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In this paper, we incorporate spatial analysis to estimate industry-level productivity in the presence of inter-sectoral linkages. Since each industry plays a role in providing intermediate goods to other sectors, the interdependence of economic activities across industries is inevitable. We exploit the linkage patterns from the input-output relationship to define cross-industry dependencies in economic space. We propose a spatial stochastic frontier model, which extends the stochastic frontier model to a spatially dependent specification. The models are estimated using quasi-maximum likelihood methods. Applying the approach to U.S. industry-level data from 1947 to 2010, we find that sectoral dependencies are the consequences of indirect effects via the supply chain network of industries resulting in larger output elasticities as well as scale effects for the networked production processes. However, productivity growth is estimated comparably across different spatial and non-spatial model specifications. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Techniques is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Do any of the known synergies existing between either geography and art or mathematics and art bridge all three of these disciplines? The geo-humanities and the math-humanities literatures describe only these two individual synergies. A new quantitative geography methodology exploits a sophisticated mathematical concept to analyze remotely sensed satellite images, which, when extended to artistic paintings, indeed spans all three disciplines. The organizing concept is spatial autocorrelation, or the tendency for dis/similar colors and their intensities to cluster in paintings. This article summarizes demonstrations of this contention, with specific applications to da Vinci, Monet, and Rembrandt paintings. Its principal contribution is that, for high geographic resolution digital versions of paintings, a replication constructed with judiciously selected and combined spatial autocorrelation components remarkably closely corresponds with a digital copy of its original source, further generalizing certain recent findings reported in the literature. [ABSTRACT FROM AUTHOR]

This study examines the determinants of cooperative banks' diversification proclivity, with consideration of the spatial dependence effect. The empirical analysis demonstrates that Italian cooperative banks operate as a network with significant spillover effects that should not be ignored. Indeed, local banks compete in the same market segment, and any shift in their diversification strategy has a cascading effect on neighbouring cooperative banks as a result of customer migration. Finally, we observe that an increase in bank market power results in a decline in local bank lending activity. [ABSTRACT FROM AUTHOR]

Most spatial models include a spatial weights matrix (W) derived from the first law of geography to adjust the spatial dependence to fulfill the independence assumption. In various fields such as epidemiological and environmental studies, the spatial dependence often shows clustering (or geographic discontinuity) due to natural or social factors. In such cases, adjustment using the first‐law‐of‐geography‐based W might be inappropriate and leads to inaccuracy estimations and loss of statistical power. In this work, we propose a series of data‐driven Ws (DDWs) built following the spatial pattern identified by the scan statistic, which can be easily carried out using existing tools such as SaTScan software. The DDWs take both the clustering (or discontinuous) and the intuitive first‐law‐of‐geographic‐based spatial dependence into consideration. Aiming at two common purposes in epidemiology studies (ie, estimating the effect value of explanatory variable X and estimating the risk of each spatial unit in disease mapping), the common spatial autoregressive models and the Leroux‐prior‐based conditional autoregressive (CAR) models were selected to evaluate performance of DDWs, respectively. Both simulation and case studies show that our DDWs achieve considerably better performance than the classic W in datasets with clustering (or discontinuous) spatial dependence. Furthermore, the latest published density‐based spatial clustering models, aiming at dealing with such clustering (or discontinuity) spatial dependence in disease mapping, were also compared as references. The DDWs, incorporated into the CAR models, still show considerable advantage, especially in the datasets for common diseases. [ABSTRACT FROM AUTHOR]

The measurement of spatial dependence within a set of observations or the residuals from a regression is one of the most common operations within spatial analysis. However, there appears to be a lack of appreciation for the fact that these measurements are generally based on an a priori definition of a spatial weights matrix and hence are limited to detecting spatial dependence at a single spatial scale. This paper highlights the scale-dependence problem with current measures of spatial dependence and defines a new, multi-scale approach to defining a spatial weights matrix based on a discrete Fourier transform. This approach is shown to be able to detect statistically significant spatial dependence which other multi-scale approaches to measuring spatial dependence cannot. The paper thus serves as a warning not to rely on traditional measures of spatial dependence and offers a more comprehensive, and scale-free, approach to measuring such dependence. [ABSTRACT FROM AUTHOR]

