Data clustering
Encyclopedia
Cluster analysis or clustering is the task of assigning a set of objects into groups (called clusters) so that the objects in the same cluster are more similar (in some sense or another) to each other than to those in other clusters.
Clustering is a main task of explorative data mining
, and a common technique for statistical
data analysis
used in many fields, including machine learning
, pattern recognition
, image analysis
, information retrieval
, and bioinformatics
.
Cluster analysis itself is not one specific algorithm
, but the general task to be solved. It can be achieved by various algorithms that differ significantly in their notion of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with low distances among the cluster members, dense areas of the data space, intervals or particular statistical distributions. The appropriate clustering algorithm and parameter settings (including values 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
that involves try and failure. It will often be necessary to modify preprocessing and 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") and typological analysis. The subtle differences are often in the usage of the results: while in data mining, the resulting groups are the matter of interest, in automatic classification primarily their discriminitative power is of interest. This often leads to misunderstandings of researchers coming from the fields of data mining and machine learning, since they use the same terms and often the same algorithms, but have different goals.
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 in:
There are also finer distinctions possible, for example:
, is based on the core idea of objects being more related to nearby objects than to objects farther away. As such, 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
, which explains where the common name "hierarchical clustering" 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 object, there are multiple candidates to compute the distance to) to use. Popular choices are known as single-linkage clustering (the minimum of object distances), complete linkage clustering (the maximum of object distances) or UPGMA
("Unweighted Pair Group Method with Arithmetic Mean", also known as average linkage clustering). Furthermore, hierarchical clustering can be computed agglomerative (starting with single elements and aggregating them into clusters) or divisive (starting with the complete data set and dividing it into partitions).
While these methods are fairly easy to understand, the results are not always easy to use, as they will not produce a unique partitioning of the data set, but a hierarchy the user still needs to choose appropriate clusters from. The methods 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
, which makes them too slow for large data sets. For some special cases, optimal efficient methods (of complexity 
) are known: SLINK for single-linkage and CLINK for complete-linkage clustering. In the data mining
community these methods are recognized as a theoretical foundation of cluster analysis, but often considered obsolete. They did however provide inspiration for many later methods such as density based clustering.
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
, and thus the common approach is to search only for approximate solutions. A particularly well known approximative method is Lloyd's algorithm
, often actually referred to as "k-means algorithm". 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
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
-
- 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 in between of clusters (which is not surprising, as the algorithm optimized cluster centers, not cluster borders).
K-means has a number of interesting theoretical properties. On one hand, it partitions the data space into a structure known as Voronoi diagram
. On the other hand, it is conceptually close to nearest neighbor classification and as such popular in machine learning
. Third, it can be seen as a variation of model based classification, and Lloyd's algorithm as a variation of the Expectation-maximization algorithm
for this model discussed below.
. Clusters can then easily be defined as objects belonging most likely to the same distribution. A nice 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
, unless constraints are put on the model complexity. A more complex model will usually always be able to explain the data better, which makes choosing the appropriate model complexity inherently difficult.
The most prominent method is known as expectation-maximization algorithm
(or short: EM-clustering). Here, the data set is usually modeled with a fixed (to avoid overfitting) number of Gaussian distributions that are initialized randomly and whose parameters are iteratively optimized to fit better to 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 is a semantically strong method, as it not only provides you with clusters, but also produces complex models for the clusters that can also capture correlation and dependence of attributes. However, using these algorithms puts an extra burden on the user: to choose appropriate data models to optimize, and for many real data sets, there may be no mathematical model available the algorithm is able to optimize (e.g. assuming Gaussian distributions is a rather strong assumption on the data).
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
for core and noise points, but not for border points) in each run, therefore there is no need to run it multiple times. OPTICS
is a generalization of DBSCAN that removes the need to choose an appropriate value for the range parameter
, 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
parameter entirely and offering performance improvements over OPTICS by using an R-tree
index.
The key drawback of DBSCAN
and OPTICS
is that they expect some kind of density drop to detect cluster borders. On data sets with e.g. 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 mixtures of Gaussians data set, they will almost every time be outperformed by methods such as EM clustering, that are able to precisely model this kind of data.
(Zhang et al., 1996). With the recent need to process larger and larger data sets (also known as big data
), the willingness to treat 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 in 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
family of algorithms) have been adopted to subspace clustering (HiSC, hierarchical subspace clustering and DiSH) and correlation clustering (HiCO, hierarchical corelation clustering, 4C using "correlation connectivity" and ERiC exploring hierarchical density-based correlation clusters).
Several different clustering systems based on mutual information
have been proposed. One is Marina Meilă's variation of information metric; another provides hierarchical clustering. Using genetic algorithms, a wide range of different fit-functions can be optimized, including mutual information.
