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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 sense) 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 Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
data analysis Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enc ...
, used in many fields, including pattern recognition,
image analysis Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as soph ...
,
information retrieval Information retrieval (IR) in computing and information science is the process of obtaining information system resources that are relevant to an information need from a collection of those resources. Searches can be based on full-text or other c ...
,
bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
, data compression,
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
and
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
. Cluster analysis itself is not one specific
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
, but the general task to be solved. 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 Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making that is concerned with ...
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 is a number of terms with similar meanings, including ''automatic classification'', ''
numerical taxonomy Numerical taxonomy is a classification system in biological systematics which deals with the grouping by numerical methods of taxonomic units based on their character states. It aims to create a taxonomy using numeric algorithms like cluster ...
'', ''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 was 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 Personality psychology is a branch of psychology that examines personality and its variation among individuals. It aims to show how people are individually different due to psychological forces. Its areas of focus include: * construction of a ...
.


Definition

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: * ''s'': for example,
hierarchical clustering In data mining and statistics, hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into t ...
builds models based on distance connectivity. * ''s'': for example, the k-means algorithm represents each cluster by a single mean vector. * ''s'': clusters are modeled using statistical distributions, such as multivariate normal distributions used by the expectation-maximization algorithm. * ''s'': for example, DBSCAN and
OPTICS Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
defines clusters as connected dense regions in the data space. * ''s'': in biclustering (also known as co-clustering or two-mode-clustering), clusters are modeled with both cluster members and relevant attributes. * ''s'': some algorithms do not provide a refined model for their results and just provide the grouping information. * ''s'': 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 In the psychology of motivation, balance theory is a theory of attitude change, proposed by Fritz Heider. It conceptualizes the cognitive consistency motive as a drive toward psychological balance. The consistency motive is the urge to maintain ...
, 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. * ''s'': the most well known unsupervised
neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
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: * ': each object belongs to a cluster or not * ' (also: '): 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: * ': each object belongs to exactly one cluster * ': objects can also belong to no cluster, and are considered outliers * ' (also: ''alternative clustering'', ''multi-view clustering''): objects may belong to more than one cluster; usually involving hard clusters * ': objects that belong to a child cluster also belong to the parent cluster * ': while an overlapping clustering, within a uniquely defined subspace, clusters are not expected to overlap


Algorithms

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." 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.


Connectivity-based clustering (hierarchical clustering)

Connectivity-based clustering, also known as ''
hierarchical clustering In data mining and statistics, hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into t ...
'', is based on the core idea of objects being more related to nearby objects than to objects farther away. These algorithms connect "objects" to form "clusters" based on their distance. A cluster can be described largely by the maximum distance needed to connect parts of the cluster. At different distances, different clusters will form, which can be represented using a
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. ...
, which explains where the common name "
hierarchical clustering In data mining and statistics, hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into t ...
" comes from: these algorithms do not provide a single partitioning of the data set, but instead provide an extensive hierarchy of clusters that merge with each other at certain distances. In a dendrogram, the y-axis marks the distance at which the clusters merge, while the objects are placed along the x-axis such that the clusters don't mix. Connectivity-based clustering is a whole family of methods that differ by the way distances are computed. Apart from the usual choice of distance functions, the user also needs to decide on the linkage criterion (since a cluster consists of multiple objects, there are multiple candidates to compute the distance) to use. Popular choices are known as single-linkage clustering (the minimum of object distances), complete linkage clustering (the maximum of object distances), and
UPGMA UPGMA (unweighted pair group method with arithmetic mean) is a simple agglomerative (bottom-up) hierarchical clustering method. The method is generally attributed to Sokal and Michener. The UPGMA method is similar to its ''weighted'' variant, the ...
or WPGMA ("Unweighted or Weighted Pair Group Method with Arithmetic Mean", also known as average linkage clustering). Furthermore, hierarchical clustering can be agglomerative (starting with single elements and aggregating them into clusters) or divisive (starting with the complete data set and dividing it into partitions). These methods will not produce a unique partitioning of the data set, but a hierarchy from which the user still needs to choose appropriate clusters. They are not very robust towards outliers, which will either show up as additional clusters or even cause other clusters to merge (known as "chaining phenomenon", in particular with single-linkage clustering). In the general case, the complexity is \mathcal(n^3) for agglomerative clustering and \mathcal(2^) for divisive clustering, which makes them too slow for large data sets. For some special cases, optimal efficient methods (of complexity \mathcal(n^2)) are known: SLINK for single-linkage and CLINK for complete-linkage clustering. File:SLINK-Gaussian-data.svg, Single-linkage on Gaussian data. At 35 clusters, the biggest cluster starts fragmenting into smaller parts, while before it was still connected to the second largest due to the single-link effect. File:SLINK-density-data.svg, Single-linkage on density-based clusters. 20 clusters extracted, most of which contain single elements, since linkage clustering does not have a notion of "noise".


