Models for Spatial Weights: A Systematic Look

General properties of spatial weights models, in particular Markovian properties, are systematically investigated. The role of stationary spatial distribution, interpretable as an importance-centrality or prominence index, is emphasized. Spatial interaction models, and among them the gravity model, are classified with respect to the time reversal and aggregation invariance properties obeyed by the associated spatial weights. Nine examples, involving connectivity, flows and distance decay analysis, integral geometry, and Dirichlet-Voronoi tessellations illustrate the main concepts, with a particular geometrical emphasis, and show how traditional, heuristic ingredients aimed at defining spatial weights can be recovered from general models.     

Using AMOEBA to Create a Spatial Weights Matrix and Identify Spatial Clusters

The creation of a spatial weights matrix by a procedure called AMOEBA, A Multidirectional Optimum Ecotope-Based Algorithm , is dependent on the use of a local spatial autocorrelation statistic. The result is (1) a vector that identifies those spatial units that are related and unrelated to contiguous spatial units and (2) a matrix of weights whose values are a function of the relationship of the ith spatial unit with all other nearby spatial units for which there is a spatial association. In addition, the AMOEBA procedure aids in the demarcation of clusters, called ecotopes, of related spatial units. Experimentation reveals that AMOEBA is an effective tool for the identification of clusters. A comparison with a scan statistic procedure (SaTScan) gives evidence of the value of AMOEBA. Total fertility rates in enumeration districts in Amman, Jordan, are used to show a real-world example of the use of AMOEBA for the construction of a spatial weights matrix and for the identification of clusters. Again, comparisons reveal the effectiveness of the AMOEBA procedure.     

Constructing the Spatial Weights Matrix Using a Local Statistic

Spatial weights matrices are necessary elements in most regression models where a representation of spatial structure is needed. We construct a spatial weights matrix, W , based on the principle that spatial structure should be considered in a two‐part framework, those units that evoke a distance effect, and those that do not. Our two‐variable local statistics model (LSM) is based on the G_i* local statistic. The local statistic concept depends on the designation of a critical distance, d_c, defined as the distance beyond which no discernible increase in clustering of high or low values exists. In a series of simulation experiments LSM is compared to well‐known spatial weights matrix specifications—two different contiguity configurations, three different inverse distance formulations, and three semi‐variance models. The simulation experiments are carried out on a random spatial pattern and two types of spatial clustering patterns. The LSM performed best according to the Akaike Information Criterion, a spatial autoregressive coefficient evaluation, and Moran's I tests on residuals. The flexibility inherent in the LSM allows for its favorable performance when compared to the rigidity of the global models.     

Estimation Bias in Spatial Models with Strongly Connected Weight Matrices

This article shows that, for both spatial lag and spatial error models with strongly connected weight matrices, maximum likelihood estimates of the spatial dependence parameter are necessarily biased downward . In addition, this bias is shown to be present in general Moran tests of spatial dependency. Thus, positive dependencies may often fail to be detected when weight matrices are strongly connected. The analysis begins with a detailed examination of downward bias for the extreme case of maximally connected weight matrices. Results for this case are then extended by continuity to a broader range of (appropriately defined) strongly connected matrices. Finally, a simulated numerical example is presented to illustrate some of the practical consequences of these biases.     

The Maximal Covering Problem with Some Negative Weights

This article presents the maximal covering problem on a network when some of the weights can be negative. Integer programming formulations are proposed and tested with ILOG CPLEX. Heuristic algorithms, an ascent algorithm, and simulated annealing are proposed and tested. The simulated annealing approach provides the best results for a data set comprising 40 problems.     

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。     

Population Forecasting with Nonstationary Multiregional Growth Matrices

Though the mathematics of multiregional population projections were defined over twenty years ago, and the methodology has seen some adoption internationally, most practitioners in the United States still use rudimentary cohort component projections techniques. Both the stationarity assumption and the implicit five‐year retrospective time scale imposed by the census migration data have probably contributed to the limited use of multiregional projections methods. This paper reviews previous attempts to overcome the stationarity assumption and proposes a decompositional approach using log linear models estimated via the ECM algorithm. The paper discusses the advantages of the decompositional approach and implements the model for intrastate migration in California.     

平泉县博物馆藏元代铜权

平泉县博物馆收藏有5件元代铜权,均为合范浇铸,权身阴刻竖排文字,分别为汉字、阿拉伯文、蒙文和波斯文,为研究元代平泉地区政治、经济的发展提供了珍贵资料。     

Spatial Cluster Detection in Spatial Flow Data

As a typical form of geographical phenomena, spatial flow events have been widely studied in contexts like migration, daily commuting, and information exchange through telecommunication. Studying the spatial pattern of flow data serves to reveal essential information about the underlying process generating the phenomena. Most methods of global clustering pattern detection and local clusters detection analysis are focused on single‐location spatial events or fail to preserve the integrity of spatial flow events. In this research we introduce a new spatial statistical approach of detecting clustering (clusters) of flow data that extends the classical local K‐function, while maintaining the integrity of flow data. Through the appropriate measurement of spatial proximity relationships between entire flows, the new method successfully upgrades the classical hot spot detection method to the stage of “hot flow” detection. Several specific aspects of the method are discussed to provide evidence of its robustness and expandability, such as the multiscale issue and relative importance control, using a real data set of vehicle theft and recovery location pairs in Charlotte, NC.     