There have been several suggestions for a measure of similarity between two clusterings. Such a measure can be used to compare how well different data clustering algorithms perform on a set of data. These measures are usually tied to the type of criterion being considered in assessing the quality of a clustering method.
The following methods can be used to assess the quality clustering algorithms based on internal criterion:
Clustering is a main task of explorative data mining
Data mining
Data mining , a relatively young and interdisciplinary field of computer science is the process of discovering new patterns from large data sets involving methods at the intersection of artificial intelligence, machine learning, statistics and database systems...
, and a common technique for statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
data analysis
Data analysis
Analysis of data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making...
used in many fields, including machine learning
Machine learning
Machine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...
, pattern recognition
Pattern recognition
In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...
, image analysis
Image analysis
Image analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques...
, information retrieval
Information retrieval
Information retrieval is the area of study concerned with searching for documents, for information within documents, and for metadata about documents, as well as that of searching structured storage, relational databases, and the World Wide Web...
, and bioinformatics
Bioinformatics
Bioinformatics is the application of computer science and information technology to the field of biology and medicine. Bioinformatics deals with algorithms, databases and information systems, web technologies, artificial intelligence and soft computing, information and computation theory, software...
.
Cluster analysis itself is not one specific algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
, but the general task to be solved. It can be achieved by various algorithms that differ significantly in their notion of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with low distances among the cluster members, dense areas of the data space, intervals or particular statistical distributions. The appropriate clustering algorithm and parameter settings (including values 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
Knowledge discovery
Knowledge discovery is a concept of the field of computer science that describes the process of automatically searching large volumes of data for patterns that can be considered knowledge about the data . It is often described as deriving knowledge from the input data...
that involves try and failure. It will often be necessary to modify preprocessing and 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
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 analysis rather than using subjective evaluation of...
, botryology (from Greek βότρυς "grape") and typological analysis. The subtle differences are often in the usage of the results: while in data mining, the resulting groups are the matter of interest, in automatic classification primarily their discriminitative power is of interest. This often leads to misunderstandings of researchers coming from the fields of data mining and machine learning, since they use the same terms and often the same algorithms, but have different goals.
Clusters and clusterings
The notion of a cluster varies between algorithms and is one of the many decisions to take when choosing the appropriate algorithm for a particular problem. At first the terminology of a cluster seems obvious: a group of data objects. However, the clusters found by different algorithms vary significantly in their properties, and understanding these cluster models is key to understanding the differences between the various algorithms. Typical cluster models include:- Connectivity models: for example hierarchical clusteringHierarchical clusteringIn statistics, hierarchical clustering is a method of cluster analysis which seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two types:...
builds models based on distance connectivity. - Centroid models: for example the k-means algorithmK-means algorithmIn statistics and data mining, k-means clustering is a method of cluster analysis which aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean...
represents each cluster by a single mean vector. - Distribution models: clusters are modeled using statistic distributions, such as multivariate normal distributions used by the Expectation-maximization algorithmExpectation-maximization algorithmIn statistics, an expectation–maximization algorithm is an iterative method for finding maximum likelihood or maximum a posteriori estimates of parameters in statistical models, where the model depends on unobserved latent variables...
. - Density models: for example DBSCANDBSCANDBSCAN is a data clustering algorithm proposed by Martin Ester, Hans-Peter Kriegel, Jörg Sander and Xiaowei Xu in 1996....
and OPTICSOpticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
defines clusters as connected dense regions in the data space. - Subspace models: in BiclusteringBiclusteringBiclustering, co-clustering, or two-mode clustering is a data mining technique which allows simultaneous clustering of the rows and columns of a matrix....
(also known as Co-clustering or two-mode-clustering), clusters are modeled with both cluster members and relevant attributes. - Group models: some algorithms (unfortunately) do not provide a refined model for their results and just provide the grouping information.
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 in:
- hard clustering: each object belongs to a cluster or not
- soft clustering (also: fuzzy clusteringFuzzy clusteringFuzzy clustering is a class of algorithms for cluster analysis in which the allocation of data points to clusters is not "hard" but "fuzzy" in the same sense as fuzzy logic.- Explanation of clustering :...
): each object belongs to each cluster to a certain degree (e.g. a likelihood of belonging to the cluster)
There are also finer distinctions possible, for example:
- strict partitioning clustering: here each object belongs to exactly one cluster
- strict partitioning clustering with outliers: object can also belong to no cluster, and are considered outlierAnomaly detectionAnomaly detection, also referred to as outlier detection refers to detecting patterns in a given data set that do not conform to an established normal behavior....
s. - overlapping clustering (also: alternative clustering, multi-view clustering): while usually a hard clustering, objects may belong to more than one cluster.
