Statistical Integration and Approximation

建议编号：0306612本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，数字网络并没有充分利用被积函数的光滑性。开发了新的随机数字网络结构，以利用被积函数中增加的光滑性。虽然数字网络和积分晶格已经随机化，但关于随机化斯莫利亚克规则的工作很少或根本没有。这个项目开发了这种随机化的斯莫利亚克规则。从理论上和计算研究中研究了所得技术的质量。第二个应用是用拟回归逼近和解释多维函数。感兴趣的主要应用是确定各种输入及其相互作用的相对重要性，以提供适合于机器学习应用的预测函数的诊断和解释。多维函数是依赖于多个输入并且通常依赖于多个输入的函数。在许多应用中，人们希望对所有可能的输入取这样一个函数的平均值。在金融学中，函数可能是证券的收益，投入是随机的未来价格，平均价格决定了公平的价格。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。最近，确定性策略抽样和随机抽样相结合的方法得到了成功的应用。这个项目开发了一些这种类型的新方法，并调查了它们的工作情况。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是最重要的，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.