Spatial Autocorrelation Parameter Research Articles

Accommodating spatio-temporal dependency in airline demand modeling

The objective of the current study is to examine monthly air passenger departures at the airport level considering spatial interactions between airports. In this study, we develop a novel spatial grouped generalized ordered probit (SGGOP) model system of monthly air passenger departures at the airport level. Specifically, we estimate two variants of spatial models including spatial lag model and spatial error model. In the presence of repeated demand measures for the airports, we also consider temporal variations of spatial correlation effects among proximally located airports by employing space and time-based weight matrix. The proposed model is estimated using monthly air passenger departures for five years for 369 airports across the US. The proposed spatial model is implemented using composite marginal likelihood (CML) approach that offers a computationally feasible framework. From the estimation results, it is evident that air passenger departures at the airport level are influenced by different factors including MSA specific demographic characteristics, built environment characteristics, airport specific factors, spatial factors, and temporal factors. Moreover, spatial autocorrelation parameter is found to be significant validating our hypothesis of the presence of common unobserved factors associated with the spatial unit of analysis. In this study, we also perform a validation analysis to examine the predictive performance of the proposed spatial models. The results highlight the superiority of spatial error model compared to spatial lag model and the independent model that ignores the spatial interactions. Finally, we undertake an elasticity analysis to quantify the impact of the independent variables.

Estimation of An Optimum Spatial Autocorrelation in Linear Mixed Models Containing Spatial Effects

Fay-Herriot model assumes that the random effects between regions (areas) are independent of each other. This allows the regions to be mutually independent so that the estimators obtained are unbiased estimators. In cases where the regions are not mutually independent, it can develop a model in which the assumptions are violated (not fulfilled) or allow the regions to be dependent. The development of this model is known as small area estimation (SAE) with spatial effects. In small area estimation with spatial effects, one of the important parameters is the spatial autocorrelation parameter. In various small area estimation with spatial effects, it is still very rare to generate estimators of the spatial autocorrelation parameter. Most of the parameter values used are known, that is by trying to enter several spatial autocorrelation parameter values to show that the addition of regional aspects can increase the accuracy of the small area estimation. In addition, they have also tried a restricted maximum likelihood approach in estimating the spatial autocorrelation and component variance but they still assume that the sampling variance is known (assigned). Therefore, this research proposes a concentrated log-likelihood function by means of numerical procedure to find an optimum estimate value for spatial autocorrelation coefficient where both a sampling variance and a component variance are unknown. Parameters estimators obtained in the models, fixed and random effects parameters, are proved to be consistent.

Uncovering a positive and negative spatial autocorrelation mixture pattern: a spatial analysis of breast cancer incidences in Broward County, Florida, 2000–2010

Spatial cancer data analyses frequently utilize regression techniques to investigate associations between cancer incidences and potential covariates. Model specification, a process of formulating an appropriate model, is a well-recognized task in the literature. It involves a distributional assumption for a dependent variable, a proper set of predictor variables (i.e., covariates), and a functional form of the model, among other things. For example, one of the assumptions of a conventional statistical model is independence of model residuals, an assumption that can be easily violated when spatial autocorrelation is present in observations. A failure to account for spatial structure can result in unreliable estimation results. Furthermore, the difficulty of describing georeferenced data may increase with the presence of a positive and negative spatial autocorrelation mixture, because most current model specifications cannot successfully explain a mixture of spatial processes with a single spatial autocorrelation parameter. Particularly, properly accounting for a spatial autocorrelation mixture is challenging. This paper empirically investigates and uncovers a possible spatial autocorrelation mixture pattern in breast cancer incidences in Broward County, Florida, during 2000–2010, employing different model specifications. The analysis results show that Moran eigenvector spatial filtering provides a flexible method to examine such a mixture.

Interval Estimation for Age-Adjusted Rate Ratios Using Bayesian Convolution Model.

Spatial correlation raises challenges in estimating confidence intervals for region specific event rates and rate ratios between geographic units that are nested. Methods have been proposed to incorporate spatial correlation by assuming various distributions for the structure of autocorrelation patterns. However, the derivation of these statistics based on approximation may have to condition on the distributional assumption underlying the data generating process, which may not hold for certain situations. This paper explores the feasibility of utilizing a Bayesian convolution model (BCM), which includes an uncorrelated heterogeneity (UH) and a conditional autoregression (CAR) component to accommodate both uncorrelated and correlated spatial heterogeneity, to estimate the 95% confidence intervals for age-adjusted rate ratios among geographic regions with existing spatial correlations. A simulation study is conducted and a BCM method is applied to two cancer incidence datasets to calculate age-adjusted rate/ratio for the counties in the State of Kentucky relative to the entire state. In comparison to three existing methods, without and with spatial correlation, the Bayesian convolution model-based estimation provides moderate shrinkage effect for the point estimates based on the neighbor structure across regions and produces a wider interval due to the inclusion of uncertainty in the spatial autocorrelation parameters. The overall spatial pattern of region incidence rate from BCM approach appears to be like the direct estimates and other methods for both datasets, even though “smoothing” occurs in some local regions. The Bayesian Convolution Model allows flexibility in the specification of risk components and can improve the accuracy of interval estimates of age-adjusted rate ratios among geographical regions as it considers spatial correlation.

