Exploring Spatial Process Dynamics Using Irregular Cellular Automaton Models

Cellular automaton models have enjoyed popularity in recent years as easily constructed models of many complex spatial processes, particularly in the natural sciences, and more recently in geography also. Most such models adopt a regular lattice (often a grid) as the basis for the spatial relations of adjacency that govern evolution of the model. A number of variations on the cellular automaton formalism have been introduced in geography but the impact of such variations on the likely behavior of the models has not been explored. This paper proposes a method for beginning to explore these issues and suggests that this is a new approach to the investigation of the relationships between spatial structure and dynamics of spatial processes. A framework for this exploration is suggested, and details of the required methods and measures are provided. In particular, a measure of spatial pattern—spatial information—based on entropy concepts is introduced. Initial results from investigation along the proposed lines are reported, which suggest that a distinction can he made between spatially robust and fragile processes. Some implications of this result and the methodology presented are briefly discussed.     

Destination Choice of Athenians: An Application of Geographically Weighted Versions of Standard and Zero Inflated Poisson Spatial Interaction Models

The main aim of this article is to combine recent developments in spatial interaction modeling to better model and explain spatial decisions. The empirical study refers to migration decisions made by internal migrants from Athens, Greece. To achieve this, geographically weighted versions of standard and zero inflated Poisson (ZIP) spatial interaction models are defined and fit. In the absence of empirical studies for the effect of potential determinants on internal migration decisions in Greece and the presence of an excessive number of zero migration flows among municipalities in Greece, this article provides empirical evidence for the power of the proposed Geographically Weighted ZIP regression method to better explain destination choices of Athenian internal migrants. We also discuss statistical inference issues in relation to the application of the proposed regression techniques.     

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.     

INTERPRETING SPATIAL ECONOMETRIC ORIGIN‐DESTINATION FLOW MODELS

Spatial interaction or gravity models have been used to model flows that take many forms, for example population migration, commodity flows, traffic flows, all of which reflect movements between origin and destination regions. We focus on how to interpret estimates from spatial autoregressive extensions to the conventional regression‐based gravity models that relax the assumption of independence between flows. These models proposed by LeSage and Pace ( 2008 , 2009 ) define spatial dependence involving flows between regions. We show how to calculate partial derivative expressions for these models that can be used to quantify these various types of effect that arise from changes in the characteristics/explanatory variables of the model.     

Analyzing Cross-classified Data with Inherent Spatial Dependence

Despite considerable recent progress in the methods available for the log-linear analysis of categorical data arising from complex sampling schemes, only a few papers have been published that deal with the parallel phenomenon of dependence induced by spatial sampling. This paper aims to add to the general awareness of this topic and suggests some new ideas for tackling the problems raised. In the paper it is shown that the method that has been proposed for the valid selection of log-linear models given spatially dependent data and some derivative methods are somewhat conservative when compared to an approach based on a model of spatial dependence outlined in section 4. The method also serves as a data exploratory technique to enhance the use of the more robust conservative approach.     

Geographically Weighted Extreme Learning Machine: A Method for Space–Time Prediction

Spatial heterogeneity has been regarded as an important issue in space–time prediction. Although some statistical methods of space–time predictions have been proposed to address spatial heterogeneity, the linear assumption makes it difficult for these methods to predict geographical processes accurately because geographical processes always involve complicated nonlinear characteristics. An extreme learning machine (ELM) has the advantage of approximating nonlinear relationships with a rapid learning speed and excellent generalization performance. However, determining how to incorporate spatial heterogeneity into an ELM to predict space–time data is an urgent problem. For this purpose, a new method called geographically weighted ELM (GWELM) is proposed to address spatial heterogeneity based on an ELM in this article. GWELM is essentially a locally varying ELM in which the parameters are regarded as functions of spatial locations, and geographically weighted least squares is applied to estimate the parameters in a local model. The proposed method is used to analyze two groups of different data sets, and the results demonstrate that the GWELM method is superior to the comparative method, which is also developed to address spatial heterogeneity.     

