A flexible and efficient spatial interpolator for radar rainfall estimation

A key challenge in rainfall estimation is spatio-temporal variablility. Weather radars are used to estimate precipitation with high spatial and temporal resolution. Due to the inherent errors in radar estimates, spatial interpolation has been often employed to calibrate the estimates. Kriging is a simple and popular spatial interpolation method, but the method has several shortcomings. In particular, the prediction is quite unstable and often fails to be performed when sample size is small. In this paper, we proposed a flexible and efficient spatial interpolator for radar rainfall estimation, with several advantages over kriging. The method is illustrated using a real-world data set.     

Spatial modeling of extreme rainfall in northeast Thailand

It is well recognized that the generalized extreme value (GEV) distribution is widely used for any extreme events. This notion is based on the study of discrete choice behavior; however, there is a limit for predicting the distribution at ungauged sites. Hence, there have been studies on spatial dependence within extreme events in continuous space using recorded observations. We model the annual maximum daily rainfall data consisting of 25 locations for the period from 1982 to 2013. The spatial GEV model that is established under observations is assumed to be mutually independent because there is no spatial dependency between the stations. Furthermore, we divide the region into two regions for a better model fit and identify the best model for each region. We show that the regional spatial GEV model reflects the spatial pattern well compared with the spatial GEV model over the entire region as the local GEV distribution. The advantage of spatial extreme modeling is that more robust return levels and some indices of extreme rainfall can be obtained for observed stations as well as for locations without observed data. Thus, the model helps to determine the effects and assessment of vulnerability due to heavy rainfall in northeast Thailand.     

The variability of rainfall acidity

Two years of rainfall acidity data for the eastern United States were analyzed. The data consist of rainfall-event pH measurements from a nine station monitoring network. A spatio-temporal stochastic model, including deterministic components for seasonal variation and rainfall washout, and stochastic components for spatial, temporal, and measurement variation, was fitted to the data. The fitted autocorrelation structure from this model was used, in the process known as Kriging, to obtain BLUE contour maps of seasonal and rainfall adjusted yearly average pH over the monitoring region.     

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.     

Childhood malaria in the Gambia: a case-study in model-based geostatistics

Summary. The paper develops a spatial generalized linear mixed model to describe the variation in the prevalence of malaria among a sample of village resident children in the Gambia. The response from each child is a binary indicator of the presence of malarial parasites in a blood sample. The model includes terms for the effects of child level covariates (age and bed net use), village level covariates (inclusion or exclusion from the primary health care system and greenness of surrounding vegetation as derived from satellite information) and separate components for residual spatial and non-spatial extrabinomial variation. The results confirm and quantify the progressive increase in prevalence with age, and the protective effects of bed nets. They also show that the extrabinomial variation is spatially structured, suggesting an environmental effect rather than variation in familial susceptibility. Neither inclusion in the primary health care system nor the greenness of the surrounding vegetation appeared to affect the prevalence of malaria. The method of inference was Bayesian using vague priors and a Markov chain Monte Carlo implementation.     

Bayesian spatial models with repeated measurements: with application to the herbaceous data analysis

This paper proposes a new statistical spatial model to analyze and predict the coverage percentage of the upland ground flora in the Missouri Ozark Forest Ecosystem Project (MOFEP). The flora coverage percentages are collected from clustered locations, which requires a new spatial model other than the traditional kriging method. The proposed model handles this special data structure by treating the flora coverage percentages collected from the clustered locations as repeated measurements in a Bayesian hierarchical setting. The correlation among the observations from the clustered locations are considered as well. The total vegetation coverage data in MOFEP is analyzed in this study. An Markov chain Monte Carlo algorithm based on the shrinkage slice sampler is developed for simulation from the posterior densities. The total vegetation coverage is modeled by three components, including the covariates, random spatial effect and correlated random errors. Prediction of the total vegetation coverage at unmeasured locations is developed.     

Spatial generalized linear mixed models in small area estimation

In survey sampling, policy decisions regarding the allocation of resources to sub‐groups of a population depend on reliable predictors of their underlying parameters. However, in some sub‐groups, called small areas due to small sample sizes relative to the population, the information needed for reliable estimation is typically not available. Consequently, data on a coarser scale are used to predict the characteristics of small areas. Mixed models are the primary tools in small area estimation (SAE) and also borrow information from alternative sources (e.g., previous surveys and administrative and census data sets). In many circumstances, small area predictors are associated with location. For instance, in the case of chronic disease or cancer, it is important for policy makers to understand spatial patterns of disease in order to determine small areas with high risk of disease and establish prevention strategies. The literature considering SAE with spatial random effects is sparse and mostly in the context of spatial linear mixed models. In this article, small area models are proposed for the class of spatial generalized linear mixed models to obtain small area predictors and corresponding second‐order unbiased mean squared prediction errors via Taylor expansion and a parametric bootstrap approach. The performance of the proposed approach is evaluated through simulation studies and application of the models to a real esophageal cancer data set from Minnesota, U.S.A. The Canadian Journal of Statistics 47: 426–437; 2019 © 2019 Statistical Society of Canada     

Approximate composite marginal likelihood inference in spatial generalized linear mixed models

Non-Gaussian spatial responses are usually modeled using spatial generalized linear mixed model with spatial random effects. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. There are numerical ways to maximize the likelihood function, such as Monte Carlo Expectation Maximization and Quadrature Pairwise Expectation Maximization algorithms. They can be applied but may in such cases be computationally very slow or even prohibitive. Gauss–Hermite quadrature approximation only suitable for low-dimensional latent variables and its accuracy depends on the number of quadrature points. Here, we propose a new approximate pairwise maximum likelihood method to the inference of the spatial generalized linear mixed model. This approximate method is fast and deterministic, using no sampling-based strategies. The performance of the proposed method is illustrated through two simulation examples and practical aspects are investigated through a case study on a rainfall data set.     

