Nonparametric Methods for Temporally Correlated and High Dimensional Data

面对不断增加的数据量所带来的挑战，现代统计方法的进步仍然至关重要，需要从重要决策过程中日益关键的数据中获取准确信息，以及计算能力增强所提供的机会。非参数统计方法，即对数据进行很少和非常弱的假设的方法，由于其广泛的适用性，在几乎每个应用领域都很有吸引力。本研究主要集中在非参数贝叶斯方法、时间序列谱方法和高维数据降维等方面的理论和应用。这是当前活跃研究统计中的三个不同领域，处理不同类型的定性数据。与所有贝叶斯方法一样，贝叶斯非参数方法中的推理是基于对感兴趣的数量从先验分布到后验分布的过渡，而在非参数情况下，感兴趣的数量不是有限维的。例如未知分布、密度函数或生存函数。这种方法的可取之处在于不需要假设未知量遵循某种参数形式，并且一旦以可行的形式获得后验分布，就可以进行广泛的推断。这些方法所面临的挑战是给出可行的后验表示所需的理论和计算，以便有效地从这些表示中得出推断。时间序列的谱方法在频域分析数据。现代光谱方法面临的一些挑战包括频率混叠问题和与非平稳时间序列相关的难题，包括时变光谱的估计和不同形式的非平稳测试。一项更为务实的挑战是将这种方法引入传统上不使用这种方法的领域，我们的重点是环境健康风险和环境流行病学。在统计学中处理高维数据的挑战是识别数据中的低维结构（通常是非线性的），然后可以使用它来给出低维表示，同时保留数据的显著特征。这里提出的工作将解决所有这些挑战，并导致高质量的原创研究，这将进一步推动非参数统计方法的理论和应用。非参数贝叶斯方法的研究将深入了解后验表示的结构，这将产生新的数据结构推理和分组的计算方法。时间序列光谱方法的工作将推进处理时间序列中的混叠效应和非平稳性的方法，这具有广泛的潜在下游影响，其中最重要的是更好地理解定义我们物理世界的过程。进一步的下游影响将是完善和提高环境健康风险和环境流行病学领域风险估计的可解释性和可靠性，这最终可能影响我们制定的与加拿大人健康有关的政策。减少高维数据的工作最终是朝着开放为低维数据设计的强大统计方法以应对日益复杂的现代数据迈出的一步。培训是该研究项目的主要组成部分，该项目将支持5名博士和5名硕士研究生，4名本科生暑期研究生和2至3名博士后。

Progress in modern statistical methods remains vital in the face of challenges put forth by ever increasing amounts of data, the need to obtain accurate information from data that is increasingly pivotal in important decision processes, and the opportunities afforded by increased computing power. Nonparametric statistical methods, which is to say methods that place few and very weak assumptions on the data, are attractive in virtually every field of application for their broad applicability.The proposed research falls in this area and centres on developing theory and applications in nonparametric Bayesian methods, spectral methods for time series, and the reduction of high dimensional data. These are three distinct areas within statistics of active current research, dealing with qualitatively different types of data. Inference in Bayesian nonparametric methods, like all Bayesian methods, is based on a passage from a prior distribution to a posterior distribution over the quantity of interest which in the nonparametric case is not finite-dimensional. Examples are an unknown distribution, density function, or survival function. The desirable features of such methods are freeness from assumptions that the unknown quantity follows some parametric form as well as the wide range of inference possible once the posterior distribution is obtained in a workable form. The challenge with such methods is the theory and computation necessary to give posterior representations that are feasible to work with so as to draw inference from these representations efficiently. Spectral methods in time series analyze the data in the frequency domain. Some of the challenges in modern spectral methods include the problem of frequency aliasing and the difficult problems associated with nonstationary time series, including estimation of time varying spectra and testing for different forms of nonstationarity. A more pragmatic challenge involves the introduction of such methods into fields where such methods are traditionally not used, and we focus on environmental health risk and environmental epidemiolgy. The challenge in processing of high-dimensional data in statistics is to identify low dimensional structure in the data (which generally will be nonlinear) that can then be used to give a low dimensional representation while retaining the salient features of the data. The The work proposed here will address all of these challenges and lead to high quality, original research that will further the theory and applications in nonparametric statistical methods. The work in nonparametric Bayesian methods will give insight into the structure of posterior representations that will give rise to new computational methods for inference and grouping of data structure. The work in spectral methods for time series will advance methods for dealing with aliasing effects and nonstationarity in time series, which has wide ranging potential downstream implications, not the least of which is a better understanding of the processes defining our physical world. A further downstream impact will be to refine and improve the interpretability and reliability of risk estimation in the fields of environmental health risk and environmental epidemiology, which can ultimately affect the policies we make that pertain to the health of Canadians. The work in reduction of high dimensional data is ultimately a step towards opening up powerful statistical methodologies designed for low dimensional data to the ever increasing complexities of modern data. Training is a major component of this research program, which will support 5 Phd and 5 MSc graduate students, 4 undergraduate summer research students, and 2 to 3 Postdoctoral Fellows in total.