- 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.
Clustering Algorithms
Clustering algorithms can be categorized based on their cluster model, as listed above. The following overview will only list the most prominent examples of clustering algorithms, as there are probably a few dozen (if not 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.Connectivity based clustering (Hierarchical clustering)
Connectivity based clustering, also known as hierarchical clusteringHierarchical clustering
In statistics, hierarchical clustering is a method of cluster analysis which seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two types:...
, is based on the core idea of objects being more related to nearby objects than to objects farther away. As such, 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
Dendrogram
A dendrogram is a tree diagram frequently used to illustrate the arrangement of the clusters produced by hierarchical clustering...
, which explains where the common name "hierarchical clustering" 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 object, there are multiple candidates to compute the distance to) to use. Popular choices are known as single-linkage clustering (the minimum of object distances), complete linkage clustering (the maximum of object distances) or UPGMA
UPGMA
UPGMA is a simple agglomerative or hierarchical clustering method used in bioinformatics for the creation of phenetic trees...
("Unweighted Pair Group Method with Arithmetic Mean", also known as average linkage clustering). Furthermore, hierarchical clustering can be computed agglomerative (starting with single elements and aggregating them into clusters) or divisive (starting with the complete data set and dividing it into partitions).
While these methods are fairly easy to understand, the results are not always easy to use, as they will not produce a unique partitioning of the data set, but a hierarchy the user still needs to choose appropriate clusters from. The methods 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
, which makes them too slow for large data sets. For some special cases, optimal efficient methods (of complexity ) are known: SLINK for single-linkage and CLINK for complete-linkage clustering. In the data miningData mining
Data mining , a relatively young and interdisciplinary field of computer science is the process of discovering new patterns from large data sets involving methods at the intersection of artificial intelligence, machine learning, statistics and database systems...
community these methods are recognized as a theoretical foundation of cluster analysis, but often considered obsolete. They did however provide inspiration for many later methods such as density based clustering.
Centroid-based clustering
In centroid-based clustering, clusters are represented by a central vector, which must not necessarily be 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 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
NP-hard
NP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...
, and thus the common approach is to search only for approximate solutions. A particularly well known approximative method is Lloyd's algorithm
Lloyd's algorithm
In computer science and electrical engineering, Lloyd's algorithm, also known as Voronoi iteration or relaxation, is an algorithm for grouping data points into a given number of categories, used for k-means clustering....
, often actually referred to as "k-means algorithm". It does however only find a local optimum
Local optimum
Local optimum is a term in applied mathematics and computer science.A local optimum of a combinatorial optimization problem is a solution that is optimal within a neighboring set of solutions...
, 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
K-medoids
The -medoids algorithm is a clustering algorithm related to the -means algorithm and the medoidshift algorithm. Both the -means and -medoids algorithms are partitional and both attempt to minimize squared error, the distance between points labeled to be in a cluster and a point designated as the...
), choosing median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
s (k-medians clustering
K-medians clustering
In statistics and machine learning, k-medians clustering is a variation of k-means clustering where instead of calculating the mean for each cluster to determine its centroid, one instead calculates the median...
), choosing the initial centers less randomly (K-means++
K-means++
In applied statistics, k-means++ is an algorithm for choosing the initial values for the k-means clustering algorithm. It was proposed in 2007 by David Arthur and Sergei Vassilvitskii, as an approximation algorithm for the NP-hard k-means problem—a way of avoiding the sometimes poor...
) or allowing a fuzzy cluster assignment (Fuzzy c-means
Fuzzy clustering
Fuzzy clustering is a class of algorithms for cluster analysis in which the allocation of data points to clusters is not "hard" but "fuzzy" in the same sense as fuzzy logic.- Explanation of clustering :...
).
Most k-means-type algorithms require the number of clusters
Determining the number of clusters in a data set
Determining the number of clusters in a data set, a quantity often labeled 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....
-
- 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 in between of clusters (which is not surprising, as the algorithm optimized cluster centers, not cluster borders).K-means has a number of interesting theoretical properties. On one hand, it partitions the data space into a structure known as Voronoi diagram
Voronoi diagram
In mathematics, a Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects in the space...
. On the other hand, it is conceptually close to nearest neighbor classification and as such popular in machine learning
Machine learning
Machine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...
. Third, it can be seen as a variation of model based classification, and Lloyd's algorithm as a variation of the Expectation-maximization algorithm
Expectation-maximization algorithm
In statistics, an expectation–maximization algorithm is an iterative method for finding maximum likelihood or maximum a posteriori estimates of parameters in statistical models, where the model depends on unobserved latent variables...
for this model discussed below.
