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.     

Determinants of Variation in Population–Employment Interaction Findings: A Quasi‐Experimental Meta‐Analysis. 人口‐就业相互作用的决定因素研究：一种准实验元分析法

This article reports about a metaregression analysis of empirical results generated using data for the northern Netherlands (1988–2002) in order to investigate the ambiguity in results in the population–employment interaction literature. Specifically, the analysis deals with the issue whether “jobs follow people” or “people follow jobs.” The article starts with introducing the basics of quasi‐experimental meta‐analysis and with identifying some advantages of using quasi‐experimental meta‐analysis as compared with the standard meta‐analysis approach. Two subsequent sections document the selection of the population–employment interaction model and salient characteristics of the data set as well as the setup of the primary analyses. A total of 4,050 quasi‐experimental empirical results for the jobs–people direction of causality are generated using different specifications and estimators for a spatial econometric interaction model. The subsequent metaregression analysis reveals that the empirical results are largely shaped by the spatial, temporal, and employment characteristics of the data sampling. The results also appear much more sensitive to different measurements of the model's key variables when compared with alternative specifications of the spatial weights matrix. The main determinant driving empirical results about jobs–people causality are differences in model specification and estimation, as revealed by an inherent bias in parameter estimates and misguided inferences for some of the commonly used specifications. Finally, suggestions for future research are identified.     

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.     

Spatial Cluster Detection in Spatial Flow Data

As a typical form of geographical phenomena, spatial flow events have been widely studied in contexts like migration, daily commuting, and information exchange through telecommunication. Studying the spatial pattern of flow data serves to reveal essential information about the underlying process generating the phenomena. Most methods of global clustering pattern detection and local clusters detection analysis are focused on single‐location spatial events or fail to preserve the integrity of spatial flow events. In this research we introduce a new spatial statistical approach of detecting clustering (clusters) of flow data that extends the classical local K‐function, while maintaining the integrity of flow data. Through the appropriate measurement of spatial proximity relationships between entire flows, the new method successfully upgrades the classical hot spot detection method to the stage of “hot flow” detection. Several specific aspects of the method are discussed to provide evidence of its robustness and expandability, such as the multiscale issue and relative importance control, using a real data set of vehicle theft and recovery location pairs in Charlotte, NC.     

Maximum Getis–Ord Statistic Adjusted for Spatially Autocorrelated Data

Local statistics test the null hypothesis of no spatial association or clustering around the vicinity of a location. To carry out statistical tests, it is assumed that the observations are independent and that they exhibit no global spatial autocorrelation. In this article, approaches to account for global spatial autocorrelation are described and illustrated for the case of the Getis–Ord statistic with binary weights. Although the majority of current applications of local statistics assume that the spatial scale of the local spatial association (as specified via weights) is known, it is more often the case that it is unknown. The approaches described here cover the cases of testing local statistics for the cases of both known and unknown weights, and they are based upon methods that have been used with aspatial data, where the objective is to find changepoints in temporal data. After a review of the Getis–Ord statistic, the article provides a review of its extension to the case where the objective is to choose the best set of binary weights to estimate the spatial scale of the local association and assess statistical significance. Modified approaches that account for spatially autocorrelated data are then introduced and discussed. Finally, the method is illustrated using data on leukemia in central New York, and some concluding comments are made.     

Application of LASSO to the Eigenvector Selection Problem in Eigenvector‐based Spatial Filtering

Eigenvector‐based spatial filtering is one of the often used approaches to model spatial autocorrelation among the observations or errors in a regression model. In this approach, a subset of eigenvectors extracted from a modified spatial weight matrix is added to the model as explanatory variables. The subset is typically specified via the selection procedure of the forward stepwise model, but it is disappointingly slow when the observations n take a large number. Hence, as a complement or alternative, the present article proposes the use of the least absolute shrinkage and selection operator (LASSO) to select the eigenvectors. The LASSO model selection procedure was applied to the well‐known Boston housing data set and simulation data set, and its performance was compared with the stepwise procedure. The obtained results suggest that the LASSO procedure is fairly fast compared with the stepwise procedure, and can select eigenvectors effectively even if the data set is relatively large (n = 10⁴), to which the forward stepwise procedure is not easy to apply.     

