A Procedure for Simultaneous Calculation of Sensitivities in Probabilistic Risk Assessments.* S. Uryasev, Brookhaven National Laboratory, Bldg. 130, Upton, NY, 11973; and A. Shlyakhter, Dept. of Physics, Harvard Univ., Cambridge, MA 02138 and Dept. of Environmental Health, Harvard School of Public Health, 665 Huntington Ave., Boston MA 02115
Sensitivity analysis of risk estimates to parameters is the first part of uncertainty analysis. It involves calculation of the derivatives which is often simple for risk estimates themselves but becomes intractable (in analytical form) for the upper percentiles of risk if the distributions other than lognormal are involved. A new general formula for calculating the derivatives of the integrals over sets given by inequalities was recently developed by Uryasev [I]. Probability and quantile functions (for example, 95 percentile) are the special cases of this general formula. Derivative of a quantile function with respect to parameter is presented as the ratio of two probability functions with the same probability density and integration set. Thus derivatives of the quantile function with respect to all parameters are reduced to calculation of the similar integrals over the same sets. These integrals can be calculated simultaneously by the Monte Carlo simulation. To illustrate this formulation, let us consider a simple risk model in the form of the ratio of two random variables, y1 and y2: R=y1/ y2. Let us assume that y1 follows a lognormal distribution with the density f1 (y1) and parameters m and s (which may be a combination of several lognormal distributions for exposure variables). Let us further assume that y2 follows a normal distribution with the density f2(y2) with parameters m and s (e.g. the distribution of bodyweights). Derivatives of the quantile value a(m,s,m,s), take the form of the ratio where the numerator and denominator are the integrals with the itegrand f1 (y1)f2(y2) in the numerator multiplied by the function a1(y1,y2) and the integrand f1(y1) f2(y2) in the denominator multiplied by the function b(y1,y2). For our example (lognormal distribution in the numerator and normal distribution in the denominator) these functions take the following form: a1(y1,y2)=(ln y1 - m)/s2, a2(y1,y2)=((m - ln y1)2 - s2)/s3, a3 (y1,y2)=(y2 - m)/s2, and a4 (y1,y2)=((m - y2)2 - s2)/s3 for the sensitivity of a(m,s,m,s) to m, s, m, and s, respectively; and b(y1,y2)= y2(m - s2 - ln y1)/y1s2. In this manner, all sensitivities of the upper percentile of risk can be obtained together with percentile a(m,s,m,s) in the same Monte Carlo simulation without appreciable additional calculations. Analytical formulas for sensitivities for the typical models used in risk assessment must be derived only once. After this is done, the results can be used in routine Monte Carlo simulations.
S.Uryasev "Derivatives of Probability Functions and Integrals Over Sets Given by Inequalities," Journal of Computational and Applied Mathematics, v.56 (1994), in press.
*Work performed under the auspices of the U.S. Department of Energy.