Multivariate Vector

As has been shown earlier (Gupta and Manohar, 2005), the construct for bivariate vector of Poisson random variables can be easily generalized for the case m > 2. The number of mutually independent

Poisson random variables can be generalized to be given by Cf + Cf, where Cf denotes combin­ation of m variables taken k at a time. Thus, for m = 3, consider six mutually independent Poisson random variables, {Ui}6=x, with parameters {Х}6=x and define

N^a!) = Ux + U4 + U5 N2(a2) = U2 + U4 + U6

N3(a3) = U3 + U5 + U6 (31)

The equations relating {Х}6=x to the moments of {Ni}3=x can be shown to be given by

(32)

It is to be noted that for m > 2, the formulation requires the evaluation of a set of integrals of the form in Equation (12) and at no stage does the order of the integrals becomes greater than that of Equation (12). In general, the number of such integrals that need to be evaluated is Cf.

Numerical Algorithm

A crucial step in the above formulation lies in evaluating integrals of the type as in Equation (21). Closed form solutions for the integrals are possible only for a limited class of problems. Here, we propose the use of Monte Carlo methods, in conjunction with importance sampling to increase the efficiency, for evaluating these integrals. The integrals in Equation (21) can be recast as

where hx (■) is the importance sampling pdf and I [■] is an indicator function taking values of unity if q(X) < 0, indicating that the sample lies within the domain of integration Qj, and zero otherwise. Since the problem is formulated into the standard normal space X, hx (■) can be taken to be Gaussian with unit standard deviation and shifted mean. The difficulty, however, lies in determining where should hx(■) be centered. An inspection of Equation (33) reveals that the form of the integrals are similar to reliability integrals which are of the form

(34)

This implies that for efficient computation of the integrals, the importance sampling pdf hx (■) may be centered around the design point for the function q(x) = 0. If q(x) is available in explicit form, first order reliability methods can be used to determine the design point. If q(x) is not available explicitly, an adaptive importance sampling strategy can be adopted to determine the design point. In certain problems, the domain of integration, characterized by q(x) = 0, may consist of multiple design points or multiple regions which contribute significantly to ij. This is especially true when

Fig. 1. Schematic diagram for numerical algorithm for evaluating multidimensional integrals; g(x,X2) = 0 is the limit surface ion the X — X2 random variable space; hy^ (yi) and hy2 (У2) are the two importance sampling pdfs; two design points at distance в from the origin.

q(X) = 0 is highly nonlinear, irregular or consists of disjointed regions. In these situations, it is necessary to construct a number of importance sampling functions, with each function centered at the various design points.

The steps for implementing the algorithm for numerical evaluation of integrals of the type in Equation (21), has been developed and discussed (Gupta and van Gelder, 2005). The sequential steps for implementing the algorithm is detailed below, with reference to the schematic diagram in Figure 1.

(1) Carry out pilot Monte Carlo simulations in the standard normal space. If there are too few samples in the failure domain, we carry out Monte Carlo simulations with a Gaussian importance sampling function with mean zero and a higher variance. On the other hand, if there are too few samples in the safe region, the variance of the importance sampling function is taken to be smaller. Repeat this step, till we have a reasonable number of samples in the failure and the safe regions.

(2) We sort the samples lying in the failure domain according to their distance from the origin. (3) A Gaussian importance sampling pdf is constructed which is centered at the sample in the failure domain lying closest to the origin. Let this point be denoted by d0 and its distance from the origin be denoted by в0. (4) We check for samples in the failure domain, within a hyper-sphere of radius ві, ві — во = є, where є is a positive number. (5) For samples lying within this hyper-sphere, we check for the sample d1, which lie closest to the origin but is not located in the vicinity of d0. This is checked by comparing the direction cosines of d1 and d0. (6) By comparing the direction cosines of all samples lying within the hyper-sphere of radius в1, we can identify the number of design points. We construct importance sampling pdfs at each of these design points. If there exist no samples with direction cosines distinctly different from d0, there is only one design point and a single importance sampling pdf is sufficient. (7) During importance sampling procedure corresponding to a design point, for each sample realization, we check if x1 and X1 expressed in terms of the random variables (Z1,…, Z2n—2) are real. The indicator function is assigned a value of unity if real, and zero otherwise. (8) An estimate of Ij is obtained from Equation (33).