Statistical Integration and Approximation

摘要PI：Art B Owen提案号：0306612该项目以多维函数的数值研究为中心。 第一个应用是这些功能的集成。 目前的数字网在被积函数上没有充分利用光滑性。 随机数字网的新结构的开发，以利用增加平滑的被积函数。 虽然数字网和积分格已经被随机化了，但很少或根本没有关于随机化Sirkak规则的工作。 这个项目开发了这样的随机化的Scorpak规则。 所产生的技术的质量进行了理论研究和计算调查。 第二个应用是用拟回归近似和解释多维函数。 主要应用是确定各种输入及其相互作用的相对重要性，以提供诊断和解释适合机器学习应用的预测函数。多维函数是依赖于多个输入并且通常依赖于许多输入的函数。 在许多应用中，人们希望在所有可能的输入上平均这样的函数。 在金融中，该函数可能是证券的收益，输入是随机的未来价格，平均值决定了公平的价格。 在计算机图形学中，图像通常是通过平均许多采样光路的贡献来生成的。 近年来，确定性策略抽样与随机抽样相结合的方法得到了成功的应用。 该项目开发了一些这种类型的新方法，并研究它们的工作情况。 第二个应用是从数值上确定一个函数的几十个输入变量中哪一个最重要，以及这些变量是单独作用还是协同作用。

AbstractPI: Art B Owenproposal number: 0306612This project centers on numerical investigation of multi-dimensional functions. The first application is to integration of these functions. At present digital nets do not exploit much smoothness in the integrand. New constructions of randomized digital nets are developed to exploit increased smoothness in the integrand. While digital nets and integration lattices have been randomized, there is little or no work on randomizing Smolyak rules. This project develops such randomized Smolyak rules. The quality of the resulting techniques is studied theoretically and in computational investigations. The second application is approximation and interpretation of multi-dimensional functions by quasi-regression. The primary application of interest is to determine the relative importance of various inputs and their interactions to provide diagnostics and interpretation of prediction functions fit in machine learning applications.A multidimensional function is one that depends on more than one input and often on many inputs. In many applications one wants to average such a function over all possible inputs. In finance, the function might be the payoff of a security, the inputs are random future prices, and the average determines a fair price. In computer graphics, images are often generated by averaging the contributions of many sampled paths of light. Recently, methods that mix deterministic strategic sampling with random sampling have been successfully applied. This project develops some new methods of this type and investigates how well they work. The second application is to determine numerically which of perhaps dozens of input variables to a function is most important and whether the variables act individually or synergistically.