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.     

Influence diagnostics in Gaussian spatial linear models

Spatial linear models have been applied in numerous fields such as agriculture, geoscience and environmental sciences, among many others. Spatial dependence structure modelling, using a geostatistical approach, is an indispensable tool to estimate the parameters that define this structure. However, this estimation may be greatly affected by the presence of atypical observations in the sampled data. The purpose of this paper is to use diagnostic techniques to assess the sensitivity of the maximum-likelihood estimators, covariance functions and linear predictor to small perturbations in the data and/or the spatial linear model assumptions. The methodology is illustrated with two real data sets. The results allowed us to conclude that the presence of atypical values in the sample data have a strong influence on thematic maps, changing the spatial dependence structure.     

Spatial bootstrapped microeconometrics: Forecasting for out-of-sample geo-locations in big data

Spatial econometric models estimated on the big geo-located point data have at least two problems: limited computational capabilities and inefficient forecasting for the new out-of-sample geo-points. This is because of spatial weights matrix W defined for in-sample observations only and the computational complexity. Machine learning models suffer the same when using kriging for predictions; thus this problem still remains unsolved. The paper presents a novel methodology for estimating spatial models on big data and predicting in new locations. The approach uses bootstrap and tessellation to calibrate both model and space. The best bootstrapped model is selected with the PAM (Partitioning Around Medoids) algorithm by classifying the regression coefficients jointly in a nonindependent manner. Voronoi polygons for the geo-points used in the best model allow for a representative space division. New out-of-sample points are assigned to tessellation tiles and linked to the spatial weights matrix as a replacement for an original point what makes feasible usage of calibrated spatial models as a forecasting tool for new locations. There is no trade-off between forecast quality and computational efficiency in this approach. An empirical example illustrates a model for business locations and firms' profitability.     

基于EM算法的改进MLM模型及参数估计

传统的分层模型假设组与组之间独立,没有考虑组之间的相关性。而以地理单元分组的数据往往具有空间依赖性,个体不仅受本地区的影响,也可能受相邻地区的影响。此时,传统分层模型层-2残差分布的假设不再成立。为了处理空间分层数据,将空间统计和空间计量经济模型的思想引入到分层模型中,既纳入分层的思想,又顾及空间相关性,提出了空间分层线性模型,并给出了其固定效应、方差协方差成分和空间回归参数的最大似然估计,在运用EM算法时,结合运用了Fisher得分算法。     

Efficient estimation of nonparametric spatial models with general correlation structures

Spatially correlated data appear in many environmental studies, and consequently there is an increasing demand for estimation methods that take account of spatial correlation and thereby improve the accuracy of estimation. In this paper we propose an iterative nonparametric procedure for modelling spatial data with general correlation structures. The asymptotic normality of the proposed estimators is established under mild conditions. We demonstrate, using both simulation and case studies, that the proposed estimators are more efficient than the traditional locally linear methods which fail to account for spatial correlation.     

An investigation of the impact of various geographical scales for the specification of spatial dependence

Ecological studies are based on characteristics of groups of individuals, which are common in various disciplines including epidemiology. It is of great interest for epidemiologists to study the geographical variation of a disease by accounting for the positive spatial dependence between neighbouring areas. However, the choice of scale of the spatial correlation requires much attention. In view of a lack of studies in this area, this study aims to investigate the impact of differing definitions of geographical scales using a multilevel model. We propose a new approach – the grid-based partitions and compare it with the popular census region approach. Unexplained geographical variation is accounted for via area-specific unstructured random effects and spatially structured random effects specified as an intrinsic conditional autoregressive process. Using grid-based modelling of random effects in contrast to the census region approach, we illustrate conditions where improvements are observed in the estimation of the linear predictor, random effects, parameters, and the identification of the distribution of residual risk and the aggregate risk in a study region. The study has found that grid-based modelling is a valuable approach for spatially sparse data while the statistical local area-based and grid-based approaches perform equally well for spatially dense data.     

空间面板数据模型设定问题分析

空间面板数据模型将空间计量经济学和面板数据方法相结合,不仅同时考虑时空特征,而且将空间效应纳入研究体系,成为当前计量经济学的热点研究领域,但其模型设定、参数估计及模型检验也更为复杂,实证研究中往往出现模型设定偏误等问题。因此,基于空间面板数据模型的前沿理论,重点探讨模型设定中的常见问题,包括空间滞后模型与空间误差模型的选择、随机效应与固定效应的选择以及模型拟合优度的选择与比较,为模型的应用和新模型的扩展提供理论依据和参考。     

Theoretical properties of tests for spatial clustering of count data

Testing for spatial clustering of count data is an important problem in spatial data analysis. Several procedures have been proposed to this end but despite their extensive use, studies of their fundamental theoretical properties are almost non‐existent. The authors suggest two conditions that any reasonable test for spatial clustering should satisfy. The latter are based on the notion that the null hypothesis should be rejected almost surely as the amount of spatial clustering tends to infinity. The authors show that the chisquared test and the Potthoff—Whittinghill V have both properties but that other classical tests do not.     