Randomized algorithms of maximum likelihood estimation with spatial autoregressive models for large-scale networks

The spatial autoregressive (SAR) model is a classical model in spatial econometrics and has become an important tool in network analysis. However, with large-scale networks, existing methods of likelihood-based inference for the SAR model become computationally infeasible. We here investigate maximum likelihood estimation for the SAR model with partially observed responses from large-scale networks. By taking advantage of recent developments in randomized numerical linear algebra, we derive efficient algorithms to estimate the spatial autocorrelation parameter in the SAR model. Compelling experimental results from extensive simulation and real data examples demonstrate empirically that the estimator obtained by our method, called the randomized maximum likelihood estimator, outperforms the state of the art by giving smaller bias and standard error, especially for large-scale problems with moderate spatial autocorrelation. The theoretical properties of the estimator are explored, and consistency results are established.

On the Statistical Distribution of the Nonzero Spatial Autocorrelation Parameter in a Simultaneous Autoregressive Model

This paper focuses on the spatial autocorrelation parameter ρ of the simultaneous autoregressive model, and furnishes its sampling distribution for nonzero values, for two regular square (rook and queen) tessellations as well as a hexagonal case with rook connectivity, using Monte Carlo simulation experiments with a large sample size. The regular square lattice directly relates to increasingly used, remotely sensed images, whereas the regular hexagonal configuration is frequently used in sampling and aggregation situations. Results suggest an asymptotic normal distribution for estimated ρ. More specifically, this paper posits functions between ρ and its variance for three adjacency structures, which makes hypothesis testing implementable and furnishes an easily-computed version of the asymptotic variance for ρ at zero for each configuration. In addition, it also presents three examples, where the first employed a simulated dataset for a zero spatial autocorrelation case, and the other two used two empirical datasets—of these, one is a census block dataset for Wuhan (with a Moran coefficient of 0.53, allowing a null hypothesis of, e.g., ρ=0.7) to illustrate a moderate spatial autocorrelation case, and the other is a remotely sensed image of the Yellow Mountain region, China (with a Moran coefficient of 0.91, allowing a null hypothesis of, e.g., ρ=0.95) to illustrate a high spatial autocorrelation case.

Multivariate Posterior Inference for Spatial Models with the Integrated Nested Laplace Approximation

SummaryThe integrated nested Laplace approximation (INLA) is a convenient way to obtain approximations to the posterior marginals for parameters in Bayesian hierarchical models when the latent effects can be expressed as a Gaussian Markov random field. In addition, its implementation in the R-INLA package for the R statistical software provides an easy way to fit models using the INLA in practice in a fraction of the time that other computer-intensive methods (e.g. Markov chain Monte Carlo methods) take to fit the same model. Although the INLA provides a fast approximation to the marginals of the model parameters, it is difficult to use it with models that are not implemented in R-INLA. It is also difficult to make multivariate posterior inference on the parameters of the model as the INLA focuses on the posterior marginals and not the joint posterior distribution. We describe how to use the INLA within the Metropolis–Hastings algorithm to fit complex spatial models and to estimate the joint posterior distribution of a small number of parameters. We illustrate the benefits of this new method with two examples. In the first, a spatial econometrics model with two auto-correlation parameters (for the response and the error term) is considered. This model is not currently available in R-INLA, and multivariate inference is often required to assess dependence between the two spatial auto-correlation parameters in the model. Furthermore, the estimation of spillover effects is based on the joint posterior distribution of a spatial auto-correlation parameter and a covariate coefficient. In the second example, a multivariate spatial model for several diseases is proposed for disease mapping. This model includes a shared specific spatial effect as well as disease-specific spatial effects. Dependence on the shared spatial effect is modulated via disease-specific weights. By inspecting the joint posterior distribution of these weights it is possible to assess which diseases have a similar spatial pattern.

Measuring bank contagion in Europe using binary spatial regression models

The recent European sovereign debt crisis clearly illustrates the importance of measuring the contagion effects of bank failures. Indeed, to better understand and monitor contagion risk, the European Central Bank has assumed the supervision of the largest banks in each of the member states. We propose a measure of contagion risk based on the spatial autocorrelation parameter of a binary spatial autoregressive model. Using different specifications of the interbank connectivity matrix, we estimate the contagion parameter for banks within the Eurozone, between 1996 and 2012. We provide evidence of high levels of systemic risk due to contagion during the European sovereign debt crisis.

