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Parameter inference for linear and non-Gaussian state space models is challenging because the likelihood function contains an intractable integral over the latent state variables. Exact inference using Markov chain Monte Carlo is computationally expensive, particularly for long time series data. Variational Bayes methods are useful when exact inference is infeasible. These methods approximate the posterior density of the parameters by a simple and tractable distribution found through optimisation. In this paper, we propose a novel sequential variational Bayes approach that makes use of the Whittle likelihood for computationally efficient parameter inference in this class of state space models. Our algorithm, which we call Recursive Variational Gaussian Approximation with the Whittle Likelihood (R-VGA-Whittle), updates the variational parameters by processing data in the frequency domain. At each iteration, R-VGA-Whittle requires the gradient and Hessian of the Whittle log-likelihood, which are available in closed form for a wide class of models. Through several examples using a linear Gaussian state space model and a univariate/bivariate non-Gaussian stochastic volatility model, we show that R-VGA-Whittle provides good approximations to posterior distributions of the parameters and is very computationally efficient when compared to asymptotically exact methods such as Hamiltonian Monte Carlo.

Simulation-based methods for statistical inference have evolved dramatically over the past 50 years, keeping pace with technological advancements. The field is undergoing a new revolution as it embraces the representational capacity of neural networks, optimisation libraries and graphics processing units for learning complex mappings between data and inferential targets. The resulting tools are amortised, in the sense that they allow rapid inference through fast feedforward operations. In this article we review recent progress in the context of point estimation, approximate Bayesian inference, summary-statistic construction, and likelihood approximation. We also cover software, and include a simple illustration to showcase the wide array of tools available for amortised inference and the benefits they offer over Markov chain Monte Carlo methods. The article concludes with an overview of relevant topics and an outlook on future research directions., Comment: 45 pages, 11 figures, 2 tables

interpretable, and well understood models that are routinely employed even though, as is revealed through prior and posterior predictive checks, these can poorly characterise the spatial heterogeneity in the underlying process of interest. Here, we propose a new, flexible class of spatial-process models, which we refer to as spatial Bayesian neural networks (SBNNs). An SBNN leverages the representational capacity of a Bayesian neural network; it is tailored to a spatial setting by incorporating a spatial ``embedding layer'' into the network and, possibly, spatially-varying network parameters. An SBNN is calibrated by matching its finite-dimensional distribution at locations on a fine gridding of space to that of a target process of interest. That process could be easy to simulate from or we may have many realisations from it. We propose several variants of SBNNs, most of which are able to match the finite-dimensional distribution of the target process at the selected grid better than conventional BNNs of similar complexity. We also show that an SBNN can be used to represent a variety of spatial processes often used in practice, such as Gaussian processes, lognormal processes, and max-stable processes. We briefly discuss the tools that could be used to make inference with SBNNs, and we conclude with a discussion of their advantages and limitations., Comment: 35 pages, 21 figures

Neural Bayes estimators are neural networks that approximate Bayes estimators in a fast and likelihood-free manner. Although they are appealing to use with spatial models, where estimation is often a computational bottleneck, neural Bayes estimators in spatial applications have, to date, been restricted to data collected over a regular grid. These estimators are also currently dependent on a prescribed set of spatial locations, which means that the neural network needs to be re-trained for new data sets; this renders them impractical in many applications and impedes their widespread adoption. In this work, we employ graph neural networks to tackle the important problem of parameter point estimation from data collected over arbitrary spatial locations. In addition to extending neural Bayes estimation to irregular spatial data, our architecture leads to substantial computational benefits, since the estimator can be used with any configuration or number of locations and independent replicates, thus amortising the cost of training for a given spatial model. We also facilitate fast uncertainty quantification by training an accompanying neural Bayes estimator that approximates a set of marginal posterior quantiles. We illustrate our methodology on Gaussian and max-stable processes. Finally, we showcase our methodology on a data set of global sea-surface temperature, where we estimate the parameters of a Gaussian process model in 2161 spatial regions, each containing thousands of irregularly-spaced data points, in just a few minutes with a single graphics processing unit.