Principal Component Regression in Spatial Lag Model: Teen Employment in the City of Rosario,Argentina

Teen employment is a very important socioeconomic phenomenon because of its consequences on human capital formation. We assess the relation between teen employment and poverty, education, and unemployment in the city of Rosario, using information from the 2010 Argentina Census disaggregated at census block level. We use two different spatial models: The spatial lag model (SLM) and a linear regression model with the spatial component filtered (filtering model, FM). Given the nature of the variables employed, multicollinearity is an issue. One of the techniques proposed in the literature to deal with multicollinearity problems is principal component regression (PCR). We develop an adaptation of such methodology to be used in the SLM. Both models are estimated using their traditional methodologies (instrumental variables for the SLM and OLS for the FM) and using PCR. Although results are similar between the two models, depending on the methodology used in the estimations they differ greatly. Under traditional methodologies estimations show high variability, instability, and contradictory outcomes, but under PCR, results behave according to the literature.     

SPATIAL COMPETITION WITH CONSISTENT CONJECTURES*

ABSTRACT Equilibrium in spatial models invariably depends on firms' conjectures about how competitors will react to their price changes. This paper analyzes spatial price and location equilibrium when firms hold consistent (i.e. correct) conjectures. Most spatial models assume an exogenous conjecture. Consistent conjectures are one method, albeit a controversial one, for endogenizing the conjecture. We show that the consistent conjecture about a competitor's reaction to a price change in the simplest case is 1/3. When demand is elastic the consistent conjecture is a decreasing function of the radius. It is always below 1/3 and can be negative. In the third model, we show that the consistent conjecture declines as the number of dimensions and the number of competitors increases.     

A Statistical Method for Analyzing the Spatial Relationship between the Distribution of Activity Points and the Distribution of Activity Continuously Distributed over a Region

This paper shows a statistical method for analyzing the spatial relationship between the distributions of two different kinds of activity in a region. One kind of activity is discretely distributed as points in a region (such as the distribution of retail stores), and the other kind of activity is continuously distributed over the region (such as the distribution of population). First, three models representing the relationship between the above two distributions are formulated. Second, statistical methods for fitting these models to data are developed and the measures of fitness are proposed. Third, using these measures, the relationship between the distributions of thirty-seven kinds of retail stores and the distribution of population is examined in a suburb of Osaka in Japan.     

Policy Coalitions in the U.S. Congress: A Spatial Duration Modeling Approach

This paper applies spatial duration models to the analysis of cosponsorship coalitions in the U.S. House of Representatives. This approach provides a unique and simultaneous statistical analysis of ideological space (specifically, coalition formation) and geographical space. Typically, duration models are associated with temporal longitudinal data, but recently have been adapted to the spatial domain (Pellegrini and Reader 1996). In this paper, spatial duration models are further adapted to examine ideological space including a consideration of unobserved sources of spatial variation (or omitted variable bias). We examine two features of cosponsorship coalitions, breadth and clustering. Breadth is defined as the ideological distance between the two most extreme members of the coalition which is an important “signal” to the rest of Congress regarding the scope and broad appeal of the proposed legislation. In contrast, clustering refers to the distance between individual members of a coalition and reveals the tendency, or not, of ideologically similar members of Congress to support various bills. To examine breadth and clustering, we employ spatial duration models of cosponsorship that permit a multivariate analysis incorporating both the characteristics of members of Congress and the geographical regions they represent. Results indicate that cosponsorship coalition patterns are primarily determined by the content of the legislation, not the actions of the coalition leadership. While the leadership characteristics of sponsors have a limited effect on cosponsorship breadth, the size of the coalition is the primary determinent. Leadership characteristics also have little effect on cosponsorship clustering. Rather, clustering is due to members' policy preferences, as measured by distance to the coalition leader. In addition, the duration analysis results suggest that geographical proximity between members of Congress “overcomes” ideological distance. Finally, the spatial duration approach is noted as a fruitful methodology for examining explicitly spatial patterns in both ideological or geographical space.     

