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High-dimensional Model Representation
High-dimensional model representation is a finite expansion for a given ''multivariable'' function. The expansion was first described by Ilya M. Sobol Ilya Meyerovich Sobol’ (russian: Илья Меерович Соболь; born 15 August 1926) is a Russian mathematician, known for his work on Monte Carlo methods. His research spans several applications, from nuclear studies to astrophysics, ... as : f(\mathbf) = f_0+ \sum_^nf_i(x_i)+ \sum_^n f_(x_,x_)+ \cdots + f_(x_1,\ldots,x_n). The method, used to determine the right hand side functions, is given in Sobol's paper. A review can be found hereHigh Dimensional Model Representation (HDMR): Concepts and Applications The underlying logic behind the HDMR is to express all variable interactions in a system in a hierarchical order. For instance f_0 represents the mean response of the model f. It can be considered as measuring what is left from the model after stripping down all variable effects. The uni-variate functions f_i( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb A nonfinite verb is a derivative form of a verb unlike finite verbs. Accordingly, nonfinite verb forms are inflected for neither number nor person, and they cannot perform action as the root ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements ( vertices, edges, etc.). Equivalently this operation can be imagined by keeping facets in the same position but reducing their size. The expansion of a regular polytope creates a uniform polytope, but the operation can be applied to any convex polytope, as demonstrated for polyhedra in Conway polyhedron notation (which represents expansion with the letter ). For polyhedra, an expanded polyhedron has all the faces of the original polyhedron, all the faces of the dual polyhedron, and new square faces in place of the original edges. Expansion of regular polytopes According to Coxeter, this multidimensional term was defined by Alicia Boole StottCoxeter, ''Regular Polytopes'' (1973), p. 123. p.210 for creating new polytopes, specifically starting from regular polytopes to construct new uniform polytopes. The ''expansion'' operation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ilya M
Ilya, Iliya, Ilia, Ilja, or Ilija (russian: Илья́, Il'ja, , or russian: Илия́, Ilija, ; uk, Ілля́, Illia, ; be, Ілья́, Iĺja ) is the East Slavic form of the male Hebrew name Eliyahu (Eliahu), meaning "My God is Yahu/ Jah." It comes from the Byzantine Greek pronunciation of the vocative (Ilía) of the Greek Elias (Ηλίας, Ilías). It is pronounced with stress on the second syllable. The diminutive form is Iliusha or Iliushen'ka. The Russian patronymic for a son of Ilya is " Ilyich", and a daughter is "Ilyinichna". People with the name Real people *Ilya (Archbishop of Novgorod), 12th-century Russian Orthodox cleric and saint * Ilya Ivanovitch Alekseyev (1772–1830), commander of the Russian Imperial Army *Ilya Borok (born 1993), Russian jiujitsu fighter *Ilya Bryzgalov (born 1980), Russian ice hockey goalie *Ilya Ehrenburg (1891–1967), Russian writer and Soviet cultural ambassador *Ilya Glazunov (1930–2017), Russian painter *Ilya Gringolts (born 198 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance-based Sensitivity Analysis
Variance-based sensitivity analysis (often referred to as the Sobol method or Sobol indices, after Ilya M. Sobol) is a form of global sensitivity analysis.Sobol,I.M. (2001), Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. ''MATH COMPUT SIMULAT'',55(1–3),271-280, Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D. Saisana, M., and Tarantola, S., 2008, ''Global Sensitivity Analysis. The Primer'', John Wiley & Sons. Working within a probabilistic framework, it decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. For example, given a model with two inputs and one output, one might find that 70% of the output variance is caused by the variance in the first input, 20% by the variance in the second, and 10% due to interactions between the two. These percentages are directly interpreted as measures of sensitivity. Variance-based measures of sen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volterra Series
The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture "memory" effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of this system depends strictly on the input at that particular time. In the Volterra series the output of the nonlinear system depends on the input to the system at ''all'' other times. This provides the ability to capture the "memory" effect of devices like capacitors and inductors. It has been applied in the fields of medicine (biomedical engineering) and biology, especially neuroscience. It is also used in electrical engineering to model intermodulation distortion in many devices, including power amplifiers and frequency mixers. Its main advantage lies in its generality: it can represent a wide range of systems. Thus it is sometimes considered a non-parametric model. In mathematics, a Volterra series de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
