arXiv subjects

Noel Cressie

Publications and source records attributed to Noel Cressie.

At least 19 recordsLinked to original sources

arXiv2026

Neural Parameter Estimation with Incomplete Data

Advances in artificial intelligence (AI) and deep learning have led to neural networks being used to generate lightning-speed answers to complex science questions, paintings in the style of Monet, or stories like those of Twain. Leveraging their computational speed and flexibility, neural networks are also being used to facilitate fast, likelihood-free statistical inference. However, it is not straightforward to use neural networks with data that for various reasons are incomplete, which precludes their use in many applications. A recently proposed approach to remedy this issue uses an appropriately padded data vector and a vector that encodes the missingness pattern as input to a neural network. While computationally efficient, this "masking" approach is not robust to the missingness mechanism and can result in statistically inefficient inferences. Here, we propose an alternative approach that is based on the Monte Carlo expectation-maximization (EM) algorithm. Our EM approach is likelihood-free, substantially faster than the conventional EM algorithm as it does not require numerical optimization at each iteration, and more statistically efficient than the masking approach. This research addresses a prototypical problem that asks how improvements could be made in AI by introducing Bayesian statistical thinking. We compare the two approaches to missingness using simulated incomplete data from a variety of spatial models. The utility of the methodology is shown on Arctic sea-ice data, analyzed using a novel hidden Potts model with an intractable likelihood.

stat.ME↗

arXiv2026

Spatio-temporal modeling and forecasting with Fourier neural operators

Spatio-temporal process models are often used for modeling dynamic physical and biological phenomena that evolve across space and time. These phenomena may exhibit environmental heterogeneity and complex interactions that are difficult to capture using traditional statistical process models such as Gaussian processes. This work proposes the use of Fourier neural operators (FNOs) for constructing statistical dynamical spatio-temporal models for forecasting. An FNO is a flexible mapping of functions that approximates the solution operator of possibly unknown linear or non-linear partial differential equations (PDEs) in a computationally efficient manner. It does so using samples of inputs and their respective outputs, and hence explicit knowledge of the underlying PDE is not required. Through simulations from a nonlinear PDE with known solution, we compare FNO forecasts to those from state-of-the-art statistical spatio-temporal-forecasting methods. Further, using sea surface temperature data over the Atlantic Ocean and precipitation data across Europe, we demonstrate the ability of FNO-based dynamic spatio-temporal (DST) statistical modeling to capture complex real-world spatio-temporal dependencies. Using collections of testing instances, we show that the FNO-DST forecasts are accurate with valid uncertainty quantification.

stat.ME↗

arXiv2025

Neural posterior inference with state-space models for calibrating ice sheet simulators

Ice sheet models are routinely used to quantify and project an ice sheet's contribution to sea level rise. In order for an ice sheet model to generate realistic projections, its parameters must first be calibrated using observational data; this is challenging due to the nonlinearity of the model equations, the high dimensionality of the underlying parameters, and limited data availability for validation. This study leverages the emerging field of neural posterior approximation for efficiently calibrating ice sheet model parameters and boundary conditions. We make use of a one-dimensional (flowline) Shallow-Shelf Approximation model in a state-space framework. A neural network is trained to infer the underlying parameters, namely the bedrock elevation and basal friction coefficient along the flowline, based on observations of ice velocity and ice surface elevation. Samples from the approximate posterior distribution of the parameters are then used within an ensemble Kalman filter to infer latent model states, namely the ice thickness along the flowline. We show through a simulation study that our approach yields more accurate estimates of the parameters and states than a state-augmented ensemble Kalman filter, which is the current state-of-the-art. We apply our approach to infer the bed elevation and basal friction along a flowline in Thwaites Glacier, Antarctica.

stat.AP↗

arXiv2025

Bayesian copula-based spatial random effects models for inference with complex spatial data