Centroid-based clustering

In centroid-based clustering, each cluster is represented by a central vector, which is not necessarily a member of the data set. When the number of clusters is fixed to ''k'', ''k''-means clustering gives a formal definition as an optimization problem: find the ''k'' cluster centers and assign the objects to the nearest cluster center, such that the squared distances from the cluster are minimized. The optimization problem itself is known to be
NP-hard In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
, and thus the common approach is to search only for approximate solutions. A particularly well known approximate method is Lloyd's algorithm, often just referred to as "''k-means algorithm''" (although another algorithm introduced this name). It does however only find a local optimum, and is commonly run multiple times with different random initializations. Variations of ''k''-means often include such optimizations as choosing the best of multiple runs, but also restricting the centroids to members of the data set ( ''k''-medoids), choosing
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
s ( ''k''-medians clustering), choosing the initial centers less randomly ( ''k''-means++) or allowing a fuzzy cluster assignment ( fuzzy c-means). Most ''k''-means-type algorithms require the number of clusters – ''k'' – to be specified in advance, which is considered to be one of the biggest drawbacks of these algorithms. Furthermore, the algorithms prefer clusters of approximately similar size, as they will always assign an object to the nearest centroid. This often leads to incorrectly cut borders of clusters (which is not surprising since the algorithm optimizes cluster centers, not cluster borders). K-means has a number of interesting theoretical properties. First, it partitions the data space into a structure known as a
Voronoi diagram In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...
. Second, it is conceptually close to nearest neighbor classification, and as such is popular in
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
. Third, it can be seen as a variation of model based clustering, and Lloyd's algorithm as a variation of the Expectation-maximization algorithm for this model discussed below. File:KMeans-Gaussian-data.svg, ''k''-means separates data into Voronoi cells, which assumes equal-sized clusters (not adequate here) File:KMeans-density-data.svg, ''k''-means cannot represent density-based clusters Centroid-based clustering problems such as ''k''-means and ''k''-medoids are special cases of the uncapacitated, metric facility location problem, a canonical problem in the operations research and computational geometry communities. In a basic facility location problem (of which there are numerous variants that model more elaborate settings), the task is to find the best warehouse locations to optimally service a given set of consumers. One may view "warehouses" as cluster centroids and "consumer locations" as the data to be clustered. This makes it possible to apply the well-developed algorithmic solutions from the facility location literature to the presently considered centroid-based clustering problem.


Distribution-based clustering

The clustering model most closely related to statistics is based on distribution models. Clusters can then easily be defined as objects belonging most likely to the same distribution. A convenient property of this approach is that this closely resembles the way artificial data sets are generated: by sampling random objects from a distribution. While the theoretical foundation of these methods is excellent, they suffer from one key problem known as
overfitting mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitt ...
, unless constraints are put on the model complexity. A more complex model will usually be able to explain the data better, which makes choosing the appropriate model complexity inherently difficult. One prominent method is known as Gaussian mixture models (using the expectation-maximization algorithm). Here, the data set is usually modeled with a fixed (to avoid overfitting) number of
Gaussian distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
s that are initialized randomly and whose parameters are iteratively optimized to better fit the data set. This will converge to a local optimum, so multiple runs may produce different results. In order to obtain a hard clustering, objects are often then assigned to the Gaussian distribution they most likely belong to; for soft clusterings, this is not necessary. Distribution-based clustering produces complex models for clusters that can capture correlation and dependence between attributes. However, these algorithms put an extra burden on the user: for many real data sets, there may be no concisely defined mathematical model (e.g. assuming Gaussian distributions is a rather strong assumption on the data). File:EM-Gaussian-data.svg, On Gaussian-distributed data, EM works well, since it uses Gaussians for modelling clusters File:EM-density-data.svg, Density-based clusters cannot be modeled using Gaussian distributions


