Beyond Moran's I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis-testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatial-dependence parameter when the parameter is close to 0. In this research, we develop an alternative closed-form measure of spatial autocorrelation, which we call APLE , because it is an approximate profile-likelihood estimator (APLE) of the SAR model's spatial-dependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation-based motivations (including comparison with the maximum-likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well-known Mercer and Hall wheat-yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data analysis.     

Exploring Spatial Patterns Using an Expanded Spatial Autocorrelation Framework

Spatial autocorrelation (SA) is regarded as an important dimension of spatial pattern. SA measures usually consist of two components: measuring the similarity of attribute values and defining the spatial relationships among observations. The latter component is often represented by a spatial weights matrix that predefines spatial relationship between observations in most measures. Therefore, SA measures, in essence, are measures of attribute similarity, conditioned by spatial relationship. Another dimension of spatial pattern can be explored by controlling observations to be compared based upon the degree of attribute similarity. The resulting measures are spatial proximity measures of observations, meeting predefined attribute similarity criteria. Proposed measures reflect degrees of clustering or dispersion for observations meeting certain levels of attribute similarity. An existing spatial autocorrelation framework is expanded to a general framework to evaluate spatial patterns and can accommodate the proposed approach measuring proximity. Analogous to the concept of variogram, clustergram is proposed to show the levels of spatial clustering over a range of attribute similarity, or attribute lags. Specific measures based on the proposed approach are formulated and applied to a hypothetical landscape and an empirical example, showing that these new measures capture spatial pattern information not reflected by traditional spatial autocorrelation measures.     

Developing Spatial Weight Matrices for Incorporation into Multiple Linear Regression Models: An Example Using Grizzly Bear Body Size and Environmental Predictor Variables

In this study, we develop spatial autoregressive (SAR) models relating grizzly bear body length to environmental predictor variables in the Alberta Rocky Mountains. We examine the ability of several different spatial neighborhoods to model spatial dependence and compare the estimated parameters and residuals from a standard linear regression model (LRM) with those from three types of SAR models: error, lag, and Durbin. Further, we examine variable selection in the presence of negative dependence by repeating the modeling process using a SAR model. Two findings are that significant negative spatial dependence was present in the residuals of the LRM and that the choice of spatial neighborhood greatly affects the ability to detect spatial dependence. The incorporation of appropriate spatial weights into SAR models improves the fit and increases the significance of the parameter estimates vis‐à‐vis the linear model. The results of this study indicate that negative dependence may not have as severe negative effects on variable selection and parameter estimation as positive dependence. An examination of spatial dependence in regression modeling appears to be an important means of exploring the appropriateness of a sampling framework, predictor variables, and model form. En este estudio desarrollamos modelos espaciales autorregresivos (SAR) que vinculan la longitud del cuerpo de osos grizzli con variables predictivas ambientales en las montañas rocosas de Alberta, Canadá. Examinamos la capacidad de varias vecindades espaciales para modelar la dependencia espacial y la comparación de los parámetros estimados, así como los residuos de un modelo de regresión lineal estándar (LRM) versus tres tipos de modelos SAR: error, retraso (lag) y Durbin. Además, se examina la selección de variables en la presencia de dependencia negativa mediante la repetición del proceso de modelado con un modelo de SAR. El estudio concluye que: 1) existe dependencia espacial negativa significativa en los residuos de la LRM y; 2) la selección de la vecindad espacial afecta en gran medida la capacidad de detectar la dependencia espacial. La incorporación de ponderaciones espaciales correspondientes a los modelos SAR mejora el ajuste y aumenta la importancia de los parámetros estimados versus el modelo lineal. Los resultados de este estudio indican que la dependencia negativa puede no tener los graves efectos negativos en la selección de variables y la estimación de parámetros si se comparan dichos efectos con = la dependencia positiva. Los autores recomiendan un examen de la dependencia espacial en modelos de regresión como medio importante para explorar la conveniencia de un marco de muestreo, de variables de predicción, y de la forma del modelo. 本文构建了阿尔伯达省落基山脉地区的灰熊体态大小与环境预测变量之间的空间自回归模型（SAR）,检验了几种以不同空间邻域矩阵拟合变量的空间相关性,并比较了标准回归模型(LRM)与几种不同类型的SAR模型（空间残差模型、空间滞后模型和空间杜宾模型）的估计参数和残差大小。进而利用一种SAR模型重复模拟过程,进一步测试变量选择对负相关性存在的影响。研究表明,显著的空间负相关存在于LRM的残差中,且空间邻域权重的选择很大程度上影响模型空间相关性的探测能力。将适当的空间权重引入SAR模型中可提高拟合精度,增加相对于线性模型参数估计的显著性。研究结果表明,负相关性在变量选择和参数估计上严重负影响的程度不如正相关性强。回归模型中空间相关性检验似乎是采样结构、预测变量和模型形式适用性分析的一个重要途径。     

