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.     

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.     

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 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.     

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.     

Exploring Spatial Patterns Using an Expanded Spatial Autocorrelation Framework

Spatial autocorrelation (SA) is regarded as an important dimension of spatial pattern. SA measures usually consist of two components: measuring the similarity of attribute values and defining the spatial relationships among observations. The latter component is often represented by a spatial weights matrix that predefines spatial relationship between observations in most measures. Therefore, SA measures, in essence, are measures of attribute similarity, conditioned by spatial relationship. Another dimension of spatial pattern can be explored by controlling observations to be compared based upon the degree of attribute similarity. The resulting measures are spatial proximity measures of observations, meeting predefined attribute similarity criteria. Proposed measures reflect degrees of clustering or dispersion for observations meeting certain levels of attribute similarity. An existing spatial autocorrelation framework is expanded to a general framework to evaluate spatial patterns and can accommodate the proposed approach measuring proximity. Analogous to the concept of variogram, clustergram is proposed to show the levels of spatial clustering over a range of attribute similarity, or attribute lags. Specific measures based on the proposed approach are formulated and applied to a hypothetical landscape and an empirical example, showing that these new measures capture spatial pattern information not reflected by traditional spatial autocorrelation measures.     

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.     

秦汉度量衡制度的考古学研究

秦汉时期是中国度量衡发展史上的重要时期,自商鞅变法实施"平斗桶、权衡、丈尺"的政策以来,秦汉帝国在全国范围内大力推行度量衡的统一。本文依据出土和传世的度量衡实物对秦汉时期的度量衡制度作了探讨,并概述其特点,如"以度审容"、进位关系、测量基准等等,由此可以看出秦汉时期奠定了中国古代度量衡制度发展的基础。     

Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity

Several diagnostics for the assessment of model misspecification due to spatial dependence and spatial heterogeneity are developed as an application of the Lagrange Multiplier principle. The starting point is a general model which incorporates spatially lagged dependent variables, spatial residual autocorrelation and heteroskedasticity. Particular attention is given to tests for spatial residual autocorrelation in the presence of spatially lagged dependent variables and in the presence of heteroskedasticity. The tests are formally derived and illustrated in a number of simple empirical examples.