Distribution-based clustering
The clustering model most closely related to statistics is based on distribution modelsProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
. Clusters can then easily be defined as objects belonging most likely to the same distribution. A nice 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
Overfitting
In statistics, overfitting occurs when a statistical model describes random error or noise instead of the underlying relationship. Overfitting generally occurs when a model is excessively complex, such as having too many parameters relative to the number of observations...
, unless constraints are put on the model complexity. A more complex model will usually always be able to explain the data better, which makes choosing the appropriate model complexity inherently difficult.
The most prominent method is known as expectation-maximization algorithm
Expectation-maximization algorithm
In statistics, an expectation–maximization algorithm is an iterative method for finding maximum likelihood or maximum a posteriori estimates of parameters in statistical models, where the model depends on unobserved latent variables...
(or short: EM-clustering). Here, the data set is usually modeled with a fixed (to avoid overfitting) number of Gaussian distributions that are initialized randomly and whose parameters are iteratively optimized to fit better to the data set. This will converge to a local optimum
Local optimum
Local optimum is a term in applied mathematics and computer science.A local optimum of a combinatorial optimization problem is a solution that is optimal within a neighboring set of solutions...
, 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 is a semantically strong method, as it not only provides you with clusters, but also produces complex models for the clusters that can also capture correlation and dependence of attributes. However, using these algorithms puts an extra burden on the user: to choose appropriate data models to optimize, and for many real data sets, there may be no mathematical model available the algorithm is able to optimize (e.g. assuming Gaussian distributions is a rather strong assumption on the data).
Density-based clustering
In density-based clustering, clusters are defined as areas of higher density than the remainder of the data set. Objects in these 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
DBSCAN
DBSCAN is a data clustering algorithm proposed by Martin Ester, Hans-Peter Kriegel, Jörg Sander and Xiaowei Xu in 1996....
. 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
Deterministic algorithm
In computer science, a deterministic algorithm is an algorithm which, in informal terms, behaves predictably. Given a particular input, it will always produce the same output, and the underlying machine will always pass through the same sequence of states...
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 algorithm
OPTICS is an algorithm for finding density-based clusters in spatial data. It was presented by Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel and Jörg Sander....
is a generalization of DBSCAN that removes the need to choose an appropriate value for the range parameter
, and produces a hierarchical result related to that of linkage clusteringHierarchical clustering
In statistics, hierarchical clustering is a method of cluster analysis which seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two types:...
. DeLi-Clu, Density-Link-Clustering combines ideas from single-linkage clustering and OPTICS, eliminating the
parameter entirely and offering performance improvements over OPTICS by using an R-treeR-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 significant use in both research and real-world applications...
index.
The key drawback of DBSCAN
DBSCAN
DBSCAN is a data clustering algorithm proposed by Martin Ester, Hans-Peter Kriegel, Jörg Sander and Xiaowei Xu in 1996....
and OPTICS
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
is that they expect some kind of density drop to detect cluster borders. On data sets with e.g. 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 mixtures of Gaussians data set, they will almost every time be outperformed by methods such as EM clustering, that are able to precisely model this kind of data.
Newer Developments
In recent years considerable effort has been put into improving algorithm performance of the existing algorithms. Among them are CLARANS (Ng and Han, 1994), and BIRCHBIRCH (data clustering)
BIRCH is an unsupervised data mining algorithm used to perform hierarchical clustering over particularly large data-sets...
(Zhang et al., 1996). With the recent need to process larger and larger data sets (also known as big data
Big data
Big data are datasets that grow so large that they become awkward to work with using on-hand database management tools. Difficulties include capture, storage, search, sharing, analytics, and visualizing...
), the willingness to treat semantic meaning of the generated clusters for performance has been increasing. This led to the development of pre-clustering methods such as canopy clustering
Canopy clustering algorithm
The canopy clustering algorithm is an unsupervised pre-clustering algorithm, often used as preprocessing step for the K-means algorithm or the Hierarchical clustering algorithm....
, 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
Curse of dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing high-dimensional spaces that do not occur in low-dimensional settings such as the physical space commonly modeled with just three dimensions.There are multiple phenomena referred to by this name in...