Density-based clustering

In density-based clustering, clusters are defined as areas of higher density than the remainder of the data set. Objects in sparse areas – that are required to separate clusters – are usually considered to be noise and border points. The most popular density based clustering method is DBSCAN. In contrast to many newer methods, it features a well-defined cluster model called "density-reachability". Similar to linkage based clustering, it is based on connecting points within certain distance thresholds. However, it only connects points that satisfy a density criterion, in the original variant defined as a minimum number of other objects within this radius. A cluster consists of all density-connected objects (which can form a cluster of an arbitrary shape, in contrast to many other methods) plus all objects that are within these objects' range. Another interesting property of DBSCAN is that its complexity is fairly low – it requires a linear number of range queries on the database – and that it will discover essentially the same results (it is
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
for core and noise points, but not for border points) in each run, therefore there is no need to run it multiple times.
OPTICS Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
is a generalization of DBSCAN that removes the need to choose an appropriate value for the range parameter \varepsilon, and produces a hierarchical result related to that of linkage clustering. DeLi-Clu, Density-Link-Clustering combines ideas from single-linkage clustering and OPTICS, eliminating the \varepsilon parameter entirely and offering performance improvements over OPTICS by using an
R-tree R-trees are tree data structures used for spatial access methods, i.e., for indexing multi-dimensional information such as geographical coordinates, rectangles or polygons. The R-tree was proposed by Antonin Guttman in 1984 and has found sig ...
index. The key drawback of DBSCAN and
OPTICS Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
is that they expect some kind of density drop to detect cluster borders. On data sets with, for example, overlapping Gaussian distributions – a common use case in artificial data – the cluster borders produced by these algorithms will often look arbitrary, because the cluster density decreases continuously. On a data set consisting of mixtures of Gaussians, these algorithms are nearly always outperformed by methods such as EM clustering that are able to precisely model this kind of data. Mean-shift is a clustering approach where each object is moved to the densest area in its vicinity, based on kernel density estimation. Eventually, objects converge to local maxima of density. Similar to k-means clustering, these "density attractors" can serve as representatives for the data set, but mean-shift can detect arbitrary-shaped clusters similar to DBSCAN. Due to the expensive iterative procedure and density estimation, mean-shift is usually slower than DBSCAN or k-Means. Besides that, the applicability of the mean-shift algorithm to multidimensional data is hindered by the unsmooth behaviour of the kernel density estimate, which results in over-fragmentation of cluster tails. File:DBSCAN-density-data.svg, Density-based clustering with DBSCAN. File:DBSCAN-Gaussian-data.svg, DBSCAN assumes clusters of similar density, and may have problems separating nearby clusters File:OPTICS-Gaussian-data.svg,
OPTICS Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
is a DBSCAN variant, improving handling of different densities clusters


Grid-based clustering

The grid-based technique is used for a multi-dimensional data set. In this technique, we create a grid structure, and the comparison is performed on grids (also known as cells). The grid-based technique is fast and has low computational complexity. There are two types of grid-based clustering methods: STING and CLIQUE. Steps involved in grid-based clustering
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
are: # Divide data space into a finite number of cells. # Randomly select a cell ‘c’, where c should not be traversed beforehand. # Calculate the density of ‘c’ # If the density of ‘c’ greater than threshold density ## Mark cell ‘c’ as a new cluster ## Calculate the density of all the neighbors of ‘c’ ## If the density of a neighboring cell is greater than threshold density then, add the cell in the cluster and repeat steps 4.2 and 4.3 till there is no neighbor with a density greater than threshold density. # Repeat steps 2,3 and 4 till all the cells are traversed. # Stop.


Recent developments

In recent years, considerable effort has been put into improving the performance of existing algorithms. Among them are ''CLARANS'', and ''
BIRCH A birch is a thin-leaved deciduous hardwood tree of the genus ''Betula'' (), in the family Betulaceae, which also includes alders, hazels, and hornbeams. It is closely related to the beech- oak family Fagaceae. The genus ''Betula'' cont ...
''. With the recent need to process larger and larger data sets (also known as
big data Though used sometimes loosely partly because of a lack of formal definition, the interpretation that seems to best describe Big data is the one associated with large body of information that we could not comprehend when used only in smaller am ...
), the willingness to trade semantic meaning of the generated clusters for performance has been increasing. This led to the development of pre-clustering methods such as canopy clustering, which can process huge data sets efficiently, but the resulting "clusters" are merely a rough pre-partitioning of the data set to then analyze the partitions with existing slower methods such as k-means clustering. For high-dimensional data, many of the existing methods fail due to the curse of dimensionality, which renders particular distance functions problematic in high-dimensional spaces. This led to new clustering algorithms for high-dimensional data that focus on subspace clustering (where only some attributes are used, and cluster models include the relevant attributes for the cluster) and correlation clustering that also looks for arbitrary rotated ("correlated") subspace clusters that can be modeled by giving a correlation of their attributes. Examples for such clustering algorithms are CLIQUE and SUBCLU. Ideas from density-based clustering methods (in particular the DBSCAN/
OPTICS Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
family of algorithms) have been adapted to subspace clustering (HiSC, hierarchical subspace clustering and DiSH) and correlation clustering (HiCO, hierarchical correlation clustering, 4C using "correlation connectivity" and ERiC exploring hierarchical density-based correlation clusters). Several different clustering systems based on
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
have been proposed. One is Marina Meilă's ''
variation of information In probability theory and information theory, the variation of information or shared information distance is a measure of the distance between two clusterings ( partitions of elements). It is closely related to mutual information; indeed, it is ...
'' metric; another provides hierarchical clustering. Using genetic algorithms, a wide range of different fit-functions can be optimized, including mutual information. Also belief propagation, a recent development in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
and statistical physics, has led to the creation of new types of clustering algorithms.


