A Note on the Extremities of Local Moran's Iis and Their Impact on Global Moran's I

By defining local Moran's I_i as a ratio of quadratic forms and making use of its overall additivity to match global Moran's I, we can identify spatial objects with a strong impact on global Moran's I. First, we concentrate on the spatial properties of local Moran's I_i expressed by the local linkage degree. Depending on whether we use the W- or C-coding of the spatial connectivity matrix, the variance of local Moran's I_i for a small local linkage degree will be either large or small. Note that spatial objects associated with a local Moran's I_i with a large variance affect the global statistic much more than spatial objects associated with a local Moran's I_i with a small variance. Counterintuitively, global Moran's I defined in the W-coding is most influenced by spatial objects with a small number of spatial neighbors. In contrast, spatial objects with a large number of spatial neighbors exert more impact on global Moran's I setup in the C-coding. Second, we investigate the impact of the empirical data on local Moran's I_i and show that local Moran's I_i will only be significant for extreme absolute residuals at and around the reference location. Clusters of average regression residuals cannot be detected by local Moran's I_i. Consequently, spatial cliques of extreme residuals contribute more to significance tests on global autocorrelation.     

The Analysis of Spatial Association by Use of Distance Statistics

Introduced in this paper is a family of statistics, G, that can be used as a measure of spatial association in a number of circumstances. The basic statistic is derived, its properties are identified, and its advantages explained. Several of the G statistics make it possible to evaluate the spatial association of a variable within a specified distance of a single point. A comparison is made between a general G statistic and Moran's I for similar hypothetical and empirical conditions. The empirical work includes studies of sudden infant death syndrome by county in North Carolina and dwelling unit prices in metropolitan San Diego by zip-code districts. Results indicate that G statistics should be used in conjunction with I in order to identify characteristics of patterns not revealed by the I statistic alone and, specifically, the G_i and G_i* statistics enable us to detect local “pockets” of dependence that may not show up when using global statistics.     

A Simulation Study to Explore Inference about Global Moran's I with Random Spatial Indexes

Inference procedures for spatial autocorrelation statistics assume that the underlying configurations of spatial units are fixed. However, sometimes this assumption can be disadvantageous, for example, when analyzing social media posts or moving objects. This article examines for the case of point geometries how a change from fixed to random spatial indexes affects inferences about global Moran's I, a popular spatial autocorrelation measure. Homogeneous and inhomogeneous Matérn and Thomas cluster processes are studied and for each of these processes, 10,000 random point patterns are simulated for investigating three aspects that are key in an inferential context: the null distributions of I when the underlying geometries are varied; the effect of the latter on critical values used to reject null hypotheses; and how the presence of point processes affects the statistical power of Moran's I. The results show that point processes affect all three characteristics. Inferences about spatial structure in relevant application contexts may therefore be different from conventional inferences when this additional source of randomness is taken into account.     

Local Spatial Autocorrelation Statistics: Distributional Issues and an Application

The statistics G_i(d) and G_i*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored. In particular, nonbinary weights are allowed and the statistics are related to Moran's autocorrelation statistic, I. The correlations between nearby values of the statistics are derived and verified by simulation. A Bonferroni criterion is used to approximate significance levels when testing extreme values from the set of statistics. An example of the use of the statistics is given using spatial-temporal data on the AIDS epidemic centering on San Francisco. Results indicate that in recent years the disease is intensifying in the counties surrounding the city.     

Modified Moran's I for Small Samples

The most common indicator used to measure spatial dependence is Moran's I proposed by statistician Patrick A. P. Moran in 1950. The index is simple to use and applies the principle of the Pearson correlation coefficient, although it incorporates a proximity measure between elements. However, Moran's I tends to underestimate real spatial autocorrelation when the number of locations are few. This study aims to present a modified version of Moran's I that can measure real spatial autocorrelation even with small samples and check for spatial dependence.     

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.     

Local Spatial Autocorrelation in a Biological Model

We review the recently developed local spatial autocorrelation statistics I_i, c_i, G_i, and G_i*. We discuss two alternative randomization assumptions, total and conditional, and then newly derive expectations and variances under conditional randomization for I_i and c_i, as well as under total randomization for c_i. The four statistics are tested by a biological simulation model from population genetics in which a population lives on a 21 × 21 lattice of stepping stones (sixty-four individuals per stone) and reproduces and disperses over a number of generations. Some designs model global spatial autocorrelation, others spatially random surfaces. We find that spatially random designs give reliable test results by permutational methods of testing significance. Globally autocorrelated designs do not fit expectations by any of the three tests we employed. Asymptotic methods of testing significance failed consistently, regardless of design. Because most biological data sets are autocorrelated, significance testing for local spatial autocorrelation is problematic. However, the statistics are informative when employed in an exploratory manner. We found that hotspots (positive local autocorrelation) and coldspots (negative local autocorrelation) are successfully distinguished in spatially autocorrelated, biologically plausible data sets.     

