Nonparametric Methods for Temporally Correlated and High Dimensional Data

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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. TheThe 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.