Spatial autoregressive models typically rely on the assumption that the spatial dependence structure is known in advance and is represented by a deterministic spatial weights matrix, although it is unknown in most empirical applications. Thus, we investigate the estimation of sparse spatial dependence structures for regular lattice data. In particular, an adaptive least absolute shrinkage and selection operator (lasso) is used to select and estimate the individual nonzero connections of the spatial weights matrix. To recover the spatial dependence structure, we propose cross‐sectional resampling, assuming that the random process is exchangeable. The estimation procedure is based on a two‐step approach to circumvent simultaneity issues that typically arise from endogenous spatial autoregressive dependencies. The two‐step adaptive lasso approach with cross‐sectional resampling is verified using Monte Carlo simulations. Eventually, we apply the procedure to model nitrogen dioxide (NO2) concentrations and show that estimating the spatial dependence structure contrary to using prespecified weights matrices improves the prediction accuracy considerably. [ABSTRACT FROM AUTHOR]

Spatial weights matrices used in quantitative geography furnish maps with their individual latent eigenvectors, whose geographic distributions portray distinct spatial autocorrelation (SA) components. These polygon patterns on maps have specific meaning, partially in terms of geographic scale, which this article describes. The goal of this description is to enable spatial analysts to better understand and interpret these maps individually, as well as mixtures of them, when accounting for SA in a spatial analysis. Linear combinations of Moran eigenvector maps supply a powerful and relatively simple tool that can explain SA in regression residuals, with an ability to render reasonably accurate reproductions of empirical geographic distributions with or without the aid of substantive covariates. The focus of this article is positive SA, the most commonly encountered nature of autocorrelation in georeferenced data. The principal innovative contribution of this article is establishing a better clarification of what the synthetic SA variates extracted from spatial weights matrices epitomize with regard to global, regional, and local clusters of similar values on a map. This article shows that the Getis-Ord Gi* statistic provides a useful tool for classifying Moran eigenvector maps into these three qualitative categories, illustrating findings with a range of specimen geographic landscapes. [ABSTRACT FROM AUTHOR]

Existing interfaces between mathematics and art, and geography and art, began overlapping in recent years. This newer overarching intersection partly is attributable to the scientific visualization of the concept of an eigenvector from the subdiscipline of matrix algebra. Spectral geometry and signal processing expanded this overlap. Today, novel applications of the statistical Moran eigenvector spatial filtering (MESF) methodology to paintings accentuates and exploits spatial autocorrelation as a fundamental element of art, further expanding this overlap. This paper studies MESF visualizations by compositing identified relevant spatial autocorrelation components, examining a particular Van Gogh painting for the first time, and more intensely re-examining several paintings already evaluated with MESF techniques. Findings include: painting replications solely based upon their spatial autocorrelation components as captured and visualized by certain eigenvectors are visibly indistinguishable from their original counterparts; and, spatial autocorrelation supplies measurements allowing a differentiation of paintings, a potentially valuable discovery for art history. [ABSTRACT FROM AUTHOR]

Conditional autoregressive distributions are commonly used to model spatial dependence between nearby geographic units in disease mapping studies. These distributions induce spatial dependence by means of a spatial weights matrix that quantifies the strength of dependence between any two neighboring spatial units. The most common procedure for defining that spatial weights matrix is using an adjacency criterion. In that case, all pairs of spatial units with adjacent borders are given the same weight (typically 1) and the remaining non-adjacent units are assigned a weight of 0. However, assuming all spatial neighbors in a model to be equally influential could be possibly a too rigid or inappropriate assumption. In this paper, we propose several adaptive conditional autoregressive distributions in which the spatial weights for adjacent areas are random variables, and we discuss their use in spatial disease mapping models. We will introduce our proposal in a multivariate context so that the spatial dependence structure between spatial units is shared and estimated from a sufficiently large set of mortality causes. As we will see, this is a key aspect for making inference on the spatial dependence structure. We show that our adaptive modeling proposal provides more flexible and accurate mortality risk estimates than traditional proposals in which spatial weights for neighboring areas are fixed to a common value. [ABSTRACT FROM AUTHOR]

