The Bearing Correlogram: A New Method of Analyzing Directional Spatial Autocorrelation

Extensions of nondirectional spatial autocorrelation techniques to two dimensions have existed for many years, but the results are difficult to compare to the traditional nondirectional techniques and often lack ease of interpretability. This paper reviews the traditional one‐ and two‐dimensional spatial autocorrelation methods and proposes a new directional method which is both easier to compare to nondirectional methods and easier to interpret than previous directional methods.     

On the Statistical Significance of the Observed Map in Spatial Diffusion

This paper examines the problem of determining the statistical significance of an observed spatial distribution vis-à-vis a simulated pattern of the “Hägerstrand type.” It is argued that the sampling distribution for a Hägerstrand type simulator is multinomial. The diffusion model considered is logically equivalent to a multinomial experiment with independent assignment of adoptions to areal cells according to a probability distribution that remains fixed for a given time interval. Conventional tests for goodness of multinomial fit are thus appropriate.     

Assessing Spatial Dependence in Count Data: Winsorized and Spatial Filter Specification Alternatives to the Auto-Poisson Model

The auto-Poisson probability model furnishes an obvious tool for modeling counts of geographically distributed rare events. Unfortunately, its original specification can accommodate only negative spatial autocorrelation, which itself is a rare event. More recent alternative reformulations, namely, the Winsorized and spatial filter specifications, circumvent this drawback. A comparison of their performances presented in this article reveals some of their relative advantages and disadvantages.     

Assessing Quality of Spatial Models Using the Structural Similarity Index and Posterior Predictive Checks

Model assessment is one of the most important aspects of statistical analysis. In geographical analysis, models represent spatial processes, where variability in mapped output results from uncertainty in parameter estimates. Slight spatial misalignments can cause inflated error scores when comparing maps of observed and predicted variables using traditional error metrics at the level of individual spatial units. We conceptualize spatial model assessment as a continuous value map comparison problem and employ methods from image analysis to score model outputs. The structural similarity index, a measure that attempts to replicate the human visual system using a local region approach, is used as an exploratory map comparison statistic. The measure is implemented within a Bayesian spatial modeling framework as a discrepancy measure in a posterior predictive check of model fit. Results are reported for simulation studies representing a variety of spatial processes in a spatial and space–time context. A case study of rainfall mapping in Sri Lanka demonstrates the proposed methodology applied to assessment of Bayesian kriging interpolations. Both simulation studies as well as the case study demonstrate that the approach reveals hidden spatial structure not uncovered by traditional methods. The spatially sensitive assessment methodology provides a diagnostic tool to support spatial modeling and analysis. La evaluación de modelos es uno de los aspectos más importantes de análisis estadístico. En el análisis geográfico, los modelos representan procesos espaciales en los que la variabilidad en los outputs es el resultado de la incertidumbre en los parámetros estimados. Leves desajustes espaciales pueden inflar los valores de error en la comparación entre los mapas de las observaciones y los mapas de las predicciones de las variables si es que se usan medidas tradicionales de medición de error al nivel de unidades espaciales individuales. Los autores conceptualizan la evaluación de modelos espaciales como un problema de comparación mapas de valor continuo y emplea métodos de análisis de imágenes para cuantificar los resultados del modelo. Se utiliza el índice de similitud estructural (SSIM), una medida que intenta replicar el sistema visual humano utilizando un enfoque de región local, como técnica de exploratoria comparación estadística de mapas. El índice es implementado dentro de un marco de modelización espacial bayesiano como medida de discrepancia en la comprobación posterior de predicción del desempeño del modelo. Los resultados se presentan para varios casos de simulación que representan una serie de procesos espaciales en un contexto espacio‐temporal y espacial. Un estudio de caso de mapeo de lluvias en Sri Lanka sirve como demostración de la metodología propuesta y su aplicación a la evaluación de las interpolaciones tipo krigeage (kriging) bayesianas. Tanto los estudios de simulación, así como el estudio de caso demuestran que el enfoque propuesto revela la estructura espacial oculta no evidenciada por métodos tradicionales. La metodología de evaluación espacialmente sensible que se presenta en este artículo proporciona una herramienta de diagnóstico para apoyar la elaboración de modelos y análisis espacial. 模型评估是统计分析中最为重要的内容之一。在地理分析中用模型表达空间过程,参数估计的不确定性会导致地图输出结果的可变性。当采用传统误差指标度量,在个体空间单位水平上进行观测和预测变量的地图比较时,微小的空间错位就可能导致误差的倍增。为此,本文通过将空间模型评估指标概念化为一个连续值图比较问题,并利用图像分析方法来评定模型输出。一种尝试以局域方法仿制人类视觉系统的度量指标——结构相似指数(SSIM),被用作为探索性地图的比较统计量。在贝叶斯空间模型框架下实现其量算,并将其作为一个偏差度量应用于模型拟合的后预测校验。仿真研究的结果显示出空间及时空环境下多类空间过程。以斯里兰卡降雨过程图为案例,展示了上述方法对贝叶斯克里格插值的准确性评估。仿真研究与实证结果均证明本文提出的方法可揭示以往传统方法掩盖的空间结构特征,空间敏感性评价为本研究的空间建模和分析提供了一个诊断工具。     

