Statistical methods for space-time processes, time-frequency methodologies, and applications

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This project considers space-time models and time-frequency methods for the analysis of space-time data observed continuously (and often sparsely) in space, but discretely (and regular) in time. This research extends the spatially-dependent filtering approach used to define space-time processes to include space-time long memory processes, non-Gaussian processes, and non-linear processes. Since methods need to be developed for these types of statistical models that are efficient to use, part of this research focuses on statistical inference. A secondary study involves spectral and wavelet methods for the analysis of space-time processes. A spectral analysis is used to explore features of a statistical process in the frequency domain in terms of a linear combination of complex exponentials (sinusoids). A wavelet analysis provides a space/time-scale (approximately a space/time-frequency) decomposition of a statistical process in terms of averages and changes of averages over different temporal or spatial scales. Developing methods ofspectral- and wavelet-based exploratory data analysis and inference are of key interest.There is a growing need in many scientific areas to be able to understand phenomena that vary jointly across space and in time. Statistical methods are required in practice because these phenomena are observed in the presence of uncertainty. For example, Paleooclimatology (the history or "archaeology" of climate) involves obtaining surrogate measures for climatic variables over space that are valid over long time scales. Important scientific questions can be answered by relating data obtained from paleoclimatology to drivers of climate variability. The use of space-time statistical models and spectral and wavelet-based space-time analyses can inform how different temporal scales affect the climate relationships observed, and to understand how these relationships vary spatially. This research is directly applicable to other scientific areas, and results will be communicated via peer-reviewed articles in subject-matter as well as statistical areas. A diverse cross-section of students (statistical and non-statistical) will be mentored in methods of time series analysis and spatial statistics (via supervision and teaching).