Spatial Causality: A Systematic Review on Spatial Causal Inference

The growing interest in causal inference in recent years has led to new causal inference methodologies and their applications across disciplines and research domains. Yet, studies on spatial causal inference are still rare. Causal inference on spatial processes is faced with additional challenges, such as spatial dependency, spatial heterogeneity, and spatial effects. These challenges can lead to spurious results and subsequently, incorrect interpretations of the outcomes of causal analyses. Recognizing the growing importance of causal inference in the spatial domain, we conduct a systematic literature review on spatial causal inference based on a formal concept mapping. To identify how to assess and control for the adverse effects of spatial influences, we assess publications relevant to spatial causal inference based on criteria relating to application discipline, methods used, and techniques applied for managing issues related to spatial processes. We thus present a snapshot of state of the art in spatial causal inference and identify methodological gaps, weaknesses and challenges of current spatial inference studies, along with opportunities for future research.     

Spatial and temporal effects in Mexican direct elections for the chamber of deputies

By using data of the elections for the Chamber of Deputies of 1997 and 2000 in Mexico, we fit spatial autologistic models with temporal effects to test the significance of spatial and temporal effects on those elections. The binary variable of interest is the one that indicates a win of the National Action Party (PAN) or the alliance that it formed. By spatial effect, we refer to the fact that neighbouring constituencies present dependence on their electoral results. The temporal effect refers to the existence of dependence, for the same constituency, of the result of the election with the result of the previous election. The model that we used to test the significance of spatial and temporal effects is the spatial autologistic model with temporal effects for which estimation is complex and requires simulation techniques. By defining an urban constituency as one that contains at least one population center of 200,000 inhabitants or more, among our principal results, we find that, for the Mexican election of 2000, the spatial effect is significant only when neighbouring constituencies are both urban. For the election of 1997, the spatial effect is significant independent of the type of neighbouring constituencies. The temporal effect is significant on both elections.     

The Importance of Context and Comparison in the Study of European Spatial Planning

The European Spatial Development Perspective (ESDP) was agreed in 1999 at Potsdam, Germany, as a non-binding framework intended to guide spatially significant policy-making at different spatial scales in order to achieve a more balanced and sustainable growth of the EU territory. This paper develops a conceptualization of the nature of transnational planning frameworks such as the ESDP and presents a framework for the investigation of the application of their policy orientations in the spatial planning systems of European states. It is argued that investigations of the application of transnational spatial development frameworks like the ESDP and the ‘Territorial Agenda of the European Union’ document adopted by EU member states in 2007, need to be sensitized to the diversity of territorial contexts in which these apply, and that a contextualized and comparative approach is therefore essential in evaluating their influence in Europe's varied territories.     

Spatial Autoregressive Models for Geographically Hierarchical Data Structures

This article discusses how standard spatial autoregressive models and their estimation can be extended to accommodate geographically hierarchical data structures. Whereas standard spatial econometric models normally operate at a single geographical scale, many geographical data sets are hierarchical in nature—for example, information about houses nested into data about the census tracts in which those houses are found. Here we outline four model specifications by combining different formulations of the spatial weight matrix W and of ways of modeling regional effects. These are (1) groupwise W and fixed regional effects; (2) groupwise W and random regional effects; (3) proximity‐based W and fixed regional effects; and (4) proximity‐based W and random regional effects. We discuss each of these model specifications and their associated estimation methods, giving particular attention to the fourth. We describe this as a hierarchical spatial autoregressive model. We view it as having the most potential to extend spatial econometrics to accommodate geographically hierarchical data structures and as offering the greatest coming together of spatial econometric and multilevel modeling approaches. Subsequently, we provide Bayesian Markov Chain Monte Carlo algorithms for implementing the model. We demonstrate its application using a two‐level land price data set where land parcels nest into districts in Beijing, China, finding significant spatial dependence at both the land parcel level and the district level.     

