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WPGMA
WPGMA (Weighted Pair Group Method with Arithmetic Mean) is a simple agglomerative (bottom-up) hierarchical clustering method, generally attributed to Sokal and Michener. The WPGMA method is similar to its ''unweighted'' variant, the UPGMA method. Algorithm The WPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise distance matrix (or a similarity matrix). At each step, the nearest two clusters, say i and j, are combined into a higher-level cluster i \cup j. Then, its distance to another cluster k is simply the arithmetic mean of the average distances between members of k and i and k and j : d_ = \frac The WPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption: it produces an ultrametric tree in which the distances from the root to every branch tip are equal. This ultrametricity assumption is called the molecular clock when the tips involve DNA, RNA and protein data. Working example This work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WPGMA Dendrogram 5S Data
WPGMA (Weighted Pair Group Method with Arithmetic Mean) is a simple agglomerative (bottom-up) hierarchical clustering method, generally attributed to Sokal and Michener. The WPGMA method is similar to its ''unweighted'' variant, the UPGMA method. Algorithm The WPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise distance matrix (or a similarity matrix). At each step, the nearest two clusters, say i and j, are combined into a higher-level cluster i \cup j. Then, its distance to another cluster k is simply the arithmetic mean of the average distances between members of k and i and k and j : d_ = \frac The WPGMA algorithm produces rooted dendrograms and requires a constant-rate assumption: it produces an ultrametric tree in which the distances from the root to every branch tip are equal. This ultrametricity assumption is called the molecular clock when the tips involve DNA, RNA and protein data. Working example This worki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 WPGMA method. Note that the unweighted term indicates that all distances contribute equally to each average that is computed and does not refer to the math by which it is achieved. Thus the simple averaging in WPGMA produces a weighted result and the proportional averaging in UPGMA produces an unweighted result ('' see the working example''). Algorithm The UPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise similarity matrix (or a dissimilarity matrix). At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters \mathcal and \mathcal, each of size (''i.e.'', cardinality) and , is taken to be the average of all distances d(x,y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UPGMA Dendrogram 5S Data
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 WPGMA method. Note that the unweighted term indicates that all distances contribute equally to each average that is computed and does not refer to the math by which it is achieved. Thus the simple averaging in WPGMA produces a weighted result and the proportional averaging in UPGMA produces an unweighted result ('' see the working example''). Algorithm The UPGMA algorithm constructs a rooted tree (dendrogram) that reflects the structure present in a pairwise similarity matrix (or a dissimilarity matrix). At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters \mathcal and \mathcal, each of size (''i.e.'', cardinality) and , is taken to be the average of all distances d(x,y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete-linkage Clustering
Complete-linkage clustering is one of several methods of agglomerative hierarchical clustering. At the beginning of the process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters until all elements end up being in the same cluster. The method is also known as farthest neighbour clustering. The result of the clustering can be visualized as a dendrogram, which shows the sequence of cluster fusion and the distance at which each fusion took place. Clustering procedure At each step, the two clusters separated by the shortest distance are combined. The definition of 'shortest distance' is what differentiates between the different agglomerative clustering methods. In complete-linkage clustering, the link between two clusters contains all element pairs, and the distance between clusters equals the distance between those two elements (one in each cluster) that are farthest away from each other. The shortest of these links that r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single-linkage Clustering
In statistics, single-linkage clustering is one of several methods of hierarchical clustering. It is based on grouping clusters in bottom-up fashion (agglomerative clustering), at each step combining two clusters that contain the closest pair of elements not yet belonging to the same cluster as each other. A drawback of this method is that it tends to produce long thin clusters in which nearby elements of the same cluster have small distances, but elements at opposite ends of a cluster may be much farther from each other than two elements of other clusters. This may lead to difficulties in defining classes that could usefully subdivide the data. Overview of agglomerative clustering methods In the beginning of the agglomerative clustering process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters, until all elements end up being in the same cluster. At each step, the two clusters separated by the shortest distance are combi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Linkage Clustering
