Monte Carlo and Quasi-Monte Carlo Methods for Statistics

计算机模拟几乎应用于科学和工程的每一个分支，因为有些问题完全超出了封闭形式的数学解决，而且因为计算机已经变得非常快。与这种趋势相反，有一种持续的感觉和数学上的支持，即策略性采样的值比随机的值能给出更好的结果。计算机生成的图像是模拟的最大用户之一。从具有经济意义的电影和电脑游戏行业到建筑渲染和科学可视化问题，对图像中光的行为的模拟。选择模拟输入点的许多相同思想也用于计算机化困难的统计决策和设计工业产品。拟蒙特卡罗采样是一种产生比随机点分布更均匀的输入的方法。据说它们的差异很小。大多数现有的低差异解决方案都是关于在单位立方体内放置点。在计算机渲染中的一些应用程序需要从三角形、单纯形或其他类似的空间中采样。这些空间的简单蒙特卡罗采样是直接的，但低差异采样是完全不同的。蒙特卡罗变换可能会破坏三角形中所得点的低差异特性。本项目将为三角形和三角形的张量积开发低差异采样方法。因子分析已经有近百年的历史了，但即使是选择在其中使用的因子的数量也仍然存在问题。本项目将开发交叉验证方法来选择因子的数量。

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.