Abstract Background When analysing spatial data, it is important to account for spatial autocorrelation. In Bayesian statistics, spatial autocorrelation is commonly modelled by the intrinsic conditional autoregressive prior distribution. At the heart of this model is a spatial weights matrix which controls the behaviour and degree of spatial smoothing. The purpose of this study is to review the main specifications of the spatial weights matrix found in the literature, and together with some new and less common specifications, compare the effect that they have on smoothing and model performance. Methods The popular BYM model is described, and a simple solution for addressing the identifiability issue among the spatial random effects is provided. Seventeen different definitions of the spatial weights matrix are defined, which are classified into four classes: adjacency-based weights, and weights based on geographic distance, distance between covariate values, and a hybrid of geographic and covariate distances. These last two definitions embody the main novelty of this research. Three synthetic data sets are generated, each representing a different underlying spatial structure. These data sets together with a real spatial data set from the literature are analysed using the models. The models are evaluated using the deviance information criterion and Moran’s I statistic. Results The deviance information criterion indicated that the model which uses binary, first-order adjacency weights to perform spatial smoothing is generally an optimal choice for achieving a good model fit. Distance-based weights also generally perform quite well and offer similar parameter interpretations. The less commonly explored options for performing spatial smoothing generally provided a worse model fit than models with more traditional approaches to smoothing, but usually outperformed the benchmark model which did not conduct spatial smoothing. Conclusions The specification of the spatial weights matrix can have a colossal impact on model fit and parameter estimation. The results provide some evidence that a smaller number of neighbours used in defining the spatial weights matrix yields a better model fit, and may provide a more accurate representation of the underlying spatial random field. The commonly used binary, first-order adjacency weights still appear to be a good choice for implementing spatial smoothing.

Discusses the interface between a GIS and spatial data analysis and further focuses on three aspects of spatial econometric analysis: (1) spatial weights matrix (2) spatial dependence in regression models (3) the estimation of spatial regression models. Includes software routines for extracting spatial weights matrices from common GIS packages (Arc/Info, packages GAUSS, LIMDEP, RATS, SHAZAM, and SPLUS.

The vast majority of spatial econometric research relies on the assumption that the spatial network structure is known a priori. This study considers a two-step estimation strategy for estimating the n(n-1) interaction effects in a spatial autoregressive panel model where the spatial dimension is potentially large. The identifying assumption is approximate sparsity of the spatial weights matrix. The proposed estimation methodology exploits the Lasso estimator and mimics two-stage least squares (2SLS) to account for endogeneity of the spatial lag. The developed two-step estimator is of more general interest. It may be used in applications where the number of endogenous regressors and the number of instrumental variables is larger than the number of observations. We derive convergence rates for the two-step Lasso estimator. Our Monte Carlo simulation results show that the two-step estimator is consistent and successfully recovers the spatial network structure for reasonable sample size, T.

This paper develops a new identification result for the causal ordering of observation units in a recursive network or directed acyclic graph. Inferences are developed for an unknown spatial weights matrix in a spatial lag model under the assumption of recursive ordering. The performance of the methods in finite sample settings is very good. Application to data on portfolio returns produces interesting new evidences on the contemporaneous lead-lag relationships between the portfolios and generates superior predictions. [ABSTRACT FROM AUTHOR]

