Non-stationary Spatial Model Research Articles

Regularization of the Ensemble Kalman Filter using a non-parametric, non-stationary spatial model

The sample covariance matrix of a random vector is a good estimate of the true covariance matrix if the sample size is much larger than the length of the vector. In high-dimensional problems, this condition is never met. As a result, in high dimensions the Ensemble Kalman Filter’s (EnKF) ensemble does not contain enough information to specify the prior covariance matrix accurately. This necessitates the need for regularization of the analysis (observation update) problem. We propose a regularization technique based on a new spatial model. The model is a constrained version of the general Gaussian process convolution model. The constraints include local stationarity and smoothness of local spectra. We regularize EnKF by postulating that its prior covariances obey the spatial model. Placing a hyperprior distribution on the model parameters and using the likelihood of the prior ensemble data allows for an optimized use of both the ensemble and the hyperprior. A linear version of the respective estimator is shown to be consistent. A more accurate nonlinear neural-Bayes implementation of the estimator is developed. In simulation experiments, the new technique led to substantially better EnKF performance than several existing techniques.

Non-stationary Bayesian spatial model for disease mapping based on sub-regions.

This paper aims to extend the Besag model, a widely used Bayesian spatial model in disease mapping, to a non-stationary spatial model for irregular lattice-type data. The goal is to improve the model's ability to capture complex spatial dependence patterns and increase interpretability. The proposed model uses multiple precision parameters, accounting for different intensities of spatial dependence in different sub-regions. We derive a joint penalized complexity prior to the flexible local precision parameters to prevent overfitting and ensure contraction to the stationary model at a user-defined rate. The proposed methodology can be used as a basis for the development of various other non-stationary effects over other domains such as time. An accompanying R package fbesag equips the reader with the necessary tools for immediate use and application. We illustrate the novelty of the proposal by modeling the risk of dengue in Brazil, where the stationary spatial assumption fails and interesting risk profiles are estimated when accounting for spatial non-stationary. Additionally, we model different causes of death in Brazil, where we use the new model to investigate the spatial stationarity of these causes.

LAND-USE FILTERING FOR NONSTATIONARY SPATIAL PREDICTION OF COLLECTIVE EFFICACY IN AN URBAN ENVIRONMENT.

Collective efficacy-the capacity of communities to exert social control toward the realization of their shared goals-is a foundational concept in the urban sociology and neighborhood effects literature. Traditionally, empirical studies of collective efficacy use large sample surveys to estimate collective efficacy of different neighborhoods within an urban setting. Such studies have demonstrated an association between collective efficacy and local variation in community violence, educational achievement, and health. Unlike traditional collective efficacy measurement strategies, the Adolescent Health and Development in Context (AHDC) Study implemented a new approach, obtaining spatially-referenced, place-based ratings of collective efficacy from a representative sample of individuals residing in Columbus, OH. In this paper we introduce a novel nonstationary spatial model for interpolation of the AHDC collective efficacy ratings across the study area, which leverages administrative data on land use. Our constructive model specification strategy involves dimension expansion of a latent spatial process and the use of a filter defined by the land-use partition of the study region to connect the latent multivariate spatial process to the observed ordinal ratings of collective efficacy. Careful consideration is given to the issues of parameter identifiability, computational efficiency of an MCMC algorithm for model fitting, and fine-scale spatial prediction of collective efficacy.

