Monte Carlo Analysis: A New Paradigm not Requiring Apriori Probability Distributions

ECS-9811051 Barmish这是一项设立一个三年研究项目的建议。这一建议的动机来自于新出现的概率稳健性领域的最新结果。人们认为，一个成功的项目将对许多依赖蒙特卡罗模拟的领域产生影响，本研究与现有文献的关键之处在于：与经典的蒙特卡罗模拟不同，所考虑的新方法几乎不需要关于不确定参数的概率分布的先验信息。在这个新的框架中，给定的数据与具有实参数不确定性的经典稳健性理论中的数据完全相同，即只假设了一个数学输入输出模型和每个不确定参数的上下界。这引发了以下问题：在蒙特卡罗环境中使用什么概率分布是有意义的？当没有可用的概率分布时，有没有一种有意义的方法来“编程”蒙特卡洛模拟的随机数生成器？在提案的正文中，对这些问题进行了更详细的研究，并提出了旨在解决这些问题的建议研究。激励建议研究的初步结果表明，不确定性的抽样通常应该以一种与经典蒙特卡罗方法截然不同的方式进行。例如，在正文中，描述了一个涉及两级放大器的案例研究。结果表明，抽样分布需要截断两个不确定电容器的概率分布。这种分布不是通常在蒙特卡罗电路分析中使用的分布。最终的结果是，使用传统的蒙特卡罗方法预测“成功操作的概率”将受到挑战。研究的主要目标之一是证明新思想在许多应用领域的适用性，并将旧的蒙特卡罗方法的结果与使用新方法获得的结果进行比较。人们认为，经典的蒙特卡罗分析经常导致过度乐观的预测，这种情况(有界可用，但没有统计)在系统科学领域是非常典型的，当一个人在稳健性背景下或通过区间分析方法来处理问题时。总而言之，拟议的研究集中在无法获得可靠概率分布的问题上。蒙特卡罗理论假设一个分布作为理论的输入，而建议的工作涉及找到作为理论输出的适当分布；即，理论首先确定适当的分布，然后才进行计算机模拟。建议研究的起点是由首席研究员和他的研究生合作开发的一种描述不确定性的新型范式。与上面的例子一致，这个新的范例只需要随机变量的百分比误差的先验界。唯一需要的另一个假设是，偏离平均值较大的可能性比较小的偏差要小。***

ECS-9811051BarmishThis is a proposal to set up a three year research project. The motivation for this proposal is derived from the recent results in the newly emergent area of Probabilistic Robustness. It is felt that a successful project would have an impact on many fields which rely on Monte Carlo simulation, The critical point distinguishing this proposed research from existing literature is as follows: Unlike classical Monte Carlo simulation, the new approach being considered requires almost no apriori information about the probability distribution for the uncertain parameters. In this new framework, the given data is exactly the same as in classical robustness theory with real parametric uncertainty; i.e., only a mathematical input-output model and upper and lower bounds for each uncertain parameter are assumed. This raises the following questions: What probability distribution makes sense to use in a Monte Carlo context? Is there a meaning way to 'prograrn" the random number generator for Monte Carlo simulation when no probability distribution is available? In the body of the proposal, these questions are examined in greater detail and the proposed research aimed at their resolution.Preliminary results motivating the proposed research suggest that sampling of uncertainty should often be carried out in a way which is quite different from classical Monte Carlo schemes. For example, in the body, a case study involving a 2-stage amplifier is described. The sampling distribution turns out to require truncation of the probability distribution for the two uncertain capacitors. Such a distribution is not one which would normally be used in a Monte Carlo circuit analysis. The end result is that a prediction of the 'probability of successful operation" using a traditional Monte Carlo scheme is subjected to challenge. One of the main objectives of the research is to demonstrate applicability of the new ideas to many application areas and compare 'old style Monte Carlo" results with the results obtained using the new approach. It is felt that classical Monte Carlo analysis often leads to predictions which are unduly optimistic.This situation (the availability of bounds but no statistics) is quite typical in the Systems Science area when one approaches a problem in a robustness context or via interval analysis methods. In summary, the proposed research concentrates on problems for which a reliable probability distribution is unavailable. Whereas Monte Carlo theory assumes a distribution as input to the theory, the proposed work involves finding the 'appropriate" distribution as the output of the theory; i.e., the theory first determines the appropriate distribution, and, only then is computer simulation carried out.The starting point for the proposed research is a new type of paradigm to describe uncertainty - developed by the Principal Investigator in collaboration with his graduate student. Consistent with the example above, this new paradigm requires only apriori bounds on the percentage errors for the random variables. The only other assumption required is that large deviations from the mean are less probable than small deviations. ***