Date of Graduation

12-2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Chakraborty, Avishek A.

Committee Member

Petris, Giovanni G.

Second Committee Member

Tipton, John R.

Third Committee Member

Kaman, Tulin

Fourth Committee Member

Robinson, Samantha E.

Keywords

Conditional autoregressive prior; Functional predictors; Graph Laplacian; Implicit Restarted Lanczos algorithm; Landsat time series; Reduced rank model

Abstract

The focus of this dissertation is development of a novel hierarchical framework, that can be used for predictive modeling of Forest Inventory and Analysis (FIA) data over large regions. This dissertation has two significant contributions. Based on a study region in north-central Wisconsin, we analyze satellite imagery, along with a sample of national forest inventory field plots, to monitor and predict changes in forest conditions over time. The auxiliary data from the satellite imagery of this region are relatively dense in space and time, and can be used to learn how forest conditions changed over that decade. However, these records have a significant proportion of missing values due to weather conditions and system failures that we fill in first using a spatio-temporal model. Subsequently, we use the complete imagery as functional predictors in a two-component mixture model to capture the spatial variation in yearly average live tree basal area, an attribute of interest measured on field plots. We further modify the regression equation to accommodate a biophysical constraint on how plot-level live tree basal area can change from one year to the next. Findings from our analysis, represented with a series of maps, match known spatial patterns across the landscape. However, due to the computational challenge in the spatio-temporal model used to fill in the missing satellite imagery, our analysis was limited to a part of the study region. Therefore, in the next part of this dissertation, we focus on spatial and spatio-temporal models for large areal data, that would enable us to extend our analysis to the entire study region.

Data observed in a group of non-overlapping spatial areal units are widespread in scientific applications. Usually, a spatial random effect is introduced to model this kind of data, and a conditional autoregressive (CAR) prior is assigned to it. However, this framework requires a sequential update of the spatial random effect in the Markov chain Monte Carlo scheme, which reduces the computational efficiency of the method inducing high correlation across posterior samples. An alternative approach, known as spectral method, has been proposed in recent literature where any areal level function is represented using a linear combination of a small number of eigenvectors associated with smaller eigenvalues of the Laplacian matrix of the adjacency structure. We show that spectral method's approach of assigning priors to the coefficients of linear combinations can potentially lead to significant overestimation of variance parameters of the spatial random effect as well as pure error. We propose a modification to their prior specification and establish that this modification (i) significantly improves the estimation of spatial variance parameter and (ii) makes our approach a reduced-rank approximation to the CAR prior. We note that, this approach still overestimates the pure error variance. To overcome this problem, we propose second approach by adding a spatially independent yet heteroscedastic error term. We discuss how to extend these specifications to spatio-temporal methods. We present several examples using synthetic and real-world datasets that validate the predictive accuracy and computational efficiency of proposed models.

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Available for download on Monday, February 17, 2025

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