Complete-linkage clustering is one of several methods of agglomerative hierarchical clustering. At the beginning of the process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters until all elements end up being in the same cluster. The method is also known as farthest neighbour clustering. The result of the clustering can be visualized as a dendrogram, which shows the sequence of cluster fusion and the distance at which each fusion took place. Clustering procedure At each step, the two clusters separated by the shortest distance are combined. The definition of 'shortest distance' is what differentiates between the different agglomerative clustering methods. In complete-linkage clustering, the link between two clusters contains all element pairs, and the distance between clusters equals the distance between those two elements (one in each cluster) that are farthest away from each other. The shortest of these links that rem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single Linkage Clustering
In statistics, single-linkage clustering is one of several methods of hierarchical clustering. It is based on grouping clusters in bottom-up fashion (agglomerative clustering), at each step combining two clusters that contain the closest pair of elements not yet belonging to the same cluster as each other. A drawback of this method is that it tends to produce long thin clusters in which nearby elements of the same cluster have small distances, but elements at opposite ends of a cluster may be much farther from each other than two elements of other clusters. This may lead to difficulties in defining classes that could usefully subdivide the data. Overview of agglomerative clustering methods In the beginning of the agglomerative clustering process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters, until all elements end up being in the same cluster. At each step, the two clusters separated by the shortest distance are com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 two categories: * Agglomerative: This is a " bottom-up" approach: Each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy. * Divisive: This is a "top-down" approach: All observations start in one cluster, and splits are performed recursively as one moves down the hierarchy. In general, the merges and splits are determined in a greedy manner. The results of hierarchical clustering are usually presented in a dendrogram. The standard algorithm for hierarchical agglomerative clustering (HAC) has a time complexity of \mathcal(n^3) and requires \Omega(n^2) memory, which makes it too slow for even medium data sets. However, for some special cases, optimal efficient agglomerative methods (of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighbor-joining
In bioinformatics, neighbor joining is a bottom-up (agglomerative) clustering method for the creation of phylogenetic trees, created by Naruya Saitou and Masatoshi Nei in 1987. Usually based on DNA or protein sequence data, the algorithm requires knowledge of the distance between each pair of taxa (e.g., species or sequences) to create the phylogenetic tree. The algorithm Neighbor joining takes a distance matrix, which specifies the distance between each pair of taxa, as input. The algorithm starts with a completely unresolved tree, whose topology corresponds to that of a star network, and iterates over the following steps, until the tree is completely resolved, and all branch lengths are known: # Based on the current distance matrix, calculate a matrix Q (defined below). # Find the pair of distinct taxa i and j (i.e. with i \neq j) for which Q(i,j) is smallest. Make a new node that joins the taxa i and j, and connect the new node to the central node. For example, in part (B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 two categories: * Agglomerative: This is a " bottom-up" approach: Each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy. * Divisive: This is a "top-down" approach: All observations start in one cluster, and splits are performed recursively as one moves down the hierarchy. In general, the merges and splits are determined in a greedy manner. The results of hierarchical clustering are usually presented in a dendrogram. The standard algorithm for hierarchical agglomerative clustering (HAC) has a time complexity of \mathcal(n^3) and requires \Omega(n^2) memory, which makes it too slow for even medium data sets. However, for some special cases, optimal efficient agglomerative methods (of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrametric
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. Formal definition An ultrametric on a set is a real-valued function :d\colon M \times M \rightarrow \mathbb (where denote the real numbers), such that for all : # ; # (''symmetry''); # ; # if then ; # (strong triangle inequality or ultrametric inequality). An ultrametric space is a pair consisting of a set together with an ultrametric on , which is called the space's associated distance function (also called a metric). If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on . In the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Analysis
Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some 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 statistics, statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. 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 understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small Distance function, distances between cluster members, dense areas of the data space, intervals or particular statistical distributions. Clustering can therefore be formulated as a multi-object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