In this article, we develop fully Bayesian, copula-based, spatial-statistical models for large, noisy, incomplete, and non-Gaussian spatial data. Our approach includes novel constructions of copulas that accommodate a spatial-random-effects structure, enabling low-rank representations and computationally efficient Bayesian inference. The spatial copula is used in a latent process model of the Bayesian hierarchical spatial-statistical model, and, conditional on the latent copula-based spatial process, the data model handles measurement errors and missing data. Our simulation studies show that a fully Bayesian approach delivers accurate and fast inference for both parameter estimation and spatial-process prediction, outperforming several benchmark methods, including fixed rank kriging (FRK). The new class of copula-based models is used to map atmospheric methane in the Bowen Basin, Queensland, Australia, from Sentinel 5P satellite data.

stat.ME↗

arXiv2025

WOMBAT v2.S: A Bayesian inversion framework for attributing global CO$_2$ flux components from multiprocess data

Contributions from photosynthesis and other natural components of the carbon cycle present the largest uncertainties in our understanding of carbon dioxide (CO$_2$) sources and sinks. While the global spatiotemporal distribution of the net flux (the sum of all contributions) can be inferred from atmospheric CO$_2$ concentrations through flux inversion, attributing the net flux to its individual components remains challenging. The advent of solar-induced fluorescence (SIF) satellite observations provides an opportunity to isolate natural components by anchoring gross primary productivity (GPP), the photosynthetic component of the net flux. Here, we introduce a novel statistical flux-inversion framework that simultaneously assimilates observations of SIF and CO$_2$ concentration, extending WOMBAT v2.0 (WOllongong Methodology for Bayesian Assimilation of Trace-gases, version 2.0) with a hierarchical model of spatiotemporal dependence between GPP and SIF processes. We call the new framework WOMBAT v2.S, and we apply it to SIF and CO$_2$ data from NASA's Orbiting Carbon Observatory-2 (OCO-2) satellite and other instruments to estimate natural fluxes over the globe during a recent six-year period. In a simulation experiment that matches OCO-2's retrieval characteristics, the inclusion of SIF improves accuracy and uncertainty quantification of component flux estimates. Comparing estimates from WOMBAT v2.S, v2.0, and the independent FLUXCOM initiative, we observe that linking GPP to SIF has little effect on net flux, as expected, but leads to spatial redistribution and more realistic seasonal structure in natural flux components.

stat.AP↗

arXiv2024

CQUESST: A dynamical stochastic framework for predicting soil-carbon sequestration

A statistical framework we call CQUESST (Carbon Quantification and Uncertainty from Evolutionary Soil STochastics), which models carbon sequestration and cycling in soils, is applied to a long-running agricultural experiment that controls for crop type, tillage, and season. The experiment, known as the Millenium Tillage Trial (MTT), ran on 42 field-plots for ten years from 2000-2010; here CQUESST is used to model soil carbon dynamically in six pools, in each of the 42 agricultural plots, and on a monthly time step for a decade. We show how CQUESST can be used to estimate soil-carbon cycling rates under different treatments. Our methods provide much-needed statistical tools for quantitatively inferring the effectiveness of different experimental treatments on soil-carbon sequestration. The decade-long data are of multiple observation types, and these interacting time series are ingested into a fully Bayesian model that has a dynamic stochastic model of multiple pools of soil carbon at its core. CQUESST's stochastic model is motivated by the deterministic RothC soil-carbon model based on nonlinear difference equations. We demonstrate how CQUESST can estimate soil-carbon fluxes for different experimental treatments while acknowledging uncertainties in soil-carbon dynamics, in physical parameters, and in observations. CQUESST is implemented efficiently in the probabilistic programming language Stan using its MapReduce parallelization, and it scales well for large numbers of field-plots, using software libraries that allow for computation to be shared over multiple nodes of high-performance computing clusters.

stat.AP↗

arXiv2024

Spatial Bayesian Neural Networks

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.

stat.ML↗

arXiv2024

Optimal prediction of positive-valued spatial processes: asymmetric power-divergence loss

This article studies the use of asymmetric loss functions for the optimal prediction of positive-valued spatial processes. We focus on the family of power-divergence loss functions due to its many convenient properties, such as its continuity, convexity, relationship to well known divergence measures, and the ability to control the asymmetry and behaviour of the loss function via a power parameter. The properties of power-divergence loss functions, optimal power-divergence (OPD) spatial predictors, and related measures of uncertainty quantification are examined. In addition, we examine the notion of asymmetry in loss functions defined for positive-valued spatial processes and define an asymmetry measure that is applied to the power-divergence loss function and other common loss functions. The paper concludes with a spatial statistical analysis of zinc measurements in the soil of a floodplain of the Meuse River, Netherlands, using OPD spatial prediction.

