Doctor of Philosophy / Department of Statistics / Juan Du / It is common to assume the spatial or spatio-temporal data are realizations of underlying random elds or stochastic processes. E ective approaches to modelling of the underlying autocorrelation structure of the same random eld and the association among multiple processes are of great demand in many areas including atmospheric sciences, meteorology and agriculture. To this end, this dissertation studies methods and application of the spatial modeling of large-scale dependence structure and spatio-temporal regression modelling. First, variogram and variogram matrix functions play important roles in modeling dependence structure among processes at di erent locations in spatial statistics. With more and more data collected on a global scale in environmental science, geophysics, and related elds, we focus on the characterizations of the variogram models on spheres of all dimensions for both stationary and intrinsic stationary, univariate and multivariate random elds. Some e cient approaches are proposed to construct a variety of variograms including simple polynomial structures. In particular, the series representation and spherical behavior of intrinsic stationary random elds are explored in both theoretical and simulation study. The applications of the proposed model and related theoretical results are demonstrated using simulation and real data analysis. Second, knowledge of the influential factors on the number of days suitable for fieldwork (DSFW) has important implications on timing of agricultural eld operations, machinery decision, and risk management. To assess how some global climate phenomena such as El Nino Southern Oscillation (ENSO) a ects DSFW and capture their complex associations in space and time, we propose various spatio-temporal dynamic models under hierarchical Bayesian framework. The Integrated Nested Laplace Approximation (INLA) is used and adapted to reduce the computational burden experienced when a large number of geo-locations and time points is considered in the data set. A comparison study between dynamics models with INLA viewing spatial domain as discrete and continuous is conducted and their pros and cons are evaluated based on multiple criteria. Finally a model with time- varying coefficients is shown to reflect the dynamic nature of the impact and lagged effect of ENSO on DSFW in US with spatio-temporal correlations accounted.

Rabies has been reported in Tanzania, mainly in the southern highland regions, since 1954. To date, rabies is endemic in all districts in Tanzania and efforts are being made to contain the disease. It was determined that mass vaccination of at least 70% of an animal population is most effective, in terms of profitability and cost, in reducing transmission of rabies. The current approach for vaccination in Tanzanian villages takes some features from the EPI method but is rather basic and unreliable. This mini-dissertation proposes using a sampling technique which incorporates the spatial component of the village data and minimises the walking distance between the sampled houses while ensuring the 70% coverage of the animal population. / Mini Dissertation(MSc)--University of Pretoria, 2017. / STATOMET The Centre for Artificial Intelligence Research (CAIR) National Research Foundation of South Africa (NRF CSUR grant number 90315) / Statistics / MSc / Unrestricted