In the study of geographical spatial autocorrelation, the spatial weights matrix is regarded as a fundamental and essential field area of research. In this paper, we firstly analyze the advantages and disadvantages of several common weights matrixes like ROOK matrix, QUEEN matrix and K-Nearest matrix. Considering the improvements for in the traditional spatial weights matrix, we put forward RLA (Ratio of Length and Area) which isthat can be applied with a stricter measurement standards and representsing the relative relationship of a spatial unit to other adjacent ones units instead of a simple true or false discrimination to improve the accuracy. For verification, we carry carried out experiments on the Mainland China viral hepatitis statistical data from 2004 to 2012 of Mainland China based on the provincial administrative divisions. The rResults indicate that the proposed weights matrix not only achieves the fundamental functions of a spatial weights matrix, but also is treated as a general definition of ROOK matrix, freely applicable to when implementing spatial autocorrelation analysis. The adoption of this RLA spatial weights matrix will further reveal the spatial relationships among different geographical units, also providing support for the prevention of epidemics. [ABSTRACT FROM AUTHOR]

Copyright of Research Papers of the Wroclaw University of Economics / Prace Naukowe Uniwersytetu Ekonomicznego we Wroclawiu is the property of Uniwersytet Ekonomiczny we Wroclawiu and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Accessibility analyses are conducted for a variety of applications, including urban planning and public health studies. These applications may aggregate data at the level of administrative units, such as provinces or municipalities. Accessibility between administrative units can be quantified by travel distance. However, modelling the distances between all administrative units in a region is computationally expensive if a large number of administrative units is considered. We propose a methodology to model accessibility between administrative units as a homogeneous Markov chain, where the administrative units are states and standardised inverse travel distances act as transition probabilities. Single transitions are allowed only between adjacent administrative units, resulting in a sparse one-step transition probability matrix (TPM). Powers of the TPM are taken to obtain transition probabilities between non-adjacent units. The methodology assumes that the Markov property holds for travel between units. We apply the methodology to administrative units within Tshwane, South Africa, considering only major roads for the sake of computation. The results are compared to those obtained using Euclidean distance, and we show that using network distance yields more reasonable results. The proposed methodology is computationally efficient and can be used to estimate accessibility between any set of administrative units connected by a road network. [ABSTRACT FROM AUTHOR]

Calculation accuracy of VaR has been a hot issue in the field of the industry and the academic. This paper first defines economic metric distance and gravitational space weight matrix, establishes generalized multidimensional space econometric model to capture space effect information, and constructs spatial value at risk (S-VaR) model to improve VaR calculation accuracy. Last, we use S&P Asia 50 index as the stock portfolio of alternative variables to test this model. The empirical results show that S-VaR can capture spatial correlation and risk spillover effect in financial market, and it raised the accuracy of VaR risk prediction. [ABSTRACT FROM AUTHOR]

Spatial econometrics is a hot area of research in economics today, but spatial weights matrix has been a bottleneck in the development of spatial econometrics. This paper introduces spatial gravitational effect of the regional economic, while combined with geographic regions and economic status indicators, constructs a new spatial weights matrix. At the background of European debt crisis, we examine sensitivity and effectiveness of gravitational effects spatial weights matrix (GESWM). Then through dynamically setting, we use it to mine contagion path and examine economic crisis contagion mode. The empirical results show that when mining multidimensional space effects of financial data, GESWM has a significant advantage. Furthermore, we find financial capital channel is the fundamental of economic crisis contagion path, economic crisis contagion is the results together of bidirectional contagion and indirectly related contagion. [ABSTRACT FROM AUTHOR]

The studies of spatial complexity have resulted in new understanding of the distance-decay functions of geography. The weight functions of spatial autocorrelation should be consistent with the functions of spatial distribution and interaction. This paper is on 'spatial autocorrelation analysis' of spatial autocorrelation. The 1-dimension spatial autocorrelation function (ACF) and partial autocorrelation function (PACF) of space series are employed to analyze four kinds of weight function in common use for the 2-dimensional spatial autocorrelation model. The aim of this study is at how to select a proper weight function to construct a spatial contiguity matrix for spatial analysis. The scopes of application of different weight functions are defined in terms of the characters of their ACFs and PACFs. [ABSTRACT FROM AUTHOR]

