Non-stationary Spatial Model Research Articles

Spatial analysis of future East Asian seasonal temperature using two regional climate model simulations

CORDEX-East Asia, a branch of the coordinated regional climate downscaling experiment (CORDEX) initiative, provides high-resolution climate simulations for the domain covering East Asia. This study analyzes temperature data from regional climate models (RCMs) participating in the CORDEX - East Asia region, accounting for the spatial dependence structure of the data. In particular, we assess similarities and dissimilarities of the outputs from two RCMs, HadGEM3-RA and RegCM4, over the region and over time. A Bayesian functional analysis of variance (ANOVA) approach is used to simultaneously model the temperature patterns from the two RCMs for the current and future climate. We exploit nonstationary spatial models to handle the spatial dependence structure of the temperature variable, which depends heavily on latitude and altitude. For a seasonal comparison, we examine changes in the winter temperature in addition to the summer temperature data. We find that the temperature increase projected by RegCM4 tends to be smaller than the projection of HadGEM3-RA for summers, and that the future warming projected by HadGEM3-RA tends to be weaker for winters. Also, the results show that there will be a warming of 1-3°C over the region in 45 years. More specifically, the warming pattern clearly depends on the latitude, with greater temperature increases in higher latitude areas, which implies that warming may be more severe in the northern part of the domain.

Spatial modeling with system of stochastic partial differential equations

To define a spatial process as the solution to a stochastic partial differential equation (SPDE) is an approach to spatial modeling that is gaining popularity. The model corresponds to a Gaussian random spatial process with Matérn covariance function. The SPDE approach allows for computational benefits and provides a framework for making valid complex models (e.g., nonstationary spatial models). Using systems of SPDEs to define spatial processes extends the class of models that can be specified as SPDEs, while the computational benefits are kept. In this study, we give an overview of the current state of spatial modeling with systems of SPDEs. Systems of SPDEs have contributed toward modeling and computational efficient inference for spatial Gaussian random field (GRF) models with oscillating covariance functions and multivariate GRF models. For multivariate GRF models special systems of SPDEs corresponding to models known from the literature are set up. Little work has been done for exploring opportunities and properties of spatial processes defined as systems of SPDEs. We also describe some of the interesting topics for further research. WIREs Comput Stat 2016, 8:112–125. doi: 10.1002/wics.1378This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

Estimation of a non-stationary model for annual precipitation in southern Norway using replicates of the spatial field

Estimation of stationary dependence structure parameters using only a single realisation of the spatial process, typically leads to inaccurate estimates and poorly identified parameters. A common way to handle this is to fix some of the parameters, or within the Bayesian framework, impose prior knowledge. In many applied settings, stationary models are not flexible enough to model the process of interest, thus non-stationary spatial models are used. However, more flexible models usually means more parameters, and the identifiability problem becomes even more challenging. We investigate aspects of estimation of a Bayesian non-stationary spatial model for annual precipitation using observations from multiple years. The model contains replicates of the spatial field, which increases precision of the estimates and makes them less prior sensitive. Using R-INLA, we analyse precipitation data from southern Norway, and investigate statistical properties of the replicate model in a simulation study. The non-stationary spatial model we explore belongs to a recently introduced class of stochastic partial differential equation (SPDE) based spatial models. This model class allows for non-stationary models with explanatory variables in the dependence structure. We derive conditions to facilitate prior specification for these types of non-stationary spatial models.

Regression‐based covariance functions for nonstationary spatial modeling

In many environmental applications involving spatially‐referenced data, limitations on the number and locations of observations motivate the need for practical and efficient models for spatial interpolation, or kriging. A key component of models for continuously‐indexed spatial data is the covariance function, which is traditionally assumed to belong to a parametric class of stationary models. However, stationarity is rarely a realistic assumption. Alternative methods that more appropriately model the nonstationarity present in environmental processes often involve high‐dimensional parameter spaces, which lead to difficulties in model fitting and interpretability. To overcome this issue, we build on the growing literature of covariate‐driven nonstationary spatial modeling. Using process convolution techniques, we propose a Bayesian model for continuously‐indexed spatial data based on a flexible parametric covariance regression structure for a convolution‐kernel covariance matrix. The resulting model is a parsimonious representation of the kernel process, and we explore properties of the implied model, including a description of the resulting nonstationary covariance function and the interpretational benefits in the kernel parameters. Furthermore, we demonstrate that our model provides a practical compromise between stationary and highly parameterized nonstationary spatial covariance functions that do not perform well in practice. We illustrate our approach through an analysis of annual precipitation data. Copyright © 2015 John Wiley & Sons, Ltd.