Evaluating Bayesian spatial methods for modelling species distributions with clumped and restricted occurrence data.

Statistical approaches for inferring the spatial distribution of taxa (Species Distribution Models, SDMs) commonly rely on available occurrence data, which is often clumped and geographically restricted. Although available SDM methods address some of these factors, they could be more directly and accurately modelled using a spatially-explicit approach. Software to fit models with spatial autocorrelation parameters in SDMs are now widely available, but whether such approaches for inferring SDMs aid predictions compared to other methodologies is unknown. Here, within a simulated environment using 1000 generated species’ ranges, we compared the performance of two commonly used non-spatial SDM methods (Maximum Entropy Modelling, MAXENT and boosted regression trees, BRT), to a spatial Bayesian SDM method (fitted using R-INLA), when the underlying data exhibit varying combinations of clumping and geographic restriction. Finally, we tested how any recommended methodological settings designed to account for spatially non-random patterns in the data impact inference. Spatial Bayesian SDM method was the most consistently accurate method, being in the top 2 most accurate methods in 7 out of 8 data sampling scenarios. Within high-coverage sample datasets, all methods performed fairly similarly. When sampling points were randomly spread, BRT had a 1–3% greater accuracy over the other methods and when samples were clumped, the spatial Bayesian SDM method had a 4%-8% better AUC score. Alternatively, when sampling points were restricted to a small section of the true range all methods were on average 10–12% less accurate, with greater variation among the methods. Model inference under the recommended settings to account for autocorrelation was not impacted by clumping or restriction of data, except for the complexity of the spatial regression term in the spatial Bayesian model. Methods, such as those made available by R-INLA, can be successfully used to account for spatial autocorrelation in an SDM context and, by taking account of random effects, produce outputs that can better elucidate the role of covariates in predicting species occurrence. Given that it is often unclear what the drivers are behind data clumping in an empirical occurrence dataset, or indeed how geographically restricted these data are, spatially-explicit Bayesian SDMs may be the better choice when modelling the spatial distribution of target species.

Spatial Autocorrelation and Uncertainty Associated with Remotely-Sensed Data

Virtually all remotely sensed data contain spatial autocorrelation, which impacts upon their statistical features of uncertainty through variance inflation, and the compounding of duplicate information. Estimating the nature and degree of this spatial autocorrelation, which is usually positive and very strong, has been hindered by computational intensity associated with the massive number of pixels in realistically-sized remotely-sensed images, a situation that more recently has changed. Recent advances in spatial statistical estimation theory support the extraction of information and the distilling of knowledge from remotely-sensed images in a way that accounts for latent spatial autocorrelation. This paper summarizes an effective methodological approach to achieve this end, illustrating results with a 2002 remotely sensed-image of the Florida Everglades, and simulation experiments. Specifically, uncertainty of spatial autocorrelation parameter in a spatial autoregressive model is modeled with a beta-beta mixture approach and is further investigated with three different sampling strategies: coterminous sampling, random sub-region sampling, and increasing domain sub-regions. The results suggest that uncertainty associated with remotely-sensed data should be cast in consideration of spatial autocorrelation. It emphasizes that one remaining challenge is to better quantify the spatial variability of spatial autocorrelation estimates across geographic landscapes.

Approximation of Gaussian spatial autoregressive models for massive regular square tessellation data

Most of the literature to date proposes approximations to the determinant of a positive definite n × n spatial covariance matrix (the Jacobian term) for Gaussian spatial autoregressive models that fail to support the analysis of massive georeferenced data sets. This paper briefly surveys this literature, recalls and refines much simpler Jacobian approximations, presents selected eigenvalue estimation techniques, summarizes validation results (for estimated eigenvalues, Jacobian approximations, and estimation of a spatial autocorrelation parameter), and illustrates the estimation of the spatial autocorrelation parameter in a spatial autoregressive model specification for cases as large as n = 37,214,101. The principal contribution of this paper is to the implementation of spatial autoregressive model specifications for any size of georeferenced data set. Its specific additions to the literature include (1) new, more efficient estimation algorithms; (2) an approximation of the Jacobian term for remotely sensed data forming incomplete rectangular regions; (3) issues of inference; and (4) timing results.

