Gong, Kaibo2022-08-292022-08-292021-10https://hdl.handle.net/11299/241390University of Minnesota Ph.D. dissertation. October 2021. Major: Statistics. Advisors: Snigdhansu Chatterjee, Nathaniel Helwig. 1 computer file (PDF); vi, 184 pages.Comparison of two different data samples, and of paired data samples, is a well known problem in Statistics. Specifically, there is a wide range of applications in the fields of climate study. In this thesis, we provide a brief review on the ensemble of climate models and the need of probabilistic evaluation of model outputs, which is equivalent to the comparison between two models. Based on recent advancements in the context of evaluating climate model outputs, we develop two different approaches for comparing two functional time series. The first one is based on wavelet decomposition and the second one by comparing the local spectral density of non-stationary series. For the last chapter, we conduct a brief review on Gaussian Process and a framework for Bayesian Optimization, which establishes a theoretical framework and algorithmic properties of t-process based spatio-temporal modeling, for further use in modeling climate and neuroscience data.enClimateGaussian processSpectral densityStatisticsTime seriesWaveletThesis or Dissertation