On spatiotemporal prediction for on-line monitoring data

Spatiotemporal prediction is of interest in many areas of applied statistics, especially in environmental monitoring with on-line data information. At first, this article reviews the approaches for spatiotemporal modeling in the context of stochastic processes and then introduces the new class of spatiotemporal dynamic linear models. Further, the methods for linear spatial data analysis, universal kriging and trend surface prediction, are related to the method of spatial linear Bayesian analysis. The Kalman filter is the preferred method for temporal linear Bayesian inferences. By combining the Kalman filter recursions with the trend surface predictor and universal kriging predictor, the prior and posterior spatiotemporal predictors for the observational process are derived, which form the main result of this article. The problem of spatiotemporal linear prediction in the case of unknown first and second order moments is treated as well.     

人口空间分布及迁移影响的实证分析——基于空间计量方法与河南省数据

在传统研究方法的基础上考虑空间相关性,运用空间计量经济学的理论与方法,并以河南省人口数据为例进行人口空间分布及迁移的实证研究。一方面,研究中运用全域空间性和局域空间性的相关知识分析河南省人口空间分布,通过定量性指标的运用得出河南省人口空间分布存在相关性,且是高值集聚,即人口密度较高的地区集中在一起;另一方面,在考虑空间相关性的基础上建立空间计量模型,主要研究各个政府支出对河南省人口迁移和分布的影响,结果发现各支出的影响显著。     

Physically constrained spatiotemporal modeling: generating clear-sky constructions of land surface temperature from sparse,remotely sensed satellite data

Satellite remote-sensing is used to collect important atmospheric and geophysical data at various spatial resolutions, providing insight into spatiotemporal surface and climate variability globally. These observations are often plagued with missing spatial and temporal information of Earth''s surface due to (1) cloud cover at the time of a satellite passing and (2) infrequent passing of polar-orbiting satellites. While many methods are available to model missing data in space and time, in the case of land surface temperature (LST) from thermal infrared remote sensing, these approaches generally ignore the temporal pattern called the ‘diurnal cycle’ which physically constrains temperatures to peak in the early afternoon and reach a minimum at sunrise. In order to infill an LST dataset, we parameterize the diurnal cycle into a functional form with unknown spatiotemporal parameters. Using multiresolution spatial basis functions, we estimate these parameters from sparse satellite observations to reconstruct an LST field with continuous spatial and temporal distributions. These estimations may then be used to better inform scientists of spatiotemporal thermal patterns over relatively complex domains. The methodology is demonstrated using data collected by MODIS on NASA''s Aqua and Terra satellites over both Houston, TX and Phoenix, AZ USA.     

Theory & Methods: Krige,Smooth, Both or Neither?

Both kriging and non-parametric regression smoothing can model a non-stationary regression function with spatially correlated errors. However comparisons have mainly been based on ordinary kriging and smoothing with uncorrelated errors. Ordinary kriging attributes smoothness of the response to spatial autocorrelation whereas non-parametric regression attributes trends to a smooth regression function. For spatial processes it is reasonable to suppose that the response is due to both trend and autocorrelation. This paper reviews methodology for non-parametric regression with autocorrelated errors which is a natural compromise between the two methods. Re-analysis of the one-dimensional stationary spatial data of Laslett (1994) and a clearly non-stationary time series demonstrates the rather surprising result that for these data, ordinary kriging outperforms more computationally intensive models including both universal kriging and correlated splines for spatial prediction. For estimating the regression function, non-parametric regression provides adaptive estimation, but the autocorrelation must be accounted for in selecting the smoothing parameter.     

Optimal sampling for spatial prediction of functional data

This paper combines optimal spatial sampling designs with geostatistical analysis of functional data. We propose a methodology and design criteria to find the set of spatial locations that minimizes the variance of the spatial functional prediction at unsampled sites for three functional predictors: ordinary kriging, simple kriging and simple cokriging. The last one is a modification of an existing predictor that uses ordinary cokriging based on the basis coefficients. Instead, we propose to use a simple cokriging predictor with the scores resulting from a representation of the functional data with the empirical functional principal components, allowing to remove restrictions and complexity of the covariance models and constraints on the estimation procedure. The methodology is applied to a network of air quality in Bogotá city, Colombia.     

