mining (ICDM 2007). IEEE. pp. 607–612. doi:10.1109/icdm.2007.73. ISBN 978-0-7695-3018-5. ...we address the problem of combining multiple clusterings without access to the underlying features of the data. This process is known in the literature as clustering ensembles, clustering aggregation, or consensus clustering. Consensus clustering yields a stable and robust final clustering that is in agreement with multiple clusterings. We find that an iterative EM-like method is remarkably effective for this problem. We present an iterative algorithm and its variations for finding clustering consensus. An extensive empirical study compares our proposed algorithms with eleven other consensus clustering methods on four data sets using three different clustering performance metrics. The experimental results show that the new ensemble clustering methods produce clusterings that are as good as, and often better than, these other methods.
Constrained clustering : In computer science, constrained clustering is a class of semi-supervised learning algorithms. Typically, constrained clustering incorporates either a set of must-link constraints, cannot-link constraints, or both, with a data clustering algorithm. A cluster in which the members conform to all must-link and cannot-link constraints is called a chunklet.
Constrained clustering : Both a must-link and a cannot-link constraint define a relationship between two data instances. Together, the sets of these constraints act as a guide for which a constrained clustering algorithm will attempt to find chunklets (clusters in the dataset which satisfy the specified constraints). A must-link constraint is used to specify that the two instances in the must-link relation should be associated with the same cluster. A cannot-link constraint is used to specify that the two instances in the cannot-link relation should not be associated with the same cluster. Some constrained clustering algorithms will abort if no such clustering exists which satisfies the specified constraints. Others will try to minimize the amount of constraint violation should it be impossible to find a clustering which satisfies the constraints. Constraints could also be used to guide the selection of a clustering model among several possible solutions.
Constrained clustering : Examples of constrained clustering algorithms include: COP K-means PCKmeans (Pairwise Constrained K-means) CMWK-Means (Constrained Minkowski Weighted K-Means) == References ==
Correlation clustering : Clustering is the problem of partitioning data points into groups based on their similarity. Correlation clustering provides a method for clustering a set of objects into the optimum number of clusters without specifying that number in advance.
Correlation clustering : In machine learning, correlation clustering or cluster editing operates in a scenario where the relationships between the objects are known instead of the actual representations of the objects. For example, given a weighted graph G = ( V , E ) where the edge weight indicates whether two nodes are similar (positive edge weight) or different (negative edge weight), the task is to find a clustering that either maximizes agreements (sum of positive edge weights within a cluster plus the absolute value of the sum of negative edge weights between clusters) or minimizes disagreements (absolute value of the sum of negative edge weights within a cluster plus the sum of positive edge weights across clusters). Unlike other clustering algorithms this does not require choosing the number of clusters k in advance because the objective, to minimize the sum of weights of the cut edges, is independent of the number of clusters. It may not be possible to find a perfect clustering, where all similar items are in a cluster while all dissimilar ones are in different clusters. If the graph indeed admits a perfect clustering, then simply deleting all the negative edges and finding the connected components in the remaining graph will return the required clusters. Davis found a necessary and sufficient condition for this to occur: no cycle may contain exactly one negative edge. But, in general a graph may not have a perfect clustering. For example, given nodes a,b,c such that a,b and a,c are similar while b,c are dissimilar, a perfect clustering is not possible. In such cases, the task is to find a clustering that maximizes the number of agreements (number of + edges inside clusters plus the number of − edges between clusters) or minimizes the number of disagreements (the number of − edges inside clusters plus the number of + edges between clusters). This problem of maximizing the agreements is NP-complete (multiway cut problem reduces to maximizing weighted agreements and the problem of partitioning into triangles can be reduced to the unweighted version).
Correlation clustering : Let G = ( V , E ) be a graph with nodes V and edges E . A clustering of G is a partition of its node set Π = ,\dots ,\pi _\ with V = π 1 ∪ ⋯ ∪ π k \cup \dots \cup \pi _ and π i ∩ π j = ∅ \cap \pi _=\emptyset for i ≠ j . For a given clustering Π , let δ ( Π ) = ∈ E ∣ ⊈ π ∀ π ∈ Π \in E\mid \\not \subseteq \pi \;\forall \pi \in \Pi \ denote the subset of edges of G whose endpoints are in different subsets of the clustering Π . Now, let w : E → R ≥ 0 _ be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − \cup E^ be a partition of the edges into attractive ( E + ) and repulsive ( E − ) edges. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . & &&\sum _\cap \delta (\Pi )w_+\sum _\setminus \delta (\Pi )w_\;.\end Here, the set E + ∩ δ ( Π ) \cap \delta (\Pi ) contains the attractive edges whose endpoints are in different components with respect to the clustering Π and the set E − ∖ δ ( Π ) \setminus \delta (\Pi ) contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π . Together these two sets contain all edges that disagree with the clustering Π . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . & &&\sum _\setminus \delta (\Pi )w_+\sum _\cap \delta (\Pi )w_\;.\end Here, the set E + ∖ δ ( Π ) \setminus \delta (\Pi ) contains the attractive edges whose endpoints are in the same component with respect to the clustering Π and the set E − ∩ δ ( Π ) \cap \delta (\Pi ) contains the repulsive edges whose endpoints are in different components with respect to the clustering Π . Together these two sets contain all edges that agree with the clustering Π . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 _ and a given partition E = E + ∪ E − \cup E^ of the edges into attractive and repulsive edges, the edge costs can be defined by c e = c_=\;\;w_&e\in E^\\-w_&e\in E^\end\end for all e ∈ E . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) of all edges that are cut is often called a multicut of G . The minimum cost multicut problem is the problem of finding a clustering Π of G such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . & &&\sum _c_\;.\end Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . & &&\sum _c_\;.\end This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives.
