arXiv subjects

Tomasz Badowski

Publications and source records attributed to Tomasz Badowski.

3 recordsLinked to original sources

arXiv2016

Adaptive importance sampling via minimization of estimators of cross-entropy, mean square, and inefficiency constant

The inefficiency of using an unbiased estimator in a Monte Carlo procedure can be quantified using an inefficiency constant, equal to the product of the variance of the estimator and its mean computational cost. We develop methods for obtaining the parameters of the importance sampling (IS) change of measure via single- and multi-stage minimization of well-known estimators of cross-entropy and the mean square of the IS estimator, as well as of new estimators of such a mean square and inefficiency constant. We prove the convergence and asymptotic properties of the minimization results in our methods. We show that if a zero-variance IS parameter exists, then, under appropriate assumptions, minimization results of the new estimators converge to such a parameter at a faster rate than such results of the well-known estimators, and a positive definite asymptotic covariance matrix of the minimization results of the cross-entropy estimators is four times such a matrix for the well-known mean square estimators. We introduce criteria for comparing the asymptotic efficiency of stochastic optimization methods, applicable to the minimization methods of estimators considered in this work. In our numerical experiments for computing expectations of functionals of an Euler scheme, the minimization of the new estimators led to the lowest inefficiency constants and variances of the IS estimators, followed by the minimization of the well-known mean square estimators, and the cross-entropy ones.

stat.CO↗

arXiv2013

Variance-based sensitivity analysis for stochastic chemical kinetics

Sensitivity analysis is a process of computing sensitivity indices, which are certain measures of importance of parameters in influencing the outputs of mathematical models. Sensitivity indices computed in variance-based sensitivity analysis yield quantitative answers to questions like how much on average the variance of model output, measuring its uncertainty, decreases, if exact values of certain unknown parameters are determined, e. g. in an experiment. We propose new schemes for estimation of variance-based sensitivity indices of outputs of stochastic models, their conditional expectations, and histograms given the parameters. Unbiased estimators obtained in these schemes can be used in a Monte Carlo (MC) procedure approximating sensitivity indices. We derive relations between variances of final estimators of MC procedures making the same number of evaluations of given function, but using different schemes, both for the newly introduced schemes and for some used before in the literature. Numerical experiment for a discrete state stochastic Markov model of a chemical reaction network (DM) shows that our method can lead to much lower error than method analogous to the one offered by Degasperi et al. Further numerical experiments demonstrate that the application of random time change (RTC) algorithm due to Rathinam et al. for simulation of DM can lead to over 30 times lower variance of estimators of certain sensitivity indices than when Gillespie's direct (GD) method is used, and that this variance may significantly depend on the order of reactions in GD method. We provide some intuitions explaining these effects. We generalize measures used for comparing dispersion of different distributions, such as coefficient of variation and Fano factor to the random parameters case, in a way that they can be computed along with variance-based sensitivity indices.

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arXiv2013

Variance-based sensitivity analysis and orthogonal approximations for stochastic models

We develop new unbiased estimators of a number of quantities defined for functions of conditional moments, like conditional expectations and variances, of functions of two independent random variables given the first variable, including certain outputs of stochastic models given the models parameters. These quantities include variance-based sensitivity indices, mean squared error of approximation with functions of the first variable, orthogonal projection coefficients, and newly defined nonlinearity coefficients. We define the above estimators and analyze their performance in Monte Carlo procedures using generalized concept of an estimation scheme and its inefficiency constant. In numerical simulations of chemical reaction networks, using the Gillespie's direct and random time change methods, the new schemes for sensitivity indices of conditional expectations in some cases outperformed the ones proposed previously, and variances of some estimators significantly depended on the simulation method being applied.

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