Making inference with spatial extremal dependence models can be computationally burdensome since they involve intractable and/or censored likelihoods. Building on recent advances in likelihood-free inference with neural Bayes estimators, that is, neural networks that approximate Bayes estimators, we develop highly efficient estimators for censored peaks-over-threshold models that {use data augmentation techniques} to encode censoring information in the neural network {input}. Our new method provides a paradigm shift that challenges traditional censored likelihood-based inference methods for spatial extremal dependence models. Our simulation studies highlight significant gains in both computational and statistical efficiency, relative to competing likelihood-based approaches, when applying our novel estimators to make inference with popular extremal dependence models, such as max-stable, $r$-Pareto, and random scale mixture process models. We also illustrate that it is possible to train a single neural Bayes estimator for a general censoring level, precluding the need to retrain the network when the censoring level is changed. We illustrate the efficacy of our estimators by making fast inference on hundreds-of-thousands of high-dimensional spatial extremal dependence models to assess extreme particulate matter 2.5 microns or less in diameter (${\rm PM}_{2.5}$) concentration over the whole of Saudi Arabia.

Models with random effects, such as generalised linear mixed models (GLMMs), are often used for analysing clustered data. Parameter inference with these models is difficult because of the presence of cluster-specific random effects, which must be integrated out when evaluating the likelihood function. Here, we propose a sequential variational Bayes algorithm, called Recursive Variational Gaussian Approximation for Latent variable models (R-VGAL), for estimating parameters in GLMMs. The R-VGAL algorithm operates on the data sequentially, requires only a single pass through the data, and can provide parameter updates as new data are collected without the need of re-processing the previous data. At each update, the R-VGAL algorithm requires the gradient and Hessian of a "partial" log-likelihood function evaluated at the new observation, which are generally not available in closed form for GLMMs. To circumvent this issue, we propose using an importance-sampling-based approach for estimating the gradient and Hessian via Fisher's and Louis' identities. We find that R-VGAL can be unstable when traversing the first few data points, but that this issue can be mitigated by using a variant of variational tempering in the initial steps of the algorithm. Through illustrations on both simulated and real datasets, we show that R-VGAL provides good approximations to the exact posterior distributions, that it can be made robust through tempering, and that it is computationally efficient.

The last decade has seen an explosion in data sources available for the monitoring and prediction of environmental phenomena. While several inferential methods have been developed that make predictions on the underlying process by combining these data, an optimal sampling design for when additional data is needed to complement those from other heterogeneous sources has not yet been developed. Here, we provide an adaptive spatial design strategy based on a utility function that combines both prediction uncertainty and risk-factor criteria. Prediction uncertainty is obtained through a spatial data fusion approach based on fixed rank kriging that can tackle data with differing spatial supports and signal-to-noise ratios. We focus on the application of low-cost portable sensors, which tend to be relatively noisy, for air pollution monitoring, where data from regulatory stations as well as numeric modeling systems are also available. Although we find that spatial adaptive sampling designs can help to improve predictions and reduce prediction uncertainty, low-cost portable sensors are only likely to be beneficial if they are sufficient in number and quality. Our conclusions are based on a multi-factorial simulation experiment, and on a realistic simulation of pollutants in the Erie and Niagara counties in Western New York., Comment: 26 pages, 9 figures

Normalizing flows are objects used for modeling complicated probability density functions, and have attracted considerable interest in recent years. Many flexible families of normalizing flows have been developed. However, the focus to date has largely been on normalizing flows on Euclidean domains; while normalizing flows have been developed for spherical and other non-Euclidean domains, these are generally less flexible than their Euclidean counterparts. To address this shortcoming, in this work we introduce a mixture-of-normalizing-flows model to construct complicated probability density functions on the sphere. This model provides a flexible alternative to existing parametric, semiparametric, and nonparametric, finite mixture models. Model estimation is performed using the expectation maximization algorithm and a variant thereof. The model is applied to simulated data, where the benefit over the conventional (single component) normalizing flow is verified. The model is then applied to two real-world data sets of events occurring on the surface of Earth; the first relating to earthquakes, and the second to terrorist activity. In both cases, we see that the mixture-of-normalizing-flows model yields a good representation of the density of event occurrence.

