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1. Nonparametric covariance estimation with shrinkage toward stationary models

   https://doi.org/10.1002/wics.1507  

   Blake, Tayler A. ; Lee, Yoonkyung ( March 2020 , WIREs Computational Statistics)

   Estimation of an unstructured covariance matrix is difficult because of the challenges posed by parameter space dimensionality and the positive‐definiteness constraint that estimates should satisfy. We consider a general nonparametric covariance estimation framework for longitudinal data using the Cholesky decomposition of a positive‐definite matrix. The covariance matrix of time‐ordered measurements is diagonalized by a lower triangular matrix with unconstrained entries that are statistically interpretable as parameters for a varying coefficient autoregressive model. Using this dual interpretation of the Cholesky decomposition and allowing for irregular sampling time points, we treat covariance estimation as bivariate smoothing and cast it in a regularization framework for desired forms of simplicity in covariance models. Viewing stationarity as a form of simplicity or parsimony in covariance, we model the varying coefficient function with components depending on time lag and its orthogonal direction separately and penalize the components that capture the nonstationarity in the fitted function. We demonstrate construction of a covariance estimator using the smoothing spline framework. Simulation studies establish the advantage of our approach over alternative estimators proposed in the longitudinal data setting. We analyze a longitudinal dataset to illustrate application of the methodology and compare our estimates to those resulting from alternative models.

   This article is categorized under:

   Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

   Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

   Algorithms and Computational Methods > Maximum Likelihood Methods

    

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2. Conjugate sparse plus low rank models for efficient Bayesian interpolation of large spatial data

   https://doi.org/10.1002/env.2748  

   Shirota, Shinichiro ; Finley, Andrew O. ; Cook, Bruce D. ; Banerjee, Sudipto ( August 2022 , Environmetrics)

   A key challenge in spatial data science is the analysis for massive spatially‐referenced data sets. Such analyses often proceed from Gaussian process specifications that can produce rich and robust inference, but involve dense covariance matrices that lack computationally exploitable structures. Recent developments in spatial statistics offer a variety of massively scalable approaches. Bayesian inference and hierarchical models, in particular, have gained popularity due to their richness and flexibility in accommodating spatial processes. Our current contribution is to provide computationally efficient exact algorithms for spatial interpolation of massive data sets using scalable spatial processes. We combine low‐rank Gaussian processes with efficient sparse approximations. Following recent work by Zhang et al. (2019), we model the low‐rank process using a Gaussian predictive process (GPP) and the residual process as a sparsity‐inducing nearest‐neighbor Gaussian process (NNGP). A key contribution here is to implement these models using exact conjugate Bayesian modeling to avoid expensive iterative algorithms. Through the simulation studies, we evaluate performance of the proposed approach and the robustness of our models, especially for long range prediction. We implement our approaches for remotely sensed light detection and ranging (LiDAR) data collected over the US Forest Service Tanana Inventory Unit (TIU) in a remote portion of Interior Alaska.

    

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3. Extrinsic Gaussian Processes for Regression and Classification on Manifolds

   https://doi.org/10.1214/18-BA1135  

   Lin, Lizhen ; Mu, Niu ; Cheung, Pokman ; Dunson, David ( January 2018 , Bayesian Analysis)

   Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications, methods, theory and algorithms related to GPs, the overwhelming majority of this literature focuses on the case in which the input domain corresponds to a Euclidean space. However, particularly in recent years with the increasing collection of complex data, it is commonly the case that the input domain does not have such a simple form. For example, it is common for the inputs to be restricted to a non-Euclidean manifold, a case which forms the motivation for this article. In particular, we propose a general extrinsic framework for GP modeling on manifolds, which relies on embedding of the manifold into a Euclidean space and then constructing extrinsic kernels for GPs on their images. These extrinsic Gaussian processes (eGPs) are used as prior distributions for unknown functions in Bayesian inferences. Our approach is simple and general, and we show that the eGPs inherit fine theoretical properties from GP models in Euclidean spaces. We consider applications of our models to regression and classification problems with predictors lying in a large class of manifolds, including spheres, planar shape spaces, a space of positive definite matrices, and Grassmannians. Our models can be readily used by practitioners in biological sciences for various regression and classification problems, such as disease diagnosis or detection. Our work is also likely to have impact in spatial statistics when spatial locations are on the sphere or other geometric spaces. 