Correlation clustering : Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters.
Correlation clustering : In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the underlying number of clusters. This analysis suggests the functional assumes a uniform prior over all possible partitions regardless of their number of clusters. Thus, a non-uniform prior over the number of clusters emerges. Several discrete optimization algorithms are proposed in this work that scales gracefully with the number of elements (experiments show results with more than 100,000 variables). The work of Bagon and Galun also evaluated the effectiveness of the recovery of the underlying number of clusters in several applications.
Correlation clustering : Correlation clustering also relates to a different task, where correlations among attributes of feature vectors in a high-dimensional space are assumed to exist guiding the clustering process. These correlations may be different in different clusters, thus a global decorrelation cannot reduce this to traditional (uncorrelated) clustering. Correlations among subsets of attributes result in different spatial shapes of clusters. Hence, the similarity between cluster objects is defined by taking into account the local correlation patterns. With this notion, the term has been introduced in simultaneously with the notion discussed above. Different methods for correlation clustering of this type are discussed in and the relationship to different types of clustering is discussed in. See also Clustering high-dimensional data. Correlation clustering (according to this definition) can be shown to be closely related to biclustering. As in biclustering, the goal is to identify groups of objects that share a correlation in some of their attributes; where the correlation is usually typical for the individual clusters. == References ==
Dendrogram : A dendrogram is a diagram representing a tree. This diagrammatic representation is frequently used in different contexts: in hierarchical clustering, it illustrates the arrangement of the clusters produced by the corresponding analyses. in computational biology, it shows the clustering of genes or samples, sometimes in the margins of heatmaps. in phylogenetics, it displays the evolutionary relationships among various biological taxa. In this case, the dendrogram is also called a phylogenetic tree. The name dendrogram derives from the two ancient greek words δένδρον (déndron), meaning "tree", and γράμμα (grámma), meaning "drawing, mathematical figure".
Dendrogram : For a clustering example, suppose that five taxa ( a to e ) have been clustered by UPGMA based on a matrix of genetic distances. The hierarchical clustering dendrogram would show a column of five nodes representing the initial data (here individual taxa), and the remaining nodes represent the clusters to which the data belong, with the arrows representing the distance (dissimilarity). The distance between merged clusters is monotone, increasing with the level of the merger: the height of each node in the plot is proportional to the value of the intergroup dissimilarity between its two daughters (the nodes on the right representing individual observations all plotted at zero height).
Dendrogram : Cladogram Distance matrices in phylogeny Hierarchical clustering MEGA, a freeware for drawing dendrograms yEd, a freeware for drawing and automatically arranging dendrograms Taxonomy
Dendrogram : Iris dendrogram - Example of using a dendrogram to visualize the 3 clusters from hierarchical clustering using the "complete" method vs the real species category (using R).
Determining the number of clusters in a data set : Determining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether. The correct choice of k is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user. In addition, increasing k without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point is considered its own cluster (i.e., when k equals the number of data points, n). Intuitively then, the optimal choice of k will strike a balance between maximum compression of the data using a single cluster, and maximum accuracy by assigning each data point to its own cluster. If an appropriate value of k is not apparent from prior knowledge of the properties of the data set, it must be chosen somehow. There are several categories of methods for making this decision.
Determining the number of clusters in a data set : The elbow method looks at the percentage of explained variance as a function of the number of clusters: One should choose a number of clusters so that adding another cluster does not give much better modeling of the data. More precisely, if one plots the percentage of variance explained by the clusters against the number of clusters, the first clusters will add much information (explain a lot of variance), but at some point the marginal gain will drop, giving an angle in the graph. The number of clusters is chosen at this point, hence the "elbow criterion". In most datasets, this "elbow" is ambiguous, making this method subjective and unreliable. Because the scale of the axes is arbitrary, the concept of an angle is not well-defined, and even on uniform random data, the curve produces an "elbow", making the method rather unreliable. Percentage of variance explained is the ratio of the between-group variance to the total variance, also known as an F-test. A slight variation of this method plots the curvature of the within group variance. The method can be traced to speculation by Robert L. Thorndike in 1953. While the idea of the elbow method sounds simple and straightforward, other methods (as detailed below) give better results.
Determining the number of clusters in a data set : In statistics and data mining, X-means clustering is a variation of k-means clustering that refines cluster assignments by repeatedly attempting subdivision, and keeping the best resulting splits, until a criterion such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC) is reached.
Determining the number of clusters in a data set : Another set of methods for determining the number of clusters are information criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), or the deviance information criterion (DIC) — if it is possible to make a likelihood function for the clustering model. For example: The k-means model is "almost" a Gaussian mixture model and one can construct a likelihood for the Gaussian mixture model and thus also determine information criterion values.
