The Saddlepoint Approximation of Moran's I's and Local Moran's Ii's Reference Distributions and Their Numerical Evaluation

Global Moran's I and local Moran's I_i are the most commonly used test statistics for spatial autocorrelation in univariate map patterns or in regression residuals. They belong to the general class of ratios of quadratic forms for whom a whole array of approximation techniques has been proposed in the statistical literature, such as the prominent saddlepoint approximation by Offer Lieberman (1994). The saddlepoint approximation outperforms other approximation methods with respect to its accuracy and computational costs. In addition, only the saddlepoint approximation is capable of handling, in analytical terms, reference distributions of Moran's I that are subject to significant underlying spatial processes. The accuracy and computational benefits of the saddlepoint approximation are demonstrated for a set of local Moran's Ii statistics under either the assumption of global spatial independence or subject to an underlying global spatial process. Local Moran's I_i is known to have an excessive kurtosis and thus void the use of the simple approximation methods of its reference distribution. The results demonstrate how well the saddlepoint approximation fits the reference distribution of local Moran's I_i. Furthermore, for local Moran's I_i under the assumption of global spatial independence several algebraic simplifications lead to substantial gains in numerical efficiency. This makes it possible to evaluate local Moran's I_i's significance in large spatial tessellations.     

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.     

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.     

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 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.     

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.     

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.     

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.     

Properties of Tests for Spatial Dependence in Linear Regression Models

Based on a large number of Monte Carlo simulation experiments on a regular lattice, we compare the properties of Moran's I and Lagrange multiplier tests for spatial dependence, that is, for both spatial error autocorrelation and for a spatially lagged dependent variable. We consider both bias and power of the tests for six sample sizes, ranging from twenty-five to 225 observations, for different structures of the spatial weights matrix, for several underlying error distributions, for misspecified weights matrices, and for the situation where boundary effects are present. The results provide an indication of the sample sizes for which the asymptotic properties of the tests can be considered to hold. They also illustrate the power of the Lagrange multiplier tests to distinguish between substantive spatial dependence (spatial lag) and spatial dependence as a nuisance (error autocorrelation).     

A Response to Fabiano Santos and Fernando Guarnieri*

This article responds to Fabiano Santos and Fernando Guarnieri's dispatch, published under the title “From Protest to Parliamentary Coup: An Overview of Brazil's Recent History” in the December 2016 issue of JLACS. I start by pointing out several errors in the authors' use of terms such as “fascism” and “coup.” I proceed to map out the omissions that allow them to portray Rousseff's impeachment in a moral and Manichean light, as a coup of malignant social actors against innocent victims. I then reconstruct the chronology of Rousseff's long-winded downward spiral of self-inflicted agony, from the electoral larceny of October 2014 through the gigantic popular protests of March 2015 to the final vote in August 2016, showing how the authors chose to omit every fact that revealed Rousseff's and the Workers' Party's own responsibilities in the economic collapse and the political crisis of the country they governed for five and a half and thirteen and a half years, respectively. I conclude by pointing out the contradiction between the authors' claim that there was a parliamentary coup in Brazil and their claim that the country's political system has remained intact.     

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.     

The Making of the Gough Map Reconsidered: a Personal View

ABSTRACT

The claim that the Gough map was originally produced in or about the 1370s, then overwritten and re-inked about fifty years later, is unconvincing. The ‘earlier’ hand seen north of Hadrian's Wall cannot be dated accurately, but is as likely to be from after 1400 as before. Outside one small area of the map, I find almost nothing to indicate rewriting of names or other text, and even in that area the rewriting proves little about date. Nor do I find evidence for the reworking or re-inking of town signs or other pictorial elements. The claimed extensive erasure of earlier writing in the area of the map which is said to be revised is not evident to me. The serious but uneven and erratic fading of the ink on the map, which is not surprising, accounts for most of the anomalies and variations that we see. Dr Solopova's claim, based on her theory of revision, that the map displays contrasting political or ‘ideological’ emphases from two different periods seems unrelated to the actual content of the map. The known history of the development of chorography in medieval Europe indicates that the appearance of a map as advanced as the Gough map in northern Europe before 1400 is almost unimaginable.     

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.     

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.     