Bayesian Spatial Modelling withR-INLA

The principles behind the interface to continuous domain spatial models in the RINLA software package for R are described. The integrated nested Laplace approximation (INLA) approach proposed by Rue, Martino, and Chopin (2009) is a computationally effective alternative to MCMC for Bayesian inference. INLA is designed for latent Gaussian models, a very wide and flexible class of models ranging from (generalized) linear mixed to spatial and spatio-temporal models. Combined with the stochastic partial differential equation approach (SPDE, Lindgren, Rue, and Lindstrom 2011), one can accommodate all kinds of geographically referenced data, including areal and geostatistical ones, as well as spatial point process data. The implementation interface covers stationary spatial models, non-stationary spatial models, and also spatio-temporal models, and is applicable in epidemiology, ecology, environmental risk assessment, as well as general geostatistics.

Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs

A nonstationary spatial two-dimensional model of sediment transport in the coastal zone of water reservoirs is considered, taking into account the following physical parameters and processes: the porosity of soil, the critical shear stress at which sediment begins moving, turbulent exchange, the dynamically changing geometry and function of the bottom elevation level, as well as wind currents and friction on the bottom. The hydrodynamic spatial two-dimensional model of sediment transport in the coastal zone of the reservoirs is also considered. For the proposed mathematical models their discretization has been fulfilled. Based on the developed algorithms a complex of programs has been built and the results of numerical experiments are presented.

Variance estimation and kriging prediction for a class of non-stationary spatial models

Variance estimation and kriging prediction for a class of non-stationary spatial models

Bayesian nonstationary spatial modeling for very large datasets

With the proliferation of modern high‐resolution measuring instruments mounted on satellites, planes, ground‐based vehicles, and monitoring stations, a need has arisen for statistical methods suitable for the analysis of large spatial datasets observed on large spatial domains. Statistical analyses of such datasets provide two main challenges: first, traditional spatial‐statistical techniques are often unable to handle large numbers of observations in a computationally feasible way; second, for large and heterogeneous spatial domains, it is often not appropriate to assume that a process of interest is stationary over the entire domain. We address the first challenge by using a model combining a low‐rank component, which allows for flexible modeling of medium‐to‐long‐range dependence via a set of spatial basis functions, with a tapered remainder component, which allows for modeling of local dependence using a compactly supported covariance function. Addressing the second challenge, we propose two extensions to this model that result in increased flexibility: first, the model is parameterized on the basis of a nonstationary Matérn covariance, where the parameters vary smoothly across space; second, in our fully Bayesian model, all components and parameters are considered random, including the number, locations, and shapes of the basis functions used in the low‐rank component. Using simulated data and a real‐world dataset of high‐resolution soil measurements, we show that both extensions can result in substantial improvements over the current state‐of‐the‐art. Copyright © 2013 John Wiley & Sons, Ltd.

Nonstationary Spatial Modeling Using Penalized Likelihood

We consider nonstationary spatial modeling using empirical orthogonal functions (EOFs) under the consideration that data may be observed only at some sparse, irregularly spaced locations with repeated measurements. Instead of obtain- ing EOFs by principal component analysis based on a class of pre-specified basis functions or a pre-smoothing step with data imputed on a regular grid, we propose a regularization approach using penalized likelihood, which provides a new EOF-type expansion in terms of a small number of functions with their degrees of smooth- ness controlled by a tuning parameter. An expectation-conditional-maximization algorithm is applied to obtain the penalized maximum likelihood estimates of the mean and the covariance parameters simultaneously. Some simulation results show that the proposed method performs well in both spatial prediction and covariance function estimation, regardless of whether the underlying spatial process is station- ary or nonstationary. In addition, the method is applied to analyze precipitation in Colorado.

