About Me

Morris, L. R., & Sibanda, N. (2022)

Te Herenga Waka—Victoria University of Wellington

Within the field of geostatistics, Gaussian processes are a staple for modelling spatial and spatio-temporal data. Statistical literature is rich with estimation methods for the mean and covariance of such processes (in both frequentist and Bayesian contexts). Considerably less attention has been paid to developing goodness-of-fit tests for assessment of model adequacy. Jun et al. (Environmetrics 25(8):584-595, 2014) introduced a statistical test that uses pivotal discrepancy measures to assess goodness-of-fit in the Bayesian context. We present a modification and generalization of their statistical test. The initial method involves spatial partitioning of the data, followed by evaluation of a pivotal discrepancy measure at each posterior draw to obtain a posterior distribution of pivotal statistics. Order statistics from this distribution are used to obtain approximate p-values. Jun et al. (Environmetrics 25(8):584-595, 2014) use arbitrary partitions based on pre-existing spatial boundaries. The partitions are made to be of equal size. Our contribution is two-fold. We use K-means clustering to create the spatial partitions and we generalise Jun et al.'s approach to incorporate unequal partition sizes. Observations from a spatial or spatio-temporal process are partitioned using an appropriate feature vector that incorporates the geographic location of the observations into subsets (not necessarily of the same size). The method's viability is illustrated in a simulation study, and in an application to hoki (Macruronus novaezelandiae) catch data from a survey of the sub-Antarctic region.

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Morris, L. R. (2021)

Te Herenga Waka—Victoria University of Wellington

Awarded Doctor of Philosophy

Spatial and spatio-temporal phenomena are commonly modelled as Gaussian processes via the geostatistical model (Gelfand & Banerjee, 2017). In the geostatistical model the spatial dependence structure is modelled using covariance functions. Most commonly, the covariance functions impose an assumption of spatial stationarity on the process. That means the covariance between observations at particular locations depends only on the distance between the locations (Banerjee et al., 2014). It has been widely recognized that most, if not all, processes manifest spatially nonstationary covariance structure Sampson (2014). If the study domain is small in area or there is not enough data to justify more complicated nonstationary approaches, then stationarity may be assumed for the sake of mathematical convenience (Fouedjio, 2017). However, relationships between variables can vary significantly over space, and a 'global' estimate of the relationships may obscure interesting geographical phenomena (Brunsdon et al., 1996; Fouedjio, 2017; Sampson & Guttorp, 1992). In this thesis, we considered three non-parametric approaches to flexibly account for non-stationarity in both spatial and spatio-temporal processes. First, we proposed partitioning the spatial domain into sub-regions using the K-means clustering algorithm based on a set of appropriate geographic features. This allowed for fitting separate stationary covariance functions to the smaller sub-regions to account for local differences in covariance across the study region. Secondly, we extended the concept of covariance network regression to model the covariance matrix of both spatial and spatio-temporal processes. The resulting covariance estimates were found to be more flexible in accounting for spatial autocorrelation than standard stationary approaches. The third approach involved geographic random forest methodology using a neighbourhood structure for each location constructed through clustering. We found that clustering based on geographic measures such as longitude and latitude ensured that observations that were too far away to have any influence on the observations near the locations where a local random forest was fitted were not selected to form the neighbourhood. In addition to developing flexible methods to account for non-stationarity, we developed a pivotal discrepancy measure approach for goodness-of-fit testing of spatio-temporal geostatistical models. We found that partitioning the pivotal discrepancy measures increased the power of the test.

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Morris, L. R. (2017)

Te Herenga Waka—Victoria University of Wellington

Awarded Master of Science

In order to carry out assessment of marine stock levels, an accurate estimate of the current year's population abundance must be formulated. Standardized catch per unit of effort (CPUE) values are, in theory, proportional to population abundance. However, this only holds if the species catchability is constant over time. In almost all cases it is not, due to the existence of spatial and temporal variation. In this thesis, we fit various models to test different combinations and structures of spatial and temporal autocorrelation within hoki (Macruronus novaezelandiae) CPUE. A Bayesian approach was taken, and the spatial and temporal components were modelled using Gaussian Markov random fields. The data was collected from summer research trawl surveys carried out by the National Institute of Water and Atmospheric Research (NIWA) and the Ministry for Primary Industries (MPI). It allowed us to model spatial distribution using both areal and point reference approaches. To fit the models, we used the software Stan (Gelman et al., 2015) which implements Hamiltonian Monte Carlo. Model comparison was carried out using the Watanabe-Akaike information criterion (WAIC, (Watanabe, 2010)). We found that trawl year was the most important factor to explain variation in research survey hoki CPUE. Furthermore, the areal approach provided better indices of abundance than the point reference approach.

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