The natural cycles of the surface-to-atmosphere fluxes of carbon dioxide (CO$_2$) and other important greenhouse gases are changing in response to human influences. These changes need to be quantified to understand climate change and its impacts, but this is difficult to do because natural fluxes occur over large spatial and temporal scales. To infer trends in fluxes and identify phase shifts and amplitude changes in flux seasonal cycles, we construct a flux-inversion system that uses a novel spatially varying time-series decomposition of the fluxes, while also accommodating physical constraints on the fluxes. We incorporate these features into the Wollongong Methodology for Bayesian Assimilation of Trace-gases (WOMBAT, Zammit-Mangion et al., Geosci. Model Dev., 15, 2022), a hierarchical flux-inversion framework that yields posterior distributions for all unknowns in the underlying model. We apply the new method, which we call WOMBAT v2.0, to a mix of satellite observations of CO$_2$ mole fraction from the Orbiting Carbon Observatory-2 (OCO-2) satellite and direct measurements of CO$_2$ mole fraction from a variety of sources. We estimate the changes to CO$_2$ fluxes that occurred from January 2015 to December 2020, and compare our posterior estimates to those from an alternative method based on a bottom-up understanding of the physical processes involved. We find substantial trends in the fluxes, including that tropical ecosystems trended from being a net source to a net sink of CO$_2$ over the study period. We also find that the amplitude of the global seasonal cycle of ecosystem CO$_2$ fluxes increased over the study period by 0.11 PgC/month (an increase of 8%), and that the seasonal cycle of ecosystem CO$_2$ fluxes in the northern temperate and northern boreal regions shifted earlier in the year by 0.4-0.7 and 0.4-0.9 days, respectively (2.5th to 97.5th posterior percentiles).

Neural point estimators are neural networks that map data to parameter point estimates. They are fast, likelihood free and, due to their amortised nature, amenable to fast bootstrap-based uncertainty quantification. In this paper, we aim to increase the awareness of statisticians to this relatively new inferential tool, and to facilitate its adoption by providing user-friendly open-source software. We also give attention to the ubiquitous problem of making inference from replicated data, which we address in the neural setting using permutation-invariant neural networks. Through extensive simulation studies we show that these neural point estimators can quickly and optimally (in a Bayes sense) estimate parameters in weakly-identified and highly-parameterised models with relative ease. We demonstrate their applicability through an analysis of extreme sea-surface temperature in the Red Sea where, after training, we obtain parameter estimates and bootstrap-based confidence intervals from hundreds of spatial fields in a fraction of a second., Comment: The American Statistician (2023)

Deep neural network models have become ubiquitous in recent years, and have been applied to nearly all areas of science, engineering, and industry. These models are particularly useful for data that have strong dependencies in space (e.g., images) and time (e.g., sequences). Indeed, deep models have also been extensively used by the statistical community to model spatial and spatio-temporal data through, for example, the use of multi-level Bayesian hierarchical models and deep Gaussian processes. In this review, we first present an overview of traditional statistical and machine learning perspectives for modeling spatial and spatio-temporal data, and then focus on a variety of hybrid models that have recently been developed for latent process, data, and parameter specifications. These hybrid models integrate statistical modeling ideas with deep neural network models in order to take advantage of the strengths of each modeling paradigm. We conclude by giving an overview of computational technologies that have proven useful for these hybrid models, and with a brief discussion on future research directions., Comment: 27 pages, 1 figure

Spatial statistics is concerned with the analysis of data that have spatial locations associated with them, and those locations are used to model statistical dependence between the data. The spatial data are treated as a single realisation from a probability model that encodes the dependence through both fixed effects and random effects, where randomness is manifest in the underlying spatial process and in the noisy, incomplete, measurement process. The focus of this review article is on the use of basis functions to provide an extremely flexible and computationally efficient way to model spatial processes that are possibly highly non-stationary. Several examples of basis-function models are provided to illustrate how they are used in Gaussian, non-Gaussian, multivariate, and spatio-temporal settings, with applications in geophysics. Our aim is to emphasise the versatility of these spatial statistical models and to demonstrate that they are now centre-stage in a number of application domains. The review concludes with a discussion and illustration of software currently available to fit spatial-basis-function models and implement spatial-statistical prediction., Comment: 30 pages, 6 figures

Understanding and predicting environmental phenomena often requires the construction of spatio-temporal statistical models, which are typically Gaussian processes. A common assumption made on Gaussian processes is that of covariance stationarity, which is unrealistic in many geophysical applications. In this article, we introduce a deep-learning-inspired approach to construct descriptive nonstationary spatio-temporal models by modeling stationary processes on warped spatio-temporal domains. The warping functions we use are constructed using several simple injective warping units which, when combined through composition, can induce complex warpings. A stationary spatio-temporal covariance function on the warped domain induces covariance nonstationarity on the original domain. Sparse linear algebraic methods are used to reduce the computational complexity when fitting the model in a big data setting. We show that our proposed nonstationary spatio-temporal model can capture covariance nonstationarity in both space and time, and provide better probabilistic predictions than conventional stationary models in both simulation studies and on a real-world data set.

