Monte Carlo and Quasi-Monte Carlo Methods for Statistics

计算机模拟几乎被用于科学和工程的每一个分支，因为有些问题根本不是封闭形式的数学解，而且因为计算机已经变得非常快。与这一趋势相反的是，人们一直认为，战略抽样的值可以比随机抽样的值产生更好的结果，这种想法得到了持续不断的感觉和数学上的支持。计算机生成的成像是模拟的最大用户之一。从经济意义重大的电影和计算机游戏行业的光线行为的模拟，到建筑渲染和科学可视化的问题。选择模拟输入点的许多想法也被用于计算机化困难的统计决策和设计工业产品。准蒙特卡罗抽样是一种产生比随机点更均匀分布的输入的方法。据说它们的差异很小。现有的大多数低偏差解决方案都是关于在单位立方体内放置点的。计算机渲染中的一些应用程序需要来自三角形、单纯形或其他此类空间的样本。这些空间的普通蒙特卡罗抽样是直接的，但低偏差抽样则有很大的不同。适用于蒙特卡罗的变换可能会破坏三角形中结果点的低偏差属性。该项目将为三角形和三角形的张量积开发低偏差抽样方法。因子分析已经有一百多年的历史了，但即使是选择要使用的因子数量也是有问题的。该项目将开发交叉验证方法来选择因素的数量。

Computer simulations are used in virtually every branch of science and engineering, because some problems are simply beyond closed form mathematical solution, and because computers have become extremely fast. Against that trend, there is the constant feeling, and mathematical support, for the idea that strategically sampled values can give even better results than random ones do. Computer generated imaging is one of the largest users of simulations. Simulations of the behavior of light underly images from the economically significant motion picture and computer game industries, to problems of architectural rendering and scientific visualization. Many of the same ideas that go into choosing input points for simulations are also used to computerize difficult statistical decisions and to design industrialproducts.Quasi-Monte Carlo sampling is a method of producing inputs that are more evenly distributed than random points are. They are said to have low discrepancy. Most of the existing low discrepancy solutions are about placing points inside the unit cube. Some applications in computer rendering require samples from the triangle, simplex or other such spaces. Plain Monte Carlo sampling of those spaces is straightforward but low discrepancy sampling is quite different. The transformations that work for Monte Carlo may destroy the low discrepancy properties of the resulting points in the triangle. This project will develop low discrepancy sampling methods for the triangle and for tensor products of triangles. Factor analysis is about hundred years old, yet it remains problematic to even choose the number of factors to use in it. This project will develop cross-validatory methods to select the number of factors.