stat.ME↗

arXiv2022

Modelling Big, Heterogeneous, Non-Gaussian Spatial and Spatio-Temporal Data using FRK

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.

stat.CO↗

arXiv2022

Inferring changes to the global carbon cycle with WOMBAT v2.0, a hierarchical flux-inversion framework

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).

physics.ao-ph↗

arXiv2022

Decisions, decisions, decisions in an uncertain environment

Decision-makers abhor uncertainty, and it is certainly true that the less there is of it the better. However, recognizing that uncertainty is part of the equation, particularly for deciding on environmental policy, is a prerequisite for making wise decisions. Even making no decision is a decision that has consequences, and using the presence of uncertainty as the reason for failing to act is a poor excuse. Statistical science is the science of uncertainty, and it should play a critical role in the decision-making process. This opinion piece focuses on the summit of the knowledge pyramid that starts from data and rises in steps from data to information, from information to knowledge, and finally from knowledge to decisions. Enormous advances have been made in the last 100 years ascending the pyramid, with deviations that have followed different routes. There has generally been a healthy supply of uncertainty quantification along the way but, in a rush to the top, where the decisions are made, uncertainty is often left behind. In my opinion, statistical science needs to be much more pro-active in evolving classical decision theory into a relevant and practical area of decision applications. This article follows several threads, building on the decision-theoretic foundations of loss functions and Bayesian uncertainty.

stat.ME↗

arXiv2022

Basis-Function Models in Spatial Statistics

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.

stat.ME↗

arXiv2021

From Many to One: Consensus Inference in a MIP

A Model Intercomparison Project (MIP) consists of teams who each estimate the same underlying quantity (e.g., temperature projections to the year 2070), and the spread of the estimates indicates their uncertainty. It recognizes that a community of scientists will not agree completely but that there is value in looking for a consensus and information in the range of disagreement. A simple average of the teams' outputs gives a consensus estimate, but it does not recognize that some outputs are more variable than others. Statistical analysis of variance (ANOVA) models offer a way to obtain a weighted consensus estimate of outputs with a variance that is the smallest possible and hence the tightest possible 'one-sigma' and 'two-sigma' intervals. Modulo dependence between MIP outputs, the ANOVA approach weights a team's output inversely proportional to its variation. When external verification data are available for evaluating the fidelity of each MIP output, ANOVA weights can also provide a prior distribution for Bayesian Model Averaging to yield a consensus estimate. We use a MIP of carbon dioxide flux inversions to illustrate the ANOVA-based weighting and subsequent consensus inferences.

stat.AP↗

arXiv2021

Spatial Statistics

Spatial statistics is an area of study devoted to the statistical analysis of data that have a spatial label associated with them. Geographers often refer to the "location information" associated with the "attribute information," whose study defines a research area called "spatial analysis." Many of the ways to manipulate spatial data are driven by algorithms with no uncertainty quantification associated with them. When a spatial analysis is statistical, that is, it incorporates uncertainty quantification, it falls in the research area called spatial statistics. The primary feature of spatial statistical models is that nearby attribute values are more statistically dependent than distant attribute values; this is a paraphrasing of what is sometimes called the First Law of Geography (Tobler, 1970).

stat.ME↗

arXiv2021

WOMBAT: A fully Bayesian global flux-inversion framework

WOMBAT (the WOllongong Methodology for Bayesian Assimilation of Trace-gases) is a fully Bayesian hierarchical statistical framework for flux inversion of trace gases from flask, in situ, and remotely sensed data. WOMBAT extends the conventional Bayesian-synthesis framework through the consideration of a correlated error term, the capacity for online bias correction, and the provision of uncertainty quantification on all unknowns that appear in the Bayesian statistical model. We show, in an observing system simulation experiment (OSSE), that these extensions are crucial when the data are indeed biased and have errors that are correlated. Using the GEOS-Chem atmospheric transport model, we show that WOMBAT is able to obtain posterior means and uncertainties on non-fossil-fuel CO$_2$ fluxes from Orbiting Carbon Observatory-2 (OCO-2) data that are comparable to those from the Model Intercomparison Project (MIP) reported in Crowell et al. (2019, Atmos. Chem. Phys., vol. 19). We also find that our predictions of out-of-sample retrievals from the Total Column Carbon Observing Network are, for the most part, more accurate than those made by the MIP participants. Subsequent versions of the OCO-2 datasets will be ingested into WOMBAT as they become available.