工业集聚与工业所有制效率的空间交互影响分析——基于空间联立方程的经验研究

运用空间联立方程考察工业集聚与工业所有制效率的相互作用机制,将所有制因素纳入到Ciccone和Hall的产出密度理论模型,构建工业集聚与工业所有制效率的交互影响理论模型,采用中国31个省(区)市2003—2012年的数据对理论模型进行验证。研究发现:工业集聚促进了工业所有制效率的提高,工业所有制效率对工业集聚水平存在反向促进作用,同时工业集聚与工业所有制效率均存在明显的空间溢出效应;新型工业化应兼顾工业集聚和工业所有制效率,区域间工业协同发展可以有效缓解当前的产能过剩和产业同构,工业的集约式发展和所有制改革需要考虑地区差异。     

Comparison of three-dimensional profiles over time

In this paper, we describe an analysis for data collected on a three-dimensional spatial lattice with treatments applied at the horizontal lattice points. Spatial correlation is accounted for using a conditional autoregressive model. Observations are defined as neighbours only if they are at the same depth. This allows the corresponding variance components to vary by depth. We use the Markov chain Monte Carlo method with block updating, together with Krylov subspace methods, for efficient estimation of the model. The method is applicable to both regular and irregular horizontal lattices and hence to data collected at any set of horizontal sites for a set of depths or heights, for example, water column or soil profile data. The model for the three-dimensional data is applied to agricultural trial data for five separate days taken roughly six months apart in order to determine possible relationships over time. The purpose of the trial is to determine a form of cropping that leads to less moist soils in the root zone and beyond. We estimate moisture for each date, depth and treatment accounting for spatial correlation and determine relationships of these and other parameters over time.     

Spatial measurement error in infectious disease models

Individual-level models (ILMs) for infectious disease can be used to model disease spread between individuals while taking into account important covariates. One important covariate in determining the risk of infection transfer can be spatial location. At the same time, measurement error is a concern in many areas of statistical analysis, and infectious disease modelling is no exception. In this paper, we are concerned with the issue of measurement error in the recorded location of individuals when using a simple spatial ILM to model the spread of disease within a population. An ILM that incorporates spatial location random effects is introduced within a hierarchical Bayesian framework. This model is tested upon both simulated data and data from the UK 2001 foot-and-mouth disease epidemic. The ability of the model to successfully identify both the spatial infection kernel and the basic reproduction number (R ₀) of the disease is tested.     

Spatial modelling of the two‐party preferred vote in Australian federal elections: 2001–2016

We examine the relationships between electoral socio‐demographic characteristics and two‐party preferences in the six Australian federal elections held between 2001 and 2016. Socio‐demographic information is derived from the Australian Census which occurs every 5 years. Since a census is not directly available for each election, an imputation method is employed to estimate census data for the electorates at the time of each election. This accounts for both spatial and temporal changes in electoral characteristics between censuses. To capture any spatial heterogeneity, a spatial error model is estimated for each election, which incorporates a spatially structured random effect vector. Over time, the impact of most socio‐demographic characteristics that affect electoral two‐party preference do not vary, with age distribution, industry of work, incomes, household mobility and relationships having strong effects in each of the six elections. Education and unemployment are among those that have varying effects. All data featured in this study have been contributed to the eechidna R package (available on CRAN).     

Generalized linear models to study spatial distribution of tree species in Argentinean arid Chaco

This work adapts some generalized linear models in order to study the spatial pattern of an important tree species. The classical multivariate Ising model, which incorporates the dependence on neighbour individuals in a regular lattice, was adapted by setting a Poisson regression with an extra variation parameter to fit over-dispersion. Because the spatial pattern is only evident to a special reference scale, plots were sampled at two different scales. Two individual presence-absence matrices were analysed for each case through over-dispersion Poisson regression and log-linear models, including binary indicators for a neighbour in the four directions in the linear predictor. The results showed that the species, in the adult stage, has a spatial distribution in patches having no more than two adult individuals.     

空间计量经济学理论体系的解析及其展望

随着空间经济理论研究出现重大突破,空间计量经济学也从边缘走向现代计量经济学的主流。重点对空间计量经济模型的设立包括空间横截面数据模型、空间面板数据模型以及空间离散数据模型进行讨论,对模型参数估计方法包括最大似然估计法、两阶段最小二乘法和矩估计法等进行分析,对模型检验方法包括Moran’s I方法和LM/RS方法等内容进行总结,最后展望了该理论研究未来的发展趋势。     

Bayesian spatial quantile regression for areal count data,with application on substitute care placements in Texas

Quantile regression (QR) allows one to model the effect of covariates across the entire response distribution, rather than only at the mean, but QR methods have been almost exclusively applied to continuous response variables and without considering spatial effects. Of the few studies that have performed QR on count data, none have included random spatial effects, which is an integral facet of the Bayesian spatial QR model for areal counts that we propose. Additionally, we introduce a simplifying alternative to the response variable transformation currently employed in the QR for counts literature. The efficacy of the proposed model is demonstrated via simulation study and on a real data application from the Texas Department of Family and Protective Services (TDFPS). Our model outperforms a comparable non-spatial model in both instances, as evidenced by the deviance information criterion (DIC) and coverage probabilities. With the TDFPS data, we identify one of four covariates, along with the intercept, as having a nonconstant effect across the response distribution.