Mitigation of spatial nonstationarity with vision transformers

Spatial nonstationarity, the location variance of features’ statistical distributions, is ubiquitous in many natural settings. For example, in geological reservoirs rock matrix porosity varies vertically due to geomechanical compaction trends, in mineral deposits grades vary due to sedimentation and concentration processes, in hydrology rainfall varies due to the atmosphere and topography interactions, and in metallurgy crystalline structures vary due to differential cooling. Conventional geostatistical modeling workflows rely on the assumption of stationarity to be able to model spatial features for geostatistical inference. Nevertheless, this is often not a realistic assumption when dealing with nonstationary spatial data and this has motivated a variety of nonstationary spatial modeling workflows such as trend and residual decomposition, cosimulation with secondary features, and spatial segmentation and independent modeling over stationary subdomains. The advent of deep learning technologies has enabled new workflows for modeling spatial relationships. However, there is a paucity of demonstrated best practice and general guidance on mitigation of spatial nonstationarity with deep learning in the geospatial context. We demonstrate the impact of two common types of geostatistical spatial nonstationarity on deep learning model prediction performance and propose the mitigation of such impacts using self-attention (vision transformer) models. We demonstrate the utility of vision transformers for the mitigation of nonstationarity with relative errors as low as 10%, exceeding the performance of alternative deep learning methods such as convolutional neural networks. We establish best practice by demonstrating the ability of self-attention networks for modeling large-scale spatial relationships in the presence of commonly observed geospatial nonstationarity.

A deep learning synthetic likelihood approximation of a non-stationary spatial model for extreme streamflow forecasting

Extreme streamflow is a key indicator of flood risk, and quantifying the changes in its distribution under non-stationary climate conditions is key to mitigating the impact of flooding events. We propose a non-stationary process mixture model (NPMM) for annual streamflow maxima over the central US (CUS) which uses downscaled climate model precipitation projections to forecast extremal streamflow. Spatial dependence for the model is specified as a convex combination of transformed Gaussian and max-stable processes, indexed by a weight parameter which identifies the asymptotic regime of the process. The weight parameter is modeled as a function of the annual precipitation for each of the two hydrologic regions within the CUS, introducing spatio-temporal non-stationarity within the model. The NPMM is flexible with desirable tail dependence properties, but yields an intractable likelihood. To address this, we embed a neural network within a density regression model which is used to learn a synthetic likelihood function using simulations from the NPMM with different parameter settings. Our model is fitted using observational data for 1972–2021, and inference carried out in a Bayesian framework. The two regions within the CUS are estimated to be in different asymptotic regimes based on the posterior distribution of the weight parameter. Annual streamflow maxima estimates based on global climate models for two representative climate pathway scenarios suggest an overall increase in the frequency and magnitude of extreme streamflow for 2006–2035 compared to the historical period of 1972–2005.

A Scalable Partitioned Approach to Model Massive Nonstationary Non-Gaussian Spatial Datasets

Nonstationary non-Gaussian spatial data are common in many disciplines, including climate science, ecology, epidemiology, and social sciences. Examples include count data on disease incidence and binary satellite data on cloud mask (cloud/no-cloud). Modeling such datasets as stationary spatial processes can be unrealistic since they are collected over large heterogeneous domains (i.e., spatial behavior differs across subregions). Although several approaches have been developed for nonstationary spatial models, these have focused primarily on Gaussian responses. In addition, fitting nonstationary models for large non-Gaussian datasets is computationally prohibitive. To address these challenges, we propose a scalable algorithm for modeling such data by leveraging parallel computing in modern high-performance computing systems. We partition the spatial domain into disjoint subregions and fit locally nonstationary models using a carefully curated set of spatial basis functions. Then, we combine the local processes using a novel neighbor-based weighting scheme. Our approach scales well to massive datasets (e.g., 2.7 million samples) and can be implemented in nimble, a popular software environment for Bayesian hierarchical modeling. We demonstrate our method to simulated examples and two massive real-world datasets acquired through remote sensing.

Partition-Based Nonstationary Covariance Estimation Using the Stochastic Score Approximation

We introduce computational methods that allow for effective estimation of a flexible nonstationary spatial model when the field size is too large to compute the multivariate normal likelihood directly. In this method, the field is defined as a weighted spatially varying linear combination of a globally stationary process and locally stationary processes. Often in such a model, the difficulty in its practical use is in the definition of the boundaries for the local processes, and therefore, we describe one such selection procedure that generally captures complex nonstationary relationships. We generalize the use of a stochastic approximation to the score equations in this nonstationary case and provide tools for evaluating the approximate score in operations and O(n) storage for data on a subset of a grid. We perform various simulations to explore the effectiveness and speed of the proposed methods and conclude by predicting average daily temperature. Supplementary materials for this article are available online.

Partitioning macroscale and microscale ecological processes using covariate-driven non-stationary spatial models.