Geographical and Temporal Weighted Regression (GTWR)

Both space and time are fundamental in human activities as well as in various physical processes. Spatiotemporal analysis and modeling has long been a major concern of geographical information science (GIScience), environmental science, hydrology, epidemiology, and other research areas. Although the importance of incorporating the temporal dimension into spatial analysis and modeling has been well recognized, challenges still exist given the complexity of spatiotemporal models. Of particular interest in this article is the spatiotemporal modeling of local nonstationary processes. Specifically, an extension of geographically weighted regression (GWR), geographical and temporal weighted regression (GTWR), is developed in order to account for local effects in both space and time. An efficient model calibration approach is proposed for this statistical technique. Using a 19‐year set of house price data in London from 1980 to 1998, empirical results from the application of GTWR to hedonic house price modeling demonstrate the effectiveness of the proposed method and its superiority to the traditional GWR approach, highlighting the importance of temporally explicit spatial modeling.     

THE SLX MODEL

We provide a comprehensive overview of the strengths and weaknesses of different spatial econometric model specifications in terms of spillover effects. Based on this overview, we advocate taking the SLX model as point of departure in case a well‐founded theory indicating which model is most appropriate is lacking. In contrast to other spatial econometric models, the SLX model also allows for the spatial weights matrix W to be parameterized and the application of standard econometric techniques to test for endogenous explanatory variables. This starkly contrasts commonly used spatial econometric specification strategies and is a complement to the critique of spatial econometrics raised in a special theme issue of the Journal of Regional Science (Volume 52, Issue 2). To illustrate the pitfalls of the standard spatial econometrics approach and the benefits of our proposed alternative approach in an empirical setting, the Baltagi and Li (2004) cigarette demand model is estimated.     

PROBIT WITH SPATIAL AUTOCORRELATION

ABSTRACT. Commonly-employed spatial autocorrelation models imply heteroskedastic errors, but heteroskedasticity causes probit to be inconsistent. This paper proposes and illustrates the use of two categories of estimators for probit models with spatial autocorrelation. One category is based on the EM algorithm, and requires repeated application of a maximum-likelihood estimator. The other category, which can be applied to models derived using the spatial expansion method, only requires weighted least squares.     

Designs for Detecting Spatial Dependence

The aim of this article is to find optimal or nearly optimal designs for experiments to detect spatial dependence that might be in the data. The questions to be answered are: how to optimally select predictor values to detect the spatial structure (if it is existent) and how to avoid to spuriously detect spatial dependence if there is no such structure. The starting point of this analysis involves two different linear regression models: (1) an ordinary linear regression model with i.i.d. error terms—the nonspatial case and (2) a regression model with a spatially autocorrelated error term, a so-called simultaneous spatial autoregressive error model. The procedure can be divided into two main parts: The first is use of an exchange algorithm to find the optimal design for the respective data collection process; for its evaluation an artificial data set was generated and used. The second is estimation of the parameters of the regression model and calculation of Moran's I , which is used as an indicator for spatial dependence in the data set. The method is illustrated by applying it to a well-known case study in spatial analysis.     

TESTING THE SPATIAL SCALE AND THE DYNAMIC STRUCTURE IN REGIONAL MODELS (A CONTRIBUTION TO SPATIAL ECONOMETRIC SPECIFICATION ANALYSIS)*

This article addresses the problem of specification uncertainty in modeling spatial economic theories in stochastic form. It is ascertained that the traditional approach to spatial econometric modeling does not adequately deal with the type and extent of specification uncertainty commonly encountered in spatial economic analyses. Two alternative spatial econometric modeling procedures proposed in the literature are reviewed and shown to be suitable for analyzing systematically two sources of specification uncertainty, viz., the level of aggregation and the spatio-temporal dynamic structure in multiregional econometric models. The usefulness of one of these specification procedures is illustrated by the construction of a simple multiregional model for The Netherlands.     

