Mathematical Sciences: Application of Interval Analysis in Statistical Computing

摘要IEEE提出的二进制浮点计算机算法标准已被大多数计算机制造商所采用。这个浮点数系统的特性包括程序员可选择的舍入模式。两种舍入模式，舍入到正无穷大或负无穷大，为区间分析计算的实现提供硬件支持。在统计学以外的实质性领域，区间分析和区间计算在研究中具有很好的优势。统计计算领域的研究人员在认识或评估区间分析的潜在效用方面进展缓慢。本研究的总体目标是确定统计计算中区间分析可以发挥良好优势的领域，并在这些领域开发所需的方法和算法。具体来说，将开发一个程序库，用于大多数常见的单变量连续概率分布和选定的多变量分布的cdf和逆cdf的自验证逼近。精度基本上是机器精度在非常广泛的区域的可变/参数空间将是目标。对于某些类型的全局优化问题的自验证计算方法也将被开发。科学计算通常是用数字计算机完成的。然而，计算机不能准确地表达构成计算问题的中间和最终结果的大多数数字。计算机算术固有的不精确性的一部分是由于每次算术运算后需要舍入到特定的数字位数。因此，对于复杂的计算任务，计算机产生的答案往往只是未知真实答案的粗略近似值，如果能够进行精确计算，就可以获得未知真实答案。这个研究项目的目标是开发替代的计算方法，在困难的统计计算应用中产生更准确的答案。提高准确性的代价是要消耗更多的计算机资源来生成答案。然而，今天有足够的计算机资源来支持这种计算密集型的方法。这项研究的主要动机是，最好花额外的时间和精力来确保好的答案，而不是满足于快速获得的糟糕答案。

9500831 Kennedy Abstract The IEEE proposed standard for binary floating-point computer arithmetic has been adopted by most computer manufacturers. Included among the features of this floating-point number system is programmer selectable rounding mode. Two rounding modes, round to plus or minus infinity, give hardware support for implementations of interval analysis computations. Substantive areas outside statistics are employing interval analysis and interval computations to good advantage in research. Research workers in statistical computing have been slow to recognize or evaluate the potential utility of interval analysis. The overall objective of this research is to identify areas in statistical computing where interval analysis can be used to good advantage, and to develop needed methods and algorithms in these areas. Specifically, a library of routines for self- validating approximation of cdf and inverse cdf for most of the usual univariate continuous probability distributions, and selected multivariate distributions will be developed. Accuracy essentially to machine precision over very broad areas of the variable/parameter space will be the goal. Self-validating computational methods for certain kinds of global optimization problems will also be developed. Scientific computations are usually done using a digital computer. Computers are not, however, capable of expressing exactly most numbers that constitute intermediate and final results for a computing problem. A part of the inexactness inherent in computer arithmetic is due to the need to round to a specific number of numeric digits after each arithmetic operation. Therefore answers produced by a computer for difficult computing tasks are often only rough approximations to the unknown true answers which would be obtained if exact computations could be performed. The objective of this research project is to develop alternative computational methods that yield more accurate answers in difficult statistical computing ap plications. The price of increased accuracy is that substantially more computer resources are consumed to produce answers. However, today there are enough computer resources available to support this very computationally intensive approach. The main motivation for the research is that it is best to spend extra time and effort to insure good answers, instead of settling for poor answers that are obtained quickly.