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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] |
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] |
Square-integrable Function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty,+\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions (with respect to Lebesgue measure) forms the ''Lp'' space with p=2. Among the ''Lp'' spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Monte Carlo Method
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences). This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers. Monte Carlo and quasi-Monte Carlo methods are stated in a similar way. The problem is to approximate the integral of a function ''f'' as the average of the function evaluated at a set of points ''x''1, ..., ''x''''N'': : \int_ f(u)\,u \approx \frac\,\sum_^N f(x_i). Since we are integrating over the ''s''-dimensional unit cube, each ''x''''i'' is a vector of ''s'' elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the ''x''''i'' are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Monte Carlo uses a pseudorandom sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sensitivity Analysis
Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty; ideally, uncertainty and sensitivity analysis should be run in tandem. The process of recalculating outcomes under alternative assumptions to determine the impact of a variable under sensitivity analysis can be useful for a range of purposes, including: * Testing the robustness of the results of a model or system in the presence of uncertainty. * Increased understanding of the relationships between input and output variables in a system or model. * Uncertainty reduction, through the identification of model input that cause significant uncertainty in the output and should therefore be the focus of attention in order to increase robustness (perhap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin Hypercube Sampling
Latin hypercube sampling (LHS) is a statistical method for generating a near-random sample of parameter values from a multidimensional distribution. The sampling method is often used to construct computer experiments or for Monte Carlo integration. LHS was described by Michael McKay of Los Alamos National Laboratory in 1979. An independently equivalent technique was proposed by Vilnis Eglājs in 1977. It was further elaborated by Ronald L. Iman and coauthors in 1981. Detailed computer codes and manuals were later published. In the context of statistical sampling, a square grid containing sample positions is a Latin square if (and only if) there is only one sample in each row and each column. A Latin hypercube is the generalisation of this concept to an arbitrary number of dimensions, whereby each sample is the only one in each axis-aligned hyperplane containing it. When sampling a function of N variables, the range of each variable is divided into M equally probable interv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobol Sequence
Sobol sequences (also called LPτ sequences or (''t'', ''s'') sequences in base 2) are an example of quasi-random low-discrepancy sequences. They were first introduced by the Russian mathematician Ilya M. Sobol (Илья Меерович Соболь) in 1967.Sobol,I.M. (1967), "Distribution of points in a cube and approximate evaluation of integrals". ''Zh. Vych. Mat. Mat. Fiz.'' 7: 784–802 (in Russian); ''U.S.S.R Comput. Maths. Math. Phys.'' 7: 86–112 (in English). These sequences use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension. Good distributions in the ''s''-dimensional unit hypercube Let ''Is'' = ,1sup>''s'' be the ''s''-dimensional unit hypercube, and ''f'' a real integrable function over ''Is''. The original motivation of Sobol was to construct a sequence ''xn'' in ''Is'' so that : \lim_ \frac \sum_^n f(x_i) = \int_ f and the convergence be as fast as possible. It is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Monte Carlo Method
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences). This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers. Monte Carlo and quasi-Monte Carlo methods are stated in a similar way. The problem is to approximate the integral of a function ''f'' as the average of the function evaluated at a set of points ''x''1, ..., ''x''''N'': : \int_ f(u)\,u \approx \frac\,\sum_^N f(x_i). Since we are integrating over the ''s''-dimensional unit cube, each ''x''''i'' is a vector of ''s'' elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the ''x''''i'' are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Monte Carlo uses a pseudorandom sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Low-discrepancy Sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of ''N'', its subsequence ''x''1, ..., ''x''''N'' has a low discrepancy. Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set ''B'' is close to proportional to the measure of ''B'', as would happen on average (but not for particular samples) in the case of an equidistributed sequence. Specific definitions of discrepancy differ regarding the choice of ''B'' ( hyperspheres, hypercubes, etc.) and how the discrepancy for every B is computed (usually normalized) and combined (usually by taking the worst value). Low-discrepancy sequences are also called quasirandom sequences, due to their common use as a replacement of uniformly distributed random numbers. The "quasi" modifier is used to denote more clearly that the values of a low-discrepancy sequence are neither random nor pseudorandom, but such sequences share so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Number Generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's ''seed'' (which may include truly random values). Although sequences that are closer to truly random can be generated using hardware random number generators, ''pseudorandom number generators'' are important in practice for their speed in number generation and their reproducibility. PRNGs are central in applications such as simulations (e.g. for the Monte Carlo method), electronic games (e.g. for procedural generation), and cryptography. Cryptographic applications require the output not to be predictable from earlier outputs, and more elaborate algorithms, which do not inherit the linearity of simpler PRNGs, are needed. Good statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Construction Of ABi Matrices In Monte Carlo Estimation Of Sensitivity Indices
Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and comes from Latin ''constructio'' (from ''com-'' "together" and ''struere'' "to pile up") and Old French ''construction''. To construct is the verb: the act of building, and the noun is construction: how something is built, the nature of its structure. In its most widely used context, construction covers the processes involved in delivering buildings, infrastructure, industrial facilities and associated activities through to the end of their life. It typically starts with planning, financing, and design, and continues until the asset is built and ready for use; construction also covers repairs and maintenance work, any works to expand, extend and improve the asset, and its eventual demolition, dismantling or decommissioning. The construction i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