Spatial heterogeneity, spatial dependence and spatial scale constitute key features of spatial analysis of housing markets. However, the common practice of modelling spatial dependence as being generated by spatial interactions through a known spatial weights matrix is often not satisfactory. While existing estimators of spatial weights matrices are based on repeat sales or panel data, this paper takes the approach to a cross-section setting. Specifically, based on an a priori definition of housing submarkets and the assumption of a multifactor model, we develop maximum likelihood methodology to estimate hedonic models that facilitate understanding of both spatial heterogeneity and spatial interactions. The methodology, based on statistical orthogonal factor analysis, applied to the urban housing market of Aveiro (Portugal) at two different spatial scales, provides exciting inferences on the spatial structure of the housing market. RÉSUMÉ L'hétérogénéité spatiale, la dépendance spatiale et l’échelle spatiale sont des caractéristiques clé de l'analyse spatiale dans les marchés de l'immobilier. Toutefois, la pratique habituelle de la modélisation de la dépendance spatiale comme étant le résultat d'interactions spatiales par le biais d'une matrice de poids spatiaux n'est souvent pas satisfaisante. Alors que les estimateurs existants des matrices de poids spatiaux sont basés sur des données de panel ou des ventes répétées, la présente communication adopte le principe d'un cadre transversal. Plus spécifiquement, sur la base d'une définition à priori des sub-marchés de l'immobilier, et de l'hypothèse d'un modèle multifactoriel, nous créons une méthodologie de probabilité maximale pour estimer des modèles hédoniques qui facilitent les connaissances de l'hétérogénéité spatiale et des interactions spatiales. Cette méthodologie, basée sur une analyse des facteurs orthogonaux, appliquée au secteur de l'immobilier urbain à Aveiro (Portugal) à deux échelles spatiales différentes, fournit des inférences excitantes en ce qui concerne la structure spatiale du secteur de l'immobilier. EXTRACTO La heterogeneidad, dependencia y escala espaciales constituyen características clave del análisis espacial de los mercados de la vivienda. No obstante, la práctica común de modelar la dependencia espacial como algo generado por interacciones espaciales a través de una matriz conocida de pesos espaciales, a menudo, no es satisfactoria. Aunque los estimadores existentes de matrices de pesos espaciales se basan en ventas repetidas o datos de panel, este estudio lleva el planteamiento a un marco de corte transversal. Específicamente, basados en una definición a priori de los submercados de la vivienda y en la presuposición de un modelo de múltiples factores, desarrollamos una metodología de probabilidad máxima para estimar modelos hedónicos, que facilita la comprensión de la heterogeneidad espacial y las interacciones espaciales. La metodología, basada en el análisis estadístico de factores ortogonales y aplicada al mercado de la vivienda urbana de Aveiro (Portugal) en dos escalas espaciales diferentes, proporciona interesantes inferencias sobre la estructura espacial del mercado de la vivienda. [ABSTRACT FROM PUBLISHER]

Until recently, considerable effort has been devoted to the estimation of panel data regression models without adequate attention being paid to the drivers of interaction amongst cross-section and spatial units. We discuss some new methodologies in this emerging area and demonstrate their use in measurement and inferences on cross-section and spatial interactions. Specifically, we highlight the important distinction between spatial dependence driven by unobserved common factors and those based on a spatial weights matrix. We argue that purely factor-driven models of spatial dependence may be inadequate because of their connection with the exchangeability assumption. The three methods considered are appropriate for different asymptotic settings; estimation under structural constraints when N is fixed and T → ∞, whilst the methods based on GMM and common correlated effects are appropriate when T ≫ N → ∞. Limitations and potential enhancements of the existing methods are discussed, and several directions for new research are highlighted. [ABSTRACT FROM AUTHOR]