Model Uncertainty in Matrix Exponential Spatial Growth Regression Models

This article considers the most important aspects of model uncertainty for spatial regression models, namely, the appropriate spatial weight matrix to be employed and the appropriate explanatory variables. We focus on the spatial Durbin model (SDM) specification in this study that nests most models used in the regional growth literature, and develop a simple Bayesian model‐averaging approach that provides a unified and formal treatment of these aspects of model uncertainty for SDM growth models. The approach expands on previous work by reducing the computational costs through the use of Bayesian information criterion model weights and a matrix exponential specification of the SDM model. The spatial Durbin matrix exponential model has theoretical and computational advantages over the spatial autoregressive specification due to the ease of inversion, differentiation, and integration of the matrix exponential. In particular, the matrix exponential has a simple matrix determinant that vanishes for the case of a spatial weight matrix with a trace of zero. This allows for a larger domain of spatial growth regression models to be analyzed with this approach, including models based on different classes of spatial weight matrices. The working of the approach is illustrated for the case of 32 potential determinants and three classes of spatial weight matrices (contiguity‐based, k‐nearest neighbor, and distance‐based spatial weight matrices), using a data set of income per capita growth for 273 European regions.     

The Application of SfM Photogrammetry Software for Extracting Artifact Provenience from Palaeolithic Excavation Surfaces

While photogrammetry has become popular in archaeology and heritage management as an effective, low-cost method for generating detailed three-dimensional models, it remains to be established that the accuracy of model-derived measurements is sufficient for analytical purposes. Based on an expedient, in-field model processing protocol, we report preliminary results concerning the accuracy of artifact provenience information derived from photogrammetry models of excavation surfaces at the Upper Palaeolithic site of Shuidonggou Locality 2 in China. Error in model-derived provenience can range easily into the centimeter scale; accuracy in some spatial axes are significantly, but weakly, affected by the size of the sampled surface. While the observed error range is larger than thresholds proposed for Palaeolithic excavations, it is arguably acceptable in settings where the analytical demand for provenience precision is lower. We identify possible sources of error and discuss how model accuracy can be improved by additional systematic testing.     

Quantifying Interpolation Errors in Urban Airborne Laser Scanning Models

Airborne laser scanning (ALS) is becoming an increasingly popular data capture technique for a variety of applications in urban surface modeling. Raw ALS data are captured and supplied as a 3D point cloud. Many applications require that these data are interpolated onto a regular grid in order that they may be processed. In this article, we identify and analyze the magnitudes and spatial patterning of residuals from ALS models of urban surfaces, at a range of different scales. Previous research has demonstrated the effects of interpolation method and scale upon the nature of error in digital surface models (DSMs), but the size and spatial patterning of such errors have not hitherto been investigated for urban surfaces. The contribution of this analysis is thus to investigate the ways in which different methods may introduce error, and to understand the uncertainty that characterizes urban surface models that are devised for a wide range of applications. The importance of the research is shown using examples of how the different methods may introduce different amounts of error and how the uncertainty information may benefit users of ALS height models. Our analysis uses a range of validation techniques, including split-sample, cross-validation, and jackknifing, to estimate the error created in DSMs of urban areas.     

Exploring scale-dependent correlations between cancer mortality rates using factorial kriging and population-weighted semivariograms

This article presents a geostatistical methodology that accounts for spatially varying population size in the processing of cancer mortality data. The approach proceeds in two steps: (1) spatial patterns are first described and modeled using population-weighted semivariogram estimators, (2) spatial components corresponding to nested structures identified on semivariograms are then estimated and mapped using a variant of factorial kriging. The main benefit over traditional spatial smoothers is that the pattern of spatial variability (i.e., direction-dependent variability, range of correlation, presence of nested scales of variability) is directly incorporated into the computation of weights assigned to surrounding observations. Moreover, besides filtering the noise in the data, the procedure allows the decomposition of the structured component into several spatial components (i.e., local versus regional variability) on the basis of semivariogram models. A simulation study demonstrates that maps of spatial components are closer to the underlying risk maps in terms of prediction errors and provide a better visualization of regional patterns than the original maps of mortality rates or the maps smoothed using weighted linear averages. The proposed approach also attenuates the underestimation of the magnitude of the correlation between various cancer rates resulting from noise attached to the data. This methodology has great potential to explore scale-dependent correlation between risks of developing cancers and to detect clusters at various spatial scales, which should lead to a more accurate representation of geographic variation in cancer risk, and ultimately to a better understanding of causative relationships.     

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.