, which renders in particular distance functions problematic in high-dimensional spaces. This led to new clustering algorithms for high-dimensional data
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 data spaces are often encountered in areas such as medicine, where DNA microarray technology can produce a large number of measurements at once, and...
that focus on subspace clustering (where only some attributes are used, and cluster models include the relevant attributes for the cluster) and correlation clustering
Correlation clustering
In machine learning, correlation clustering or cluster editing operates in a scenario where the relationship between the objects are known instead of the actual representation of the objects...
that also looks for arbitrary rotated ("correlated") subspace clusters that can be modeled by giving a correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
of their attributes. Examples for such clustering algorithms are CLIQUE and SUBCLU
SUBCLU
SUBCLU is an algorithm for clustering high-dimensional data by Karin Kailing, Hans-Peter Kriegel and Peer Kröger. It is a subspace clustering algorithm that builds on the density-based clustering algorithm DBSCAN...
.
Ideas from density-based clustering methods (in particular the DBSCAN
DBSCAN
DBSCAN is a data clustering algorithm proposed by Martin Ester, Hans-Peter Kriegel, Jörg Sander and Xiaowei Xu in 1996....
/OPTICS
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
family of algorithms) have been adopted to subspace clustering (HiSC, hierarchical subspace clustering and DiSH) and correlation clustering (HiCO, hierarchical corelation clustering, 4C using "correlation connectivity" and ERiC exploring hierarchical density-based correlation clusters).
Several different clustering systems based on mutual information
Mutual information
In probability theory and information theory, the mutual information of two random variables is a quantity that measures the mutual dependence of the two random variables...
have been proposed. One is Marina Meilă's variation of information metric; another provides hierarchical clustering. Using genetic algorithms, a wide range of different fit-functions can be optimized, including mutual information.
Evaluation of Clustering Results
Evaluation of clustering results sometimes is referred to as cluster validation.There have been several suggestions for a measure of similarity between two clusterings. Such a measure can be used to compare how well different data clustering algorithms perform on a set of data. These measures are usually tied to the type of criterion being considered in assessing the quality of a clustering method.
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 interal criterion will likely overrate the resulting clustering.The following methods can be used to assess the quality clustering algorithms based on internal criterion:
- Davies–Bouldin indexDavies–Bouldin indexThe Davies–Bouldin index 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 done is made using quantities and features inherent to the dataset...
- The Davies–Bouldin indexDavies–Bouldin indexThe Davies–Bouldin index 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 done is made using quantities and features inherent to the dataset...
can be calculated by the following formula:
- where n is the number of clusters,
is the centroidCentroidIn geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
of cluster
, 
is the average distance of all elements in cluster 
to centroid 
, and 
is the distance between centroids 
and 
. 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 indexDavies–Bouldin indexThe Davies–Bouldin index 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 done is made using quantities and features inherent to the dataset...
is considered the best algorithm based on this criterion.
- Dunn index (J. C. Dunn 1974)
- 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 :
- where
represents the distance between clusters 
and 
, and 
measures the intra-cluster distance of cluster 
. The inter-cluster distance 
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 
may be measured in a variety ways, such as the maximal distance between any pair of elements in cluster 
. 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.
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 human (experts). Thus, the benchmark sets can be thought of as a gold standardGold standard (test)In medicine and statistics, gold standard test refers to a diagnostic test or benchmark that is the best available under reasonable conditions. It does not have to be necessarily the best possible test for the condition in absolute terms...
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 anomaliesAnomaly detectionAnomaly detection, also referred to as outlier detection refers to detecting patterns in a given data set that do not conform to an established normal behavior....
. Additionally, from a knowledge discoveryKnowledge discoveryKnowledge discovery is a concept of the field of computer science that describes the process of automatically searching large volumes of data for patterns that can be considered knowledge about the data . It is often described as deriving knowledge from the input data...
point of view, the reproduction of known knowledge may not necessarily be the intended result.
Some of the measures of quality of a cluster algorithm using external criterion include:- Rand measure (William M. Rand 1971)
- The Rand index computes how similar the clusters (returned by the clustering algorithm) are to the benchmark classifications. One can also view the Rand index as a measure of the percentage of correct decisions made by the algorithm. It can be computed using the following formula:
- where
is the number of true positives, 
is the number of true negatives, 
is the number of false positives, and 
is the number of false negatives. One issue with the Rand indexRand indexThe Rand index or Rand measure in statistics, and in particular in data clustering, is a measure of the similarity between two data clusterings...
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.- F-measure
- The F-measure can be used to balance the contribution of false negatives by weighting recall through a parameter
. Let precision and recall be defined as follows:
- where
is the precision rate and 
is the recall rate. We can calculate the F-measure by using the following formula :
- Notice that when
, 
. In other words, recall has no impact on the F-measure when 
, and increasing 
allocates an increasing amount of weight to recall in the final F-measure.
- Pair-counting F-Measure is the F-Measure applied to the set of object pairs, where objects are paired with each other when they are part of the same cluster. This measure is able to compare clusterings with different numbers of clusters.
