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Calinski–Harabasz Index
The Calinski–Harabasz index (CHI), also known as the Variance Ratio Criterion (VRC), is a metric for evaluating clustering algorithms, introduced by Tadeusz Caliński and Jerzy Harabasz in 1974. It is an internal evaluation metric, where the assessment of the clustering quality is based solely on the dataset and the clustering results, and not on external, ground-truth labels. Definition Given a data set of ''n'' points: , and the assignment of these points to ''k'' clusters: , the Calinski–Harabasz (CH) Index is defined as the ratio of the between-cluster separation (BCSS) to the within-cluster dispersion (WCSS), normalized by their number of degrees of freedom: CH = \frac BCSS (Between-Cluster Sum of Squares) is the weighted sum of squared Euclidean distances between each cluster centroid (mean) and the overall data centroid (mean): BCSS = \sum_^ n_i , , \mathbf_i - \mathbf, , ^2 where ''ni'' is the number of points in cluster ''Ci'', c''i'' is the centroid of ''Ci'', a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clustering Algorithm
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 similar (in some specific sense defined by the analyst) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. Cluster analysis refers to a family of algorithms and tasks rather than one specific algorithm. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small distances between cluster members, dense areas of the data space, intervals or particular statistical distributions. Clustering can ... [...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] |
Centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-dimensional Euclidean space. In geometry, one often assumes uniform mass density, in which case the '' barycenter'' or ''center of mass'' coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin. In physics, if variations in gravity are considered, then a '' center of gravity'' can be defined as the weighted mean of all points weighted by their specific weight. In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center. History The term "centroid" was coined in 1814. It is used as a substitute for the older terms "center of grav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-means Clustering
''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] |
F-test
An F-test is a statistical test that compares variances. It is used to determine if the variances of two samples, or if the ratios of variances among multiple samples, are significantly different. The test calculates a Test statistic, statistic, represented by the random variable F, and checks if it follows an F-distribution. This check is valid if the null hypothesis is true and standard assumptions about the errors (ε) in the data hold. F-tests are frequently used to compare different statistical models and find the one that best describes the population (statistics), population the data came from. When models are created using the least squares method, the resulting F-tests are often called "exact" F-tests. The F-statistic was developed by Ronald Fisher in the 1920s as the variance ratio and was later named in his honor by George W. Snedecor. Common examples Common examples of the use of ''F''-tests include the study of the following cases * The hypothesis that the Arithme ... [...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] |
Dunn Index
The Dunn index, introduced by Joseph C. Dunn in 1974, is a metric for evaluating clustering algorithms. This is part of a group of validity indices including the Davies–Bouldin index or Silhouette index, in that it is an internal evaluation scheme, where the result is based on the clustered data itself. As do all other such indices, the aim is to identify sets of clusters that are compact, with a small variance between members of the cluster, and well separated, where the means of different clusters are sufficiently far apart, as compared to the within cluster variance. For a given assignment of clusters, a higher Dunn index indicates better clustering. One of the drawbacks of using this is the computational cost as the number of clusters and dimensionality of the data increase. Preliminaries There are many ways to define the size or diameter of a cluster. It could be the distance between the farthest two points inside a cluster, it could be the mean of all the pairwise distance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Scikit-learn
scikit-learn (formerly scikits.learn and also known as sklearn) is a free and open-source machine learning library for the Python programming language. It features various classification, regression and clustering algorithms including support-vector machines, random forests, gradient boosting, ''k''-means and DBSCAN, and is designed to interoperate with the Python numerical and scientific libraries NumPy and SciPy. Scikit-learn is a NumFOCUS fiscally sponsored project. Overview The scikit-learn project started as scikits.learn, a Google Summer of Code project by French data scientist David Cournapeau. The name of the project derives from its role as a "scientific toolkit for machine learning", originally developed and distributed as a third-party extension to SciPy. The original codebase was later rewritten by other developers. In 2010, contributors Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort and Vincent Michel, from the French Institute for Research in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
R (programming Language)
R is a programming language for statistical computing and Data and information visualization, data visualization. It has been widely adopted in the fields of data mining, bioinformatics, data analysis, and data science. The core R language is extended by a large number of R package, software packages, which contain Reusability, reusable code, documentation, and sample data. Some of the most popular R packages are in the tidyverse collection, which enhances functionality for visualizing, transforming, and modelling data, as well as improves the ease of programming (according to the authors and users). R is free and open-source software distributed under the GNU General Public License. The language is implemented primarily in C (programming language), C, Fortran, and Self-hosting (compilers), R itself. Preprocessor, Precompiled executables are available for the major operating systems (including Linux, MacOS, and Microsoft Windows). Its core is an interpreted language with a na ... [...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] |
DBCV Index
Density-Based Clustering Validation (DBCV) is a metric designed to assess the quality of clustering solutions, particularly for density-based clustering algorithms like DBSCAN, Mean shift, and OPTICS. This metric is particularly suited for identifying concave and nested clusters, where traditional metrics such as the Silhouette coefficient, Davies–Bouldin index, or Calinski–Harabasz index often struggle to provide meaningful evaluations. Unlike traditional validation measures, which often rely on compact and well-separated clusters, DBCV index evaluates how well clusters are defined in terms of local density variations and structural coherence. This metric was introduced in 2014 by David Moulavi and colleagues in their work. It utilizes density connectivity principles to quantify clustering structures, making it especially effective at detecting arbitrarily shaped clusters in concave datasets, where traditional metrics may be less reliable. The DBCV index has been employe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
