Mathematical Sciences: Investigations Into Computationally Intensive Statistical Methods

9404594 欧文 该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中的神经网络问题。关于求积的工作将发展等分布方法和蒙特卡罗方法的混合，以便联合收割机结合每种方法的最佳特性。等分布方法通常提供更精确的积分估计，如渐近计算和计算物理文献中的一些例子所证明的。蒙特卡罗方法，使其更容易评估估计积分的准确性。该混合动力车是通过随机化一类均匀分布的方法由福雷和Niederreiter。预计由此产生的方法将产生准确的答案，其准确性可以从用于生成它们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因子来预测和分类响应。与许多传统的统计工具相比，它们能够更好地估计复杂结构。当给定纯随机数据进行训练时，它们也更容易找到结构。这里考虑的问题是，在噪声环境中，神经网络将学习多少结构，并构建在噪声中找到较少结构的网络，同时保持对真实结构的敏感性。 这里考虑的积分可以被认为是一个“输出”量的平均值，因为可能有十个或二十个“输入”量在其可能的值上变化。这些平均值在化学、物理、金融和统计问题中很有意义。计算这些平均值的一种方法是基于挑选一系列代表性的输入设置，均匀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，很难确切地告诉答案有多准确。第二种方法使用随机选择的输入设置列表。这种方法通常不太准确，但有一些方法可以使用随机性来对答案进行概率准确性陈述。拟议的研究结合了这些想法，采取了一个有代表性的输入设置列表，并随机打乱它的方式，保留了代表性，但仍然应该允许概率的准确性声明作出。人工神经网络通常用于统计问题，例如预测对象属于哪一组，给出它的一些测量特征，或者预测给定一些输入数字的输出数字。它们被称为神经网络，这是基于它们的结构与大脑的结构之间的类比。它们通常在包含真实输入和输出的一组数据上进行训练，并且在许多问题中，即使在输入-输出关系非常复杂的情况下，它们也能有效地学习从未来输入预测未来输出。拟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地学习随机模式的程度，并确定哪种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.