Evaluation and assessment

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.


Internal evaluation

When a clustering result is evaluated based on the data that was clustered itself, this is called internal evaluation. These methods usually assign the best score to the algorithm that produces clusters with high similarity within a cluster and low similarity between clusters. One drawback of using internal criteria in cluster evaluation is that high scores on an internal measure do not necessarily result in effective information retrieval applications. Additionally, this evaluation is biased towards algorithms that use the same cluster model. For example, k-means clustering naturally optimizes object distances, and a distance-based internal criterion will likely overrate the resulting clustering. Therefore, the internal evaluation measures are best suited to get some insight into situations where one algorithm performs better than another, but this shall not imply that one algorithm produces more valid results than another. Validity as measured by such an index depends on the claim that this kind of structure exists in the data set. An algorithm designed for some kind of models has no chance if the data set contains a radically different set of models, or if the evaluation measures a radically different criterion. For example, k-means clustering can only find convex clusters, and many evaluation indexes assume convex clusters. On a data set with non-convex clusters neither the use of ''k''-means, nor of an evaluation criterion that assumes convexity, is sound. More than a dozen of internal evaluation measures exist, usually based on the intuition that items in the same cluster should be more similar than items in different clusters. For example, the following methods can be used to assess the quality of clustering algorithms based on internal criterion: *
Davies–Bouldin index The Davies–Bouldin index (DBI), introduced by David L. Davies and Donald W. Bouldin in 1979, is a metric for evaluating clustering algorithms. This is an internal evaluation scheme, where the validation of how well the clustering has been ...
:The
Davies–Bouldin index The Davies–Bouldin index (DBI), introduced by David L. Davies and Donald W. Bouldin in 1979, is a metric for evaluating clustering algorithms. This is an internal evaluation scheme, where the validation of how well the clustering has been ...
can be calculated by the following formula: :: DB = \frac \sum_^ \max_\left(\frac \right) :where ''n'' is the number of clusters, c_i is the centroid of cluster i, \sigma_i is the average distance of all elements in cluster i to centroid c_i, and d(c_i,c_j) is the distance between centroids c_i and c_j. Since algorithms that produce clusters with low intra-cluster distances (high intra-cluster similarity) and high inter-cluster distances (low inter-cluster similarity) will have a low Davies–Bouldin index, the clustering algorithm that produces a collection of clusters with the smallest
Davies–Bouldin index The Davies–Bouldin index (DBI), introduced by David L. Davies and Donald W. Bouldin in 1979, is a metric for evaluating clustering algorithms. This is an internal evaluation scheme, where the validation of how well the clustering has been ...
is considered the best algorithm based on this criterion. * Dunn index :The Dunn index aims to identify dense and well-separated clusters. It is defined as the ratio between the minimal inter-cluster distance to maximal intra-cluster distance. For each cluster partition, the Dunn index can be calculated by the following formula: :: D = \frac \,, :where ''d''(''i'',''j'') represents the distance between clusters ''i'' and ''j'', and ''d'' '(''k'') measures the intra-cluster distance of cluster ''k''. The inter-cluster distance ''d''(''i'',''j'') between two clusters may be any number of distance measures, such as the distance between the centroids of the clusters. Similarly, the intra-cluster distance ''d'' '(''k'') may be measured in a variety ways, such as the maximal distance between any pair of elements in cluster ''k''. Since internal criterion seek clusters with high intra-cluster similarity and low inter-cluster similarity, algorithms that produce clusters with high Dunn index are more desirable. * Silhouette coefficient :The silhouette coefficient contrasts the average distance to elements in the same cluster with the average distance to elements in other clusters. Objects with a high silhouette value are considered well clustered, objects with a low value may be outliers. This index works well with ''k''-means clustering, and is also used to determine the optimal number of clusters.