Comparison of Moran's I and Geary's c in Multivariate Spatial Pattern Analysis

This article compares multivariate spatial analysis methods that include not only multivariate covariance, but also spatial dependence of the data explicitly and simultaneously in model design by extending two univariate autocorrelation measures, namely Moran's I and Geary's c. The results derived from the simulation datasets indicate that the standard Moran component analysis is preferable to Geary component analysis as a tool for summarizing multivariate spatial structures. However, the generalized Geary principal component analysis developed in this study by adding variance into the optimization criterion and solved as a trace ratio optimization problem performs as well as, if not better than its counterpart the Moran principal component analysis does. With respect to the sensitivity in detecting subtle spatial structures, the choice of the appropriate tool is dependent on the correlation and variance of the spatial multivariate data. Finally, the four techniques are applied to the Social Determinants of Health dataset to analyze its multivariate spatial pattern. The two generalized methods detect more urban areas and higher autocorrelation structures than the other two standard methods, and provide more obvious contrast between urban and rural areas due to the large variance of the spatial component.     

Local Indicators of Spatial Association—LISA

The capabilities for visualization, rapid data retrieval, and manipulation in geographic information systems (GIS) have created the need for new techniques of exploratory data analysis that focus on the “spatial” aspects of the data. The identification of local patterns of spatial association is an important concern in this respect. In this paper, I outline a new general class of local indicators of spatial association (LISA) and show how they allow for the decomposition of global indicators, such as Moran's I, into the contribution of each observation. The LISA statistics serve two purposes. On one hand, they may be interpreted as indicators of local pockets of nonstationarity, or hot spots, similar to the G_i and G*_i statistics of Getis and Ord (1992). On the other hand, they may be used to assess the influence of individual locations on the magnitude of the global statistic and to identify “outliers,” as in Anselin's Moran scatterplot (1993a). An initial evaluation of the properties of a LISA statistic is carried out for the local Moran, which is applied in a study of the spatial pattern of conflict for African countries and in a number of Monte Carlo simulations.     

The Size-adjusted Critical Region of Moran's I Test Statistics for Spatial Autocorrelation and Its Application to Geographical Areas

In this article, we explore the expression of the asymptotic approximation of the distribution function of Moran's I test statistic for the check of spatial autocorrelation, and we derive a more accurate critical value, which gives the rejection region similar to significant level α to the order of N^?1 (N = sample size). We show that in some cases our procedure effectively finds the significance of positive spatial autocorrelation in the problem testing for the lack of the spatial autocorrelation. Compared with our method, the testing procedure of Cliff and Ord (1971) is clearly ad hoc and should not be applied blindly, as they pointed out. Our procedure is derived from the theory of asymptotic expansion. We numerically analyze four types of county systems with rectangular lattices and three regional areas with irregular lattices.     

Extending Moran's Index for Measuring Spatiotemporal Clustering of Geographic Events

Moran's Index for spatial autocorrelation and localized index for spatial association have been widely applied in many research fields as the first step to explore and assess the spatial dependency in a set of geographic events. This article presents extensions to the equations for calculating global and localized spatial autocorrelation so to include the temporal attribute values of the geographic events being analyzed. The extended equations were successfully implemented and tested with a real world data set. In addition, simulated data sets were used to reveal how the extended equations performed. Beyond the usefulness of the extended equations, we suggest that care be taken with regard to assessing spatiotemporal patterns under the normality and randomization assumptions as different outcomes from different assumptions would require different approaches for interpretation.     

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.     

A Geospatial Analysis of China's Exports, 1991-2008

Three Taiwan-based economists employ a range of exploratory spatial data analysis tools (e.g., Moran's I and LISA statistics) to investigate trends in the growth of China's exports over the period 1991-2008. A particular focus is on the detection of spatial correlations between China and 40 export destination countries in major world regions. Emphasis in the paper on the key years of 1991, 2001, 2006, and 2008 has enabled the authors to analyze the impacts on China's trade of such major events as the country's accession to the World Trade Organization and the global economic crisis of 2008-2009. The results of the spatial analysis reveal the continuing importance of the U.S. and Asian countries in China's export trade (despite changes in the character of trade relations) and identify the spatial outliers (e.g., in Latin America) that may serve as the basis for new export markets for China in the future.     