Doctoral Theses in Progress

Abstract

Jean Bonfa's 1696 map offered the most accurate portrait in its day of the Comtat Venaissin. Commissioned when France threatened the region economically and politically, the map's topographical detail and allegorical themes exhibit a political geography favouring papal interests. Yet the map's coordinates for Avignon reflect neither Bonfa's skill as a positional astronomer nor his collaboration with the Paris Observatory. Finally, because local authorities controlled the distribution of Bonfa's map, it was not available to other cartographers.     

Controlling the Waters of Granducal Florence: A New Look at Stefano Bonsignori's View of the City (1584)

ABSTRACT

A striking feature of Stefano Bonsignori's Nova pulcherrimae civitatis Florentiae topographia accuratissime delineata, printed in 1584, is the pre-eminence of the Arno River and the detailed depiction of a variety of often quite minor water-related structures. This large (nine-sheet) map was dedicated to Francesco I de' Medici, Granduke of Florence. Contemporary initiatives and legislation as well as works of art and literature reveal that water management had been an important aspect of the policies of Francesco's father, Cosimo I, whose achievements had transformed the city's landscape and whose efforts earned the Medici ruler the title of Granduke of Tuscany in 1569. Bonsignori's portrayal of urban structures was created as a celebration of Cosimo's architectural legacy, and the depiction of the Arno, with its embankments and riverside structures, along with some of the city's fountains and wells, acknowledged granducal ambition to control its waters.     

The World Map as an Emblem: Abraham Ortelius and the Stoic Contemplation

Although the close association of word and image in medieval cartography is widely acknowledged, the significance of the relationship after the rediscovery of Ptolemy's Geography and throughout the Renaissance has been overlooked, despite Abraham Ortelius's choice of the term ‘Reader’ for users of the Theatrum orbis terrarum (1570). In this paper, the map of the world, which (as in Ptolemy's Geography) opens Ortelius's Theatrum, is analysed to show how Ortelius's concept of space was very different from Ptolemy's. Attention is drawn to the content of the texts on the map, to Ortelius's notion of geography as the eye of history, and to the importance in the Renaissance of the emblem as a conceit, or device, in the system of acquisition and transmission of knowledge. As in emblems, the words on Ortelius's map are not there to explain or to comment on what is seen but to give the image meaning; the purpose of the map is to invite contemplation of God's world. The map is contradictory, however; for Ortelius's accurate and up‐to‐date presentation of the physical world is qualified by a verbal statement that the world is ‘nothing’, a mere pinpoint in the immensity of the universe. It is concluded that Ortelius was not a geographer in the same way Ptolemy was, and that Ortelius was using geography as a philosopher and his world map as an illustration of his moral and religious thinking.     

Caspar Vopel's Ventures in Sixteenth-Century Celestial Cartography

ABSTRACT

This paper concerns the undertakings in celestial cartography of the sixteenth-century Cologne cartographer Caspar Vopel. Copies of his printed celestial globe and of the celestial maps included on his world map are also described. Vopel's celestial mappings display his extraordinary interest in astronomical myths through a series of conspicuous iconographic features. In particular, Vopel's introduction of the images of Antinous and Coma Berenices is revealed to have been inspired by a humanist edition of the Ptolemaic star catalogue. Finally, a study of the celestial maps on the copies of Vopel's world map by Valvassore (1558) and by Van den Putte (1570) shows that these represent different editions of Vopel's world map and that the celestial maps on the world map of Matteo Pagano were in turn copied from those on the world map of Valvassore.     

Feeding fears: competing discourses of interdependency,sovereignty, and China's food security

Following the release of the 1994 report ‘Who will feed China?’ by the Worldwatch Institute, there has been much debate over the implications of China's growing demand for grain. The question of China's food production has elicited a variety of responses. While for some it raises the specter of regional and global instability as China becomes an environmental threat, for others the entrance of China into the world market promises increased trade and profits. In this paper I explore the responses in China and the US to the different notions of interdependence which have shaped the debate. I first turn to how concerns over China's food supply have, despite appeals to the concepts of global environmental and economic interdependence, become linked to classical state-centered geopolitical concerns such as ‘sovereignty’ and ‘containment.’ I then look at how the debate has also been actively distanced from national security concerns through the invocation of an alternative interdependence founded on the logic of commerce. I conclude by arguing for the need within critical geopolitics to further examine the circulation of strategic texts between and within states, particularly in the analysis of texts that map worlds beyond the boundaries of North America and Europe.