External evaluation

In external evaluation, clustering results are evaluated based on data that was not used for clustering, such as known class labels and external benchmarks. Such benchmarks consist of a set of pre-classified items, and these sets are often created by (expert) humans. Thus, the benchmark sets can be thought of as a
gold standard A gold standard is a monetary system in which the standard economic unit of account is based on a fixed quantity of gold. The gold standard was the basis for the international monetary system from the 1870s to the early 1920s, and from th ...
for evaluation. These types of evaluation methods measure how close the clustering is to the predetermined benchmark classes. However, it has recently been discussed whether this is adequate for real data, or only on synthetic data sets with a factual ground truth, since classes can contain internal structure, the attributes present may not allow separation of clusters or the classes may contain anomalies. Additionally, from a knowledge discovery point of view, the reproduction of known knowledge may not necessarily be the intended result. In the special scenario of constrained clustering, where meta information (such as class labels) is used already in the clustering process, the hold-out of information for evaluation purposes is non-trivial. A number of measures are adapted from variants used to evaluate classification tasks. In place of counting the number of times a class was correctly assigned to a single data point (known as true positives), such ''pair counting'' metrics assess whether each pair of data points that is truly in the same cluster is predicted to be in the same cluster. As with internal evaluation, several external evaluation measures exist, for example: * Purity: Purity is a measure of the extent to which clusters contain a single class. Its calculation can be thought of as follows: For each cluster, count the number of data points from the most common class in said cluster. Now take the sum over all clusters and divide by the total number of data points. Formally, given some set of clusters M and some set of classes D, both partitioning N data points, purity can be defined as: :: \frac\sum_\max_ :This measure doesn't penalize having many clusters, and more clusters will make it easier to produce a high purity. A purity score of 1 is always possible by putting each data point in its own cluster. Also, purity doesn't work well for imbalanced data, where even poorly performing clustering algorithms will give a high purity value. For example, if a size 1000 dataset consists of two classes, one containing 999 points and the other containing 1 point, then every possible partition will have a purity of at least 99.9%. * Rand index :The Rand index computes how similar the clusters (returned by the clustering algorithm) are to the benchmark classifications. It can be computed using the following formula: :: RI = \frac :where TP is the number of true positives, TN is the number of true negatives, FP is the number of false positives, and FN is the number of false negatives. The instances being counted here are the number of correct ''pairwise'' assignments. That is, TP is the number of pairs of points that are clustered together in the predicted partition and in the ground truth partition, FP is the number of pairs of points that are clustered together in the predicted partition but not in the ground truth partition etc. If the dataset is of size N, then TP + TN + FP + FN = \binom. One issue with the Rand index is that false positives and false negatives are equally weighted. This may be an undesirable characteristic for some clustering applications. The F-measure addresses this concern, as does the chance-corrected adjusted Rand index. * F-measure :The F-measure can be used to balance the contribution of false negatives by weighting
recall Recall may refer to: * Recall (bugle call), a signal to stop * Recall (information retrieval), a statistical measure * ''ReCALL'' (journal), an academic journal about computer-assisted language learning * Recall (memory) * ''Recall'' (Overwatc ...
through a parameter \beta \geq 0. Let precision and
recall Recall may refer to: * Recall (bugle call), a signal to stop * Recall (information retrieval), a statistical measure * ''ReCALL'' (journal), an academic journal about computer-assisted language learning * Recall (memory) * ''Recall'' (Overwatc ...
(both external evaluation measures in themselves) be defined as follows: :: P = \frac :: R = \frac :where P is the precision rate and R is the
recall Recall may refer to: * Recall (bugle call), a signal to stop * Recall (information retrieval), a statistical measure * ''ReCALL'' (journal), an academic journal about computer-assisted language learning * Recall (memory) * ''Recall'' (Overwatc ...
rate. We can calculate the F-measure by using the following formula: :: F_ = \frac :When \beta=0, F_=P. In other words,
recall Recall may refer to: * Recall (bugle call), a signal to stop * Recall (information retrieval), a statistical measure * ''ReCALL'' (journal), an academic journal about computer-assisted language learning * Recall (memory) * ''Recall'' (Overwatc ...
has no impact on the F-measure when \beta=0, and increasing \beta allocates an increasing amount of weight to recall in the final F-measure. :Also TN is not taken into account and can vary from 0 upward without bound. * Jaccard index :The Jaccard index is used to quantify the similarity between two datasets. The Jaccard index takes on a value between 0 and 1. An index of 1 means that the two dataset are identical, and an index of 0 indicates that the datasets have no common elements. The Jaccard index is defined by the following formula: :: J(A,B) = \frac = \frac :This is simply the number of unique elements common to both sets divided by the total number of unique elements in both sets. :Also TN is not taken into account and can vary from 0 upward without bound. * Dice index :The Dice symmetric measure doubles the weight on TP while still ignoring TN: :: DSC = \frac * Fowlkes–Mallows index :The Fowlkes–Mallows index computes the similarity between the clusters returned by the clustering algorithm and the benchmark classifications. The higher the value of the Fowlkes–Mallows index the more similar the clusters and the benchmark classifications are. It can be computed using the following formula: :: FM = \sqrt :where TP is the number of true positives, FP is the number of false positives, and FN is the number of false negatives. The FM index is the geometric mean of the precision and
recall Recall may refer to: * Recall (bugle call), a signal to stop * Recall (information retrieval), a statistical measure * ''ReCALL'' (journal), an academic journal about computer-assisted language learning * Recall (memory) * ''Recall'' (Overwatc ...
P and R, and is thus also known as the G-measure, while the F-measure is their harmonic mean. Moreover, precision and
recall Recall may refer to: * Recall (bugle call), a signal to stop * Recall (information retrieval), a statistical measure * ''ReCALL'' (journal), an academic journal about computer-assisted language learning * Recall (memory) * ''Recall'' (Overwatc ...
are also known as Wallace's indices B^I and B^. Chance normalized versions of recall, precision and G-measure correspond to Informedness, Markedness and Matthews Correlation and relate strongly to
Kappa Kappa (uppercase Κ, lowercase κ or cursive ; el, κάππα, ''káppa'') is the 10th letter of the Greek alphabet, representing the voiceless velar plosive sound in Ancient and Modern Greek. In the system of Greek numerals, has a value ...
. * The
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
is an information theoretic measure of how much information is shared between a clustering and a ground-truth classification that can detect a non-linear similarity between two clusterings. Normalized mutual information is a family of corrected-for-chance variants of this that has a reduced bias for varying cluster numbers. * Confusion matrix :A confusion matrix can be used to quickly visualize the results of a classification (or clustering) algorithm. It shows how different a cluster is from the gold standard cluster.


