High dimensional spatial modeling of extremes with applications to United States rainfalls Public Deposited

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  • March 21, 2019
Creator
  • Zhou, Jie
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
Abstract
  • Spatial statistical models are used to predict unobserved variables based on observed variables and to estimate unknown model parameters. Extreme value theory(EVT) is used to study large or small observations from a random phenomenon. Both spatial statistics and extreme value theory have been studied in a lot of areas such as agriculture, finance, industry and environmental science. This dissertation proposes two spatial statistical models which concentrate on non-Gaussian probability densities with general spatial covariance structures. The two models are also applied in analyzing United States Rainfalls and especially, rainfall extremes. When the data set is not too large, the first model is used. The model constructs a generalized linear mixed model(GLMM) which can be considered as an extension of Diggle's model-based geostatistical approach(Diggle et al. 1998). The approach improves conventional kriging with a form of generalized linear mixed structure. As for high dimensional problems, two different methods are established to improve the computational efficiency of Markov Chain Monte Carlo(MCMC) implementation. The first method is based on spectral representation of spatial dependence structures which provides good approximations on each MCMC iteration. The other method embeds high dimensional covariance matrices in matrices with block circulant structures. The eigenvalues and eigenvectors of block circulant matrices can be calculated exactly by Fast Fourier Transforms(FFT). The computational efficiency is gained by transforming the posterior matrices into lower dimensional matrices. This method gives us exact update on each MCMC iteration. Future predictions are also made by keeping spatial dependence structures fixed and using the relationship between present days and future days provided by some Global Climate Model(GCM). The predictions are refined by sampling techniques. Both ways of handling high dimensional covariance matrices are novel to analyze large data sets with extreme value distributions involved. One of the main outcomes of this model is for producing N-year return values and return years for a given value for precipitation at a single location given climate model projections based on a grid. This is very important, because in many applications, detailed precipitation information on pointwise locations is more important that predictions averaged over grids. The second model can be applied to those large data sets and is based on transformed Gaussian processes. These processes are thresholded due to the emphasis on rainfall extremes. Keywords: Block Circulant Matrix; Extreme value theory;Fast Fourier Transform; Generalized Linear Mixed Model; Kriging; Markov Chain Monte Carlo; Spectral Representation; Spatial statistics
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Advisor
  • Smith, Richard L.
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