Ecological inference requires integrating information across scales. This integration creates a complex spatial dependence structure that is most accurately represented by fully non-stationary models. However, ecologists rarely use these models because they are difficult to estimate and interpret. Here, we facilitate the use of fully non-stationary models in ecology by improving the interpretability of a recently developed non-stationary model and applying it to improve our understanding of the spatial processes driving lake eutrophication. We reformulated a model that incorporates non-stationary correlation by adding environmental predictors to the covariance function, thereby building on the intuition of mean regression. We created ellipses to visualize how data at a given site correlate with their surroundings (i.e., the range and directionality of underlying spatial processes). We applied this model to describe the spatial dependence structure of variables related to lake eutrophication across two different regions: a Midwestern United States region with highly agricultural landscapes, and a Northeastern United States region with heterogeneous land use. For the Midwest, increases in forest cover increased the homogeneity of the residual spatial structure of total phosphorus, indicating that macroscale processes dominated this nutrient's spatial structure. Conversely, high forest cover and baseflow reduced the spatial homogeneity of chlorophyll a residuals, indicating that microscale processes dominated for chlorophyll a in the Midwest. In the Northeast, increases in urban land use and baseflow decreased the homogeneity of phosphorus concentrations indicating the dominance of microscale processes, but none of our covariates were strongly associated with the residual spatial structure of chlorophyll a. Our model showed that the spatial dependence structure of environmental response variables shifts across space. It also helped to explain this structure using ecologically relevant covariates from different scales whose effects can be interpreted intuitively. This provided novel insight into the processes that lead to eutrophication, a complex and pervasive environmental issue.

Non-stationary spatial autoregressive modeling for the prediction of lattice data

Spatial autoregressive models are usually used for stationary lattice random fields with a zero or fixed mean. However, many lattice random fields are non-stationary, because they have a non-fixed mean, a non-fixed covariance function, or both. In non-stationary time series, subtracting a fitted trend and differencing are two methods to reach a stationary model. In this paper, these methods have been generalized for non-stationary spatial lattice data. Then, we provide a spatial prediction for each method. By using a simulation study and real data set, we compare the prediction accuracy of the two methods. The results show that predictions made by the trend estimation method are better than differencing method.

Modelling Nonstationary Spatial Lag Models with Hidden Markov Random Fields

One of the basic assumptions in spatial statistic is second-order stationarity, which implies homogeneity and isotropy. However, when using a spatial random field framework to model socio-economical or epidemiological data – just to mention two examples – it is often unreasonable to believe that the relationship between variables could be modelled as a realization of a unique stationary process. In order to provide a more realistic representation, we introduce a latent process which drives the value of the coefficients in a Cliff-Ord-type spatial autoregressive linear model identifying groups of observations with a similar behaviour. The latent process evolves as a Hidden Markov Random Field. This structure allows the topology of the problem to be taken into account when identifying groups. A simulation exercise is performed to investigate the influence of parameter values – estimated via a Markov chain Monte Carlo procedure – on the accuracy of the results. Criteria to perform model comparison in order to establish the optimal number of clusters are also provided. A case study referred to hedonic house prices in Boston illustrates the advantages of the proposed modelling strategy.

An Application of an Embedded Model Estimator to a Synthetic Nonstationary Reservoir Model With Multiple Secondary Variables.

A method (Ember) for nonstationary spatial modeling with multiple secondary variables by combining Geostatistics with Random Forests is applied to a three-dimensional Reservoir Model. It extends the Random Forest method to an interpolation algorithm retaining similar consistency properties to both Geostatistical algorithms and Random Forests. It allows embedding of simpler interpolation algorithms into the process, combining them through the Random Forest training process. The algorithm estimates a conditional distribution at each target location. The family of such distributions is called the model envelope. An algorithm to produce stochastic simulations from the envelope is demonstrated. This algorithm allows the influence of the secondary variables, as well as the variability of the result to vary by location in the simulation.

