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The antithetic-variates technique (Hammersley and Morton, 1956) achieves the variance-reduction goal by attempting to generate random variates that would induce a negative correlation for the quantity of interest between separate sim­ulation runs. Consider that Gq and ©2 are two unbiased estimators of an un­known quantity в to be estimated. The two estimators can be combined […]

The importance sampling technique concentrates the distribution of sampling points in the part of the domain that is most “important” for the task rather than spreading them out evenly (Marshall, 1956). Refer to the problem of evaluating an integral in Eq. (6.49) by the sample-mean method. The importance sampling technique attempts to generate M sampling […]

Since Monte Carlo simulation is a sampling procedure, results obtained from the procedure inevitably involve sampling errors, which decrease as the sam­ple size increases. Increasing the sample size to achieve a higher precision generally means an increase in computer time for generating random vari­ates and data processing. Variance-reduction techniques aim at obtaining high accuracy for […]

Referring to Monte Carlo integration, different algorithms yield different esti­mators for the integral. A relevant issue is which algorithm is more efficient. The efficiency issue can be examined from the statistical properties of the esti­mator from a given algorithm and its computational aspects. Rubinstein (1981) showed a practical measure of the efficiency of an algorithm […]

Consider the reliability computation involving a multidimensional integral as Eq. (6.48). Without losing generality, the following discussions assume that the stochastic variables in the original X-space have been transformed to the independent standard normal Z’-space (see Sec. 2.7.2). Consequently, the orig­inal performance function W(X) can be expressed as W(Z’). In terms of Z Eq. (6.48) […]

The sample-mean Monte Carlo integration is based on the idea that the com­putation of the integral by Eq. (6.49) alternatively can be carried out by in which fx(x) > 0 is a PDF defined over a < x < b. The transformed in­tegral given by Eq. (6.49) is equivalent to the computation of expectation of […]

Referring to Fig. 6.6, a rectangular region Ш = {(x, y)a < x < b, 0 < y < c} is superimposed to enclose the area Ф = {(x, y)a < x < b, 0 < y = g(x) < c} represented by Eq. (6.49). By the hit-and-miss method, the rectangular region Ш containing the […]

In reliability analysis, computations of system and/or component reliability and other related quantities, such as mean time to failure, essentially involve inte­gration operations. A simple example is the time-to-failure analysis in which the reliability of a system within a time interval (0, t) is obtained from where ft (t) is the failure density function. A […]

Procedures described in Sec. 6.5.2 are for generating multivariate normal (Gaussian) random variables without imposing constraints or restriction on the values of variates. The procedures under this category are also called uncon­ditional (or nonconditional) simulation (Borgman and Faucette, 1993; Chiles and Delfiner, 1999). In hydrosystems modeling, random variables often exist for which, in addition to […]

In many practical hydrosystems engineering problems, random variables often are statistically and physically dependent. Furthermore, distribution types for the random variables involved can be a mixture of different distributions, of which the corresponding joint PDF or CDF is difficult to establish. As a practical alternative, to replicate such systems properly, the Monte Carlo simulation should […]