A Spatial Price Determination Model for Multi‐Market Heterogeneously Distributed Resources: An Application to Forestry Markets in Sweden

The spatial dimension is a key paradigm in price determination, as attested by recent studies in the literature that highlighted the differential in market behavior between spatial and non‐spatial pricing settings. In this paper, we develop a model of spatial pricing for multi‐market heterogeneously distributed resources, with an application to the Swedish forestry sector. The focus of the model is to estimate the impact of spatial interaction on the demand for resources in terms of resource allocation, competition, and pricing. In its core, the pricing mechanism relies on a supply–demand framework. Using disaggregated data at the gridcell level for forest feedstock supply and harvesting costs in Sweden, we construct regional supply curves for each gridcell assuming a maximum transportation distance to delimit the potential market. Demand nodes are exogenously determined and are adjusted using a distance‐decay model to assess demand pressure across locations. We apply the model empirically to assess the impact on forest feedstock prices of a 20 TWh increase in biofuel production.     

Exploring Spatial Process Dynamics Using Irregular Cellular Automaton Models

Cellular automaton models have enjoyed popularity in recent years as easily constructed models of many complex spatial processes, particularly in the natural sciences, and more recently in geography also. Most such models adopt a regular lattice (often a grid) as the basis for the spatial relations of adjacency that govern evolution of the model. A number of variations on the cellular automaton formalism have been introduced in geography but the impact of such variations on the likely behavior of the models has not been explored. This paper proposes a method for beginning to explore these issues and suggests that this is a new approach to the investigation of the relationships between spatial structure and dynamics of spatial processes. A framework for this exploration is suggested, and details of the required methods and measures are provided. In particular, a measure of spatial pattern—spatial information—based on entropy concepts is introduced. Initial results from investigation along the proposed lines are reported, which suggest that a distinction can he made between spatially robust and fragile processes. Some implications of this result and the methodology presented are briefly discussed.     

The Role of Spatial Scale in Geomorphic Systems

Geomorphic systems are characterized by numerous, complex interrelationships between system components, and by processes and controls which may operate over different spatial scales. Factors operating at any given spatial scale can be viewed as an abstracted subset of all relationships operating at all scales. The theory that relationships which operate over spatial scales an order of magnitude different are effectively independent of each other is formally stated in terms of abstracted systems. An example is given to illustrate the use of spatial statistics to determine what constitutes a significant spatial scale difference in controls over hydraulic geometry of a desert wash.     

"The Problem of Spatial Autocorrelation" and Local Spatial Statistics

This article examines the relationship between spatial dependency and spatial heterogeneity, two properties unique to spatial data. The property of spatial dependence has led to a large body of research into spatial autocorrelation and also, largely independently, into geostatistics. The property of spatial heterogeneity has led to a growing awareness of the limitation of global statistics and the value of local statistics and local statistical models. The article concludes with a discussion of how the two properties can be accommodated within the same modelling framework.     

Spatial Analysis in Physical Geography and the Challenge of Deterministic Uncertainty

The application of complex and nonlinear dynamical systems (NDS) theory in physical geography and geosciences has proceeded through several stages, and has recently entered a phase where field-testable hypotheses and historical or mechanistic explanations are being generated. However, there are some fundamental challenges. It seems clear that chaos and dynamical self-organization are present, and may be common in earth surface systems, and that these phenomena have spatial manifestations in the landscape. However, NDS theory and methods have been formulated primarily in the temporal domain and are typically ill-suited to real-world spatial data. Spatial analytical methods are not generally capable of distinguishing deterministic complexity and uncertainty from noise. Thus, the detection of the signals of complex deterministic dynamics in real landscapes and spatial data is a major challenge. Entropy-based methods of spatial analysis can be directly linked to nonlinear dynamics, and are at present the best available method to approach this problem. However, there is evidence in the spatial analysis literature suggesting that development of techniques to detect deterministic uncertainty is possible. Pending such a break-through, three general approaches are described, based on spatial analysis of chronosequences, the characteriziation of changes in spatial structure over time, and the spatial-domain testing of specific hypotheses relevant to deterministic uncertainty. Current trends generally suggest a shift in mathematical modeling and spatial analysis in physical geography away from traditional determinism toward approaches that incorporate locational, historical, and scale contingency.     