Determining the number of clusters in a data set : Rate distortion theory has been applied to choosing k called the "jump" method, which determines the number of clusters that maximizes efficiency while minimizing error by information-theoretic standards. The strategy of the algorithm is to generate a distortion curve for the input data by running a standard clustering algorithm such as k-means for all values of k between 1 and n, and computing the distortion (described below) of the resulting clustering. The distortion curve is then transformed by a negative power chosen based on the dimensionality of the data. Jumps in the resulting values then signify reasonable choices for k, with the largest jump representing the best choice. The distortion of a clustering of some input data is formally defined as follows: Let the data set be modeled as a p-dimensional random variable, X, consisting of a mixture distribution of G components with common covariance, Γ. If we let c 1 … c K \ldots c_ be a set of K cluster centers, with c X the closest center to a given sample of X, then the minimum average distortion per dimension when fitting the K centers to the data is: d K = 1 p min c 1 … c K E [ ( X − c X ) T Γ − 1 ( X − c X ) ] =\min _\ldots c_)^\Gamma ^(X-c_)] This is also the average Mahalanobis distance per dimension between X and the closest cluster center c X . Because the minimization over all possible sets of cluster centers is prohibitively complex, the distortion is computed in practice by generating a set of cluster centers using a standard clustering algorithm and computing the distortion using the result. The pseudo-code for the jump method with an input set of p-dimensional data points X is: JumpMethod(X): Let Y = (p/2) Init a list D, of size n+1 Let D[0] = 0 For k = 1 ... n: Cluster X with k clusters (e.g., with k-means) Let d = Distortion of the resulting clustering D[k] = d^(-Y) Define J(i) = D[i] - D[i-1] Return the k between 1 and n that maximizes J(k) The choice of the transform power Y = ( p / 2 ) is motivated by asymptotic reasoning using results from rate distortion theory. Let the data X have a single, arbitrarily p-dimensional Gaussian distribution, and let fixed K = ⌊ α p ⌋ \rfloor , for some α greater than zero. Then the distortion of a clustering of K clusters in the limit as p goes to infinity is α − 2 . It can be seen that asymptotically, the distortion of a clustering to the power ( − p / 2 ) is proportional to α p , which by definition is approximately the number of clusters K. In other words, for a single Gaussian distribution, increasing K beyond the true number of clusters, which should be one, causes a linear growth in distortion. This behavior is important in the general case of a mixture of multiple distribution components. Let X be a mixture of G p-dimensional Gaussian distributions with common covariance. Then for any fixed K less than G, the distortion of a clustering as p goes to infinity is infinite. Intuitively, this means that a clustering of less than the correct number of clusters is unable to describe asymptotically high-dimensional data, causing the distortion to increase without limit. If, as described above, K is made an increasing function of p, namely, K = ⌊ α p ⌋ \rfloor , the same result as above is achieved, with the value of the distortion in the limit as p goes to infinity being equal to α − 2 . Correspondingly, there is the same proportional relationship between the transformed distortion and the number of clusters, K. Putting the results above together, it can be seen that for sufficiently high values of p, the transformed distortion d K − p / 2 ^ is approximately zero for K < G, then jumps suddenly and begins increasing linearly for K ≥ G. The jump algorithm for choosing K makes use of these behaviors to identify the most likely value for the true number of clusters. Although the mathematical support for the method is given in terms of asymptotic results, the algorithm has been empirically verified to work well in a variety of data sets with reasonable dimensionality. In addition to the localized jump method described above, there exists a second algorithm for choosing K using the same transformed distortion values known as the broken line method. The broken line method identifies the jump point in the graph of the transformed distortion by doing a simple least squares error line fit of two line segments, which in theory will fall along the x-axis for K < G, and along the linearly increasing phase of the transformed distortion plot for K ≥ G. The broken line method is more robust than the jump method in that its decision is global rather than local, but it also relies on the assumption of Gaussian mixture components, whereas the jump method is fully non-parametric and has been shown to be viable for general mixture distributions.
Determining the number of clusters in a data set : 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. A silhouette close to 1 implies the datum is in an appropriate cluster, while a silhouette close to −1 implies the datum is in the wrong cluster. Optimization techniques such as genetic algorithms are useful in determining the number of clusters that gives rise to the largest silhouette. It is also possible to re-scale the data in such a way that the silhouette is more likely to be maximized at the correct number of clusters.
Determining the number of clusters in a data set : One can also use the process of cross-validation to analyze the number of clusters. In this process, the data is partitioned into v parts. Each of the parts is then set aside at turn as a test set, a clustering model computed on the other v − 1 training sets, and the value of the objective function (for example, the sum of the squared distances to the centroids for k-means) calculated for the test set. These v values are calculated and averaged for each alternative number of clusters, and the cluster number selected such that further increase in number of clusters leads to only a small reduction in the objective function.
Determining the number of clusters in a data set : When clustering text databases with the cover coefficient on a document collection defined by a document by term D matrix (of size m×n, where m is the number of documents and n is the number of terms), the number of clusters can roughly be estimated by the formula m n t where t is the number of non-zero entries in D. Note that in D each row and each column must contain at least one non-zero element.
Determining the number of clusters in a data set : Kernel matrix defines the proximity of the input information. For example, in Gaussian radial basis function, it determines the dot product of the inputs in a higher-dimensional space, called feature space. It is believed that the data become more linearly separable in the feature space, and hence, linear algorithms can be applied on the data with a higher success. The kernel matrix can thus be analyzed in order to find the optimal number of clusters. The method proceeds by the eigenvalue decomposition of the kernel matrix. It will then analyze the eigenvalues and eigenvectors to obtain a measure of the compactness of the input distribution. Finally, a plot will be drawn, where the elbow of that plot indicates the optimal number of clusters in the data set. Unlike previous methods, this technique does not need to perform any clustering a-priori. It directly finds the number of clusters from the data.
Determining the number of clusters in a data set : Robert Tibshirani, Guenther Walther, and Trevor Hastie proposed estimating the number of clusters in a data set via the gap statistic. The gap statistics, based on theoretical grounds, measures how far is the pooled within-cluster sum of squares around the cluster centers from the sum of squares expected under the null reference distribution of data. The expected value is estimated by simulating null reference data of characteristics of the original data, but lacking any clusters in it. The optimal number of clusters is then estimated as the value of k for which the observed sum of squares falls farthest below the null reference. Unlike many previous methods, the gap statistics can tell us that there is no value of k for which there is a good clustering, but the reliability depends on how plausible the assumed null distribution (e.g., a uniform distribution) is on the given data. This tends to work well in synthetic settings, but cannot handle difficult data sets with, e.g., uninformative attributes well because it assumes all attributes to be equally important. The gap statistics is implemented as the clusGap function in the cluster package in R.
