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Elbow Method (clustering)
In cluster analysis, the elbow method is a heuristic used in determining the number of clusters in a data set. The method consists of plotting the explained variation as a function of the number of clusters and picking the elbow of the curve as the number of clusters to use. The same method can be used to choose the number of parameters in other data-driven models, such as the number of principal components to describe a data set. The method can be traced to speculation by Robert L. Thorndike in 1953. Intuition Using the "elbow" or "knee of a curve" as a cutoff point is a common heuristic in mathematical optimization to choose a point where diminishing returns are no longer worth the additional cost. In clustering, this means one should choose a number of clusters so that adding another cluster doesn't give much better modeling of the data. The intuition is that increasing the number of clusters will naturally improve the fit (explain more of the variation), since there are mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diminishing Returns
In economics, diminishing returns are the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal ( ceteris paribus). The law of diminishing returns (also known as the law of diminishing marginal productivity) states that in productive processes, increasing a factor of production by one unit, while holding all other production factors constant, will at some point return a lower unit of output per incremental unit of input. The law of diminishing returns does not cause a decrease in overall production capabilities, rather it defines a point on a production curve whereby producing an additional unit of output will result in a loss and is known as negative returns. Under diminishing returns, output remains positive, however productivity and efficiency decrease. The modern understanding of the law adds the dimension of holding other outputs equal, since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determining The Number Of Clusters In A Data Set
Determining the number of clusters in a data set, a quantity often labelled ''k'' as in the ''k''-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular ''k''-means, ''k''-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as ''k'' that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether. The correct choice of ''k'' is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user. In addition, increasing ''k'' without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NeuroImage
''NeuroImage'' is a peer-reviewed scientific journal covering research on neuroimaging, including functional neuroimaging and functional human brain mapping. The current Editor in Chief is Michael Breakspear. Abstracts from the annual meeting of the Organization for Human Brain Mapping have been published as supplements to the journal. Members of the Organization for Human Brain Mapping are eligible for reduced subscription rates. In 2012, Elsevier launched an online-only, open access sister journal to ''NeuroImage'', entitled '' NeuroImage: Clinical''. Among prolific authors publishing numerous articles in the journal are Karl J. Friston, Arthur W. Toga, Paul M. Thompson and Karl Zilles. Related journals are Human Brain Mapping, Frontiers in Human Neuroscience, Cerebral Cortex, Magnetic Resonance in Medicine, Journal of Neuroscience and Journal of Cognitive Neuroscience. Abstracting and indexing The journal is abstracted and indexed in Scopus, Science Citation Index, Curre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-test Statistic
An ''F''-test is any statistical test in which the test statistic has an ''F''-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact "''F''-tests" mainly arise when the models have been fitted to the data using least squares. The name was coined by George W. Snedecor, in honour of Ronald Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s. Common examples Common examples of the use of ''F''-tests include the study of the following cases: * The hypothesis that the means of a given set of normally distributed populations, all having the same standard deviation, are equal. This is perhaps the best-known ''F''-test, and plays an important role in the analysis of variance (ANOVA). * The hypothesis that a proposed regression model fits the data well. See Lack-of-fit sum of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ANOVA
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the ''t''-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means. History While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler. These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis testing i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strategic Management Journal
The Strategic Management Society (SMS) is a professional society for the advancement of strategic management. The society consists of nearly 3,000 members representing various backgrounds and perspectives from more than eighty different countries. Membership is composed of academics, business practitioners, and consultants. The society has been credited with being a factor in the development of strategic management as a legitimate field of scholarly endeavor. The SMS publishes the ''Strategic Management Journal'', ''Strategic Entrepreneurship Journal'' and the ''Global Strategy Journal''. History The Strategic Management Society was founded at an initial meeting in London in 1981. Founding officers were elected at a second conference held in Montreal in 1982, and the founding constitution was drawn and approved at the third meeting in Paris in 1983. There were 459 original founding members of the society. Former presidents *Russel Coff, 2017-2018 *Marjorie Lyles, 2015-2016 *Bob H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Over-fitting
mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitted model is a mathematical model that contains more parameters than can be justified by the data. The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e., the noise) as if that variation represented underlying model structure. Underfitting occurs when a mathematical model cannot adequately capture the underlying structure of the data. An under-fitted model is a model where some parameters or terms that would appear in a correctly specified model are missing. Under-fitting would occur, for example, when fitting a linear model to non-linear data. Such a model will tend to have poor predictive performance. The possibility of over-fitting exists because the criterion used for selecting the model is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, opti ... [...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] |