Cluster tendency

To measure cluster tendency is to measure to what degree clusters exist in the data to be clustered, and may be performed as an initial test, before attempting clustering. One way to do this is to compare the data against random data. On average, random data should not have clusters. * Hopkins statistic :There are multiple formulations of the Hopkins statistic. A typical one is as follows. Let X be the set of n data points in d dimensional space. Consider a random sample (without replacement) of m \ll n data points with members x_i. Also generate a set Y of m uniformly randomly distributed data points. Now define two distance measures, u_i to be the distance of y_i \in Y from its nearest neighbor in X and w_i to be the distance of x_i \in X from its nearest neighbor in X. We then define the Hopkins statistic as: :: H=\frac \,, :With this definition, uniform random data should tend to have values near to 0.5, and clustered data should tend to have values nearer to 1. :However, data containing just a single Gaussian will also score close to 1, as this statistic measures deviation from a ''uniform'' distribution, not multimodality, making this statistic largely useless in application (as real data never is remotely uniform).


Applications


Biology, computational biology and bioinformatics

;
Plant Plants are predominantly photosynthetic eukaryotes of the kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all current definitions of Plantae excl ...
and
animal Animals are multicellular, eukaryotic organisms in the biological kingdom Animalia. With few exceptions, animals consume organic material, breathe oxygen, are able to move, can reproduce sexually, and go through an ontogenetic stage ...
ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
:Cluster analysis is used to describe and to make spatial and temporal comparisons of communities (assemblages) of organisms in heterogeneous environments. It is also used in plant systematics to generate artificial
phylogenies A phylogenetic tree (also phylogeny or evolutionary tree Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA.) is a branching diagram or a tree showing the evolutionary relationships among various biological spec ...
or clusters of organisms (individuals) at the species, genus or higher level that share a number of attributes. ; Transcriptomics :Clustering is used to build groups of
genes In biology, the word gene (from , ; "...Wilhelm Johannsen coined the word gene to describe the Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a ba ...
with related expression patterns (also known as coexpressed genes) as in HCS clustering algorithm. Often such groups contain functionally related proteins, such as
enzyme Enzymes () are proteins that act as biological catalysts by accelerating chemical reactions. The molecules upon which enzymes may act are called substrates, and the enzyme converts the substrates into different molecules known as products ...
s for a specific pathway, or genes that are co-regulated. High throughput experiments using expressed sequence tags (ESTs) or
DNA microarray A DNA microarray (also commonly known as DNA chip or biochip) is a collection of microscopic DNA spots attached to a solid surface. Scientists use DNA microarrays to measure the expression levels of large numbers of genes simultaneously or to ...
s can be a powerful tool for genome annotationa general aspect of
genomics Genomics is an interdisciplinary field of biology focusing on the structure, function, evolution, mapping, and editing of genomes. A genome is an organism's complete set of DNA, including all of its genes as well as its hierarchical, three-dim ...
. ; Sequence analysis :
Sequence clustering In bioinformatics, sequence clustering algorithms attempt to group biological sequences that are somehow related. The sequences can be either of genomic, " transcriptomic" ( ESTs) or protein origin. For proteins, homologous sequences are typicall ...
is used to group homologous sequences into
gene families A gene family is a set of several similar genes, formed by duplication of a single original gene, and generally with similar biochemical functions. One such family are the genes for human hemoglobin subunits; the ten genes are in two clusters on ...
. This is a very important concept in
bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
, and
evolutionary biology Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life ...
in general. See evolution by gene duplication. ; High-throughput genotyping platforms :Clustering algorithms are used to automatically assign genotypes. ; Human genetic clustering :The similarity of genetic data is used in clustering to infer population structures.