Multivariate spatial nonparametric modelling via kernel processes mixing

In this paper we develop a nonparametric multivariate spatial model that avoids specifying a Gaussian distribution for spatial random effects. Our nonparametric model extends the stick-breaking (SB) prior of Sethuraman (1994), which is frequently used in Bayesian modelling to capture uncertainty in the parametric form of an outcome. The stick-breaking prior is extended here to the spatial setting by assigning each location a different, unknown distribution, and smoothing the distributions in space with a series of space-dependent kernel functions that have a space-varying bandwidth parameter. This results in a flexible non-stationary spatial model, as different kernel functions lead to different relationships between the distributions at nearby locations. This approach is the first to allow both the probabilities and the point mass values of the SB prior to depend on space. Thus, there is no need for replications and we obtain a continuous process in the limit. We extend the model to the multivariate setting by having for each process a different kernel function, but sharing the location of the kernel knots across the different processes. The resulting covariance for the multivariate process is in general nonstationary and nonseparable. The modelling framework proposed here is also computationally efficient because it avoids inverting large matrices and calculating determinants, which often hinders the spatial analysis of large data sets. We study the theoretical properties of the proposed multivariate spatial process. The methods are illustrated using simulated examples and an air pollution application to model components of fine particulate matter.

A note on a non-stationary point source spatial model

A point source, non-stationary covariance structure model is proposed, having only one additional parameter over a standard, stationary covariance structure, spatial model. Additionally, the proposed model is demonstrated to fit better than the three extra parameter, point source, non-stationary spatial model proposed by Ecker and De Oliveira (Commun Stat Theory Methods 37:2066–2078, 2008). The proposed model is fit from a Bayesian perspective and illustrated using a house sales dataset from Cedar Falls, Iowa.

Mapping malaria risk among children in Côte d'Ivoire using Bayesian geo-statistical models.

BackgroundIn Côte d’Ivoire, an estimated 767,000 disability-adjusted life years are due to malaria, placing the country at position number 14 with regard to the global burden of malaria. Risk maps are important to guide control interventions, and hence, the aim of this study was to predict the geographical distribution of malaria infection risk in children aged <16 years in Côte d’Ivoire at high spatial resolution.MethodsUsing different data sources, a systematic review was carried out to compile and geo-reference survey data on Plasmodium spp. infection prevalence in Côte d’Ivoire, focusing on children aged <16 years. The period from 1988 to 2007 was covered. A suite of Bayesian geo-statistical logistic regression models was fitted to analyse malaria risk. Non-spatial models with and without exchangeable random effect parameters were compared to stationary and non-stationary spatial models. Non-stationarity was modelled assuming that the underlying spatial process is a mixture of separate stationary processes in each ecological zone. The best fitting model based on the deviance information criterion was used to predict Plasmodium spp. infection risk for entire Côte d’Ivoire, including uncertainty.ResultsOverall, 235 data points at 170 unique survey locations with malaria prevalence data for individuals aged <16 years were extracted. Most data points (n = 182, 77.4%) were collected between 2000 and 2007. A Bayesian non-stationary regression model showed the best fit with annualized rainfall and maximum land surface temperature identified as significant environmental covariates. This model was used to predict malaria infection risk at non-sampled locations. High-risk areas were mainly found in the north-central and western area, while relatively low-risk areas were located in the north at the country border, in the north-east, in the south-east around Abidjan, and in the central-west between two high prevalence areas.ConclusionThe malaria risk map at high spatial resolution gives an important overview of the geographical distribution of the disease in Côte d’Ivoire. It is a useful tool for the national malaria control programme and can be utilized for spatial targeting of control interventions and rational resource allocation.

Integrated response plot designs for indicators of desertification

The improvement of land management practices on lands susceptible to desertification requires information on the status and condition of the existing resources as well as any change occurring in the resource condition over time. The Environmental Monitoring and Assessment Program (EMAP) of the U.S. Environmental Protection Agency has developed a statistical survey design for monitoring the condition of ecological resources on large spatial scales. EMAP-Rangelands used a uniformity sampling study in 1993 to evaluate response plot designs for three categories of indicators (soils, vegetation, and spectral reflectance) to be used for monitoring ecological condition of a site. The response plot design study was developed to integrate on-site measurements for the three indicator categories. The study was conducted on the Colorado Plateau in southern Utah in three rangeland resource classes (grassland, desertscrub, and conifer woodland) of differing productivity levels in an attempt to develop a common plot design for all three resource classes. Basic measurement units were developed to facilitate integration of data collection. Preliminary spatial analysis of the sampling study found considerable differences in variation patterns among the study sites and measurement categories for the indicator classes used by EMAP-Rangelands. Evidence of substantial trends in the indicator measurements on monitoring sites relative to regional trends leads to the conclusion that nonstationary spatial models for biological processes on a monitoring site may be needed to fulfill the requirements for developing plot designs and indicator criteria.