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4. Geostatistical modeling of positive‐definite matrices: An application to diffusion tensor imaging

   https://doi.org/10.1111/biom.13445  

   Lan, Zhou ; Reich, Brian J. ; Guinness, Joseph ; Bandyopadhyay, Dipankar ; Ma, Liangsuo ; Moeller, F. Gerard ( June 2022 , Biometrics)

   Geostatistical modeling for continuous point‐referenced data has extensively been applied to neuroimaging because it produces efficient and valid statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging technique characterizing the brain's anatomical structure, produces a positive‐definite (p.d.) matrix for each voxel. Currently, only a few geostatistical models for p.d. matrices have been proposed because introducing spatial dependence among p.d. matrices properly is challenging. In this paper, we use the spatial Wishart process, a spatial stochastic process (random field), where each p.d. matrix‐variate random variable marginally follows a Wishart distribution, and spatial dependence between random matrices is induced by latent Gaussian processes. This process is valid on an uncountable collection of spatial locations and is almost‐surely continuous, leading to a reasonable way of modeling spatial dependence. Motivated by a DTI data set of cocaine users, we propose a spatial matrix‐variate regression model based on the spatial Wishart process. A problematic issue is that the spatial Wishart process has no closed‐form density function. Hence, we propose an approximation method to obtain a feasible Cholesky decomposition model, which we show to be asymptotically equivalent to the spatial Wishart process model. A local likelihood approximation method is also applied to achieve fast computation. The simulation studies and real data application demonstrate that the Cholesky decomposition process model produces reliable inference and improved performance, compared to other methods.

    

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5. Efficient and Adaptive Decentralized Sparse Gaussian Process Regression for Environmental Sampling Using Autonomous Vehicles

   Norton, Tanner A. ( June 2022 , BYU Scholars Archive)

   In this thesis, I present a decentralized sparse Gaussian process regression (DSGPR) model with event-triggered, adaptive inducing points. This DSGPR model brings the advantages of sparse Gaussian process regression to a decentralized implementation. Being decentralized and sparse provides advantages that are ideal for multi-agent systems (MASs) performing environmental modeling. In this case, MASs need to model large amounts of information while having potential intermittent communication connections. Additionally, the model needs to correctly perform uncertainty propagation between autonomous agents and ensure high accuracy on the prediction. For the model to meet these requirements, a bounded and efficient real-time sparse Gaussian process regression (SGPR) model is needed. I improve real-time SGPR models in these regards by introducing an adaptation of the mean shift and fixed-width clustering algorithms called radial clustering. Radial clustering enables real-time SGPR models to have an adaptive number of inducing points through an efficient inducing point selection process. I show how this clustering approach scales better than other seminal Gaussian process regression (GPR) and SGPR models for real-time purposes while attaining similar prediction accuracy and uncertainty reduction performance. Furthermore, this thesis addresses common issues inherent in decentralized frameworks such as high computation costs, inter-agent message bandwidth restrictions, and data fusion integrity. These challenges are addressed in part through performing maximum consensus between local agent models which enables the MAS to gain the advantages of decentralization while keeping data fusion integrity. The inter-agent communication restrictions are addressed through the contribution of two message passing heuristics called the covariance reduction heuristic and the Bhattacharyya distance heuristic. These heuristics enable user to reduce message passing frequency and message size through the Bhattacharyya distance and properties of spatial kernels. The entire DSGPR framework is evaluated on multiple simulated random vector fields. The results show that this framework effectively estimates vector fields using multiple autonomous agents. This vector field is assumed to be a wind field; however, this framework may be applied to the estimation of other scalar or vector fields (e.g., fluids, magnetic fields, electricity, etc.). Keywords: Sparse Gaussian process regression, clustering, event-triggered, decentralized, sensor fusion, uncertainty propagation, inducing points 

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