In this talk we introduce a class of statistics for spatial data that is observed on an irregular set of locations. Our aim is to obtain a unified framework for inference and the statistics we consider include both parametric and nonparametric estimators of the spatial covariance function, Whittle likelihood estimation, goodness of fit tests and a test for second order spatial stationarity. To ensure that the statistics are computationally feasible they are defined within the Fourier domain, and in most cases can be expressed as a quadratic form of a discrete Fourier-type transform of the spatial data. Evaluation of such statistic is computationally tractable, requiring $O(nb)$ operations, where $b$ are the number Fourier frequencies used in the definition of the statistic (which varies according to the application) and $n$ is the sample size. The asymptotic sampling properties of the statistics are derived using mixed spatial asymptotics, where the number of locations grows at a faster rate than the size of the spatial domain and under the assumption that the spatial random field is stationary and the irregular design of the locations are independent, identically distributed random variables. We show that there are quite intriguing differences in the behaviour of the statistic when the spatial process is Gaussian and non-Gaussian. In particular, the choice of the number of frequencies $b$ in the construction of the statistic depends on whether the spatial process is Gaussian or not. If time permits we describe how the results can also be used in variance estimation. And if we still have time some simulations and real data will be presented.[-]

With the growing capabilities of Geographic Information Systems (GIS) and user-friendly software, statisticians today routinely encounter geographically referenced data containing observations from a large number of spatial locations and time points. Over the last decade, hierarchical spatiotemporal process models have become widely deployed statistical tools for researchers to better understand the complex nature of spatial and temporal variability. However, fitting hierarchical spatiotemporal models often involves expensive matrix computations with complexity increasing in cubic order for the number of spatial locations and temporal points. This renders such models unfeasible for large data sets. I will present a focused review of two methods for constructing well-defined highly scalable spatiotemporal stochastic processes. Both these processes can be used as ``priors" for spatiotemporal random fields. The first approach constructs a low-rank process operating on a lower-dimensional subspace. The second approach constructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparse precision matrices for its finite realizations. Both processes can be exploited as a scalable prior embedded within a rich hierarchical modeling framework to deliver full Bayesian inference. These approaches can be described as model-based solutions for big spatiotemporal datasets. The models ensure that the algorithmic complexity has n floating point operations (flops), where n is the number of spatial locations (per iteration). We compare these methods and provide some insight into their methodological underpinnings.[-]

Arctic sea-ice extent has been of considerable interest to scientists in recent years, mainly due to its decreasing trend over the past 20 years. In this talk, I propose a hierarchical spatio-temporal generalized linear model (GLM) for binary Arctic-sea-ice data, where data dependencies are introduced through a latent, dynamic, spatio-temporal mixed-effects model. By using a fixed number of spatial basis functions, the resulting model achieves both dimension reduction and non-stationarity for spatial fields at different time points. An EM algorithm is used to estimate model parameters, and an MCMC algorithm is developed to obtain the predictive distribution of the latent spatio-temporal process. The methodology is applied to spatial, binary, Arctic-sea-ice data for each September over the past 20 years, and several posterior summaries are computed to detect changes of Arctic sea-ice cover. The fully Bayesian version is under development awill be discussed.[-]

Averages of proper scoring rules are often used to rank probabilistic forecasts. In many cases, the individual observations and their predictive distributions in these averages have variable scale (variance). I will show that some of the most popular proper scoring rules, such as the continuous ranked probability score (CRPS), up-weight observations with large uncertainty which can lead to unintuitive rankings. We have developed a new scoring rule, scaled CRPS (SCRPS), this new proper scoring rule is locally scale invariant and therefore works in the case of varying uncertainty.
I will demonstrate this how this affects model selection through parameter estimation in spatial statitics.[-]