Model of pneumohydraulic launch system

A reactive pneumohydraulic displacement system is proposed as an inclined launch system for an aircraft. The advantages of the proposed transport system in relation to the known analogs of comparable energy are stated. A complex non-stationary spatial model of the system is presented. The mathematical apparatus for constructing a model is considered. The calculation results are presented in the form of continual mappings of physical fields and integral characteristics of the launch system. The analysis of the starting characteristics for different proportions of equipment – the ratio of water and air. The conclusion is made about the effectiveness of the proposed launch system and the best equipment conditions.

Nonstationary Modeling With Sparsity for Spatial Data via the Basis Graphical Lasso

Many modern spatial models express the stochastic variation component as a basis expansion with random coefficients. Low rank models, approximate spectral decompositions, multiresolution representations, stochastic partial differential equations, and empirical orthogonal functions all fall within this basic framework. Given a particular basis, stochastic dependence relies on flexible modeling of the coefficients. Under a Gaussianity assumption, we propose a graphical model family for the stochastic coefficients by parameterizing the precision matrix. Sparsity in the precision matrix is encouraged using a penalized likelihood framework—we term this approach the basis graphical lasso. Computations follow from a majorization-minimization (MM) approach, a byproduct of which is a connection to the standard graphical lasso. The result is a flexible nonstationary spatial model that is adaptable to very large datasets with multiple realizations. We apply the model to two large and heterogeneous spatial datasets in statistical climatology and recover physically sensible graphical structures. Moreover, the model performs competitively against the popular LatticeKrig model in predictive cross-validation but improves the Akaike information criterion score and a log score for the quality of the joint predictive distribution.

A Nonstationary Spatial Covariance Model for Processes Driven by Point Sources

We introduce a new nonstationary spatial covariance model for analyzing geostatistical point-referenced data that contain point sources (i.e., known locations that impact the outcome). Our model is based on viewing the spatial domain on the polar coordinate scale, with the point source representing the reference location. As a result, we incorporate distances from the point source and angles of the separation vector with respect to the point source into the covariance model definition in order to describe complex correlation patterns that may be induced by the point source. We apply the new model and several competing options to analyze the impact of a hog lot on house sales prices in Cedar Falls, Iowa. We find that the new model offers improved model fit and predictive ability through Watanabe–Akaike information criterion and cross-validation, respectively. Additionally, we design a simulation study to determine the impact that mean misspecification has on each model’s ability to produce quality predictions. Overall, the new model is shown to consistently outperform the competitors and is useful even when the point source has no impact on the outcome.

Spatial modeling with R‐INLA: A review

Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. Writing fast inference code for a complex spatial model with realistically‐sized datasets from scratch is time‐consuming, and if changes are made to the model, there is little guarantee that the code performs well. The key advantages of R‐INLA are the ease with which complex models can be created and modified, without the need to write complex code, and the speed at which inference can be done even for spatial problems with hundreds of thousands of observations. R‐INLA handles latent Gaussian models, where fixed effects, structured and unstructured Gaussian random effects are combined linearly in a linear predictor, and the elements of the linear predictor are observed through one or more likelihoods. The structured random effects can be both standard areal model such as the Besag and the BYM models, and geostatistical models from a subset of the Matérn Gaussian random fields. In this review, we discuss the large success of spatial modeling with R‐INLA and the types of spatial models that can be fitted, we give an overview of recent developments for areal models, and we give an overview of the stochastic partial differential equation (SPDE) approach and some of the ways it can be extended beyond the assumptions of isotropy and separability. In particular, we describe how slight changes to the SPDE approach leads to straight‐forward approaches for nonstationary spatial models and nonseparable space–time models.This article is categorized under:Statistical and Graphical Methods of Data Analysis > Bayesian Methods and TheoryStatistical Models > Bayesian ModelsData: Types and Structure > Massive Data

A fully non-stationary linear coregionalization model for multivariate random fields

This paper introduces an extension of the traditional stationary linear coregionalization model to handle the lack of stationarity. Under the proposed model, coregionalization matrices are spatially dependent, and basic univariate spatial dependence structures are non-stationary. A parameter estimation procedure of the proposed non-stationary linear coregionalization model is developed under the local stationarity framework. The proposed estimation procedure is based on the method of moments and involves a matrix-valued local stationary variogram kernel estimator, a weighted local least squares method in combination with a kernel smoothing technique. Local parameter estimates are knitted together for prediction and simulation purposes. The proposed non-stationary multivariate spatial modeling approach is illustrated using two real bivariate data examples. Prediction performance comparison is carried out with the classical stationary multivariate spatial modeling approach. According to several criteria, the prediction performance of the proposed non-stationary multivariate spatial modeling approach appears to be significantly better.