- 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:
- This is simply the number of unique elements common to both sets divided by the total number of unique elements in both sets.
- Fowlkes–Mallows indexFowlkes–Mallows IndexFowlkes-Mallows index is an external evaluation method that used to determine the similarity between two clusterings...
(E. B. Fowlkes & C. L. Mallows 1983)
- 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:
- where
is the number of true positives, 
is the number of false positives, and 
is the number of false negatives.
- Confusion matrixConfusion matrixIn the field of artificial intelligence, a confusion matrix is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one . Each column of the matrix represents the instances in a predicted class, while each row represents the...
- A confusion matrix can be used to quickly visualize the results of a classification (or clustering) algorithm. It shows how different a cluster is different from the gold standard cluster.
- The Mutual InformationMutual informationIn probability theory and information theory, the mutual information of two random variables is a quantity that measures the mutual dependence of the two random variables...
is an information theoreticInformation theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
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. Adjusted mutual informationAdjusted Mutual InformationAdjusted mutual information is used for comparing clusterings. It corrects the effect of agreement solely due to chance between clusterings, similar to the way the adjusted rand index corrects the Rand index. It is closely related to variation of information: when a similar adjustment is made to...
is the corrected-for-chance variant of this that has a reduced bias for varying cluster numbers.
- The Mutual Information
Applications
BiologyBiologyBiology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
, computational biology and bioinformaticsBioinformaticsBioinformatics is the application of computer science and information technology to the field of biology and medicine. Bioinformatics deals with algorithms, databases and information systems, web technologies, artificial intelligence and soft computing, information and computation theory, software...- PlantPlantPlants are living organisms belonging to the kingdom Plantae. Precise definitions of the kingdom vary, but as the term is used here, plants include familiar organisms such as trees, flowers, herbs, bushes, grasses, vines, ferns, mosses, and green algae. The group is also called green plants or...
and animalAnimalAnimals are a major group of multicellular, eukaryotic organisms of the kingdom Animalia or Metazoa. Their body plan eventually becomes fixed as they develop, although some undergo a process of metamorphosis later on in their life. Most animals are motile, meaning they can move spontaneously and...
ecologyEcologyEcology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
- cluster anaysis 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 systematicsSystematicsBiological systematics is the study of the diversification of terrestrial life, both past and present, and the relationships among living things through time. Relationships are visualized as evolutionary trees...
to generate artificial phylogenies or clusters of organisms (individuals) at the species, genus or higher level that share a number of attribute - TranscriptomicsTranscriptomeThe transcriptome is the set of all RNA molecules, including mRNA, rRNA, tRNA, and other non-coding RNA produced in one or a population of cells.-Scope:...
- clustering is used to build groups of genesGênesGênes is the name of a département of the First French Empire in present Italy, named after the city of Genoa. It was formed in 1805, when Napoleon Bonaparte occupied the Republic of Genoa. Its capital was Genoa, and it was divided in the arrondissements of Genoa, Bobbio, Novi Ligure, Tortona and...
with related expression patterns (also known as coexpressed genes). Often such groups contain functionally related proteins, such as enzymeEnzymeEnzymes are proteins that catalyze chemical reactions. In enzymatic reactions, the molecules at the beginning of the process, called substrates, are converted into different molecules, called products. Almost all chemical reactions in a biological cell need enzymes in order to occur at rates...
s for a specific pathwayMetabolic pathwayIn biochemistry, metabolic pathways are series of chemical reactions occurring within a cell. In each pathway, a principal chemical is modified by a series of chemical reactions. Enzymes catalyze these reactions, and often require dietary minerals, vitamins, and other cofactors in order to function...
, or genes that are co-regulated. High throughput experiments using expressed sequence tagExpressed sequence tagAn expressed sequence tag or EST is a short sub-sequence of a cDNA sequence. They may be used to identify gene transcripts, and are instrumental in gene discovery and gene sequence determination. The identification of ESTs has proceeded rapidly, with approximately 65.9 million ESTs now available in...
s (ESTs) or DNA microarrayDNA microarrayA DNA microarray 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 genotype multiple regions of a genome...
s can be a powerful tool for genome annotation, a general aspect of genomicsGenomicsGenomics is a discipline in genetics concerning the study of the genomes of organisms. The field includes intensive efforts to determine the entire DNA sequence of organisms and fine-scale genetic mapping efforts. The field also includes studies of intragenomic phenomena such as heterosis,... - Sequence analysisSequence analysisIn bioinformatics, the term sequence analysis refers to the process of subjecting a DNA, RNA or peptide sequence to any of a wide range of analytical methods to understand its features, function, structure, or evolution. Methodologies used include sequence alignment, searches against biological...
