Assessment of probability models using the bootstrap sampling method

Abstract

The application of probabilistic methods to the analysis and design of mechanical systems is becoming more commonplace. This is due to a number of reasons, among them are: (1) modern requirements to design systems much closer to expected maximum load tolerances; (2) continuing improvements in the ability to specify random variations in systems, materials, and excitations; and (3) the desire to quantify the risk associated with a given application. Monte Carlo sampling is the traditional technique for probabilistic system analysis and is often employed as the standard to which other techniques are compared for accuracy. Yet, because of the natural limitations on the (finite) number of samples that can be generated for subsequent use in a Monte Carlo analysis, probabilistic information that is estimated, such as ordinates in a cumulative distribution function (CDF) or probability density function (PDF), are approximations with accuracy, which is a function both of the number of samples and the formula used to estimate the statistic of interest, that is usually not addressed in a given analysis. This accuracy limitation can be particularly acute when the quantity of interest is the output of an analysis that involves substantial and time-consuming numerical computation for which only a relatively small number of computational results can be made available. A consequence of this limitation is that the results of other, analytically-based probabilistic analyses are frequently not checked extensively. Bootstrap sampling is a technique for the analysis of measured data arising from non-Gaussian sources where the amount of data may be limited. The bootstrap offers a method for the efficient use of limited Monte Carlo data for the assessment of the accuracy of approximate, analytical probabilistic techniques, regardless of the probability distribution of the phenomenon under consideration. It can be used to check the accuracy of statistics of probability models generated using the analytical techniques. Specifically, Monte Carlo data could be used in conjunction with the bootstrap method to estimate confidence intervals on statistics of the data. These same statistics could be evaluated from the approximate analytical model. If the statistics from the analytical model fall outside the corresponding confidence intervals obtained using the sample data, then the validity of the approximate, analytical model is rejected. Otherwise, based on the judgment of the analyst, the analytical model may be accepted. In this paper, a brief discussion of probabilistic models and the bootstrap sampling method is given. Then, the use of the bootstrap in assessing the validity of the analytical probabilistic models is outlined. Finally, the results of two numerical examples involving the probabilistic analysis of a complex system are presented.