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.     

空间回归模型选择的反思

空间计量经济学存在两种最基本的模型:空间滞后模型和空间误差模型,这里旨在重新思考和探讨这两种空间回归模型的选择,结论为:Moran’s I指数可以用来判断回归模型后的残差是否存在空间依赖性;在实证分析中,采用拉格朗日乘子检验判断两种模型优劣是最常见的做法。然而,该检验仅仅是基于统计推断而忽略了理论基础,因此,可能导致选择错误的模型;在实证分析中,空间误差模型经常被选择性遗忘,而该模型的适用性较空间滞后模型更为广泛;实证分析大多缺乏空间回归模型设定的探讨,Anselin提出三个统计量,并且,如果模型设定正确,应该遵从Wald统计量>Log likelihood统计量>LM统计量的排列顺序。     

A non-stationary spatial generalized linear mixed model approach for studying plant diversity

We analyze the multivariate spatial distribution of plant species diversity, distributed across three ecologically distinct land uses, the urban residential, urban non-residential, and desert. We model these data using a spatial generalized linear mixed model. Here plant species counts are assumed to be correlated within and among the spatial locations. We implement this model across the Phoenix metropolis and surrounding desert. Using a Bayesian approach, we utilized the Langevin–Hastings hybrid algorithm. Under a generalization of a spatial log-Gaussian Cox model, the log-intensities of the species count processes follow Gaussian distributions. The purely spatial component corresponding to these log-intensities are jointly modeled using a cross-convolution approach, in order to depict a valid cross-correlation structure. We observe that this approach yields non-stationarity of the model ensuing from different land use types. We obtain predictions of various measures of plant diversity including plant richness and the Shannon–Weiner diversity at observed locations. We also obtain a prediction framework for plant preferences in urban and desert plots.     

BAYESIAN PREDICTION FOR SPATIAL GENERALISED LINEAR MIXED MODELS WITH CLOSED SKEW NORMAL LATENT VARIABLES

Spatial generalised linear mixed models are used commonly for modelling non‐Gaussian discrete spatial responses. In these models, the spatial correlation structure of data is modelled by spatial latent variables. Most users are satisfied with using a normal distribution for these variables, but in many applications it is unclear whether or not the normal assumption holds. This assumption is relaxed in the present work, using a closed skew normal distribution for the spatial latent variables, which is more flexible and includes normal and skew normal distributions. The parameter estimates and spatial predictions are calculated using the Markov Chain Monte Carlo method. Finally, the performance of the proposed model is analysed via two simulation studies, followed by a case study in which practical aspects are dealt with. The proposed model appears to give a smaller cross‐validation mean square error of the spatial prediction than the normal prior in modelling the temperature data set.     

Childhood mortality spatial distribution in Ethiopia

The risk of a child dying before completing five years of age is highest in Sub-Saharan African countries. But Child mortality rates have shown substantial decline in Ethiopia. For this study, the 2000, 2005 and 2011 Ethiopian Demographic Survey (EDHS) was used. Generalized linear mixed model with spatial covariance structure was adapted. The model allowed for spatial correlation, and leads to the more realistic estimate for under-five mortality risk factors. The analysis showed that the risk of under-five mortality shows decline in years. But, some regions showed increase in years. The study highlights the need to implement better education for family planning and child care to improve the under-five mortality situation in some administrative areas.     

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.     

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.     

Functional approaches for predicting land use with the temporal evolution of coarse resolution remote sensing data

The sensor SPOT 4/Végétation gives every day satellite images of Europe with medium spatial resolution, each pixel corresponding to an area of 1 r km 2 1 r km. Such data are useful to characterize the development of the vegetation at a large scale. The pixels, named "mixed' pixels, aggregate information of different crops and thus different themes of interest (wheat, corn, forest, …). We aim at estimating the land use when observing the temporal evolution of reflectances of mixed pixels. The statistical problem is to predict proportions with longitudinal covariates. We compared two functional approaches. The first relies on varying-time regression models and the second is an extension of the multilogit model for functional data. The comparison is achieved on a small area on which the land use is known. Satellite data were collected between March and August 1998. The functional multilogit model gives better predictions and the use of composite vegetation index is more efficient.     

Statistical Inference on the Parametric Component in Partially Linear Spatial Autoregressive Models

A statistical test procedure is proposed to check whether the parameters in the parametric component of the partially linear spatial autoregressive models satisfy certain linear constraint conditions, in which a residual-based bootstrap procedure is suggested to derive the p-value of the test. Some simulations are conducted to assess the performance of the test and the results show that the bootstrap approximation to the null distribution of the test statistic is valid and the test is of satisfactory power. Furthermore, a real-world example is given to demonstrate the application of the proposed test.     

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.     

An Applied Comparison of Area-Level Linear Mixed Models in Small Area Estimation

This article reviews four area-level linear mixed models that borrow strength by exploiting the possible correlation among the neighboring areas or/and past time periods. Its main goal is to study if there are efficiency gains when a spatial dependence or/and a temporal autocorrelation among random-area effects are included into the models. The Fay–Herriot estimator is used as benchmark. A design-based simulation study based on real data collected from a longitudinal survey conducted by a statistical office is presented. Our results show that models that explore both spatial and chronological association considerably improve the efficiency of small area estimates.