Mathematical Sciences: Investigations Into Computationally Intensive Statistical Methods

9404594欧文这个项目考虑了现代计算密集型统计方法，重点关注高维数值求积和噪声环境下的神经网络问题。求积的工作将发展等分布方法和蒙特卡罗方法的混合，以结合每种方法的最佳特征。均匀分布方法通常提供更准确的积分估计，渐近计算和计算物理文献中的一些例子证明了这一点。蒙特卡罗方法使评估估计积分的准确性变得更容易。该杂交是通过在Faure和Niederreiter开发的一类均匀分布方法中随机化而形成的。预计所产生的方法将产生准确的答案，其准确性可以从用于生成这些答案的相同数据中可靠地衡量。人工神经网络被广泛用于基于一组预测器对响应进行预测和分类。与许多传统统计工具相比，它们能够更好地估计复杂的结构。当给予纯粹随机的数据作为训练依据时，他们也更容易找到结构。这里考虑的问题是猜测神经网络在噪声环境中能学习多少结构，以及构建在噪声中找到较少结构同时对真实结构保持敏感的网络。这里考虑的积分可以被认为是一个“输出”量的平均值，因为大约有十个或二十个“输入”量不同于它们可能的值。这些平均值对化学、物理、金融和统计学的问题很感兴趣。计算这些平均值的一种方法是基于挑选具有代表性的输入设置的列表，均匀地分布在可能的输入值中，然后对相应的输出值进行平均。对于许多问题，这种方法是相当准确的，但对于任何给定的问题，都很难准确地说出答案的准确性。第二种方法使用随机选择的输入输入设置列表。这种方法通常不太准确，但有一些方法可以利用随机性来做出关于答案的概率准确性声明。拟议的研究将这些想法结合在一起，采用具有代表性的输入设置列表，并以一种保留代表性但仍应允许做出准确性的概率陈述的方式对其进行随机置乱。人工神经网络经常被用于统计问题，例如在给定对象的某些测量特征的情况下预测对象所属的组，或者在给定一些输入数的情况下预测输出数。它们被称为神经网络，基于它们的结构与大脑的结构之间的类比。他们通常接受包含真实输入和输出的一组数据的训练，在许多问题中，他们在学习根据未来输入预测未来输出方面是有效的，即使在输入-输出关系非常复杂的情况下也是如此。建议的工作是研究人工神经网络从输入与输出无关的数据中错误学习随机模式的程度，并确定哪些类型的神经网络不太容易出现这个问题。

9404594 Owen This project considers modern computationally intensive statistical methods, focussing on problems of numerical quadrature in high dimensions and neural networks in noisy settings. The work on quadrature will develop hybrids of equidistribution methods and Monte Carlo methods, in order to combine the best features of each. Equidistribution methods commonly provide more accurate estimates of integrals, as borne out by asymptotic calculations and some examples in the computational physics literature. Monte Carlo methods, make it easier to assess the accuracy of an estimated integral. The hybrid is formed by randomizing within a class of equidistribution methods developed by Faure and Niederreiter. It is expected that the resulting methods will produce accurate answers whose accuracy can be reliably gauged from the same data used to generate them. Artificial neural networks are widely used to predict and classify responses based on a set of predictors. They are better able to estimate complicated structures than many traditional statistical tools. They are also more prone to finding structures when given purely random data to train on. The problems considered here are guaging how much structure a neural network will learn in a noisy setting, and constructing networks that find less structure in the noise while remaining sensitive to true structure. The integrals considered here may be thought of as averages of one "output" quantity as perhaps ten or twenty "input" quantities vary over their possible values. These averages are of interest in problems from chemistry, physics, finance and statistics. One approach to calculating these averages is based on picking a list of representative input settings, evenly spread through the possible input values, and then averaging the corresponding output values. For many problems this method is quite accurate, but on any given problem it can be hard to tell exactly how accurate the answer is. A second approach uses a randomly chosen list of in put settings. This approach is usually less accurate but there are ways of using the randomness to make probabilistic accuracy statements about the answer. The proposed research combines these ideas by taking a representative list of input settings and randomly scrambling it in a way that preserves the representativeness but should still allow probabilistic statements of accuracy to be made. Artificial neural networks are often used in statistical problems such as predicting what group an object belongs to, given some measured features of it, or predicting an output number given some input numbers. They are called neural networks based on an analogy between their structure and that of a brain. They are usually trained on a set of data containing the true inputs and outputs and in many problems are effective at learning to predict future outputs from future inputs, even when the input-output relationship is very complicated. The proposed work is to study the extent to which artificial neural networks mistakenly learn random patterns from data in which the inputs are irrelevant to the outputs, and to identify which sorts of neural networks are less prone to this problem.