Determining the number of clusters in a data set : Clustergram – cluster diagnostic plot – for visual diagnostics of choosing the number of (k) clusters (R code) Eight methods for determining an optimal k value for k-means analysis – Answer on stackoverflow containing R code for several methods of computing an optimal value of k for k-means cluster analysis
Farthest-first traversal : In computational geometry, the farthest-first traversal of a compact metric space is a sequence of points in the space, where the first point is selected arbitrarily and each successive point is as far as possible from the set of previously-selected points. The same concept can also be applied to a finite set of geometric points, by restricting the selected points to belong to the set or equivalently by considering the finite metric space generated by these points. For a finite metric space or finite set of geometric points, the resulting sequence forms a permutation of the points, also known as the greedy permutation. Every prefix of a farthest-first traversal provides a set of points that is widely spaced and close to all remaining points. More precisely, no other set of equally many points can be spaced more than twice as widely, and no other set of equally many points can be less than half as far to its farthest remaining point. In part because of these properties, farthest-point traversals have many applications, including the approximation of the traveling salesman problem and the metric k-center problem. They may be constructed in polynomial time, or (for low-dimensional Euclidean spaces) approximated in near-linear time.
Farthest-first traversal : A farthest-first traversal is a sequence of points in a compact metric space, with each point appearing at most once. If the space is finite, each point appears exactly once, and the traversal is a permutation of all of the points in the space. The first point of the sequence may be any point in the space. Each point p after the first must have the maximum possible distance to the set of points earlier than p in the sequence, where the distance from a point to a set is defined as the minimum of the pairwise distances to points in the set. A given space may have many different farthest-first traversals, depending both on the choice of the first point in the sequence (which may be any point in the space) and on ties for the maximum distance among later choices. Farthest-point traversals may be characterized by the following properties. Fix a number k, and consider the prefix formed by the first k points of the farthest-first traversal of any metric space. Let r be the distance between the final point of the prefix and the other points in the prefix. Then this subset has the following two properties: All pairs of the selected points are at distance at least r from each other, and All points of the metric space are at distance at most r from the subset. Conversely any sequence having these properties, for all choices of k, must be a farthest-first traversal. These are the two defining properties of a Delone set, so each prefix of the farthest-first traversal forms a Delone set.
Farthest-first traversal : Rosenkrantz, Stearns & Lewis (1977) used the farthest-first traversal to define the farthest-insertion heuristic for the travelling salesman problem. This heuristic finds approximate solutions to the travelling salesman problem by building up a tour on a subset of points, adding one point at a time to the tour in the ordering given by a farthest-first traversal. To add each point to the tour, one edge of the previous tour is broken and replaced by a pair of edges through the added point, in the cheapest possible way. Although Rosenkrantz et al. prove only a logarithmic approximation ratio for this method, they show that in practice it often works better than other insertion methods with better provable approximation ratios. Later, the same sequence of points was popularized by Gonzalez (1985), who used it as part of greedy approximation algorithms for two problems in clustering, in which the goal is to partition a set of points into k clusters. One of the two problems that Gonzalez solve in this way seeks to minimize the maximum diameter of a cluster, while the other, known as the metric k-center problem, seeks to minimize the maximum radius, the distance from a chosen central point of a cluster to the farthest point from it in the same cluster. For instance, the k-center problem can be used to model the placement of fire stations within a city, in order to ensure that every address within the city can be reached quickly by a fire truck. For both clustering problems, Gonzalez chooses a set of k cluster centers by selecting the first k points of a farthest-first traversal, and then creates clusters by assigning each input point to the nearest cluster center. If r is the distance from the set of k selected centers to the next point at position k + 1 in the traversal, then with this clustering every point is within distance r of its center and every cluster has diameter at most 2r. However, the subset of k centers together with the next point are all at distance at least r from each other, and any k-clustering would put some two of these points into a single cluster, with one of them at distance at least r/2 from its center and with diameter at least r. Thus, Gonzalez's heuristic gives an approximation ratio of 2 for both clustering problems. Gonzalez's heuristic was independently rediscovered for the metric k-center problem by Dyer & Frieze (1985), who applied it more generally to weighted k-center problems. Another paper on the k-center problem from the same time, Hochbaum & Shmoys (1985), achieves the same approximation ratio of 2, but its techniques are different. Nevertheless, Gonzalez's heuristic, and the name "farthest-first traversal", are often incorrectly attributed to Hochbaum and Shmoys. For both the min-max diameter clustering problem and the metric k-center problem, these approximations are optimal: the existence of a polynomial-time heuristic with any constant approximation ratio less than 2 would imply that P = NP. As well as for clustering, the farthest-first traversal can also be used in another type of facility location problem, the max-min facility dispersion problem, in which the goal is to choose the locations of k different facilities so that they are as far apart from each other as possible. More precisely, the goal in this problem is to choose k points from a given metric space or a given set of candidate points, in such a way as to maximize the minimum pairwise distance between the selected points. Again, this can be approximated by choosing the first k points of a farthest-first traversal. If r denotes the distance of the kth point from all previous points, then every point of the metric space or the candidate set is within distance r of the first k − 1 points. By the pigeonhole principle, some two points of the optimal solution (whatever it is) must both be within distance r of the same point among these first k − 1 chosen points, and (by the triangle inequality) within distance 2r of each other. Therefore, the heuristic solution given by the farthest-first traversal is within a factor of two of optimal. Other applications of the farthest-first traversal include color quantization (clustering the colors in an image to a smaller set of representative colors), progressive scanning of images (choosing an order to display the pixels of an image so that prefixes of the ordering produce good lower-resolution versions of the whole image rather than filling in the image from top to bottom), point selection in the probabilistic roadmap method for motion planning, simplification of point clouds, generating masks for halftone images, hierarchical clustering, finding the similarities between polygon meshes of similar surfaces, choosing diverse and high-value observation targets for underwater robot exploration, fault detection in sensor networks, modeling phylogenetic diversity, matching vehicles in a heterogenous fleet to customer delivery requests, uniform distribution of geodetic observatories on the Earth's surface or of other types of sensor network, generation of virtual point lights in the instant radiosity computer graphics rendering method, and geometric range searching data structures.