Categorical Wombling: Detecting Regions of Significant Change in Spatially Located Categorical Variables

Wombling is a method for discovering boundaries in a collection of continuous variables observed at the same geographic localities. We extend this method to categorical data. A categorical wombling statistic C_i, which identifies areas of rapid change, is defined for every pair i = 1,…, n of adjacent localities, and is equal to the average number of category mismatches at i. We use both simulation and theory to consider the order statistics of C_i under null hypotheses of randomness, and of spatial autocorrelation for each variable, but independence between variables. Graph-theoretical statistics derived from C_i investigate whether areas of rapid change resemble boundaries. Computer simulation is used to study the distributions of these under the two null hypotheses. The methods are applied to linguistic data in three European areas. Other potential applications exist in biology, linguistics, anthropology, and other social sciences.     

Network Autocorrelation in Transport Network and Flow Systems

The use of Moran's I to assess the existence of network autocorrelation in flows over the arcs of real (tangible) and abstract (intangible) networks is examined. Residuals of a migration model developed here reveal the presence of such autocorrelation or dependence. Two approaches for removing the observed dependence are examined.     

A New Perspective about Moran's Coefficient: Spatial Autocorrelation as a Linear Regression Problem. Moran系数的新视角：空间自相关视为线性回归问题

The computation of Moran's index of spatial autocorrelation requires the definition of a spatial weighting matrix. The eigendecomposition of this doubly centered matrix (i.e., one that forces the sums of all rows and columns to equal zero) has interesting properties that have been exploited in various contexts: distribution properties of the Moran coefficient (MC), spatial filtering in linear models, generalized linear models, and multivariate analysis. In this article, this eigendecomposition is used to propose a new view of MC based on its interpretation in the simple context of linear regression. I use this interpretation to demonstrate the different properties of MC and also the inefficiency of this index in some situations involving simultaneous positive and negative spatial autocorrelation. I propose some new statistics and procedures for testing spatial autocorrelation, and conduct a simulation study to evaluate these new approaches.     

Applying Local Indicators of Spatial Association to Analyze Longitudinal Data: The Absolute Perspective

Local Indicators of Spatial Association (LISA) are a class of spatial statistical methods that have been widely applied in various scientific fields. When applying LISA to make longitudinal comparisons of spatial data, a common way is to run LISA analysis at each time point, then compare the results to infer the distributional dynamics of spatial processes. Given that LISA hinges on the global mean value that often varies across time, the LISA result generated at time T_i reflects the spatial patterns strictly with respect to T_i. Therefore, the typical comparative cross-sectional analysis with LISA can only characterize the relative distributional dynamics. However, the relative perspective alone is inadequate to comprehend the full picture, as the patterns are not directly associated with the changes of the spatial process’s intensity. We argue that it is important to obtain the absolute distribution dynamics to complement the relative perspective, especially for tracking how spatial processes evolve across time at the local level. We develop a solution that modifies the significance test when implementing LISA analysis of longitudinal data to reveal and visualize the absolute distribution dynamics. Experiments were conducted with Mongolian livestock data and Rwanda population data.     

Map Resolution and Spatial Autocorrelation

Moran's I, a measure of spatial autocorrelation, is affected by map resolution and map scale. This study uses a geographic information system (GIS) to examine the resolution effects. Empirical distribution of wildland fires in Idyllwild, California, and hypothetical distributions of ordered patterns are analyzed. The results indicate that Moran's I increases systematically with the resolution level. The resolution effects can be summarized by a log-linear function relating the I coefficients to resolution levels. Empirical tests that compare the distribution of fire activity in a vegetation map and in a topographic map confirm the resolution effects observed.     

Globalising Economic Governance,Political Projects,and Spatial Imaginaries: Insights from Four Australasian1 Cities

This paper explores how territorial economic governance is discursively constituted in a globalising and neoliberalising world. It acknowledges both the increasingly recognised formative role of spatial imaginaries in economic interventions and the workings of co‐constitutive political projects that link particular imaginaries with political ambitions and policy strategies. Embracing recent calls for comparative ethnographic urban research at the local‐global interface, it explores currently dominant spatial imaginaries across the four Australasian cities of Auckland, Sydney, Melbourne, and Perth. Based on multiple qualitative methods, this study claims that a considerable number of actor's spatial associations and reference points can be related to particular city‐specific governmental projects; Auckland's Super‐City, Sydney's Global and Green City, Melbourne's Liveable City, and Perth's Vibrant City. It is demonstrated how discursive governance techniques such as ‘story‐telling’, benchmarking, and policy transfer have been pivotal in the activation, circulation, and performance of those spatial imaginaries and their transformation into temporarily dominant visions for strategic urban interventions aimed at repositioning urban actors, spaces, and activities. While spatial imaginaries can be related to differently framed global aspirations, the effects of spatial political projects on urban governance and investment trajectories differ significantly across space. Theoretically, the paper stresses the importance of particular conceptions of territorial relations and time‐ and place‐specific institutional mediation in shaping context‐dependent discursive material governance alignments.     

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。