Neural Information Processing Systems http://nips.cc/

Accurate spatial interpolation of the air quality index (AQI), computed from concentrations of multiple air pollutants, is essential for regulatory decision-making, yet AQI fields are inherently non-Gaussian and often exhibit complex nonlinear spatial structure. Classical spatial prediction methods such as kriging are linear and rely on Gaussian assumptions, which limits their ability to capture these features and to provide reliable predictive distributions. In this study, we propose \textit{deep classifier kriging} (DCK), a flexible, distribution-free deep learning framework for estimating full predictive distribution functions for univariate and bivariate spatial processes, together with a \textit{data fusion} mechanism that enables modeling of non-collocated bivariate processes and integration of heterogeneous air pollution data sources. Through extensive simulation experiments, we show that DCK consistently outperforms conventional approaches in predictive accuracy and uncertainty quantification. We further apply DCK to probabilistic spatial prediction of AQI by fusing sparse but high-quality station observations with spatially continuous yet biased auxiliary model outputs, yielding spatially resolved predictive distributions that support downstream tasks such as exceedance and extreme-event probability estimation for regulatory risk assessment and policy formulation.

A key objective in spatial statistics is to simulate from the distribution of a spatial process at a selection of unobserved locations conditional on observations (i.e., a predictive distribution) to enable spatial prediction and uncertainty quantification. However, exact conditional simulation from this predictive distribution is intractable or inefficient for many spatial process models. In this paper, we propose neural conditional simulation (NCS), a general method for spatial conditional simulation that is based on neural diffusion models. Specifically, using spatial masks, we implement a conditional score-based diffusion model that evolves Gaussian noise into samples from a predictive distribution when given a partially observed spatial field and spatial process parameters as inputs. The diffusion model relies on a neural network that only requires unconditional samples from the spatial process for training. Once trained, the diffusion model is amortized with respect to the observations in the partially observed field, the number and locations of those observations, and the spatial process parameters, and can therefore be used to conditionally simulate from a broad class of predictive distributions without retraining the neural network. We assess the NCS-generated simulations against simulations from the true conditional distribution of a Gaussian process model, and against Markov chain Monte Carlo (MCMC) simulations from a Brown--Resnick process model for spatial extremes. In the latter case, we show that it is more efficient and accurate to conditionally simulate using NCS than classical MCMC techniques implemented in standard software. We conclude that NCS enables efficient and accurate conditional simulation from spatial predictive distributions that are challenging to sample from using traditional methods.

A spatial point process can be characterized by an intensity function which predicts the number of events that occur across space.

Multi-robot systems are essential for environmental monitoring, particularly for tracking spatial phenomena like pollution, soil minerals, and water salinity, and more. This study addresses the challenge of deploying a multi-robot team for optimal coverage in environments where the density distribution, describing areas of interest, is unknown and changes over time. We propose a fully distributed control strategy that uses Gaussian Processes (GPs) to model the spatial field and balance the trade-off between learning the field and optimally covering it. Unlike existing approaches, we address a more realistic scenario by handling time-varying spatial fields, where the exploration-exploitation trade-off is dynamically adjusted over time. Each robot operates locally, using only its own collected data and the information shared by the neighboring robots. To address the computational limits of GPs, the algorithm efficiently manages the volume of data by selecting only the most relevant samples for the process estimation. The performance of the proposed algorithm is evaluated through several simulations and experiments, incorporating real-world data phenomena to validate its effectiveness.

In recent decades, statisticians have been increasingly encountering spatial data that exhibit non-Gaussian behaviors such as asymmetry and heavy-tailedness. As a result, the assumptions of symmetry and fixed tail weight in Gaussian processes have become restrictive and may fail to capture the intrinsic properties of the data. To address the limitations of the Gaussian models, a variety of skewed models has been proposed, of which the popularity has grown rapidly. These skewed models introduce parameters that govern skewness and tail weight. Among various proposals in the literature, unified skewed distributions, such as the Unified Skew-Normal (SUN), have received considerable attention. In this work, we revisit a more concise and intepretable re-parameterization of the SUN distribution and apply the distribution to random fields by constructing a generalized unified skew-normal (GSUN) spatial process. We demonstrate that the GSUN is a valid spatial process by showing its vanishing correlation in large distances and provide the corresponding spatial interpolation method. In addition, we develop an inference mechanism for the GSUN process using the concept of neural Bayes estimators with deep graphical attention networks (GATs) and encoder transformer. We show the superiority of our proposed estimator over the conventional CNN-based architectures regarding stability and accuracy by means of a simulation study and application to Pb-contaminated soil data. Furthermore, we show that the GSUN process is different from the conventional Gaussian processes and Tukey g-and-h processes, through the probability integral transform (PIT).

Spatial data is ubiquitous, encompassing a wide range of applications from environmental observations and biological measurements to more recent fields like computer vision. A critical challenge in the analysis of spatial data is spatial prediction, which involves estimating unobserved values based on nearby observations under the assumption of certain correlations. Among parametric algorithms, Kriging is particularly notable ((Matheron (1963))). Described as the best linear unbiased estimator (BLUE), Kriging employs a weighted average of nearby observations, with weights determined by a covariance function typically presumed to be stationary. However, this assumption does not hold in many real-world scenarios, such as data from satellites, monitoring stations, and urban streets, which tend to exhibit nonstationarity (Katzfuss (2013)).

In spatial statistics, fast and accurate parameter estimation, coupled with a reliable means of uncertainty quantification, can be challenging when fitting a spatial process to real-world data because the likelihood function might be slow to evaluate or wholly intractable. In this work, we propose using convolutional neural networks to learn the likelihood function of a spatial process. Through a specifically designed classification task, our neural network implicitly learns the likelihood function, even in situations where the exact likelihood is not explicitly available. Once trained on the classification task, our neural network is calibrated using Platt scaling which improves the accuracy of the neural likelihood surfaces. To demonstrate our approach, we compare neural likelihood surfaces and the resulting maximum likelihood estimates and approximate confidence regions with the equivalent for exact or approximate likelihood for two different spatial processes: a Gaussian process and a Brown-Resnick process which have computationally intensive and intractable likelihoods, respectively. We conclude that our method provides fast and accurate parameter estimation with a reliable method of uncertainty quantification in situations where standard methods are either undesirably slow or inaccurate. The method is applicable to any spatial process on a grid from which fast simulations are available.

Statistical models for spatial processes play a central role in statistical analyses of spatial data. Yet, it is the simple, 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 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 a single SBNN can be used to represent a variety of spatial processes often used in practice, such as Gaussian processes and lognormal 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.

Geographic Information Systems (GIS) and related technologies have generated substantial interest among statisticians with regard to scalable methodologies for analyzing large spatial datasets. A variety of scalable spatial process models have been proposed that can be easily embedded within a hierarchical modeling framework to carry out Bayesian inference. While the focus of statistical research has mostly been directed toward innovative and more complex model development, relatively limited attention has been accorded to approaches for easily implementable scalable hierarchical models for the practicing scientist or spatial analyst. This article discusses how point-referenced spatial process models can be cast as a conjugate Bayesian linear regression that can rapidly deliver inference on spatial processes. The approach allows exact sampling directly (avoids iterative algorithms such as Markov chain Monte Carlo) from the joint posterior distribution of regression parameters, the latent process and the predictive random variables, and can be easily implemented on statistical programming environments such as R.