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Discrepancies
{{disambiguation ...
Discrepancy may refer to: Mathematics * Discrepancy of a sequence * Discrepancy theory in structural modelling * Discrepancy of hypergraphs, an area of discrepancy theory * Discrepancy (algebraic geometry) Statistics * Discrepancy function in the context of structural equation models * Deviance (statistics) * Deviation (statistics) * Divergence (statistics) See also * Deviance (other) * Deviation (other) Deviation may refer to: Mathematics and engineering * Allowance (engineering), an engineering and machining allowance is a planned deviation between an actual dimension and a nominal or theoretical dimension, or between an intermediate-stage dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Of A Sequence
In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration. Definition A sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ''equidistributed'' on a non-degenerate interval 'a'', ''b''if for every subinterval 'c'', ''d''of 'a'', ''b''we have :\lim_= . (Here, the notation , ∩ 'c'', ''d'' denotes the number of elements, out of the first ''n'' elements of the sequence, that are between ''c'' and ''d''.) For example, if a sequence is equidistributed in , 2 since the interval .5, 0.9occupies 1/5 of the length of the interval , 2 as ''n'' becomes large, the proportion of the first ''n'' members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity. A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval. Theorems Discrepancy theory is based on the following classic theorems: * Geometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Of Hypergraphs
Discrepancy of hypergraphs is an area of discrepancy theory that studies the discrepancy of general set systems. Definitions In the classical setting, we aim at partitioning the vertices of a hypergraph \mathcal=(V, \mathcal) into two classes in such a way that ideally each hyperedge contains the same number of vertices in both classes. A partition into two classes can be represented by a coloring \chi \colon V \rightarrow \. We call −1 and +1 ''colors''. The color-classes \chi^(-1) and \chi^(+1) form the corresponding partition. For a hyperedge E \in \mathcal, set :\chi(E) := \sum_ \chi(v). The ''discrepancy of \mathcal with respect to \chi'' and the ''discrepancy of \mathcal'' are defined by :\operatorname(\mathcal,\chi) := \; \max_ , \chi(E), , :\operatorname(\mathcal) := \min_ \operatorname(\mathcal, \chi). These notions as well as the term 'discrepancy' seem to have appeared for the first time in a paper of Beck.J. Beck: "Roth's estimate of the discrepancy of integer seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy (algebraic Geometry)
In algebraic geometry, given a pair (''X'', ''D'') consisting of a normal variety ''X'' and a \mathbb-divisor ''D'' on ''X'' (e.g., canonical divisor The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical examp ...), the discrepancy of the pair (''X'', ''D'') measures the degree of the singularity of the pair. See also * Canonical singularity * Crepant resolution References * Algebraic geometry {{Algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Function
In structural equation modeling, a discrepancy function is a mathematical function which describes how closely a structural model conforms to observed data; it is a measure of goodness of fit. Larger values of the discrepancy function indicate a poor fit of the model to data. In general, the parameter estimates for a given model are chosen so as to make the discrepancy function for that model as small as possible. Analogous concepts in statistics are known as goodness of fit or statistical distance, and include deviance and divergence. Examples There are several basic types of discrepancy functions, including maximum likelihood (ML), generalized least squares (GLS), and ordinary least squares (OLS), which are considered the "classical" discrepancy functions. Discrepancy functions all meet the following basic criteria: *They are non-negative, i.e., always greater than or equal to zero. *They are zero only if the fit is perfect, i.e., if the model and parameter estimates perfectly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deviance (statistics)
In statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ..., deviance is a goodness-of-fit statistic for a statistical model; it is often used for statistical hypothesis testing. It is a generalization of the idea of using the sum of squares of residuals (SSR) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood. It plays an important role in exponential dispersion models and generalized linear models. Deviance can be related to Kullback-Leibler divergence. Definition The unit deviance d(y,\mu) is a bivariate function that satisfies the following conditions: * d(y,y) = 0 * d(y,\mu) > 0 \quad\forall y \neq \mu The total deviance D(\mathbf,\hat) of a model with predictions \hat of the observation \mathbf is the sum of its unit devian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deviation (statistics)
In mathematics and statistics, deviation serves as a measure to quantify the disparity between an observed value of a variable and another designated value, frequently the mean of that variable. Deviations with respect to the sample mean and the population mean (or "true value") are called errors and residuals, ''errors'' and ''residuals'', respectively. The Sign (mathematics), sign of the deviation reports the direction of that difference: the deviation is positive when the observed value exceeds the reference value. The absolute value of the deviation indicates the size or magnitude of the difference. In a given sample (statistics), sample, there are as many deviations as sample points. Summary statistics can be derived from a set of deviations, such as the ''standard deviation'' and the ''mean absolute deviation'', measures of statistical dispersion, dispersion, and the ''mean signed deviation'', a measure of bias of an estimator, bias. The deviation of each data point is calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence (statistics)
In information geometry, a divergence is a kind of statistical distance: a binary function which establishes the separation from one probability distribution to another on a statistical manifold. The simplest divergence is squared Euclidean distance (SED), and divergences can be viewed as generalizations of SED. The other most important divergence is relative entropy (also called Kullback–Leibler divergence), which is central to information theory. There are numerous other specific divergences and classes of divergences, notably ''f''-divergences and Bregman divergences (see ). Definition Given a differentiable manifold M of dimension n, a divergence on M is a C^2-function D: M\times M\to [0, \infty) satisfying: # D(p, q) \geq 0 for all p, q \in M (non-negativity), # D(p, q) = 0 if and only if p=q (positivity), # At every point p\in M, D(p, p+dp) is a positive-definite quadratic form for infinitesimal displacements dp from p. In applications to statistics, the manifold M i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deviance (other)
Deviance may refer to: * Deviance (sociology), actions or behaviors that violate social norms * Deviancy amplification spiral, a cognitive bias (error in judgement) and a deviancy amplification term used by interactionist sociologists * Deviance (statistics), a quality of fit statistic for a model * Positive deviance, an approach to behavioral and social change * Sexual deviance (historical term) or paraphilia, recurring or intense sexual arousal to atypical things * Deviance or bid'ah, innovations and deviant acts or groups from orthodox Islamic law (Sharia) See also * Deviant (other) * Deviation (other) * Discrepancy (other) Discrepancy may refer to: Mathematics * Discrepancy of a sequence * Discrepancy theory in structural modelling * Discrepancy of hypergraphs, an area of discrepancy theory * Discrepancy (algebraic geometry) Statistics * Discrepancy function i ... * Divergence (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
