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Discrepancy Principle
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) {{disambiguation ... [...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 (mathematics), 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 [0, 2], since the interval [0.5, 0.9] occupies 1/5 of the length of the interval [0, 2], as ''n'' becomes large, the proportion of the first ''n'' members of the sequence which fa ... [...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. __NOTOC__ Theorems Discrepancy theory is based on the following classic theorems: * T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Of Hypergraphs
Discrepancy of hypergraphs is an area of discrepancy theory. 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 sequences is nearly sharp", page 319-325. Combinatorica, ... [...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 discrepancy of the pair (''X'', ''D'') measures the degree of the singularity of the pair. See also * Canonical singularity *Crepant resolution In algebraic geometry, a crepant resolution of a singularity is a resolution that does not affect the canonical class of the manifold. The term "crepant" was coined by by removing the prefix "dis" from the word "discrepant", to indicate that t ... 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, 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. 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 deviances: D(\mathbf,\hat) = \sum_i d(y_i, \hat_i). The (total) deviance for a model ''M''0 with estimates \hat = E \hat_0/math>, based on a dataset ''y'', may be constructed by its likelihood as:McCullagh and Nelder (1989): page 17 D(y,\hat) = 2 \left(\log \left (y\mid\hat \theta_s)\right- \log \left p(y\mid\hat \theta_0)\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deviation (statistics)
In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude of the value indicates the size of the difference. Types A deviation that is a difference between an observed value and the ''true value'' of a quantity of interest (where ''true value'' denotes the Expected Value, such as the population mean) is an error. A deviation that is the difference between the observed value and an ''estimate'' of the true value (e.g. the sample mean; the Expected Value of a sample can be used as an estimate of the Expected Value of the population) is a residual. These concepts are applicable for data at the interval and ratio levels of measurement. Unsigned or absolute deviation In statistics, the absolute deviation of an element of a ... [...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 (Kullback–Leibler divergence, KL 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 manifol ... [...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): a deviancy amplification term used by interactionist sociologist to refer to the way levels of deviance or crime can be increased by the societal reaction to deviance itself. * Deviance (statistics), a quality of fit statistic for a model * Positive deviance * Paraphilia, historically referred to as sexual deviance * Bid‘ah, Islamic term for innovations and deviations acts or groups from orthodox Islamic law (Sharia) See also * Deviant (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 ... * Discrepancy (other) * Divergence (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
