Conditional Distribution Inverse Method in Generating Uniform Random Vectors Over a Simplex

This paper develops an algorithm for uniform random generation over a constrained simplex, which is the intersection of a standard simplex and a given set. Uniform sampling from constrained simplexes has numerous applications in different fields, such as portfolio optimization, stochastic multi-criteria decision analysis, experimental design with mixtures and decision problems involving discrete joint distributions with imprecise probabilities. The proposed algorithm is developed by combining the acceptancerejection and conditional methods along with the use of optimization tools. The acceptance rate of the algorithm is analytically compared to that of a crude acceptance-rejection algorithm, which generates points over the simplex and then rejects any points falling outside the intersecting set. Finally, using convex optimization, the setup phase of the algorithm is detailed for the special cases where the intersecting set is a general convex set, a convex set defined by a finite number of convex constraints or a polyhedron.Keywords Uniform random variate generation over simplexes · Convex optimization · Simulation-based portfolio optimization · Stochastic multi-criteria decision analysis · Distributions with imprecise probabilities ·

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“…Algorithm 3 (Fang and Yang 2000;Moeini et al 2011) 1. Generate n − 1 independent random numbers Z 1 , .…”

Section: Uniform Random Variate Generation Over Standard Simplexesmentioning

confidence: 99%

“…He also developed a map for converting the points uniformly generated over the standard simplex to an arbitrary simplex. Fang and Yang (2000) and Moeini et al (2011) proposed a new algorithm based on different ways of using the conditional method. Rubin (1984) developed a triangulation algorithm for a general n-dimensional polytope.…”

Section: Introductionmentioning

confidence: 99%

An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes

Ahmadi‐Javid

Moeini

2015

Stat Comput

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“…To sample the continuum of all process options, the weights need to be independently and identically distributed (iid). Therefore, random numbers r i are sampled from the uniform distribution, U[0,1], and transformed into the weights following the approach described by Moeini et al (2011). N − 1 such random numbers are required for N weights of competing options.…”

Section: Artificial Benchmark Model Setupsmentioning

confidence: 99%

Simultaneously determining global sensitivities of model parameters and model structure

Mai

Craig

Tolson

2020

Hydrol. Earth Syst. Sci.

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Abstract. Model structure uncertainty is known to be one of the three main sources of hydrologic model uncertainty along with input and parameter uncertainty. Some recent hydrological modeling frameworks address model structure uncertainty by supporting multiple options for representing hydrological processes. It is, however, still unclear how best to analyze structural sensitivity using these frameworks. In this work, we apply the extended Sobol' sensitivity analysis (xSSA) method that operates on grouped parameters rather than individual parameters. The method can estimate not only traditional model parameter sensitivities but is also able to provide measures of the sensitivities of process options (e.g., linear vs. non-linear storage) and sensitivities of model processes (e.g., infiltration vs. baseflow) with respect to a model output. Key to the xSSA method's applicability to process option and process sensitivity is the novel introduction of process option weights in the Raven hydrological modeling framework. The method is applied to both artificial benchmark models and a watershed model built with the Raven framework. The results show that (1) the xSSA method provides sensitivity estimates consistent with those derived analytically for individual as well as grouped parameters linked to model structure. (2) The xSSA method with process weighting is computationally less expensive than the alternative aggregate sensitivity analysis approach performed for the exhaustive set of structural model configurations, with savings of 81.9 % for the benchmark model and 98.6 % for the watershed case study. (3) The xSSA method applied to the hydrologic case study analyzing simulated streamflow showed that model parameters adjusting forcing functions were responsible for 42.1 % of the overall model variability, while surface processes cause 38.5 % of the overall model variability in a mountainous catchment; such information may readily inform model calibration and uncertainty analysis. (4) The analysis of time-dependent process sensitivities regarding simulated streamflow is a helpful tool for understanding model internal dynamics over the course of the year.

show abstract

“…For sampling the continuum of all process options, the weights need to be independently and identically distributed (iid). Therefore, random numbers r i are sampled from the uniform distribution, U[0,1], and transformed into the weights following the approach described by Moeini et al (2011). N − 1 such random numbers are required for N weights of competing options.…”

Section: Artificial Benchmark Model Setupsmentioning

confidence: 99%

“…The sampling strategy introduced by Moeini et al (2011) can be summarized as follows: For each set of weights N +1 needed, first generate a vector of random numbers (r 1 , r 2 , . .…”

Section: Appendix A: Generating Weightsmentioning

confidence: 99%

Simultaneously Determining Global Sensitivities of Model Parameters and Model Structure

Mai¹,

Craig²,

Tolson³

2020

Preprint

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Abstract. Model structure uncertainty is known to be one of the three main sources of hydrologic model uncertainty along with input and parameter uncertainty. Some recent hydrological modeling frameworks address model structure uncertainty by supporting multiple options for representing hydrological processes. It is, however, still unclear how best to analyze structural sensitivity using these frameworks. In this work, we apply an Extended Sobol' Sensitivity Analysis (xSSA) method that operates on grouped parameters rather than individual parameters. The method can estimate not only traditional model parameter sensitivities but is also able to provide measures of the sensitivities of process options (e.g., linear vs. non-linear storage) and sensitivities of model processes (e.g., infiltration vs. baseflow) with respect to a model output. Key to the xSSA method's applicability to process option and process sensitivity is the novel introduction of process option weights in the Raven hydrological modeling framework. The method is applied to both artificial benchmark models and a watershed model built with the Raven framework. The results show that: (1) The xSSA method provides sensitivity estimates consistent with those derived analytically for individual as well as grouped parameters linked to model structure. (2) The xSSA method with process weighting is computationally less expensive than the alternative aggregate sensitivity analysis approach performed for the exhaustive set of structural model configurations, with savings of 81.9 % for the benchmark model and 98.6 % for the watershed case study. (3) The xSSA method applied to the hydrologic case study analyzing simulated streamflow showed that model parameters adjusting forcing functions were responsible for 42.1 % of the overall model variability while surface processes cause 38.5 % of the overall model variability in a mountainous catchment; such information may readily inform model calibration. (4) The analysis of time dependent process sensitivities regarding simulated streamflow is a helpful tool to understand model internal dynamics over the course of the year.

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Conditional Distribution Inverse Method in Generating Uniform Random Vectors Over a Simplex

Cited by 9 publications

References 5 publications

An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes

An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes

Simultaneously determining global sensitivities of model parameters and model structure

Simultaneously Determining Global Sensitivities of Model Parameters and Model Structure

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