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عنوان فارسی:

پیشرفت در آمار فضایی و روش های استنتاج برای مدل های جمعیت مارکوف

عنوان انگلیسی:

Advances in Spatial Statistics and Inference Methods for Markov Population Models

Abstract

Spatial generalized linear mixed models (SGLMMs) commonly rely on Gaussian random fields (GRFs) to capture spatially correlated error. We investigate the results of replacing Gaussian processes with Laplace moving averages (LMAs) in SGLMMs. We demonstrate that LMAs offer improved predictive power when the data exhibits localized spikes in the response. SGLMMs with LMAs are shown to maintain analogous parameter inference and similar computing to Gaussian SGLMMs. We propose a novel discrete space LMA model for irregular lattices and construct conjugate samplers for LMAs with georeferenced and areal support. We provide a Bayesian analysis of SGLMMs with LMAs and GRFs over multiple data support and response types

We develop methods for privatizing spatial location data, such as spatial locations of individual disease cases. We propose two novel Bayesian methods for generating synthetic location data based on log-Gaussian Cox processes (LGCPs). We show that conditional predictive ordinate (CPO) estimates can easily be obtained for point process data. We construct a novel risk metric that utilizes CPO estimates to evaluate individual disclosure risks. We adapt the propensity mean square error (pMSE) data utility metric for LGCPs. We demonstrate that our synthesis methods offer an improved risk vs. utility balance in comparison to radial synthesis with a case study of Dr. John Snow’s cholera outbreak data.

We demonstrate how to perform inference on Markov population processes with Laplace approximations. We derive a sparse covariance structure for the linear noise approximation (LNA) which offers a joint Gaussian likelihood for Markov population models based solely on the solution to a set of deterministic equations. We show that Laplace approximations allow inference with LNAs to be parallelized and require no stochastic infill. We also demonstrate that our method offers comparable accuracy to MCMC on a simulated Susceptible-Infected-Susceptible data set. We use Laplace approximations to fit a stochastic susceptible-exposed-infected-recovered system to the Princess Diamond COVID-19 cruise ship data set.

عنوان فارسی:

آمار طیفی نمودارهای تصادفی d-Regular

عنوان انگلیسی:

Spectral Statistics of Random d-Regular Graphs

Abstract

In this thesis we study the uniform random d-regular graphs on N vertices from a random matrix theory point of view.

In the first part of this thesis, we focus on uniform random d-regular graphs with large but fixed degree. In the bulk of the spectrum down to the optimal spectral scale, we prove that the Green's functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten--McKay law holds for the empirical eigenvalue density down to the smallest scale and the bulk eigenvectors are completely delocalized. Our method is based on estimating the Green's function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs.

In the second part of this thesis, we prove, for 1 « d « N2/3, in the bulk of the spectrum the local eigenvalue correlation functions and the distribution of the gaps between consecutive eigenvalues coincide with those of the Gaussian orthogonal ensemble. In order to show this, we interpolate between the adjacent matrices of random d-regular graphs and the Gaussian orthogonal ensemble using a constrained version of Dyson Brownian motion.