A Spatial Prior for Bayesian Vector Autoregressive Models

In this paper we develop a Bayesian prior motivated by cross-sectional spatial autoregressive models for use in time-series vector autoregressive forecasting involving spatial variables. We compare forecast accuracy of the proposed spatial prior to that from a vector autoregressive model relying on the Minnesota prior and find a significant improvement. In addition to a spatially motivated prior variance as in LeSage and Pan (1995) we develop a set of prior means based on spatial contiguity. A Theil-Goldberger estimator may be used for the proposed model making it easy to implement.     

Bayesian Estimation of Limited Dependent Variable Spatial Autoregressive Models

A Gibbs sampling (Markov chain Monte Carlo) method for estimating spatial autoregressive limited dependent variable models is presented. The method can accommodate data sets containing spatial outliers and general forms of non‐constant variance. It is argued that there are several advantages to the method proposed here relative to that proposed and illustrated in McMillen (1992) for spatial probit models.     

Applying Measures of Spatial Autocorrelation: Computation and Simulation

Knowing about the challenges and opportunities of spatial autocorrelation is one thing, but applying the measures to one's own data is another matter entirely. While manual computation of the measures for toy data sets is possible, applying them to small data sets required the use of computers and thus software. This article will shed some light on how the measures were and are implemented in software and on implementation issues that are still not fully resolved.     

速览葡萄牙 游走天涯海角的旅行

<正>当飞机缓缓降落在葡萄牙里斯本国际机场时,我按捺不住内心的激动与兴奋。尽管这已不是我第一次踏上欧罗巴的土地,但对葡萄牙的遐想与渴望,对里斯本的向往与好奇,常常萦绕于怀,总想一探究竟。终于在2018年冬季,开启了葡萄牙的浪漫之旅。葡萄牙位于欧洲西南部,有"天涯海角"和"葡萄王国"之称。它的全称叫葡萄牙共和国,面积为9.22万平方公里,到2017年底,人口有1029.1万。里斯本是葡萄牙的首都,也是我们这次在葡萄牙唯一游赏的城市。古老的建筑,历史的     

Identifying Areas with Elevated Disease Incidence Rates Using Empirical Bayes Estimators

Using maps of observed disease incidence rates to identify regions with potentially elevated risk may be misleading due to the instability of the observed rates in regions with small populations. We use a simulation study to examine the use of maps based on observed incidence rates in identifying such high-risk areas as compared to maps based on empirical Bayes and constrained empirical Bayes rate estimates. In addition, because the existence of clusters of areas with elevated risk violates the usual distributional assumptions underlying the empirical Bayes approach, we also examine the robustness of the estimates and the impact of incorrect assumptions on identification of high-risk regions. The simulation results indicate that the observed incidence rates were quite sensitive in terms of identifying areas with truly elevated rates. However, due to the instability of the observed rates, maps based on observed incidence incorrectly identified areas as high risk more frequently than did maps based on the estimators. The standard and constrained empirical Bayes estimates were more stable than the observed rates even when based on incorrect distributional assumptions. The standard empirical Bayes estimates, however, were oversmoothed in that too few areas were identified as having elevated risk. The constrained empirical Bayes approach provided sensitivity closer to that of the observed rates yet with the percentage of areas incorrectly identified as high risk equal to that of the standard empirical Bayes estimates. We illustrate an application of these results with an analysis of the geographic distribution of brain cancer mortality among counties in Ohio.     

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.     

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 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.     

"The Problem of Spatial Autocorrelation" and Local Spatial Statistics

This article examines the relationship between spatial dependency and spatial heterogeneity, two properties unique to spatial data. The property of spatial dependence has led to a large body of research into spatial autocorrelation and also, largely independently, into geostatistics. The property of spatial heterogeneity has led to a growing awareness of the limitation of global statistics and the value of local statistics and local statistical models. The article concludes with a discussion of how the two properties can be accommodated within the same modelling framework.     

Estimation of the Original Number of Individuals from Paired Bone Counts Using Estimators of the Krantz Type

Abstract

In 1968, G. S. Krantz proposed a method for estimating the original number of animals deposited at an archaeological site when bone counts have been reduced, whether by attritional processes or by a failure to find all the bones. His formula involved using the number of unmatched left, unmatched right, and matched pairs of a class of paired bone. In this paper we investigate estimators derived using Krantz's assumptions from both theoretical and practical viewpoints and make a number of recommendations; for example, that Krantz's estimators are useful only in relatively large assemblages. The method is applied to fish bones obtained from three sites in New Zealand, and practical difficulties with the procedure are described.     

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.     

Exploring Goodness-of-Fit and Spatial Correlation Using Components of Tango's Index of Spatial Clustering

The ability to detect anomalies such as spatial clustering in data sets plays an important role in spatial data analysis, leading to interest in test statistics identifying patterns exhibiting significant levels of clustering. Toward this end, Tango (1995) proposed a statistic (and its associated distribution under a null hypothesis of no clustering) assessing overall patterns of spatial clustering in a set of observed regional counts. Rogerson (1999) observed that Tango's index may be decomposed into the summation of two distinct statistics, the first mirroring standard tests of goodness-of-fit (GOF), and the second an index of spatial association (SA) similar to Moran's I . In this article, we investigate the effectiveness of Rogerson's expression of Tango's statistic in separating GOF from SA in data sets containing clusters. We simulate data under the null hypothesis of no clustering as well as two alternative hypotheses. The first alternative hypothesis induces a poor fit from the null hypothesis while maintaining independent observations and the second alternative hypothesis induces spatial dependence while maintaining fit. Using Rogerson's decomposition and leukemia incidence data from upstate New York, we show graphically that one is unable to statistically distinguish poor fit from autocorrelation.