Medicine Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...

;
Medical imaging Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to re ...
:On
PET scan Positron emission tomography (PET) is a functional imaging technique that uses radioactive substances known as radiotracers to visualize and measure changes in metabolic processes, and in other physiological activities including blood flo ...
s, cluster analysis can be used to differentiate between different types of tissue in a three-dimensional image for many different purposes. ; Analysis of antimicrobial activity :Cluster analysis can be used to analyse patterns of antibiotic resistance, to classify antimicrobial compounds according to their mechanism of action, to classify antibiotics according to their antibacterial activity. ; IMRT segmentation :Clustering can be used to divide a fluence map into distinct regions for conversion into deliverable fields in MLC-based Radiation Therapy.


Business and marketing

;
Market research Market research is an organized effort to gather information about target markets and customers: know about them, starting with who they are. It is an important component of business strategy and a major factor in maintaining competitiveness. Ma ...
:Cluster analysis is widely used in market research when working with multivariate data from surveys and test panels. Market researchers use cluster analysis to partition the general
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction usi ...
of
consumer A consumer is a person or a group who intends to order, or uses purchased goods, products, or services primarily for personal, social, family, household and similar needs, who is not directly related to entrepreneurial or business activities. ...
s into market segments and to better understand the relationships between different groups of consumers/potential customers, and for use in
market segmentation In marketing, market segmentation is the process of dividing a broad consumer or business market, normally consisting of existing and potential customers, into sub-groups of consumers (known as ''segments'') based on some type of shared charact ...
, product positioning,
new product development In business and engineering, new product development (NPD) covers the complete process of bringing a new product to market, renewing an existing product or introducing a product in a new market. A central aspect of NPD is product design, along ...
and selecting test markets. ; Grouping of shopping items :Clustering can be used to group all the shopping items available on the web into a set of unique products. For example, all the items on eBay can be grouped into unique products (eBay does not have the concept of a SKU).


World Wide Web The World Wide Web (WWW), commonly known as the Web, is an information system enabling documents and other web resources to be accessed over the Internet. Documents and downloadable media are made available to the network through web ...

; Social network analysis :In the study of
social network A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods fo ...
s, clustering may be used to recognize communities within large groups of people. ; Search result grouping :In the process of intelligent grouping of the files and websites, clustering may be used to create a more relevant set of search results compared to normal search engines like
Google Google LLC () is an American Multinational corporation, multinational technology company focusing on Search Engine, search engine technology, online advertising, cloud computing, software, computer software, quantum computing, e-commerce, ar ...
. There are currently a number of web-based clustering tools such as
Clusty Yippy was a metasearch engine that grouped searched results into clusters. It was originally developed and released by Vivísimo in 2004 under the name Clusty, before Vivisimo was later acquired by IBM and Yippy was sold in 2010 to a company now c ...
. It also may be used to return a more comprehensive set of results in cases where a search term could refer to vastly different things. Each distinct use of the term corresponds to a unique cluster of results, allowing a ranking algorithm to return comprehensive results by picking the top result from each cluster. ; Slippy map optimization : Flickr's map of photos and other map sites use clustering to reduce the number of markers on a map. This makes it both faster and reduces the amount of visual clutter.


Computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

;
Software evolution Software evolution is the continual development of a piece of software after its initial release to address changing stakeholder and/or market requirements. Software evolution is important because organizations invest large amounts of money in the ...
:Clustering is useful in software evolution as it helps to reduce legacy properties in code by reforming functionality that has become dispersed. It is a form of restructuring and hence is a way of direct preventative maintenance. ; Image segmentation :Clustering can be used to divide a
digital Digital usually refers to something using discrete digits, often binary digits. Technology and computing Hardware *Digital electronics, electronic circuits which operate using digital signals ** Digital camera, which captures and stores digital ...
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
into distinct regions for border detection or object recognition.Bewley, A., & Upcroft, B. (2013). Advantages of Exploiting Projection Structure for Segmenting Dense 3D Point Clouds. In Australian Conference on Robotics and Automatio

/ref> ; Evolutionary algorithms :Clustering may be used to identify different niches within the population of an evolutionary algorithm so that reproductive opportunity can be distributed more evenly amongst the evolving species or subspecies. ; Recommender systems :Recommender systems are designed to recommend new items based on a user's tastes. They sometimes use clustering algorithms to predict a user's preferences based on the preferences of other users in the user's cluster. ; Markov chain Monte Carlo methods :Clustering is often utilized to locate and characterize extrema in the target distribution. ; Anomaly detection :Anomalies/outliers are typically – be it explicitly or implicitly – defined with respect to clustering structure in data. ;
Natural language processing Natural language processing (NLP) is an interdisciplinary subfield of linguistics, computer science, and artificial intelligence concerned with the interactions between computers and human language, in particular how to program computers to proc ...
:Clustering can be used to resolve lexical ambiguity. ; DevOps :Clustering has been used to analyse the effectiveness of DevOps teams.
/ref>


Social science

; Sequence analysis in social sciences :Cluster analysis is used to identify patterns of family life trajectories, professional careers, and daily or weekly time use for example. ; Crime analysis :Cluster analysis can be used to identify areas where there are greater incidences of particular types of crime. By identifying these distinct areas or "hot spots" where a similar crime has happened over a period of time, it is possible to manage law enforcement resources more effectively. ; Educational data mining :Cluster analysis is for example used to identify groups of schools or students with similar properties. ; Typologies :From poll data, projects such as those undertaken by the Pew Research Center use cluster analysis to discern typologies of opinions, habits, and demographics that may be useful in politics and marketing.


Others

; Field robotics :Clustering algorithms are used for robotic situational awareness to track objects and detect outliers in sensor data. ; Mathematical chemistry :To find structural similarity, etc., for example, 3000 chemical compounds were clustered in the space of 90 topological indices. ;
Climatology Climatology (from Greek , ''klima'', "place, zone"; and , '' -logia'') or climate science is the scientific study of Earth's climate, typically defined as weather conditions averaged over a period of at least 30 years. This modern field of stu ...
:To find weather regimes or preferred sea level pressure atmospheric patterns. ; Finance :Cluster analysis has been used to cluster stocks into sectors. ;Petroleum geology :Cluster analysis is used to reconstruct missing bottom hole core data or missing log curves in order to evaluate reservoir properties. ; Geochemistry :The clustering of chemical properties in different sample locations.


See also


Specialized types of cluster analysis

* Automatic clustering algorithms * Balanced clustering *
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 ca ...
*
Conceptual clustering Conceptual clustering is a machine learning paradigm for unsupervised classification that has been defined by Ryszard S. Michalski in 1980 (Fisher 1987, Michalski 1980) and developed mainly during the 1980s. It is distinguished from ordinary dat ...
* Consensus clustering * Constrained clustering * Community detection *
Data stream clustering In computer science, data stream clustering is defined as the clustering of data that arrive continuously such as telephone records, multimedia data, financial transactions etc. Data stream clustering is usually studied as a streaming algorithm an ...
* HCS clustering *
Sequence clustering In bioinformatics, sequence clustering algorithms attempt to group biological sequences that are somehow related. The sequences can be either of genomic, " transcriptomic" ( ESTs) or protein origin. For proteins, homologous sequences are typicall ...
* Spectral clustering


Techniques used in cluster analysis

*
Artificial neural network Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains. An ANN is based on a collection of connected unit ...
(ANN) * Nearest neighbor search * Neighbourhood components analysis *
Latent class analysis In statistics, a latent class model (LCM) relates a set of observed (usually discrete) multivariate variables to a set of latent variables. It is a type of latent variable model. It is called a latent class model because the latent variable is disc ...
* Affinity propagation


Data projection and preprocessing

* Dimension reduction * Principal component analysis * Multidimensional scaling


Other

* Cluster-weighted modeling * Curse of dimensionality *
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 ...
* Parallel coordinates * Structured data analysis


References

{{DEFAULTSORT:Cluster Analysis Data mining Geostatistics