Farthest-first traversal : Lloyd's algorithm, a different method for generating evenly spaced points in geometric spaces == References ==
Frequent pattern discovery : Frequent pattern discovery (or FP discovery, FP mining, or Frequent itemset mining) is part of knowledge discovery in databases, Massive Online Analysis, and data mining; it describes the task of finding the most frequent and relevant patterns in large datasets. The concept was first introduced for mining transaction databases. Frequent patterns are defined as subsets (itemsets, subsequences, or substructures) that appear in a data set with frequency no less than a user-specified or auto-determined threshold.
Frequent pattern discovery : Techniques for FP mining include: market basket analysis cross-marketing catalog design clustering classification recommendation systems For the most part, FP discovery can be done using association rule learning with particular algorithms Eclat, FP-growth and the Apriori algorithm. Other strategies include: Frequent subtree mining Structure mining Sequential pattern mining and respective specific techniques. Implementations exist for various machine learning systems or modules like MLlib for Apache Spark. == References ==
Geographical cluster : A geographical cluster is a localized anomaly, usually an excess of something given the distribution or variation of something else. Often it is considered as an incidence rate that is unusual in that there is more of some variable than might be expected. Examples would include: a local excess disease rate, a crime hot spot, areas of high unemployment, accident blackspots, unusually high positive residuals from a model, high concentrations of flora or fauna, areas with high levels of creative activity, physical features or events like earthquake epicenters etc... Identifying these extreme regions may be useful in that there could be implicit geographical associations with other variables that can be identified and would be of interest. Pattern detection via the identification of such geographical clusters is a very simple and generic form of geographical analysis that has many applications in many different contexts. The emphasis is on localized clustering or patterning because this may well contain the most useful information. A geographical cluster is different from a high concentration as it is generally second order, involving the factoring in of the distribution of something else.
Geographical cluster : Identifying geographical clusters can be an important stage in a geographical analysis. Mapping the locations of unusual concentrations may help identify causes of these. Some techniques include the Geographical Analysis Machine and Besag and Newell's cluster detection method. == References ==
GeWorkbench : geWorkbench (genomics Workbench) is an open-source software platform for integrated genomic data analysis. It is a desktop application written in the programming language Java. geWorkbench uses a component architecture. As of 2016, there are more than 70 plug-ins available, providing for the visualization and analysis of gene expression, sequence, and structure data. geWorkbench is the Bioinformatics platform of MAGNet, the National Center for the Multi-scale Analysis of Genomic and Cellular Networks, one of the 8 National Centers for Biomedical Computing funded through the NIH Roadmap (NIH Common Fund). Many systems and structure biology tools developed by MAGNet investigators are available as geWorkbench plugins.
GeWorkbench : Computational analysis tools such as t-test, hierarchical clustering, self-organizing maps, regulatory network reconstruction, BLAST searches, pattern-motif discovery, protein structure prediction, structure-based protein annotation, etc. Visualization of gene expression (heatmaps, volcano plot), molecular interaction networks (through Cytoscape), protein sequence and protein structure data (e.g., MarkUs). Integration of gene and pathway annotation information from curated sources as well as through Gene Ontology enrichment analysis. Component integration through platform management of inputs and outputs. Among data that can be shared between components are expression datasets, interaction networks, sample and marker (gene) sets and sequences. Dataset history tracking - complete record of data sets used and input settings. Integration with 3rd party tools such as GenePattern, Cytoscape, and Genomespace. Demonstrations of each feature described can be found at GeWorkbench-web Tutorials.
GeWorkbench : geWorkbench is open-source software that can be downloaded and installed locally. A zip file of the released version Java source is also available. Prepackaged installer versions also exist for Windows, Macintosh, and Linux.
GeWorkbench : Official website, includes installation, tutorials, FAQs, known issues - geworkbench release downloads - geWorkbench plugins
Latent space : A latent space, also known as a latent feature space or embedding space, is an embedding of a set of items within a manifold in which items resembling each other are positioned closer to one another. Position within the latent space can be viewed as being defined by a set of latent variables that emerge from the resemblances from the objects. In most cases, the dimensionality of the latent space is chosen to be lower than the dimensionality of the feature space from which the data points are drawn, making the construction of a latent space an example of dimensionality reduction, which can also be viewed as a form of data compression. Latent spaces are usually fit via machine learning, and they can then be used as feature spaces in machine learning models, including classifiers and other supervised predictors. The interpretation of the latent spaces of machine learning models is an active field of study, but latent space interpretation is difficult to achieve. Due to the black-box nature of machine learning models, the latent space may be completely unintuitive. Additionally, the latent space may be high-dimensional, complex, and nonlinear, which may add to the difficulty of interpretation. Some visualization techniques have been developed to connect the latent space to the visual world, but there is often not a direct connection between the latent space interpretation and the model itself. Such techniques include t-distributed stochastic neighbor embedding (t-SNE), where the latent space is mapped to two dimensions for visualization. Latent space distances lack physical units, so the interpretation of these distances may depend on the application.