- clustering is used to group homologous sequences into gene families. This is a very important concept in bioinformatics, and evolutionary biology in general. See evolution by gene duplicationGene duplicationGene duplication is any duplication of a region of DNA that contains a gene; it may occur as an error in homologous recombination, a retrotransposition event, or duplication of an entire chromosome.The second copy of the gene is often free from selective pressure — that is, mutations of it have no...
- High-throughput genotypingGenotypeThe genotype is the genetic makeup of a cell, an organism, or an individual usually with reference to a specific character under consideration...
platforms - clustering algorithms are used to automatically assign genotypes
- Human genetic clusteringHuman genetic clusteringHuman genetic clustering analysis uses mathematical cluster analysis of the degree of similarity of genetic data between individuals and groups to infer population structures and assign individuals to groups that often correspond with their self-identified geographical ancestry...
- The similarity of genetic data is used in clustering to infer population structures
MedicineMedicineMedicine is the science and art of healing. It encompasses a variety of health care practices evolved to maintain and restore health by the prevention and treatment of illness....- Medical imagingMedical imagingMedical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...
- On PET scans, cluster analysis can be used to differentiate between different types of tissueTissue (biology)Tissue is a cellular organizational level intermediate between cells and a complete organism. A tissue is an ensemble of cells, not necessarily identical, but from the same origin, that together carry out a specific function. These are called tissues because of their identical functioning...
and bloodBloodBlood is a specialized bodily fluid in animals that delivers necessary substances such as nutrients and oxygen to the cells and transports metabolic waste products away from those same cells....
in a three dimensional image. In this application, actual position does not matter, but the voxelVoxelA voxel is a volume element, representing a value on a regular grid in three dimensional space. This is analogous to a pixel, which represents 2D image data in a bitmap...
intensity is considered as a vectorCoordinate vectorIn linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
, with a dimension for each image that was taken over time. This technique allows, for example, accurate measurement of the rate a radioactive tracer is delivered to the area of interest, without a separate sampling of arterial blood, an intrusive technique that is most common today - 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 marketingMarketingMarketing is the process used to determine what products or services may be of interest to customers, and the strategy to use in sales, communications and business development. It generates the strategy that underlies sales techniques, business communication, and business developments...- Market researchMarket researchMarket research is any organized effort to gather information about markets or customers. It is a very important component of business strategy...
- Cluster analysis is widely used in market research when working with multivariate data from surveysStatistical surveySurvey methodology is the field that studies surveys, that is, the sample of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies....
and test panels. Market researchers use cluster analysis to partition the general populationPopulationA population is all the organisms that both belong to the same group or species and live in the same geographical area. The area that is used to define a sexual population is such that inter-breeding is possible between any pair within the area and more probable than cross-breeding with individuals...
of consumerConsumerConsumer is a broad label for any individuals or households that use goods generated within the economy. The concept of a consumer occurs in different contexts, so that the usage and significance of the term may vary.-Economics and marketing:...
s into market segments and to better understand the relationships between different groups of consumers/potential customers, and for use in market segmentation, Product positioningPositioning (marketing)In marketing, positioning has come to mean the process by which marketers try to create an image or identity in the minds of their target market for its product, brand, or organization....
, New product developmentNew product developmentIn business and engineering, new product development is the term used to describe the complete process of bringing a new product to market. A product is a set of benefits offered for exchange and can be tangible or intangible...
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 doesn't have the concept of a SKU
World wide webWorld Wide WebThe World Wide Web is a system of interlinked hypertext documents accessed via the Internet...- Social network analysis
- In the study of social networkSocial networkA social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
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 GoogleGoogleGoogle Inc. is an American multinational public corporation invested in Internet search, cloud computing, and advertising technologies. Google hosts and develops a number of Internet-based services and products, and generates profit primarily from advertising through its AdWords program...
. There are currently a number of web based clustering tools such as ClustyClustyClusty was a metasearch engine developed by Vivísimo which offered clusters of results. It is now known as Yippy. Vivísimo is a company built on Web search technology developed by Carnegie Mellon University researchers, much like Lycos was a decade earlier. Clusty added new features and a new... - Slippy map optimization
- FlickrFlickrFlickr is an image hosting and video hosting website, web services suite, and online community that was created by Ludicorp in 2004 and acquired by Yahoo! in 2005. In addition to being a popular website for users to share and embed personal photographs, the service is widely used by bloggers to...
'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 scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...- Software evolutionSoftware evolutionSoftware evolution is the term used in software engineering to refer to the process of developing software initially, then repeatedly updating it for various reasons.-General introduction:...