The Improbable Nature of the Implied Correlation Matrix from Spatial Regression Models

Spatial lag dependence in a regression model is similar to the inclusion of a serially autoregressive term for the dependent variable in a time-series context. However, unlike in the time-series model, the implied covariance structure matrix from the spatial autoregressive model can have a very counterintuitive and improbable structure. A single value of spatial autocorrelation parameter can imply a large band of values of pair-wise correlations among different observations of the dependent variable, when the weight matrix for the spatial model is specified exogenously. This is illustrated using cigarette sales data (1963-1992) of 46 US states. It can be seen that that two close neighbours can have very low implied correlations compared to distant neighbours when the weighting scheme is the first-order contiguity matrix. However, if the weight matrix can capture the underlying dependence structure of the observations, then this unintuitive behaviour of implied correlation is corrected to a large extent. From this, the possibility of constructing the weight matrix (or the overall spatial dependence in the data) that is consistent with the underlying correlation structure of the dependent variable is explored. The suggested procedures produced very positive results indicating further research.

Modeling the spatial distribution of commercially important reef fishes on the West Florida Shelf

Understanding the spatial distribution of a species is an important precondition to successfully managing marine populations. For reef fishes, this is of particular importance due to the patchy nature of reef communities. This study estimated the spatial distribution of five reef fish species on the West Florida Shelf: gag grouper (Mycteroperca microlepis), mutton snapper (Lutjanus analis), red grouper (Epinephelus morio), red snapper (Lutjanus campechanus), and vermilion snapper (Rhomboplites aurorubens). Estimation was done by combining large-scale fishery-dependent catch per unit of fishing effort with small scale fishery-independent video survey observation. Catch per unit of fishing effort was obtained from vessel logbook data while video observations of reef fish presence–absence and relative abundance were made by stratified random sampling on known reef and hard-bottom habitat in the Gulf of Mexico. Relative abundance estimates showed different abundance patterns for the five species depending on depth and geographical area. Variogram modeling suggested that gag grouper, mutton snapper, and red grouper were spatially autocorrelated on reef or hard-bottom habitats at short ranges of between 0.87 and 0.95km, while red and vermilion snapper were found to be randomly distributed. Range estimates for gag grouper, mutton snapper and red grouper were supported by variograms of depth soundings in the Gulf which produced range estimates of between 1.56 and 6.34km. Combining the relative estimates of abundance from catch per unit of effort with the spatial autocorrelation parameters from video surveys allowed us to provide stochastic fish abundance estimates at scales of about 2km2.

Bayesian Inference for General Gaussian Graphical Models With Application to Multivariate Lattice Data

We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcomes and the spatial correlation structure, thereby allowing for adaptive smoothing and spatial autocorrelation parameters. Our methods are illustrated using a simulated example and a real-world application which concerns cancer mortality surveillance. Supplementary materials with computer code and the datasets needed to replicate our numerical results together with additional tables of results are available online.

On the quality of likelihood-based estimators in spatial autoregressive models when the data dependence structure is misspecified

This paper seeks to help recast and extend existing results for the specification and the estimation of spatial statistical autoregressive models. In particular, a graph-theoretic formulation of the neighborhood structure allows an intuitive, but general, specification of the dependence structure among data. Moreover, issues of asymptotic efficiency and consistency of maximum likelihood (ML) estimators (MLE) are addressed in terms of correlational structure specification error. Parameters of interest are those for the mean response, the variance, and the spatial autocorrelation. Findings reported frequently differ from those derived for independent observations, and from those derived for time series correlation structures. Some findings are in keeping with those traditionally derived for generalized least squares (GLS) estimators. Moreover, theorems presented in this paper reveal that the quality of ML spatial autoregressive model parameter estimates for Gaussian geo-referenced data distributed across a lattice is such that, under specification errors, (1) the GLS mean response ML estimator is unbiased, consistent, and asymptotically efficient in selected situations, (2) the GLS error variance ML estimator is consistent in selected situations, but usually inefficient, and (3) spatial autocorrelation parameters MLEs fail to converge to their true values under increasing domain asymptotics. Finally, a model taking into account partial information about the data neighborhood structure is presented. © 1998 Elsevier Science B.V. All rights reserved.

The pyramid system for multiscale raster analysis

Geographical research requires the management and analysis of spatial data at multiple scales. As part of the U.S. Geological Survey's global change research program a software system has been developed that reads raster data (such as an image or digital elevation model) and produces a pyramid of aggregated lattices as well as various measurements of spatial complexity. For a given raster dataset the system uses the pyramid to report: (1) mean, (2) variance, (3) a spatial autocorrelation parameter based on multiscale analysis of variance, and (4) a monofractal scaling parameter based on the analysis of isoline lengths. The system is applied to 1-km digital elevation model (DEM) data for a 256-km 2 region of central California, as well as to 64 partitions of the region. PYRAMID, which offers robust descriptions of data complexity, also is used to describe the behavior of topographic aspect with scale.