Latent space : Several embedding models have been developed to perform this transformation to create latent space embeddings given a set of data items and a similarity function. These models learn the embeddings by leveraging statistical techniques and machine learning algorithms. Here are some commonly used embedding models: Word2Vec: Word2Vec is a popular embedding model used in natural language processing (NLP). It learns word embeddings by training a neural network on a large corpus of text. Word2Vec captures semantic and syntactic relationships between words, allowing for meaningful computations like word analogies. GloVe: GloVe (Global Vectors for Word Representation) is another widely used embedding model for NLP. It combines global statistical information from a corpus with local context information to learn word embeddings. GloVe embeddings are known for capturing both semantic and relational similarities between words. Siamese Networks: Siamese networks are a type of neural network architecture commonly used for similarity-based embedding. They consist of two identical subnetworks that process two input samples and produce their respective embeddings. Siamese networks are often used for tasks like image similarity, recommendation systems, and face recognition. Variational Autoencoders (VAEs): VAEs are generative models that simultaneously learn to encode and decode data. The latent space in VAEs acts as an embedding space. By training VAEs on high-dimensional data, such as images or audio, the model learns to encode the data into a compact latent representation. VAEs are known for their ability to generate new data samples from the learned latent space.
Latent space : Multimodality refers to the integration and analysis of multiple modes or types of data within a single model or framework. Embedding multimodal data involves capturing relationships and interactions between different data types, such as images, text, audio, and structured data. Multimodal embedding models aim to learn joint representations that fuse information from multiple modalities, allowing for cross-modal analysis and tasks. These models enable applications like image captioning, visual question answering, and multimodal sentiment analysis. To embed multimodal data, specialized architectures such as deep multimodal networks or multimodal transformers are employed. These architectures combine different types of neural network modules to process and integrate information from various modalities. The resulting embeddings capture the complex relationships between different data types, facilitating multimodal analysis and understanding.
Latent space : Embedding latent space and multimodal embedding models have found numerous applications across various domains: Information retrieval: Embedding techniques enable efficient similarity search and recommendation systems by representing data points in a compact space. Natural language processing: Word embeddings have revolutionized NLP tasks like sentiment analysis, machine translation, and document classification. Computer vision: Image and video embeddings enable tasks like object recognition, image retrieval, and video summarization. Recommendation systems: Embeddings help capture user preferences and item characteristics, enabling personalized recommendations. Healthcare: Embedding techniques have been applied to electronic health records, medical imaging, and genomic data for disease prediction, diagnosis, and treatment. Social systems: Embedding techniques can be used to learn latent representations of social systems such as internal migration systems, academic citation networks, and world trade networks.
Latent space : == References ==
Medoid : Medoids are representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be members of the data set. Medoids are most commonly used on data when a mean or centroid cannot be defined, such as graphs. They are also used in contexts where the centroid is not representative of the dataset like in images, 3-D trajectories and gene expression (where while the data is sparse the medoid need not be). These are also of interest while wanting to find a representative using some distance other than squared euclidean distance (for instance in movie-ratings). For some data sets there may be more than one medoid, as with medians. A common application of the medoid is the k-medoids clustering algorithm, which is similar to the k-means algorithm but works when a mean or centroid is not definable. This algorithm basically works as follows. First, a set of medoids is chosen at random. Second, the distances to the other points are computed. Third, data are clustered according to the medoid they are most similar to. Fourth, the medoid set is optimized via an iterative process. Note that a medoid is not equivalent to a median, a geometric median, or centroid. A median is only defined on 1-dimensional data, and it only minimizes dissimilarity to other points for metrics induced by a norm (such as the Manhattan distance or Euclidean distance). A geometric median is defined in any dimension, but unlike a medoid, it is not necessarily a point from within the original dataset.
Medoid : Let X := :=\,x_,\dots ,x_\ be a set of n points in a space with a distance function d. Medoid is defined as x medoid = arg ⁡ min y ∈ X ∑ i = 1 n d ( y , x i ) . =\arg \min _\sum _^d(y,x_).
Medoid : Medoids are a popular replacement for the cluster mean when the distance function is not (squared) Euclidean distance, or not even a metric (as the medoid does not require the triangle inequality). When partitioning the data set into clusters, the medoid of each cluster can be used as a representative of each cluster. Clustering algorithms based on the idea of medoids include: Partitioning Around Medoids (PAM), the standard k-medoids algorithm Hierarchical Clustering Around Medoids (HACAM), which uses medoids in hierarchical clustering
Medoid : From the definition above, it is clear that the medoid of a set X can be computed after computing all pairwise distances between points in the ensemble. This would take O ( n 2 ) ) distance evaluations (with n = \| X \| \| ). In the worst case, one can not compute the medoid with fewer distance evaluations. However, there are many approaches that allow us to compute medoids either exactly or approximately in sub-quadratic time under different statistical models. If the points lie on the real line, computing the medoid reduces to computing the median which can be done in O ( n ) by Quick-select algorithm of Hoare. However, in higher dimensional real spaces, no linear-time algorithm is known. RAND is an algorithm that estimates the average distance of each point to all the other points by sampling a random subset of other points. It takes a total of O ( n log ⁡ n ϵ 2 ) \right) distance computations to approximate the medoid within a factor of ( 1 + ϵ Δ ) with high probability, where Δ is the maximum distance between two points in the ensemble. Note that RAND is an approximation algorithm, and moreover Δ may not be known apriori. RAND was leveraged by TOPRANK which uses the estimates obtained by RAND to focus on a small subset of candidate points, evaluates the average distance of these points exactly, and picks the minimum of those. TOPRANK needs O ( n 5 3 log 4 3 ⁡ n ) \log ^n) distance computations to find the exact medoid with high probability under a distributional assumption on the average distances. trimed presents an algorithm to find the medoid with O ( n 3 2 2 Θ ( d ) ) 2^) distance evaluations under a distributional assumption on the points. The algorithm uses the triangle inequality to cut down the search space. Meddit leverages a connection of the medoid computation with multi-armed bandits and uses an upper-Confidence-bound type of algorithm to get an algorithm which takes O ( n log ⁡ n ) distance evaluations under statistical assumptions on the points. Correlated Sequential Halving also leverages multi-armed bandit techniques, improving upon Meddit. By exploiting the correlation structure in the problem, the algorithm is able to provably yield drastic improvement (usually around 1-2 orders of magnitude) in both number of distance computations needed and wall clock time.