Basis Function Models for Animal Movement

ABSTRACT Advances in satellite-based data collection techniques have served as a catalyst for new statistical methodology to analyze these data. In wildlife ecological studies, satellite-based data and methodology have provided a wealth of information about animal space use and the investigation of individual-based animal–environment relationships. With the technology for data collection improving dramatically over time, we are left with massive archives of historical animal telemetry data of varying quality. While many contemporary statistical approaches for inferring movement behavior are specified in discrete time, we develop a flexible continuous-time stochastic integral equation framework that is amenable to reduced-rank second-order covariance parameterizations. We demonstrate how the associated first-order basis functions can be constructed to mimic behavioral characteristics in realistic trajectory processes using telemetry data from mule deer and mountain lion individuals in western North America. Our approach is parallelizable and provides inference for heterogenous trajectories using nonstationary spatial modeling techniques that are feasible for large telemetry datasets. Supplementary materials for this article are available online.

Small area prediction of counts under a non-stationary spatial model

There is a growing need for current and reliable counts at small area level. The empirical predictor under a generalised linear mixed model (GLMM) is often used for small area estimation (SAE) of such counts. However, the fixed effect parameters of a GLMM are spatially invariant and do not account for the presence of spatial nonstationarity in the population of interest. A geographically weighted regression extension of the GLMM is developed, extending this model to allow for spatial nonstationarity, and SAE based on this spatially nonstationary model (NSGLMM) is described. The empirical predictor for small area counts (NSEP) under an area level NSGLMM is proposed. Analytic and bootstrap approaches to estimating the mean squared error of the NSEP are also developed, and a parametric approach to testing for spatial nonstationarity is described. The approach is illustrated by applying it to a study of poverty mapping using socio-economic survey data from India.

A Non-Stationary Spatial Model for Temperature Interpolation Applied to the State of Rio De Janeiro

SummaryWe propose a model for spatial and temporal interpolation and prediction for a set of monthly minimum temperature measurements from 37 stations in the state of Rio de Janeiro, Brazil, from 1961 to 2010. The model is based on a hierarchical specification where data are modelled with a regression structure for the mean and a non-stationary spatial representation for the spatially structured noise term. The regression term is also allowed to contain spatial dependence through coefficients and its covariates. A novel structure is assumed for the spatial non-stationarity, based on a generalization of a currently proposed convolution of stationary models. It allows more flexibility in the model specification and automatically selects the number of model components. This model provides superior performance when compared with two special cases: one stationary, and one non-stationary, provided by a sum of locally stationary processes. The results show a growth in the spread of temperature, with largely urbanized, warmest areas growing warmer, and largely forested, coolest areas growing cooler.

Local Likelihood Estimation for Covariance Functions with Spatially-Varying Parameters: The convoSPAT Package for R

In spite of the interest in and appeal of convolution-based approaches for nonstationary spatial modeling, off-the-shelf software for model fitting does not as of yet exist. Convolution-based models are highly flexible yet notoriously difficult to fit, even with relatively small data sets. The general lack of pre-packaged options for model fitting makes it difficult to compare new methodology in nonstationary modeling with other existing methods, and as a result most new models are simply compared to stationary models. Using a convolution-based approach, we present a new nonstationary covariance function for spatial Gaussian process models that allows for efficient computing in two ways: first, by representing the spatially-varying parameters via a discrete or mixture model, and second, by estimating the component parameters through a local likelihood approach. In order to make computations for a convolutionbased nonstationary spatial model readily available, this paper also presents and describes the convoSPAT package for R. The nonstationary model is fit to both a synthetic data set and a real data application involving annual precipitation to demonstrate the capabilities of the package.