Lava flow hazards are one of the few constant themes across the wide spectrum of volcanic research in the solar system. These dynamic hazards are controlled by the location of the eruption, the topography and material properties of the land upon which the flow spreads, and the properties of the lava (e.g., volume, temperature, and rheology). Understanding the influences on eruption location and how lava flows modify the landscape are important steps to accurately forecast volcanic hazards. Three studies are presented in this dissertation that address di˙erent aspects of modeling and assessing vent opening and lava flow hazards. The first study uses hierarchical clustering to explore the distribution of activity at Craters of the Moon (COM) lava field on the eastern Snake River Plain (ESRP). Volcanism at COM is characterized by 53 mapped eruptive vents and 60+ lava flows over the last 15 ka. Temporal, spatial, and spatio-temporal clustering methods that examine different aspects of the distribution of volcanic vents are introduced. The sensitivity of temporal clustering to different criteria that capture the age range of magma generation and ascent is examined Spatial clustering is dictated by structures on the ESRP that attempt to capture the footprint of an emplacing dike. A combined spatio-temporal is the best approach to understanding the distribution of linked eruptive centers and can also provide insight into the evolution of volcanism for the region. Spatial density estimation is used to visualize the differences between these models. The goal of this work is to improve vent opening forecasting tools for use in assessing lava flow hazards. The second study presents a new probabilistic lava flow hazard assessment for the U.S. Department of Energy’s Idaho National Laboratory (INL) nuclear facility that (1) explores the way eruptions are defined and modeled, (2) stochastically samples lava flow parameters from observed values for use in MOLASSES, a lava flow simulator, (3) calculates the likelihood of a new vent opening within the boundaries of INL, (4) determines probabilities of lava flow inundation for INL through Monte Carlo simulation, and (5) couples inundation probabilities with recurrence rates to determine the annual likelihood of lava flow inundation for INL. Results show a 30% probability of partial inundation of the INL given an e˙usive eruption on the ESRP, with an annual inundation probability of 8.4×10^−5 to 1.8×10^−4. An annual probability of 6.2×10^−5 to 1.2×10^−4 is estimated for the opening of a new eruptive center within INL boundaries. The third study models thermo-mechanical erosion of a pyroclastic substrate by flow-ing lava on Volcán Momotombo, Nicaragua. It describes the unique morphology of a lava flow channel using TanDEM-X/TerraSAR-X and terrestrial radar digital elevation models. New methods for modeling paleotopography on steep-sided cones are introduced to mea-sure incision depths and document cross-channel profiles. The channel is incised ~35 m into the edifice at the summit and transitions into a constructional feature halfway down the ~1,300 m high cone. An eroded volume of ~4×10^5 m3 was calculated. It is likely that a lava flow eroded into the cone as it emplaced during an eruption in 1905. There is not suÿcient energy to thermally erode this volume, given the observed morphology of the flow. Models are tested that explore the relationship of shearing and material properties of the lava and substrate against measured erosion depths and find that thermo-mechanical erosion is the most likely mode of channel formation. Additionally, it is likely that all forms of erosion via lava flow are impacted by thermal conditions due to the relationship between temperature and substrate hardness. The evolution of these structures (their creation and subsequent infilling) plays an important role in the growth of young volcanoes and also controls future lava flows hazards, as seen by the routing of the 2015 flow into the 1905 channel.

Utilizing statistical methods as a risk assessment tool can lead to potentially effective solutions and policies that address various social issues. One usage for such methods is in observation of crime trends within a municipality. Cluster and hotspot analysis is often practiced in criminal statistics to delineate potential areas at-risk of recurring criminal activity. Two approaches to this analytical method are Nearest Neighbor Hierarchical Clustering (NNHC) and Kernel Density Estimation (KDE). Kernel Density Estimation fits incidence points on a grid based on a kernel and bandwidth determined by the analyst. Nearest Neighbor Hierarchical Clustering, a less common and less quantitative method, derives clusters based on the distance between observed points and the expected distance for points of a random distribution. Crime data originated from a public web map and database service that acquires data from the Johnson City Police Department, where each incident is organized into one of many broad categories such as assault, theft, etc. Preliminary analysis of raw volume data shows trends of high crime volume in expected locales; highly trafficked areas such as downtown, the Mall, both Walmarts, as well as low-income residential areas of town. The two methods, KDE and NNHC, dispute the size and location of many clusters. A more in-depth analysis of normalized data with refined parameters may provide further insight on crime in Johnson City.

Thesis (Ph.D. (Statistics)) -- University of Limpopo, 2021 / In spatial statistics, several methods have been developed to measure the extent of local and global spatial dependence (clustering) in measured data across areas in a region of research interest. These methods are now routinely implemented in most Geographical Information Systems (GIS) and statistical computer packages. However, spatial statistics for measuring joint spatial dependence of multiple spatial measurement and outcome data have not been well developed. A naive analysis would simply apply univariate spatial dependence methods to each data separately. Though this is simple and straightforward, it ignores possible relationships between multiple spatial data because they may be measuring the same phenomena. Limited work has been done on extending the Moran’s index, a commonly used and applied univariate measure of spatial clustering, to bivariate Moran’s index in order to assess spatial dependence for two spatial data. The overall aim of this PhD was to develop multivariate spatial clustering methods for multiple spatial data, especially in the health sciences. Our proposed multivariate spatial clustering statistic is based on the fundamental theory regarding canonical correlations. We firstly reviewed and applied univariate and bivariate Moran’s indexes to spatial analyses of multiple non-communicable diseases and related risk factors in South Africa. Then we derived our proposed multivariate spatial clustering method, which was evaluated by simulation studies and applied to a spatial analysis of multiple non-communicable diseases and related risk factors in South Africa. Simulation studies showed that our proposed multivariate spatial statistic was able to identify correctly clusters of areas with high risks as well as clusters with low risk.