An Angular Version of Spatial Correlations,with Exact Significance Tests

Spatial correlations are replaced by ordinary correlations by projecting the two-dimensional coordinates of a map onto a single axis making angle θ with the horizontal (east-west) axis. This calculation permits the use of exact test statistics. Moreover, the correlations can be plotted in polar coordinate form and the significance of the maximum correlation can be tested. The plot is a useful aid in assessing the reasonableness of the isotropy assumption.     

Evaluation of a Spatial Relationship by the Concept of Intrinsic Spatial Distance

We propose the concept of intrinsic spatial distance (ISD) for the study of a spatial relationship between any two points in space. ISD is a distance measure that takes into account the separation of two points with respect to their physical and attribute closeness. We construct an algorithm to implement this concept as an ISD measurement. Based on the ISD concept, two points in space are related through a transitional path linking one to the other. As an ISD measurement decreases, the spatial relationship between two points becomes increasingly stronger. We argue theoretically and demonstrate empirically that the ISD concept is not predisposed in favor of the first law of geography, but directly considers variance of nearness in physical distance and attribute distance to derive the extent to which two points are spatially associated. Specifically, in single attribute cases, the information uncovered by an ISD measurement is more elaborate than that revealed by Moran's I, local Moran's I, and a semivariogram, giving a meticulous account of relatedness in both local and global contexts. The ISD concept is also sufficiently general to be used to study multiple attributes of relationships. Proponemos el concepto de distancia espacial intrínseca (intrinsic spatial distance ‐ISD) para el estudio de la relación espacial entre dos puntos en el espacio. La ISD es una medición de distancia que tiene en cuenta la separación de dos puntos con respecto a su cercanía física y de atributo. Construimos un algoritmo para aplicar este concepto y crear una medición ISD. De acuerdo con en el concepto ISD, dos puntos en el espacio están relacionadas a través de un camino de transición que los vincula uno al otro. A medida que la medición ISD disminuye, la relación espacial entre dos puntos se refuerza. Argumentamos teóricamente y demostramos empíricamente que el concepto ISD no está predispuesto a favor de la primera ley de la geografía (de Tobler), pero considera directamente la varianza de la cercanía de la distancia física y la distancia del valor del atributo para derivar una medición que cuantifica el grado en el que dos puntos están asociados espacialmente. En concreto, en los casos de atributos individuales, la información revelada por una medición ISD es más sofisticada que la que proporcionada por el índice de Moran, el índice de Moran local, y el semivariograma, pues da cuanta minuciosa de la relación espacial y de atributo en contextos locales y globales. El concepto de ISD también es lo suficientemente general para ser usado en el estudio de relaciones entre múltiples atributos. 本文提出了内蕴空间距离(ISD)来研究空间中任意两点之间的空间关系。ISD是考虑空间上分离的两点各自物理和属性接近度的一种距离度量。通过构造一个算法将该概念转变为ISD可度量方式。基于ISD概念,空间上的两点通过连接两者迁移路径相关联。当ISD度量减小时,两点之间空间关系变得越来越强。理论论证和实例演示表明ISD概念并不倾向于符合地理学第一定律,但直接考虑两点间物理距离和属性距离临近性的变化可以推导两点在空间上的联系程度。特别是在单一属性案例中,相比于Moran’s I,局部Moran’s I和半方差图,ISD度量揭示了全局与局部环境中详尽的空间相关信息。同时,ISD概念在研究多属性的空间关系中也是足够通用的。     

Spatial-Temporal Spread of the AIDS Epidemic, 1982–1990: A Correlogram Analysis of Four Regions of the United States

We applied correlogram analysis to county-level AIDS data of four regions—the Northeast (Delaware, the District of Columbia, Maryland, New Jersey, New York, and Pennsylvania), California, Florida, and Louisiana—for the period 1982–1990 to characterize the spatial-temporal spread of the AIDS epidemic. Correlograms computed from yearly incidence rates differ substantially among these four regions, revealing regional differences in the spatial patterns and intensity of AIDS spread. A general trend of increasing spread to rural America, however, can still be detected. Contagious spread was predominant in the Northeast throughout the nine-year period, whereas California was dominated by hierarchical spread through time. The spatial-temporal changes of AIDS incidence patterns were most drastic in Florida, where the correlograms show hierarchical spread in the early years and then contagious spread in the later years. As a representative region for most other states in the United States, Louisiana has low spatial autocorrelation and no definite spatial pattern of spread. Grouping data into three-year periods for states with low yearly incidence rates such as Louisiana should help identify the dominant trends for these states. The correlogram results could provide useful insights into the specification of spatial models for AIDS forecasting.     

'Geographical Expeditions': Assessing the benefits of a student-driven fieldwork method

A student-driven fieldwork method, focused in a local metropolitan area, is described and assessed. Developed for a second-level course in cultural geography, it was inspired by the 'Geographical Expeditions' originally devised by William Bunge in the 1960s. It is one solution to maintaining fieldwork in the syllabus for a large class, at the same time as providing benefits for participants in terms of challenge and autonomy. Student responses are compared with the merits of fieldwork as claimed in the literature. In view of concerns expressed about increasing student 'disengagement' from commitment to university study, active participation in a 'Geographical Expedition' is a strong motivating factor.     

Nonparametric Significance Test for Discovery of Network‐Constrained Spatial Co‐Location Patterns

Spatial co‐location patterns are useful for understanding positive spatial interactions among different geographical phenomena. Existing methods for detecting spatial co‐location patterns are mostly developed based on planar space assumption; however, geographical phenomena related to human activities are strongly constrained by road networks. Although these methods can be simply modified to consider the constraints of networks by using the network distance or network partitioning scheme, user‐specified parameters or priori assumptions for determining prevalent co‐location patterns are still subjective. As a result, some co‐location patterns may be wrongly reported or omitted. Therefore, a nonparametric significance test without priori assumptions about the distributions of the spatial features is proposed in this article. Both point‐dependent and location‐dependent network‐constrained summary statistics are first utilized to model the distribution characteristics of the spatial features. Then, by using these summary statistics, a network‐constrained pattern reconstruction method is developed to construct the null model of the test, and the prevalence degree of co‐location patterns is modeled as the significance level. The significance test is evaluated using the facility points‐of‐interest data sets. Experiments and comparisons show that the significance test can effectively detect network‐constrained spatial co‐location patterns with less priori knowledge and outperforms two state‐of‐the‐art methods in excluding spurious patterns.     

Spurios Spatial Regression: Some Monte Carlo Results with a Spatial Unit Root and Spatial Cointegration

In this paper I introduce the concepts of spatial unit roots and spatial cointegration, and via Monte-Carlo simulation I illustrate their implications for spatial regression. It is shown that spatial unit roots lead to spurious (spatial) regression, as in the well-known case involving time-series. Spatial cointegration is similar to its time-series counterpart, although I demonstrate that OLS estimation of spatial error-correction models is not consistent.     

Assessing a Computer-aided Instructional Strategy in a World Geography Course

An experiment was conducted to compare student achievement under two differing instructional strategies: a small-group and computer-aided strategy versus lecture instruction. Evaluation was extended in several ways beyond comparison of overall student scores. Results show significant improvements in student achievement produced by the computer-aided strategy with reference to gender, ethnicity and levels of cognitive learning. The study concludes that experimental evaluations of CAI serve best as formative tools to match instructional strategies to specific types of content material and different types of students, and that instructional media are best evaluated along with the instructional method in which they are embedded.     

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.     

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.     

Assessing the distribution of African Palaeolithic sites: a predictive model of collagen degradation

In order to assess the distribution of African Palaeolithic sites and hominin remains, we need an understanding of the factors that may affect the preservation of archaeological material in the fossil record. One of the main factors that is thought to affect seriously the global distribution of fossil remains is taphonomy, but it would seem that taphonomic processes may themselves be overprinted by collagen degradation.The rate of collagen loss is highly temperature dependent, and it has been shown that bone is more likely to be preserved in areas of lower temperature and so lower collagen loss. Hence, it would be expected that fossil fauna remains would be distributed in areas of low collagen loss, i.e. low temperature. Conversely, lithics will not be subject to a temperature bias in their preservation. We have tested this hypothesis through the use of archaeological material.Our results show that the distributions of both lithic and faunal assemblages are non-random across the African continent, with all archaeological sites being found in areas of relatively low collagen degradation. This implies that bone collagen degradation is not the only factor affecting the distribution of African Palaeolithic sites and hominin remains. We suggest that the site distribution is not a reflection of bias in excavation history, but is a real phenomenon reflecting hominin habitat choice.     

Assessing the Empirical Impact of Environmental Federalism

Abstract Many theoretical models analyze the effects of decentralized environmental policymaking. The predictions range from a race to the top, a race to the bottom, or no effect. However, little empirical evidence exists to resolve this ambiguity. This paper fills the void by examining the impact of decentralized environmental policymaking in the U.S. under Presidents Ronald Reagan and George H.W. Bush. For abatement expenditures, Reagan's decentralization had no discernible impact before the mid‐1980s, but by the mid‐1980s the data are consistent with decentralization leading to a race to the top. No statistically significant effect is found on nitrogen oxide or sulfur dioxide emissions.