Bivariate Correlation with Spatial Data

The paper examines the use of the Pearson product moment correlation coefficient and the Spearman rank correlation coefficient on map data that are spatially correlated. Research has shown that in this case standard estimators underestimate the true sampling variance of the Pearson coefficient. The Clifford and Richardson “effective sample size” solution to this problem is described, methods for implementing the procedure are considered, and properties of the technique examined. The method is also used to adjust the Spearman test. Two applications are given and the method of Clifford and Richardson contrasted with the method of data “prewhitening.”     

W Function: A New Distance‐Based Measure of Spatial Distribution of Economic Activities

Distance‐based methods are applied in various fields of research. In this paper, a new relative distance‐based method, the W function, is introduced. This method contributes to the assessment of spatial patterns of economic activities using the stochastic Monte Carlo simulation, and supplements the typology of distance‐based methods recently drawn up by Marcon and Puech. The capability of the W function is compared with results from the Kd and the recently defined m function methods, which are widely used for monitoring the spatial distribution of economic activities by considering several theoretical and empirical examples. The W function appears to provide more precise estimations of the density of economic activities compared to the m and Kd functions, particularly in cases of complex patterns such as double clustered distribution. It also appears to provide a more accurate evaluation of dispersion.     

Exploring Goodness-of-Fit and Spatial Correlation Using Components of Tango's Index of Spatial Clustering

The ability to detect anomalies such as spatial clustering in data sets plays an important role in spatial data analysis, leading to interest in test statistics identifying patterns exhibiting significant levels of clustering. Toward this end, Tango (1995) proposed a statistic (and its associated distribution under a null hypothesis of no clustering) assessing overall patterns of spatial clustering in a set of observed regional counts. Rogerson (1999) observed that Tango's index may be decomposed into the summation of two distinct statistics, the first mirroring standard tests of goodness-of-fit (GOF), and the second an index of spatial association (SA) similar to Moran's I . In this article, we investigate the effectiveness of Rogerson's expression of Tango's statistic in separating GOF from SA in data sets containing clusters. We simulate data under the null hypothesis of no clustering as well as two alternative hypotheses. The first alternative hypothesis induces a poor fit from the null hypothesis while maintaining independent observations and the second alternative hypothesis induces spatial dependence while maintaining fit. Using Rogerson's decomposition and leukemia incidence data from upstate New York, we show graphically that one is unable to statistically distinguish poor fit from autocorrelation.     

A Coherent Measure

Abstract

According to many contemporary scholars, an originalist reading of the Fourteenth Amendment is difficult, if not impossible, because the Amendment did not have a coherent original meaning. Its provisions, it is argued, were ambiguous and vague--and deliberately so. But as will be set forth in this article, a review of the commentary from the drafting and ratification of the Amendment provides substantial evidence that proponents of the Amendment held a coherent understanding of Section 1. Further this evidence indicates that the major interpretive disagreements arose not among supporters but between supporters and opponents, and even these were generally limited to disputes on only two (albeit major) interpretive issues: (1) whether the “privileges and immunities of citizens of the United States” included intermarriage and political rights, (2) whether these “privileges and immunities” would be subject to congressional redefinition.     

A Shape‐Based Local Spatial Association Measure (LISShA): A Case Study in Maritime Anomaly Detection

The explicit consideration of the shape of geographic features has been largely ignored in existing spatial association measures. The primary contribution of this work is the development of a new local spatial association measure—a Local Indicator of Spatial and Shape Association (LISShA). The LISShA measure is modeled after local Geary's Spatial Autocorrelation measure with distance between shapes, calculated using the Small–Le metric, replacing difference between attribute values and the spatial neighborhood defined by Fréchet distance. We provide some explanation of these metrics and show, in detail, how the LISShA and proposed moments are calculated in a one‐dimensional context in a case study of maritime anomaly detection.     