Copyright of Revue d'Économie Régionale & urbaine is the property of Librairie Armand Colin and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In contrast to classical econometric approaches which are based on prespecified isotropic weighting schemes, we suggest that the spatial weighting matrix in the presence of directional dependencies should be estimated. We identify this direction based on different candidate neighbourhood sets. In this paper, we consider two different types of processes – namely spatial autoregressive and spatial autoregressive conditional heteroscedastic processes – and derive the consistency of the corresponding maximum likelihood estimates in the presence of directional dependencies. Moreover, Monte Carlo simulation results indicate that the model's performance improves with sample size and with smaller neighbourhood subset sizes. Finally, we apply this approach to aerosol observations over the North Atlantic region and show that their spatial dependence matches the direction of the trade winds in this area. [ABSTRACT FROM AUTHOR]

• A set of 3-D spatial weights alternatives are compared. • The cube contiguity weights scheme can achieve best model performance. • A 3-D Autoregressive Combined model is used to account for spatial interactions. • The polluted urban river exerts negative impacts on apartment prices. • Heterogeneous environmental impacts in 3-D space have been revealed. The vertical dimension of the residential apartment market has been largely overlooked in spatial hedonic studies. To adequately quantify the environmental impacts brought by various urban natural elements on apartment prices using spatial hedonic models, the selection of an appropriate weights matrix, which could adequately approximate complex spatial relationship amongst apartment units in high-rise housing markets, is an important methodological challenge. Using a micro high-rise neighbourhood in Guangzhou (China) as a case, we compare the model performance of a set of 3-D spatial weights matrices in capturing the environmental impacts of a polluted river running by the residential neighbourhood on apartment transaction prices. Three 3-D spatial weights schemes are examined, including distance-based, contiguity-based, and a hybrid distance-contiguity weights matrix. Our Monte Carlo simulations and empirical comparison consistently confirm that contiguity-based weights matrices outperform distance-based and hybrid weights alternatives. Adopting the appropriate second-order queen's cube contiguity matrix in a 3-D Spatial Autoregressive Combined (SAC) model, the spatial heterogeneity of urban river's environmental impacts along both horizontal and vertical dimensions is revealed. We find significant negative impacts associated with the proximity and visibility of the polluted river, which are strengthened for apartments located on the 10th floor and below. Our findings highlight the urgency of restoring polluted urban rivers and provide evidence-based information for local authorities and private housing developers to restore polluted rivers in order to improve environmental amenities. [ABSTRACT FROM AUTHOR]

Background: When analysing spatial data, it is important to account for spatial autocorrelation. In Bayesian statistics, spatial autocorrelation is commonly modelled by the intrinsic conditional autoregressive prior distribution. At the heart of this model is a spatial weights matrix which controls the behaviour and degree of spatial smoothing. The purpose of this study is to review the main specifications of the spatial weights matrix found in the literature, and together with some new and less common specifications, compare the effect that they have on smoothing and model performance., Methods: The popular BYM model is described, and a simple solution for addressing the identifiability issue among the spatial random effects is provided. Seventeen different definitions of the spatial weights matrix are defined, which are classified into four classes: adjacency-based weights, and weights based on geographic distance, distance between covariate values, and a hybrid of geographic and covariate distances. These last two definitions embody the main novelty of this research. Three synthetic data sets are generated, each representing a different underlying spatial structure. These data sets together with a real spatial data set from the literature are analysed using the models. The models are evaluated using the deviance information criterion and Moran's I statistic., Results: The deviance information criterion indicated that the model which uses binary, first-order adjacency weights to perform spatial smoothing is generally an optimal choice for achieving a good model fit. Distance-based weights also generally perform quite well and offer similar parameter interpretations. The less commonly explored options for performing spatial smoothing generally provided a worse model fit than models with more traditional approaches to smoothing, but usually outperformed the benchmark model which did not conduct spatial smoothing., Conclusions: The specification of the spatial weights matrix can have a colossal impact on model fit and parameter estimation. The results provide some evidence that a smaller number of neighbours used in defining the spatial weights matrix yields a better model fit, and may provide a more accurate representation of the underlying spatial random field. The commonly used binary, first-order adjacency weights still appear to be a good choice for implementing spatial smoothing.