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Silhouette (clustering)
Silhouette is a method of interpretation and validation of consistency within Cluster analysis, clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987. The silhouette value is a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The silhouette ranges from −1 to +1, where a high value indicates that the object is well matched to its own cluster and poorly matched to neighboring clusters. If most objects have a high value, then the clustering configuration is appropriate. If many points have a low or negative value, then the clustering configuration may have too many or too few clusters. A clustering with an average silhouette width of over 0.7 is considered to be "strong", a value over 0.5 "reasonable" and over 0.25 "weak", but with increasing dimensionality of the data, it becomes difficult t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Analysis
Cluster analysis or clustering is the data analyzing technique in which task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more Similarity measure, similar (in some specific sense defined by the analyst) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for 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 refers to a family of algorithms and tasks rather than one specific algorithm. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small Distance function, distances between cluster members, dense areas of the data space, intervals or pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Rousseeuw
Peter J. Rousseeuw (born 13 October 1956) is a Belgian statistician known for his work on robust statistics and cluster analysis. He obtained his PhD in 1981 at the Vrije Universiteit Brussel, following research carried out at the ETH in Zurich, which led to a book on influence functions. Later he was professor at the Delft University of Technology, The Netherlands, at the University of Fribourg, Switzerland, and at the University of Antwerp, Belgium. Next he was a senior researcher at Renaissance Technologies. He then returned to Belgium as professor at KU Leuven, until becoming emeritus in 2022. His former PhD students include Annick Leroy, Hendrik Lopuhaä, Geert Molenberghs, Christophe Croux, Mia Hubert, Stefan Van Aelst, Tim Verdonck and Jakob Raymaekers. Research Rousseeuw has constructed and published many useful techniques. He proposed the Least Trimmed Squares method and S-estimators for robust regression, which can resist outliers in the data. He also introduced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curse Of Dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression was coined by Richard E. Bellman when considering problems in dynamic programming. The curse generally refers to issues that arise when the number of datapoints is small (in a suitably defined sense) relative to the intrinsic dimension of the data. Dimensionally cursed phenomena occur in domains such as numerical analysis, sampling, combinatorics, machine learning, data mining and databases. The common theme of these problems is that when the dimensionality increases, the volume of the space increases so fast that the available data become sparse. In order to obtain a reliable result, the amount of data needed often grows exponentially with the dimensionality. Also, organizing and searching data often relies on detecting a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between string (computer science), strings of text) or a degree of separation (as exemplified by distance (graph theory), distance between people in a social network). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space. In the social sciences, distance can refer to a qualitative measurement of separation, such as social distance or psychological distance. Distances in physics and geometry The distance between physical locations can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's ''Elements'', distances were not represented as numbers but line segments of the same length, which were considered "equal". The notion of distance is inherent in the compass tool used to draw a circle, whose points all have the same distance from a common center point. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century. The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or Metric (mathematics), metric) called the ''taxicab distance'', ''Manhattan distance'', or ''city block distance''. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length. The taxicab distance is also sometimes known as ''rectilinear distance'' or distanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-medoids
-medoids is a classical partitioning technique of clustering that splits the data set of objects into clusters, where the number of clusters assumed known ''a priori'' (which implies that the programmer must specify k before the execution of a -medoids algorithm). The "goodness" of the given value of can be assessed with methods such as the silhouette method. The name of the clustering method was coined by Leonard Kaufman and Peter J. Rousseeuw with their PAM (Partitioning Around Medoids) algorithm. The medoid of a cluster is defined as the object in the cluster whose sum (and, equivalently, the average) of dissimilarities to all the objects in the cluster is minimal, that is, it is a most centrally located point in the cluster. Unlike certain objects used by other algorithms, the medoid is an actual point in the cluster. Algorithms In general, the -medoids problem is NP-hard to solve exactly. As such, multiple heuristics to optimize this problem exist. Partitioning Arou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-means
''k''-means clustering is a method of vector quantization, originally from signal processing, that aims to partition of a set, partition ''n'' observations into ''k'' clusters in which each observation belongs to the cluster (statistics), cluster with the nearest mean (cluster centers or cluster centroid), serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells. ''k''-means clustering minimizes within-cluster variances (squared Euclidean distances), but not regular Euclidean distances, which would be the more difficult Weber problem: the mean optimizes squared errors, whereas only the geometric median minimizes Euclidean distances. For instance, better Euclidean solutions can be found using k-medians clustering, ''k''-medians and k-medoids, ''k''-medoids. The problem is computationally difficult (NP-hardness, NP-hard); however, efficient heuristic algorithms converge quickly to a local optimum. These are usually similar to the ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Medoids
Medoids are representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be members of the data set. Medoids are most commonly used on data when a mean or centroid cannot be defined, such as graphs. They are also used in contexts where the centroid is not representative of the dataset like in images, 3-D trajectories and gene expression (where while the data is sparse the medoid need not be). These are also of interest while wanting to find a representative using some distance other than squared euclidean distance (for instance in movie-ratings). For some data sets there may be more than one medoid, as with medians. A common application of the medoid is the k-medoids clustering algorithm, which is similar to the k-means algorithm but works when a mean or centroid is not definable. This algorithm basically work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 done is made using quantities and features inherent to the dataset. This has a drawback that a good value reported by this method does not imply the best information retrieval. Preliminaries Given ''n'' dimensional points, let ''C''''i'' be a cluster of data points. Let ''X''''j'' be an ''n''-dimensional feature vector assigned to cluster ''C''''i''. : S_i = \left(\frac \sum_^ \right)^ Here A_i is the centroid of ''C''''i'' and ''T''''i'' is the size of the cluster ''i''. S_i is the ''q''th root of the ''q''th moment of the points in cluster ''i'' about the mean. If q=1 then S_i is the average distance between the feature vectors in cluster ''i'' and the centroid of the cluster. Usually the value of ''p'' is 2, which makes the dista ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