Recent years have seen an increased interest in the application of methods and techniques commonly associated with machine learning and artificial intelligence to spatial statistics. Here, in a celebration of the ten-year anniversary of the journal Spatial Statistics, we bring together normalizing flows, commonly used for density function estimation in machine learning, and spherical point processes, a topic of particular interest to the journal's readership, to present a new approach for modeling non-homogeneous Poisson process intensity functions on the sphere. The central idea of this framework is to build, and estimate, a flexible bijective map that transforms the underlying intensity function of interest on the sphere into a simpler, reference, intensity function, also on the sphere. Map estimation can be done efficiently using automatic differentiation and stochastic gradient descent, and uncertainty quantification can be done straightforwardly via nonparametric bootstrap. We investigate the viability of the proposed method in a simulation study, and illustrate its use in a proof-of-concept study where we model the intensity of cyclone events in the North Pacific Ocean. Our experiments reveal that normalizing flows present a flexible and straightforward way to model intensity functions on spheres, but that their potential to yield a good fit depends on the architecture of the bijective map, which can be difficult to establish in practice., Comment: 23 pages, 5 figures

Flux inversion is the process by which sources and sinks of a gas are identified from observations of gas mole fraction. The inversion often involves running a Lagrangian particle dispersion model (LPDM) to generate sensitivities between observations and fluxes over a spatial domain of interest. The LPDM must be run backward in time for every gas measurement, and this can be computationally prohibitive. To address this problem, here we develop a novel spatio-temporal emulator for LPDM sensitivities that is built using a convolutional variational autoencoder (CVAE). With the encoder segment of the CVAE, we obtain approximate (variational) posterior distributions over latent variables in a low-dimensional space. We then use a spatio-temporal Gaussian process emulator on the low-dimensional space to emulate new variables at prediction locations and time points. Emulated variables are then passed through the decoder segment of the CVAE to yield emulated sensitivities. We show that our CVAE-based emulator outperforms the more traditional emulator built using empirical orthogonal functions and that it can be used with different LPDMs. We conclude that our emulation-based approach can be used to reliably reduce the computing time needed to generate LPDM outputs for use in high-resolution flux inversions., Comment: 25 pages, 8 figures, 2 tables, data & code available

Bayesian methods for modelling and inference are being increasingly used in the cryospheric sciences, and glaciology in particular. Here, we present a review of recent works in glaciology that adopt a Bayesian approach when conducting an analysis. We organise the chapter into three categories: i) Gaussian-Gaussian models, ii) Bayesian hierarchical models, and iii) Bayesian calibration approaches. In addition, we present two detailed case studies that involve the application of Bayesian hierarchical models in glaciology. The first case study is on the spatial prediction of surface mass balance across the Icelandic mountain glacier Langj\"okull, and the second is on the prediction of sea-level rise contributions from the Antactcic ice sheet. This chapter is presented in such a way that it is accessible to both statisticians as well as earth scientists., Comment: This work will appear as a chapter of the forthcoming book titled Statistical Modelling Using Latent Gaussian Models - with Applications in Geophysics and Environmental Sciences, expected to be published by Springer in 2022

Non-Gaussian spatial and spatio-temporal data are becoming increasingly prevalent, and their analysis is needed in a variety of disciplines. FRK is an R package for spatial/spatio-temporal modelling and prediction with very large data sets that, to date, has only supported linear process models and Gaussian data models. In this paper, we describe a major upgrade to FRK that allows for non-Gaussian data to be analysed in a generalised linear mixed model framework. These vastly more general spatial and spatio-temporal models are fitted using the Laplace approximation via the software TMB. The existing functionality of FRK is retained with this advance into non-Gaussian models; in particular, it allows for automatic basis-function construction, it can handle both point-referenced and areal data simultaneously, and it can predict process values at any spatial support from these data. This new version of FRK also allows for the use of a large number of basis functions when modelling the spatial process, and is thus often able to achieve more accurate predictions than previous versions of the package in a Gaussian setting. We demonstrate innovative features in this new version of FRK, highlight its ease of use, and compare it to alternative packages using both simulated and real data sets., Comment: 38 pages, 15 figures