Medoid : An implementation of RAND, TOPRANK, and trimed can be found here. An implementation of Meddit can be found here and here. An implementation of Correlated Sequential Halving can be found here.
Medoid : Medoids can be applied to various text and NLP tasks to improve the efficiency and accuracy of analyses. By clustering text data based on similarity, medoids can help identify representative examples within the dataset, leading to better understanding and interpretation of the data.
Medoid : As a versatile clustering method, medoids can be applied to a variety of real-world issues in numerous fields, stretching from biology and medicine to advertising and marketing, and social networks. Its potential to handle complex data sets with a high degree of perplexity makes it a powerful device in modern-day data analytics.
Medoid : A common problem with k-medoids clustering and other medoid-based clustering algorithms is the "curse of dimensionality," in which the data points contain too many dimensions or features. As dimensions are added to the data, the distance between them becomes sparse, and it becomes difficult to characterize clustering by Euclidean distance alone. As a result, distance based similarity measures converge to a constant and we have a characterization of distance between points which may not be reflect our data set in meaningful ways. One way to mitigate the effects of the curse of dimensionality is by using spectral clustering. Spectral clustering achieves a more appropriate analysis by reducing the dimensionality of then data using principal component analysis, projecting the data points into the lower dimensional subspace, and then running the chosen clustering algorithm as before. One thing to note, however, is that as with any dimension reduction we lose information, so it must be weighed against clustering in advanced how much reduction is necessary before too much data is lost. High dimensionality doesn't only affect distance metrics however, as the time complexity also increases with the number of features. k-medoids is sensitive to initial choice of medoids, as they are usually selected randomly. Depending on how such medoids are initialized, k-medoids may converge to different local optima, resulting in different clusters and quality measures, meaning k-medoids might need to run multiple times with different initializations, resulting in a much higher run time. One way to counterbalance this is to use k-medoids++, an alternative to k-medoids similar to its k-means counterpart, k-means++ which chooses initial medoids to begin with based on a probability distribution, as a sort of "informed randomness" or educated guess if you will. If such medoids are chosen with this rationale, the result is an improved runtime and better performance in clustering. The k-medoids++ algorithm is described as follows: The initial medoid is chosen randomly among all of the spatial points. For each spatial point 𝑝, compute the distance between 𝑝 and the nearest medoids which is termed as D(𝑝) and sum all the distances to 𝑆 . The next medoid is determined by using weighted probability distribution. Specifically, a random number 𝑅 between zero and the summed distance 𝑆 is chosen and the corresponding spatial point is the next medoid. Step (2) and Step (3) are repeated until 𝑘 medoids have been chosen. Now that we have appropriate first selections for medoids, the normal variation of k-medoids can be run.
Medoid : StatQuest k-means video used for visual reference in #Visualization_of_the_medoid-based_clustering_process section
Mixture model : In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information. Mixture models are used for clustering, under the name model-based clustering, and also for density estimation. Mixture models should not be confused with models for compositional data, i.e., data whose components are constrained to sum to a constant value (1, 100%, etc.). However, compositional models can be thought of as mixture models, where members of the population are sampled at random. Conversely, mixture models can be thought of as compositional models, where the total size reading population has been normalized to 1.
Mixture model : Identifiability refers to the existence of a unique characterization for any one of the models in the class (family) being considered. Estimation procedures may not be well-defined and asymptotic theory may not hold if a model is not identifiable.
Mixture model : Parametric mixture models are often used when we know the distribution Y and we can sample from X, but we would like to determine the ai and θi values. Such situations can arise in studies in which we sample from a population that is composed of several distinct subpopulations. It is common to think of probability mixture modeling as a missing data problem. One way to understand this is to assume that the data points under consideration have "membership" in one of the distributions we are using to model the data. When we start, this membership is unknown, or missing. The job of estimation is to devise appropriate parameters for the model functions we choose, with the connection to the data points being represented as their membership in the individual model distributions. A variety of approaches to the problem of mixture decomposition have been proposed, many of which focus on maximum likelihood methods such as expectation maximization (EM) or maximum a posteriori estimation (MAP). Generally these methods consider separately the questions of system identification and parameter estimation; methods to determine the number and functional form of components within a mixture are distinguished from methods to estimate the corresponding parameter values. Some notable departures are the graphical methods as outlined in Tarter and Lock and more recently minimum message length (MML) techniques such as Figueiredo and Jain and to some extent the moment matching pattern analysis routines suggested by McWilliam and Loh (2009).
Mixture model : In a Bayesian setting, additional levels can be added to the graphical model defining the mixture model. For example, in the common latent Dirichlet allocation topic model, the observations are sets of words drawn from D different documents and the K mixture components represent topics that are shared across documents. Each document has a different set of mixture weights, which specify the topics prevalent in that document. All sets of mixture weights share common hyperparameters. A very common extension is to connect the latent variables defining the mixture component identities into a Markov chain, instead of assuming that they are independent identically distributed random variables. The resulting model is termed a hidden Markov model and is one of the most common sequential hierarchical models. Numerous extensions of hidden Markov models have been developed; see the resulting article for more information.