A Local Model for Multivariate Analysis: Extending Wartenberg’s Multivariate Spatial Correlation

Local analysis can provide specific information about individual observations that is often useful in understanding nonstationary interactions among variables. This paper extends the application of Wartenberg’s Multivariate Spatial Correlation (MSC) method to a local setting. The original MSC can be considered as an adaptation of Principal Component Analysis for spatial effects with respect to spatial autocorrelation. The extended MSC method described in this paper, however, further incorporates another spatial effect, spatial heterogeneity, by the addition of geographic weights in standardizing the data and in calculating the spatial association weight matrix. The extension allows more local analysis and facilitates additional visualization of the results. The geographically weighted MSC is illustrated and justified using the classic dataset collected by André-Michel Guerry on moral statistics in 1830s France.     

Comparative Spatial Filtering in Regression Analysis

One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables’ total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis’ filtering approach is based on the autocorrelation observed with the use of the G_i local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context.     

The Network Approach: Dutch Spatial Planning between Substratum and Infrastructure Networks

This paper investigates the substratum and infrastructure networks as relevant components of spatial planning. Since 2001, in Dutch government documents on spatial planning the Layers Approach has been presented, with three layers: substratum; networks; occupation patterns. The Layers Approach assumes that the characteristics of the substratum provide conditions for infrastructure networks and that these infrastructure networks then condition the occupation patterns, including the locations for housing, business activities and related services. These more or less hierarchical relationships are supposed to produce clear ground rules for the spatial planner. The Layers Approach seems to offer a robust methodology for spatial planning. This contribution presents the background of the Layers Approach, adds some critical notes and proposes an amendment: the Network Approach, connecting different spatial scales. After the presentation of a typology of networks a preliminary overview of the dynamics of transport networks is given. The spatial relevance of information and communication technology (ICT) networks, that support transport networks, is discussed. Nodes are presented as links between infrastructure networks and occupation patterns. The paper concludes that actors, dealing with regional spatial plans in a decentralized planning system, have to acquire an in-depth knowledge of the technology, economy and governance of current networks and the qualities of the substratum, which form a conditioning and stimulating framework for the spatial planning of urban and regional areas.     