Optimal Weights for Focused Tests of Clustering Using the Local Moran Statistic

Local spatial statistics measure and test for spatial association for a variable or variables of interest in a geographic neighborhood surrounding a predefined location. Most applications adopt a single scale of analysis but give little attention to the scale of the process generating the data. Alternatively, when the researcher is uncertain about the process scale, local statistics may examine a number of scales. In these cases, it is important to include a correction for multiple testing when evaluating the statistical significance of each local statistic, something that is rarely done. Consequently, local statistics are more likely to identify significant relationships, even when no meaningful spatial association exists. In this article, we develop a methodology for the local Moran statistic that provides both an empirical estimate of the spatial scale of association and an assessment of the significance of the statistic for that scale. The key idea is to test a number of possible choices for the statistic's weight matrix and then account for the multiple testing associated with the number of weight matrices examined. Unlike previous research, our statistic avoids the use of simulation to determine statistical significance in the presence of multiple testing. To test the validity of our approach, we constructed a numerical example to assess the statistic's performance and conducted an empirical study using leukemia data from central New York state. The developed statistic addresses the need for the empirical determination of weights and spatial scale. The test therefore addresses the common weakness of many applications, where weights are defined exogenously, with little or no thought given to either the definition or its implications. Los indicadores locales (local spatial statistics) evalúan la asociación espacial de una o varias variables de interés dada un área predefinida y sus áreas vecinas. La mayoría de dichas medidas utilizan una escala única de análisis y prestan poca atención a la escala del proceso de generación de los datos. En los casos en los que el investigador no está seguro de la escala del proceso, las los indicadores locales pueden ser evaluados a varias escalas. En dichos casos, cuando se hace la evaluación de la significancia estadística de cada indicador local, es importante incorporar una corrección para pruebas múltiples (multiple tests), un ajuste que raramente se realiza en la gran mayoría de estudios. Debido al problema de pruebas múltiples, los indicadores locales son más propensos a identificar relaciones significativas, incluso cuando no existe asociación espacial significativa alguna. En este artículo los autores desarrollan una metodología que produce un índice local de Moran que proporciona tanto una estimación empírica de la escala espacial de la asociación así como una evaluación de la importancia del indicador para dicha escala. La idea clave es poner a prueba una serie de opciones posibles para la definición de la matriz de pesos espaciales (spatial weight matrix) del índice y luego tomar en cuenta las pruebas múltiples asociadas con el número de matrices de peso examinadas. A diferencia de métodos anteriores, el indicador local propuesto evita el uso de simulaciones para determinar la significancia estadística con pruebas múltiples. Para probar la validez del enfoque propuesto, se construyó un ejemplo numérico con el fin de evaluar el desempeño del nuevo índice y se llevó a cabo un estudio comparativo a partir de datos del centro de leucemia del estado de Nueva York. El índice desarrollado responde a la necesidad de definir las ponderaciones (pesos) empíricamente y la escala espacial. De esta forma el método propuesto supera limitaciones comúnmente halladas de muchas aplicaciones en las cuales los pesos son definidos exógenamente, con poca o ninguna atención a su definición o su implicancias. 局部空间统计量可用于度量和检验预定地理区域周围邻域的空间关联。大多数情况下仅采用单一尺度的分析而较少关注数据生成过程的尺度。而当其过程尺度无法确定时,局部统计量却可能检测出多个尺度。在这些案例中,对单个局部统计量统计显著性评估建立多重检验的修正是重要的,而这却鲜有实施。因此,即使存在无意义的空间关联时,局部统计也更可能识别出显著的相关性。 本文发展了一种基于局部Moran统计的方法,提供了空间尺度关联性的经验估计以及对该尺度下统计显著性的评估。其核心思想是测试统计权重矩阵的可能选择,然后考虑与权重矩阵检验数量数目相关的多重检验。与以往研究不同,该方法在多重检验情况中避免了采用模拟来确定统计显著性。为检验其有效性,采用了数值案例来评估其统计性能,并基于纽约州中部的血癌数据进行比较研究。该方法解决了权重和空间尺度确定经验估计的需求,通过验证也相应地解决了很多应用中的普遍弱点,即权重被定义成外生变量,而很少或根本没有考虑其定义或含义。     

Spatial Entropy

A major problem in information theory concerns the derivation of a continuous measure of entropy from the discrete measure. Many analysts have shown that Shannon's treatment of this problem is incomplete, but few have gone on to rework his analysis. In this paper, it is suggested that a new measure of discrete entropy which incorporates interval size explicitly is required; such a measure is fundamental to geography and this statistic has been called spatial entropy. The use of the measure is first illustrated by application to one-and two-dimensional aggregation problems, and then the implications of this statistic for Wilson's entropy-maximizing method are traced. Theil's aggregation statistic is reinterpreted in spatial terms, and finally, some heuristics are suggested for the design of real and idealized spatial systems in which entropy is at a maximum.     

Spatial Autoregression

The paper by Cliff and Ord (1969) caused us to think more about the W matrix and the concept of spatial autoregression. This article reviews some applications of W , reinforcing the point that autoregression in time series is similar to, but not the same as, autoregression in spatial series.     

Substitutability,Spatial Structure,and Spatial Interaction

This paper attempts to distill the relationship among spatial structure, consumer preferences, and spatial interaction. It introduces the notion of substitutability. Destinations are substitutable in both economic-behavioral and physical-locational senses. Inadequate treatment of this can lead to variations in the estimated parameters of conventional spatial interaction models.