Mixture model : Mixture distributions and the problem of mixture decomposition, that is the identification of its constituent components and the parameters thereof, has been cited in the literature as far back as 1846 (Quetelet in McLachlan, 2000) although common reference is made to the work of Karl Pearson (1894) as the first author to explicitly address the decomposition problem in characterising non-normal attributes of forehead to body length ratios in female shore crab populations. The motivation for this work was provided by the zoologist Walter Frank Raphael Weldon who had speculated in 1893 (in Tarter and Lock) that asymmetry in the histogram of these ratios could signal evolutionary divergence. Pearson's approach was to fit a univariate mixture of two normals to the data by choosing the five parameters of the mixture such that the empirical moments matched that of the model. While his work was successful in identifying two potentially distinct sub-populations and in demonstrating the flexibility of mixtures as a moment matching tool, the formulation required the solution of a 9th degree (nonic) polynomial which at the time posed a significant computational challenge. Subsequent works focused on addressing these problems, but it was not until the advent of the modern computer and the popularisation of Maximum Likelihood (MLE) parameterisation techniques that research really took off. Since that time there has been a vast body of research on the subject spanning areas such as fisheries research, agriculture, botany, economics, medicine, genetics, psychology, palaeontology, electrophoresis, finance, geology and zoology.
Mixture model : Nielsen, Frank (23 March 2012). "K-MLE: A fast algorithm for learning statistical mixture models". 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). pp. 869–872. arXiv:1203.5181. Bibcode:2012arXiv1203.5181N. doi:10.1109/ICASSP.2012.6288022. ISBN 978-1-4673-0046-9. S2CID 935615. The SOCR demonstrations of EM and Mixture Modeling Mixture modelling page (and the Snob program for Minimum Message Length (MML) applied to finite mixture models), maintained by D.L. Dowe. PyMix – Python Mixture Package, algorithms and data structures for a broad variety of mixture model based data mining applications in Python sklearn.mixture – A module from the scikit-learn Python library for learning Gaussian Mixture Models (and sampling from them), previously packaged with SciPy and now packaged as a SciKit GMM.m Matlab code for GMM Implementation GPUmix C++ implementation of Bayesian Mixture Models using EM and MCMC with 100x speed acceleration using GPGPU. [2] Matlab code for GMM Implementation using EM algorithm [3] jMEF: A Java open source library for learning and processing mixtures of exponential families (using duality with Bregman divergences). Includes a Matlab wrapper. Very Fast and clean C implementation of the Expectation Maximization (EM) algorithm for estimating Gaussian Mixture Models (GMMs). mclust is an R package for mixture modeling. dpgmm Pure Python Dirichlet process Gaussian mixture model implementation (variational). Gaussian Mixture Models Blog post on Gaussian Mixture Models trained via Expectation Maximization, with an implementation in Python.
WACA clustering algorithm : WACA is a clustering algorithm for dynamic networks. WACA (Weighted Application-aware Clustering Algorithm) uses a heuristic weight function for self-organized cluster creation. The election of clusterheads is based on local network information only. == References ==
Calibration (statistics) : There are two main uses of the term calibration in statistics that denote special types of statistical inference problems. Calibration can mean a reverse process to regression, where instead of a future dependent variable being predicted from known explanatory variables, a known observation of the dependent variables is used to predict a corresponding explanatory variable; procedures in statistical classification to determine class membership probabilities which assess the uncertainty of a given new observation belonging to each of the already established classes. In addition, calibration is used in statistics with the usual general meaning of calibration. For example, model calibration can be also used to refer to Bayesian inference about the value of a model's parameters, given some data set, or more generally to any type of fitting of a statistical model. As Philip Dawid puts it, "a forecaster is well calibrated if, for example, of those events to which he assigns a probability 30 percent, the long-run proportion that actually occurs turns out to be 30 percent."
Calibration (statistics) : Calibration in classification means transforming classifier scores into class membership probabilities. An overview of calibration methods for two-class and multi-class classification tasks is given by Gebel (2009). A classifier might separate the classes well, but be poorly calibrated, meaning that the estimated class probabilities are far from the true class probabilities. In this case, a calibration step may help improve the estimated probabilities. A variety of metrics exist that are aimed to measure the extent to which a classifier produces well-calibrated probabilities. Foundational work includes the Expected Calibration Error (ECE). Into the 2020s, variants include the Adaptive Calibration Error (ACE) and the Test-based Calibration Error (TCE), which address limitations of the ECE metric that may arise when classifier scores concentrate on narrow subset of the [0,1] range. A 2020s advancement in calibration assessment is the introduction of the Estimated Calibration Index (ECI). The ECI extends the concepts of the Expected Calibration Error (ECE) to provide a more nuanced measure of a model's calibration, particularly addressing overconfidence and underconfidence tendencies. Originally formulated for binary settings, the ECI has been adapted for multiclass settings, offering both local and global insights into model calibration. This framework aims to overcome some of the theoretical and interpretative limitations of existing calibration metrics. Through a series of experiments, Famiglini et al. demonstrate the framework's effectiveness in delivering a more accurate understanding of model calibration levels and discuss strategies for mitigating biases in calibration assessment. An online tool has been proposed to compute both ECE and ECI. The following univariate calibration methods exist for transforming classifier scores into class membership probabilities in the two-class case: Assignment value approach, see Garczarek (2002) Bayes approach, see Bennett (2002) Isotonic regression, see Zadrozny and Elkan (2002) Platt scaling (a form of logistic regression), see Lewis and Gale (1994) and Platt (1999) Bayesian Binning into Quantiles (BBQ) calibration, see Naeini, Cooper, Hauskrecht (2015) Beta calibration, see Kull, Filho, Flach (2017)
Calibration (statistics) : The calibration problem in regression is the use of known data on the observed relationship between a dependent variable and an independent variable to make estimates of other values of the independent variable from new observations of the dependent variable. This can be known as "inverse regression"; there is also sliced inverse regression. The following multivariate calibration methods exist for transforming classifier scores into class membership probabilities in the case with classes count greater than two: Reduction to binary tasks and subsequent pairwise coupling, see Hastie and Tibshirani (1998) Dirichlet calibration, see Gebel (2009)

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