Balancing Spatial and Non-Spatial Variation in Varying Coefficient Modeling: A Remedy for Spurious Correlation

This study discusses the importance of balancing spatial and non-spatial variation in spatial regression modeling. Unlike spatially varying coefficients (SVC) modeling, which is popular in spatial statistics, non-spatially varying coefficients (NVC) modeling has largely been unexplored in spatial fields. Nevertheless, as we will explain, consideration of non-spatial variation is needed not only to improve model accuracy but also to reduce spurious correlation among varying coefficients, which is a major problem in SVC modeling. We consider a Moran eigenvector approach modeling spatially and non-spatially varying coefficients (S&NVC). A Monte Carlo simulation experiment comparing our S&NVC model with existing SVC models suggests both modeling accuracy and computational efficiency for our approach. Beyond that, somewhat surprisingly, our approach identifies true and spurious correlations among coefficients nearly perfectly, even when usual SVC models suffer from severe spurious correlations. It implies that S&NVC model should be used even when the analysis purpose is modeling SVCs. Finally, our S&NVC model is employed to analyze a residential land price data set. Its results suggest existence of both spatial and non-spatial variation in regression coefficients in practice. The S&NVC model is now implemented in the R package spmoran.     

A Note on the Spatial Correlation Structure of County-Level Growth in the U.S.

This paper examines the spatial correlation structure of county-level growth across the contiguous United States. Estimated spatial correlograms using data on four different measures of aggregate economic activity—population, employment, income, and earnings—over the period 1984–1994 indicate that cross-county interdependence is limited to relatively short ranges of distance. For each of the measures, the average correlation between the growth rates of two counties approaches zero within a range of approximately 200 miles. Moreover, the rate at which correlations decline with distance is not uniform. Inside of roughly 40 miles correlations show only a very slow rate of decline whereas beyond this range they drop off at a substantially higher rate.     

Large Sample-Size Distribution of Statistics Used In Testing for Spatial Correlation

Test statistics for testing for spatial correlation in continuous variables have been given by both Moran and Geary and have subsequently been generalized. It has been conjectured for a long time that under the hypothesis of no spatial correlations all these statistics are normally distributed when the sample size is large. This paper proves a very general theorem on the large sample normality of quadratic forms. As corollaries to the theorem the asymptotic normality, under the hypothesis, of all the above-mentioned statistics is established. The necessary conditions are quite unrestrictive. It is also shown, by means of a counter example, that the conditions given in a similar theorem (Cliff and Ord) are inadequate to ensure normality.     

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.     

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。     

A Methodology for Neural Spatial Interaction Modeling

This paper attempts to develop a mathematically rigid and unified framework for neural spatial interaction modeling. Families of classical neural network models, but also less classical ones such as product unit neural network ones are considered for the cases of unconstrained and singly constrained spatial interaction flows. Current practice appears to suffer from least squares and normality assumptions that ignore the true integer nature of the flows and approximate a discrete‐valued process by an almost certainly misrepresentative continuous distribution. To overcome this deficiency we suggest a more suitable estimation approach, maximum likelihood estimation under more realistic distributional assumptions of Poisson processes, and utilize a global search procedure, called Alopex, to solve the maximum likelihood estimation problem. To identify the transition from underfitting to overfitting we split the data into training, internal validation, and test sets. The bootstrapping pairs approach with replacement is adopted to combine the purity of data splitting with the power of a resampling procedure to overcome the generally neglected issue of fixed data splitting and the problem of scarce data. In addition, the approach has power to provide a better statistical picture of the prediction variability. Finally, a benchmark comparison against the classical gravity models illustrates the superiority of both, the unconstrained and the origin constrained neural network model versions in terms of generalization performance measured by Kullback and Leibler's information criterion.     

Estimating Spatial Models Within A Disequilibrium Framework

Modeling demand in a spatial context requires careful handling of regional interactions. In situations where there are constraints in some markets that lead to spillovers to others it is useful to build this explicitly into the model. In this paper I present a theoretical model that is related to the disequilibrium and the spatial econometric literature. Under certain conditions the model may be estimable and an appropriate estimation technique that uses the EM algorithm, is put forward. A data set from the U.K. market for secondary school places provides a test for the procedure.