- 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 directly preventative maintenance
- Image segmentation
- Clustering can be used to divide a digitalDigitalA digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...
imageImageAn image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...
into distinct regions for border detection or object recognitionObject recognitionObject recognition in computer vision is the task of finding a given object in an image or video sequence. Humans recognize a multitude of objects in images with little effort, despite the fact that the image of the objects may vary somewhat in different view points, in many different sizes / scale... - 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
Social science- 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 miningEducational data miningEducational Data Mining is an emerging discipline, concerned with developing methods for exploring the unique types of data that come from educational settings, and using those methods to better understand students, and the settings which they learn in. A key area of EDM is mining computer logs of...
- Cluster analysis is for example used to identify groups of schools or students with similar properties
Others- Mathematical chemistryMathematical chemistryMathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena...
- To find structural similarity, etc., for example, 3000 chemical compounds were clustered in the space of 90 topological indicesTopological indexIn the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters...
- ClimatologyClimatologyClimatology is the study of climate, scientifically defined as weather conditions averaged over a period of time, and is a branch of the atmospheric sciences...
- To find weather regimes or preferred sea level pressure atmospheric patterns
- Petroleum Geology
- Cluster Analysis is used to reconstruct missing bottom hole core data or missing log curves in order to evaluate reservoir properties
- Physical Geography
- The clustering of chemical properties in different sample locations
Related topics
- Clustering high-dimensional dataClustering high-dimensional dataClustering high-dimensional data is the cluster analysis of data with anywhere from a few dozen to many thousands of dimensions. Such high-dimensional data spaces are often encountered in areas such as medicine, where DNA microarray technology can produce a large number of measurements at once, and...
- Curse of dimensionalityCurse of dimensionalityThe curse of dimensionality refers to various phenomena that arise when analyzing and organizing high-dimensional spaces that do not occur in low-dimensional settings such as the physical space commonly modeled with just three dimensions.There are multiple phenomena referred to by this name in...
- Data stream clusteringData stream clusteringIn computer science, data stream clustering is defined as the clustering of data that arrive continuously such as telephone records, multimedia data, financial transactions etc...
- Dimension reduction
- SilhouetteSilhouette (clustering)Silhouette refers to a method of interpretation and validation of clusters of data. The technique provides a succinct graphical representation of how well each object lies within its cluster. It was first described by Peter J. Rousseeuw in 1986.- Method :...
- Parallel coordinatesParallel coordinatesParallel coordinates is a common way of visualizing high-dimensional geometry and analyzing multivariate data.To show a set of points in an n-dimensional space, a backdrop is drawn consisting of n parallel lines, typically vertical and equally spaced...
Related methods
- Artificial neural networkArtificial neural networkAn artificial neural network , usually called neural network , is a mathematical model or computational model that is inspired by the structure and/or functional aspects of biological neural networks. A neural network consists of an interconnected group of artificial neurons, and it processes...
(ANN) - Cluster-weighted modelingCluster-weighted modelingIn statistics, cluster-weighted modeling is an algorithm-based approach to non-linear prediction of outputs from inputs based on density estimation using a set of models that are each notionally appropriate in a sub-region of the input space...
- Consensus clusteringConsensus clusteringClustering is the assignment of objects into groups so that objects from the same cluster are more similar to each other than objects from different clusters. Often similarity is assessed according to a distance measure...
- Constrained clusteringConstrained clusteringIn 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. Both a must-link and a cannot-link constraint...
- Latent Class AnalysisLatent class modelIn statistics, a latent class model relates a set of observed 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 discrete...
- Multidimensional scalingMultidimensional scalingMultidimensional scaling is a set of related statistical techniques often used in information visualization for exploring similarities or dissimilarities in data. MDS is a special case of ordination. An MDS algorithm starts with a matrix of item–item similarities, then assigns a location to each...
- Nearest neighbor searchNearest neighbor searchNearest neighbor search , also known as proximity search, similarity search or closest point search, is an optimization problem for finding closest points in metric spaces. The problem is: given a set S of points in a metric space M and a query point q ∈ M, find the closest point in S to q...
- Neighbourhood components analysisNeighbourhood components analysisNeighbourhood components analysis is a supervised learning method for clustering multivariate data into distinct classes according to a given distance metric over the data...
- Paired difference testPaired difference testIn statistics, a paired difference test is a type of location test that is used when comparing two sets of measurements to assess whether their population means differ...
- Principal Component Analysis
- Structured data analysis (statistics)Structured data analysis (statistics)Structured data analysis is the statistical data analysis of structured data. This can arise either in the form of an a priori structure such as multiple-choice questionnaires or in situations with the need to search for structure that fits the given data, either exactly or approximately...
- where
- Notice that when
- where
- where
- where
- where n is the number of clusters,