High incidences of slope movement are observed throughout Cuyahoga River watershed in northeast Ohio, USA. The major type of slope failure involves rotational movement in steep stream walls where erosion of the banks creates over-steepened slopes. The occurrence of landslides in the area depends on a complex interaction of natural as well as human induced factors, including: rock and soil strength, slope geometry, permeability, precipitation, presence of old landslides, proximity to streams and flood-prone areas, land use patterns, excavation of lower slopes and/or increasing the load on upper slopes, alteration of surface and subsurface drainage. These factors were used to evaluate the landslide-induced hazard in Cuyahoga River watershed using logistic regression analysis, and a landslide susceptibility map was produced in ArcGIS. The map classified land into four categories of landslide susceptibility: low, moderate, high, and very high. The susceptibility map was validated using known landslide locations within the watershed area. The landslide susceptibility map produced by the logistic regression model can be efficiently used to monitor potential landslide-related problems, and, in turn, can help to reduce hazards associated with landslides.

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FMRI datasets allow scientists to assess functionality of the brain by measuring the response of blood flow to a stimulus. Since the responses from neighboring locations within the brain are correlated, simple linear models that assume independence of measurements across locations are inadequate. Mixed models can be used to model the spatial correlation between observations, however selecting the correct covariance structure is difficult. Information criteria, such as AIC are often used to choose among covariance structures. Once the covariance structure is selected, significance tests can be used to determine if a region of interest within the brain is significantly active. Through the use of simulations, this project explores the performance of AIC in selecting the covariance structure. Type I error rates are presented for the fixed effects using the the AIC chosen covariance structure. Power of the fixed effects are also discussed.

Lifelines such as highways, pipelines, telecommunication lines, and powerlines provide communities with vital services, and their functionality is dependent upon the foundation soil that supports them. However, when designing the infrastructure, it can be difficult to know where to test the soil in order to give spatially representative sampling, particularly for long, lifeline structures. Finding this distance requires knowledge of the spatial correlation and/or the spatial variability of the soil parameter (stiffness, cohesion, etc.). But this correlation distance is not typically found in practice because it requires large amounts of data and the costs of retrieving that data can be high. Lack of representative sampling can lead to an overly conservative design and too much sampling can create an overly expensive sampling program. In this study, multiple tests using the geophysical method of spatial autocorrelation (SPAC) were conducted to find the soil stiffness along a 310 meter long profile. SPAC records passive surface waves which sample the underlying soil, and these surface waves can be used to create a shear wave velocity profile of the site. The spatial continuity of the stiffness (the soil velocity values) was then found using geostatistics. The geostastical tool primarily used in this study was the (semi-)variogram, but the covariance function and the correlogram are also shown. By using these tools, the spatial correlation/variability can give an estimate of the how far apart to test the foundation soil so that the data is spatially representative. In other words, finding the distance that the soil parameter is minimally correlated with itself. This study found the distance (the range of the semi-variogram) to be 70 meters for 5 meters depth, 100 meters for 10 to 15 meters depth, and 90 meters for 30 meters depth.

Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable or generalized Pareto processes. These models are suitable when asymptotic dependence is present. However, recent environmental data applications suggest that asymptotic independence is equally important. Therefore, development of flexible subasymptotic models is in pressing need. This dissertation consists of four major contributions to subasymptotic modeling of multivariate and spatial extremes. Firstly, the dissertation proposes a new spatial copula model for extremes based on the multivariate generalized hyperbolic distribution. The extremal dependence of this distribution is revisited and a corrected theoretical description is provided. Secondly, the dissertation thoroughly investigates the extremal dependence of stochastic processes driven by exponential-tailed Lévy noise. It shows that the discrete approximation models, which are linear transformations of a random vector with independent components, bridge asymptotic independence and asymptotic dependence in a novel way, whilst the exact stochastic processes exhibit only asymptotic independence. Thirdly, the dissertation explores two different notions of optimal prediction for extremes, and compares the classical linear kriging predictor and the conditional mean predictor for certain non-Gaussian models. Finally, the dissertation proposes a multivariate skew-elliptical link model for correlated highly-imbalanced (extreme) binary responses, and shows that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior.