Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models

In studies that produce data with spatial structure, it is common that covariates of interest vary spatially in addition to the error. Because of this, the error and covariate are often correlated. When this occurs, it is difficult to distinguish the covariate effect from residual spatial variation. In an i.i.d. normal error setting, it is well known that this type of correlation produces biased coefficient estimates, but predictions remain unbiased. In a spatial setting, recent studies have shown that coefficient estimates remain biased, but spatial prediction has not been addressed. The purpose of this paper is to provide a more detailed study of coefficient estimation from spatial models when covariate and error are correlated and then begin a formal study regarding spatial prediction. This is carried out by investigating properties of the generalized least squares estimator and the best linear unbiased predictor when a spatial random effect and a covariate are jointly modelled. Under this setup, we demonstrate that the mean squared prediction error is possibly reduced when covariate and error are correlated.     

A Bayesian hierarchical spatio-temporal rainfall model

A Bayesian hierarchical spatio-temporal rainfall model is presented and analysed. The model has the ability to deal with extensive missing or null values, uses a sophisticated variance stabilising rainfall pre-transformation, incorporates a new elevation model and can provide sub-catchment rainfall estimation and interpolation using a sequential kriging scheme. The model uses a vector autoregressive stochastic process to represent the time dependence of the rainfall field and an exponential covariogram to model the spatial correlation of the rainfall field. The model can be readily generalised to other types of stochastic processes. In this paper, some results of applying the model to a particular rainfall catchment are presented.     

Estimating the probability of simultaneous rainfall extremes within a region: a spatial approach

In this paper we investigate the impact of model mis-specification, in terms of the dependence structure in the extremes of a spatial process, on the estimation of key quantities that are of interest to hydrologists and engineers. For example, it is often the case that severe flooding occurs as a result of the observation of rainfall extremes at several locations in a region simultaneously. Thus, practitioners might be interested in estimates of the joint exceedance probability of some high levels across these locations. It is likely that there will be spatial dependence present between the extremes, and this should be properly accounted for when estimating such probabilities. We compare the use of standard models from the geostatistics literature with max-stables models from extreme value theory. We find that, in some situations, using an incorrect spatial model for our extremes results in a significant under-estimation of these probabilities which – in flood defence terms – could lead to substantial under-protection.     

Computational aspects of the EM algorithm for spatial econometric models with missing data

Maximum likelihood (ML) estimation with spatial econometric models is a long-standing problem that finds application in several areas of economic importance. The problem is particularly challenging in the presence of missing data, since there is an implied dependence between all units, irrespective of whether they are observed or not. Out of the several approaches adopted for ML estimation in this context, that of LeSage and Pace [Models for spatially dependent missing data. J Real Estate Financ Econ. 2004;29(2):233–254] stands out as one of the most commonly used with spatial econometric models due to its ability to scale with the number of units. Here, we review their algorithm, and consider several similar alternatives that are also suitable for large datasets. We compare the methods through an extensive empirical study and conclude that, while the approximate approaches are suitable for large sampling ratios, for small sampling ratios the only reliable algorithms are those that yield exact ML or restricted ML estimates.     

Threshold knot selection for large-scale spatial models with applications to the Deepwater Horizon disaster

Large spatial datasets are typically modelled through a small set of knot locations; often these locations are specified by the investigator by arbitrary criteria. Existing methods of estimating the locations of knots assume their number is known a priori, or are otherwise computationally intensive. We develop a computationally efficient method of estimating both the location and number of knots for spatial mixed effects models. Our proposed algorithm, Threshold Knot Selection (TKS), estimates knot locations by identifying clusters of large residuals and placing a knot in the centroid of those clusters. We conduct a simulation study showing TKS in relation to several comparable methods of estimating knot locations. Our case study utilizes data of particulate matter concentrations collected during the course of the response and clean-up effort from the 2010 Deepwater Horizon oil spill in the Gulf of Mexico.     

Generalized linear spatial models in epidemiology: A case study of zoonotic cutaneous leishmaniasis in Tunisia

Generalized linear spatial models (GLSM) are used here to study spatial characters of zoonotic cutaneous leishmaniasis (ZCL) in Tunisia. The response variable stands for the number of affected by district during the period 2001–2002. The model covariates are: climates (temperature and rainfall), humidity and surrounding vegetation status. As the environmental and weather data are not available for all the studied districts, Kriging based on linear interpolation was used to estimate the missing data. To account for unexplained spatial variation in the model, we include a stationary Gaussian process S with a powered exponential spatial correlation function. Moran coefficient, DIC criterion and residuals variograms are used to show the high goodness-of-fit of the GLSM. When compared with the statistical tools used in the previous ZCL studies, the optimal GLSM found here yields a better assessment of the impact of the risk factors, a better prediction of ZCL evolution and a better comprehension of the disease transmission. The statistical results show the progressive increase in the number of affected in zones with high temperature, low rainfall and high surrounding vegetation index. Relative humidity does not seem to affect the distribution of the disease in Tunisia. The results of the statistical analyses stress the important risk of misleading epidemiological conclusions when non-spatial models are used to analyse spatially structured data.     

Spatial estimation and rescaled spatial bootstrap approach for finite population

In this study, an attempt has been made to improve the sampling strategy incorporating spatial dependency at estimation stage considering usual aerial sampling scheme, such as simple random sampling, when the underlying population is finite and spatial in nature. Using the distances between spatial units, an improved method of estimation, viz. spatial estimation procedure, has been proposed for the estimation of finite population mean. Further, rescaled spatial bootstrap (RSB) methods have been proposed for approximately unbiased estimation of variance of the proposed spatial estimator (SE). The properties of the proposed SE and its corresponding RSB methods were studied empirically through simulation.     

Functional regression concurrent model with spatially correlated errors: application to rainfall ground validation

In this paper, we give an extension of the functional regression concurrent model to the case of spatially correlated errors. We propose estimating the spatial correlation structure by using functional geostatistics. The estimation of the regression parameters is carried out by feasible generalized least squares. This modeling approach is motivated by the problem of validating rainfall data retrieved from satellite sensors. In this sense, we use the methodology to study the relationship between satellite and ground rainfall time series recorded in 82 weather stations from Department of Valle del Cauca, Colombia. The model obtained allows predicting pentadal rainfall curves in many sites of the region of interest by using as input the satellite information. A residual analysis shows a good performance of the methodology proposed.     

A study on geographically weighted spatial autoregression models with spatial autoregressive disturbances

Abstract

Spatial heterogeneity and correlation are both considered in the geographical weighted spatial autoregressive model. At present, this kind of model has aroused the attention of some scholars. For the estimation of the model, the existing research is based on the assumption that the error terms are independent and identically distributed. In this article we use a computationally simple procedure for estimating the model with spatially autoregressive disturbance terms, both the estimates of constant coefficients and variable coefficients are obtained. Finally, we give the large sample properties of the estimators under some ordinary conditions. In addition, application study of the estimation methods involved will be further explored in a separate study.     

A Monte Carlo comparison of GMM and QMLE estimators for short dynamic panel data models with spatial errors

We suggest a generalized spatial system GMM (SGMM) estimation for short dynamic panel data models with spatial errors and fixed effects when n is large and T is fixed (usually small). Monte Carlo studies are conducted to evaluate the finite sample properties with the quasi-maximum likelihood estimation (QMLE). The results show that, QMLE, with a proper approximation for initial observation, performs better than SGMM in general cases. However, it performs poorly when spatial dependence is large. QMLE and SGMM perform better for different parameters when there is unknown heteroscedasticity in the disturbances and the data are highly persistent. Both estimates are not sensitive to the treatment of initial values. Estimation of the spatial autoregressive parameter is generally biased when either the data are highly persistent or spatial dependence is large. Choices of spatial weights matrices and the sign of spatial dependence do affect the performance of the estimates, especially in the case of the heteroscedastic disturbance. We also give empirical guidelines for the model.     

Multivariate spatial interpolation and exposure to air pollutants

We develop and apply an approach to the spatial interpolation of a vector-valued random response field. The Bayesian approach we adopt enables uncertainty about the underlying models to be représentés in expressing the accuracy of the resulting interpolants. The methodology is particularly relevant in environmetrics, where vector-valued responses are only observed at designated sites at successive time points. The theory allows space-time modelling at the second level of the hierarchical prior model so that uncertainty about the model parameters has been fully expressed at the first level. In this way, we avoid unduly optimistic estimates of inferential accuracy. Moreover, the prior model can be upgraded with any available new data, while past data can be used in a systematic way to fit model parameters. The theory is based on the multivariate normal and related joint distributions. Our hierarchical prior models lead to posterior distributions which are robust with respect to the choice of the prior (hyperparameters). We illustrate our theory with an example involving monitoring stations in southern Ontario, where monthly average levels of ozone, sulphate, and nitrate are available and between-station response triplets are interpolated. In this example we use a recently developed method for interpolating spatial correlation fields.     

Exploring spatial dependence in area-level random effect model for disaggregate-level crop yield estimation

This paper describes an application of small area estimation (SAE) techniques under area-level spatial random effect models when only area (or district or aggregated) level data are available. In particular, the SAE approach is applied to produce district-level model-based estimates of crop yield for paddy in the state of Uttar Pradesh in India using the data on crop-cutting experiments supervised under the Improvement of Crop Statistics scheme and the secondary data from the Population Census. The diagnostic measures are illustrated to examine the model assumptions as well as reliability and validity of the generated model-based small area estimates. The results show a considerable gain in precision in model-based estimates produced applying SAE. Furthermore, the model-based estimates obtained by exploiting spatial information are more efficient than the one obtained by ignoring this information. However, both of these model-based estimates are more efficient than the direct survey estimate. In many districts, there is no survey data and therefore it is not possible to produce direct survey estimates for these districts. The model-based estimates generated using SAE are still reliable for such districts. These estimates produced by using SAE will provide invaluable information to policy-analysts and decision-makers.     

A spatial random-effects model for interzone flows: commuting in Northern Ireland

Government policy on employment, transport, and housing often depends on reliable information about spatial variation in commuting flows across a region. Simple commuting rates summarising inter-area flows may not provide a full perspective on the underlying levels of commuting attractivity of different areas (as destinations), or the varying dependence of different areas (as origins) on outside employment. Areas also vary in the degree of commuting self-containment, as expressed in intra-area flows. This paper uses a spatial random-effects model to develop indices of attractivity, extra-dependence, and self-containment using a latent factor method. The methodology allows consideration of the degree to which different explanatory influences (e.g. socioeconomic structure, characteristics of road networks, employment density) affect these aspects of commuting. The particular application is to commuting flows in Northern Ireland, using 139 zones that aggregate smaller areas (wards), so avoiding undue sparsity in the flow matrix. The analysis involves Bayesian estimation, with the outputs comprising full densities for extra-dependence, and attractivity scores and scores for intra-area containment of zones. Spatial patterning in these aspects of commuting is allowed for in the model used. One key pattern is the difference in latent effect estimates for urban (in particular, Belfast) and rural areas reflecting variable job opportunities in these areas.     

An extended Gaussian max-stable process model for spatial extremes

The extremes of environmental processes are often of interest due to the damage that can be caused by extreme levels of the processes. These processes are often spatial in nature and modelling the extremes jointly at many locations can be important. In this paper, an extension of the Gaussian max-stable process is developed, enabling data from a number of locations to be modelled under a more flexible framework than in previous applications. The model is applied to annual maximum rainfall data from five sites in South-West England. For estimation we employ a pairwise likelihood within a Bayesian analysis, incorporating informative prior information.     

Comparisons of computational methods for clustered binary data

Clustered binary data are common in medical research and can be fitted to the logistic regression model with random effects which belongs to a wider class of models called the generalized linear mixed model. The likelihood-based estimation of model parameters often has to handle intractable integration which leads to several estimation methods to overcome such difficulty. The penalized quasi-likelihood (PQL) method is the one that is very popular and computationally efficient in most cases. The expectation–maximization (EM) algorithm allows to estimate maximum-likelihood estimates, but requires to compute possibly intractable integration in the E-step. The variants of the EM algorithm to evaluate the E-step are introduced. The Monte Carlo EM (MCEM) method computes the E-step by approximating the expectation using Monte Carlo samples, while the Modified EM (MEM) method computes the E-step by approximating the expectation using the Laplace's method. All these methods involve several steps of approximation so that corresponding estimates of model parameters contain inevitable errors (large or small) induced by approximation. Understanding and quantifying discrepancy theoretically is difficult due to the complexity of approximations in each method, even though the focus is on clustered binary data. As an alternative competing computational method, we consider a non-parametric maximum-likelihood (NPML) method as well. We review and compare the PQL, MCEM, MEM and NPML methods for clustered binary data via simulation study, which will be useful for researchers when choosing an estimation method for their analysis.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。     

Data skeletons: simultaneous estimation of multiple quantiles for massive streaming datasets with applications to density estimation

We consider the problem of density estimation when the data is in the form of a continuous stream with no fixed length. In this setting, implementations of the usual methods of density estimation such as kernel density estimation are problematic. We propose a method of density estimation for massive datasets that is based upon taking the derivative of a smooth curve that has been fit through a set of quantile estimates. To achieve this, a low-storage, single-pass, sequential method is proposed for simultaneous estimation of multiple quantiles for massive datasets that form the basis of this method of density estimation. For comparison, we also consider a sequential kernel density estimator. The proposed methods are shown through simulation study to perform well and to have several distinct advantages over existing methods.     

A latent Gaussian Markov random-field model for spatiotemporal rainfall disaggregation

Summary. Rainfall data are often collected at coarser spatial scales than required for input into hydrology and agricultural models. We therefore describe a spatiotemporal model which allows multiple imputation of rainfall at fine spatial resolutions, with a realistic dependence structure in both space and time and with the total rainfall at the coarse scale consistent with that observed. The method involves the transformation of the fine scale rainfall to a thresholded Gaussian process which we model as a Gaussian Markov random field. Gibbs sampling is then used to generate realizations of rainfall efficiently at the fine scale. Results compare favourably with previous, less elegant methods.     

Space–time calibration of radar rainfall data

Motivated by a specific problem concerning the relationship between radar reflectance and rainfall intensity, the paper develops a space–time model for use in environmental monitoring applications. The model is cast as a high dimensional multivariate state space time series model, in which the cross-covariance structure is derived from the spatial context of the component series, in such a way that its interpretation is essentially independent of the particular set of spatial locations at which the data are recorded. We develop algorithms for estimating the parameters of the model by maximum likelihood, and for making spatial predictions of the radar calibration parameters by using realtime computations. We apply the model to data from a weather radar station in Lancashire, England, and demonstrate through empirical validation the predictive performance of the model.     

A model for analyzing spatially correlated binary data clustered in uncorrelated lattices

In recent years, the spatial lattice data has been a motivating issue for researches. Modeling of binary variables observed at locations on a spatial lattice has been sufficiently investigated and the autologistic model is a popular tool for analyzing these data. But, there are many situations where binary responses are clustered in several uncorrelated lattices, and only a few studies were found to investigate the modeling of binary data distributed in such spatial structure. Besides, due to spatial dependency in data exact likelihood analyses is not possible. Bayesian inference, for the autologistic function due to intractability of its normalizing-constant, often has limitations and difficulties. In this study, spatially correlated binary data clustered in uncorrelated lattices are modeled via autologistic regression and IBF (inverse Bayes formulas) sampler with help of introducing latent variables, is extended for posterior analysis and parameter estimation. The proposed methodology is illustrated using simulated and real observations.