stat.AP↗

arXiv2021

A few statistical principles for data science

In any other circumstance, it might make sense to define the extent of the terrain (Data Science) first, and then locate and describe the landmarks (Principles). But this data revolution we are experiencing defies a cadastral survey. Areas are continually being annexed into Data Science. For example, biometrics was traditionally statistics for agriculture in all its forms but now, in Data Science, it means the study of characteristics that can be used to identify an individual. Examples of non-intrusive measurements include height, weight, fingerprints, retina scan, voice, photograph/video (facial landmarks and facial expressions), and gait. A multivariate analysis of such data would be a complex project for a statistician, but a software engineer might appear to have no trouble with it at all. In any applied-statistics project, the statistician worries about uncertainty and quantifies it by modelling data as realisations generated from a probability space. Another approach to uncertainty quantification is to find similar data sets, and then use the variability of results between these data sets to capture the uncertainty. Both approaches allow 'error bars' to be put on estimates obtained from the original data set, although the interpretations are different. A third approach, that concentrates on giving a single answer and gives up on uncertainty quantification, could be considered as Data Engineering, although it has staked a claim in the Data Science terrain. This article presents a few (actually nine) statistical principles for data scientists that have helped me, and continue to help me, when I work on complex interdisciplinary projects.

stat.OT↗

arXiv2021

Modeling Nonstationary and Asymmetric Multivariate Spatial Covariances via Deformations

Multivariate spatial-statistical models are often used when modeling environmental and socio-demographic processes. The most commonly used models for multivariate spatial covariances assume both stationarity and symmetry for the cross-covariances, but these assumptions are rarely tenable in practice. In this article we introduce a new and highly flexible class of nonstationary and asymmetric multivariate spatial covariance models that are constructed by modeling the simpler and more familiar stationary and symmetric multivariate covariances on a warped domain. Inspired by recent developments in the univariate case, we propose modeling the warping function as a composition of a number of simple injective warping functions in a deep-learning framework. Importantly, covariance-model validity is guaranteed by construction. We establish the types of warpings that allow for cross-covariance symmetry and asymmetry, and we use likelihood-based methods for inference that are computationally efficient. The utility of this new class of models is shown through two data illustrations: a simulation study on nonstationary data and an application on ocean temperatures at two different depths.

stat.ME↗

arXiv2020

False Discovery Rates to Detect Signals from Incomplete Spatially Aggregated Data

There are a number of ways to test for the absence/presence of a spatial signal in a completely observed fine-resolution image. One of these is a powerful nonparametric procedure called Enhanced False Discovery Rate (EFDR). A drawback of EFDR is that it requires the data to be defined on regular pixels in a rectangular spatial domain. Here, we develop an EFDR procedure for possibly incomplete data defined on irregular small areas. Motivated by statistical learning, we use conditional simulation (CS) to condition on the available data and simulate the full rectangular image at its finest resolution many times (M, say). EFDR is then applied to each of these simulations resulting in M estimates of the signal and M statistically dependent p-values. Averaging over these estimates yields a single, combined estimate of a possible signal, but inference is needed to determine whether there really is a signal present. We test the original null hypothesis of no signal by combining the M p-values into a single p-value using copulas and a composite likelihood. If the null hypothesis of no signal is rejected, we use the combined estimate. We call this new procedure EFDR-CS and, to demonstrate its effectiveness, we show results from a simulation study; an experiment where we introduce aggregation and incompleteness into temperature-change data in the Asia-Pacific; and an application to total-column carbon dioxide from satellite remote sensing data over a region of the Middle East, Afghanistan, and the western part of Pakistan.

stat.ME↗