A Spatial Autoregressive Poisson Gravity Model

In this article, a Poisson gravity model is introduced that incorporates spatial dependence of the explained variable without relying on restrictive distributional assumptions of the underlying data‐generating process. The model comprises a spatially filtered component—including the origin‐, destination‐, and origin‐destination‐specific variables—and a spatial residual variable that captures origin‐ and destination‐based spatial autocorrelation. We derive a two‐stage nonlinear least‐squares (NLS) estimator (2NLS) that is hetero‐scedasticity‐robust and, thus, controls for the problem of over‐ or underdispersion that often is present in the empirical analysis of discrete data or, in the case of overdispersion, if spatial autocorrelation is present. This estimator can be shown to have desirable properties for different distributional assumptions, like the observed flows or (spatially) filtered component being either Poisson or negative binomial. In our spatial autoregressive (SAR) model specification, the resulting parameter estimates can be interpreted as the implied total impact effects defined as the sum of direct and indirect spatial feedback effects. Monte Carlo results indicate marginal finite sample biases in the mean and standard deviation of the parameter estimates and convergence to the true parameter values as the sample size increases. In addition, this article illustrates the model by analyzing patent citation flows data across European regions. En el presente artículo, se introduce un modelo de gravedad Poisson, que incorpora la dependencia espacial de la variable explicada, sin apoyarse en presunciones de distribución restrictivas del proceso subyacente de generación de datos. El modelo comprende de un componente espacialmente filtrado, que incluye las variables de origen, destino y origen‐destino específico; y una variable espacial residual que captura la auto‐correlación espacial basada en el origen y destino. Se deriva del calculador (2NLS) de dos etapas no lineales de mínimos cuadrados (NLS), el cual es robusto en heterocedasticidad, y por ello controla el problema de sobre‐dispersión o baja‐dispersión (over and under dispersion), que a menudo se presenta en el análisis empírico de datos discretos; o, en el caso de de sobre‐dispersión, cuando se presenta la auto correlación espacial. Este calculador puede demostrar tener propiedades deseables para diferentes supuestos distribucionales, como los flujos observados un componente (espacialmente) filtrado, ya sea Poisson o binomial negativo. En nuestra especificación de modelo espacial auto regresivo (SAR), las estimaciones de los parámetros resultantes se pueden interpretar como los efectos de impacto total implícitos, definidos como la suma de efectos espaciales, directos o indirectos, de retroalimentación (feedback). Los resultados Monte Carlo indican sesgos marginales de muestras finitas en la media y la desviación estándar de los parámetros estimados, y la convergencia de los valores de los parámetros reales, a medida que aumenta el tamaño de muestra. Este artículo ilustra el modelo mediante el análisis de flujos de datos de citas de patentes, a través de las regiones europeas. 本文提出了一种蕴含空间依赖的泊松引力模型,该模型中解释变量无需依赖潜在数据生成过程的限制性分布假设。该模型由包含起点、终点、起点‐终点特定变量的空间滤波组分和空间残差变量组成,能捕捉到基于起点和终点的空间自相关。我们推导出一个二阶非线性最小二乘（NLS）估计（2NLS）,它对异方差具有鲁棒性,从而可控制对于离散或过离散数据经验性分析中经常出现的过离散和低离散问题。如果空间自相关存在,过离散数据分析就是一个例子。对于不同的分布假设,如或泊松分布或是负二项式分布的观测流或（空间）滤波组分,该估计量显示出令人满意的性能。在本文的空间自回归（SAR）模型设定中,参数估计结果可解释为隐含的全局影响效应,并可被定义为直接和间接的空间反馈效应之和。蒙特卡罗结果给出了参数估计中均值、标准差的临界有限样本偏差,且随样本量增大收敛于真正参数值。此外,本文基于欧洲地区专利引用的流数据进行了模型验证。     

The Spatial Association of Demographic and Population Health Characteristics with COVID-19 Prevalence Across Districts in India

In less-developed countries, the lack of granular data limits the researcher's ability to study the spatial interaction of different factors on the COVID-19 pandemic. This study designs a novel database to examine the spatial effects of demographic and population health factors on COVID-19 prevalence across 640 districts in India. The goal is to provide a robust understanding of how spatial associations and the interconnections between places influence disease spread. In addition to the linear Ordinary Least Square regression model, three spatial regression models—Spatial Lag Model, Spatial Error Model, and Geographically Weighted Regression are employed to study and compare the variables explanatory power in shaping geographic variations in the COVID-19 prevalence. We found that the local GWR model is more robust and effective at predicting spatial relationships. The findings indicate that among the demographic factors, a high share of the population living in slums is positively associated with a higher incidence of COVID-19 across districts. The spatial variations in COVID-19 deaths were explained by obesity and high blood sugar, indicating a strong association between pre-existing health conditions and COVID-19 fatalities. The study brings forth the critical factors that expose the poor and vulnerable populations to severe public health risks and highlight the application of geographical analysis vis-a-vis spatial regression models to help explain those associations.     

Analyzing Cross-classified Data with Inherent Spatial Dependence

Despite considerable recent progress in the methods available for the log-linear analysis of categorical data arising from complex sampling schemes, only a few papers have been published that deal with the parallel phenomenon of dependence induced by spatial sampling. This paper aims to add to the general awareness of this topic and suggests some new ideas for tackling the problems raised. In the paper it is shown that the method that has been proposed for the valid selection of log-linear models given spatially dependent data and some derivative methods are somewhat conservative when compared to an approach based on a model of spatial dependence outlined in section 4. The method also serves as a data exploratory technique to enhance the use of the more robust conservative approach.