MODELING SPATIAL AUTOCORRELATION IN SPATIAL INTERACTION DATA: AN APPLICATION TO PATENT CITATION DATA IN THE EUROPEAN UNION*

ABSTRACT Spatial interaction models of the gravity type are widely used to model origin–destination flows. They draw attention to three types of variables to explain variation in spatial interactions across geographic space: variables that characterize an origin region of a flow, variables that characterize a destination region of a flow, and finally variables that measure the separation between origin and destination regions. This paper outlines and compares two approaches, the spatial econometric and the eigenfunction‐based spatial filtering approach, to deal with the issue of spatial autocorrelation among flow residuals. An example using patent citation data that capture knowledge flows across 112 European regions serves to illustrate the application and the comparison of the two approaches.     

A Local Model for Multivariate Analysis: Extending Wartenberg’s Multivariate Spatial Correlation

Local analysis can provide specific information about individual observations that is often useful in understanding nonstationary interactions among variables. This paper extends the application of Wartenberg’s Multivariate Spatial Correlation (MSC) method to a local setting. The original MSC can be considered as an adaptation of Principal Component Analysis for spatial effects with respect to spatial autocorrelation. The extended MSC method described in this paper, however, further incorporates another spatial effect, spatial heterogeneity, by the addition of geographic weights in standardizing the data and in calculating the spatial association weight matrix. The extension allows more local analysis and facilitates additional visualization of the results. The geographically weighted MSC is illustrated and justified using the classic dataset collected by André-Michel Guerry on moral statistics in 1830s France.     

Does Econometric Methodology Matter? An Analysis of Public Policy Using Spatial Econometric Techniques

A popular approach to examining the effects of public policy has been to rely on a spatial data sample of border counties as in Holmes (1998)—border counties from a sample of states that are used in conjunction with least‐squares estimation techniques in an attempt to isolate the policy impact while controlling for spatial dependence that often arises from latent or unobserved variables. This technique is in the spirit of control‐group methodologies from the laboratory sciences. This paper contrasts border‐county estimation results from Holmes' (1998) approach and those from a related methodology set forth in Holcombe and Lacombe (2003), with estimates from a spatial autoregressive model explicitly accounting for within‐state and between‐state public policy effects. As an illustration, the paper examines the effects of Aid to Families with Dependent Children (AFDC) and Food Stamp payments on female‐headed households and female labor force participation using the three different methods.     

A Geostatistical Linear Regression Model for Small Area Data. 一种适用于小区域数据的地统计线性回归模型

We present a new linear regression model for use with aggregated, small area data that are spatially autocorrelated. Because these data are aggregates of individual‐level data, we choose to model the spatial autocorrelation using a geostatistical model specified at the scale of the individual. The autocovariance of observed small area data is determined via the natural aggregation over the population. Unlike lattice‐based autoregressive approaches, the geostatistical approach is invariant to the scale of data aggregation. We establish that this geostatistical approach also is a valid autoregressive model; thus, we call this approach the geostatistical autoregressive (GAR) model. An asymptotically consistent and efficient maximum likelihood estimator is derived for the GAR model. Finite sample evidence from simulation experiments demonstrates the relative efficiency properties of the GAR model. Furthermore, while aggregation results in less efficient estimates than disaggregated data, the GAR model provides the most efficient estimates from the data that are available. These results suggest that the GAR model should be considered as part of a spatial analyst's toolbox when aggregated, small area data are analyzed. More important, we believe that the GAR model's attention to the individual‐level scale allows for a more flexible and theory‐informed specification than the existing autoregressive approaches based on an area‐level spatial weights matrix. Because many spatial process models, both in geography and in other disciplines, are specified at the individual level, we hope that the GAR covariance specification will provide a vehicle for a better informed and more interdisciplinary use of spatial regression models with area‐aggregated data.     

Measuring Impacts of Condition Variables in Rule‐Based Models of Space‐Time Choice Behavior: Method and Empirical Illustration

The use of rule‐based systems for modeling space‐time choice has gained increasing research interests over the last years. The potential advantage of the rule‐based approach is that it can handle interactions between a large set of predictors. Decision tree induction methods are available and have been explored for deriving rules from data. However, the complexity of the structures that are generated by such knowledge discovery methods hampers an interpretation of the rule‐set in behavioral terms with as a consequence that the models typically remain a black box. To solve this problem, this paper develops a method for measuring the size and direction of the impact of condition variables on the choice variable as predicted by the model. The paper illustrates the method based on location and transport‐mode choice models that are part of Albatross model—an activity‐based model of space‐time choice.     

Accessibility Measurement in Transportation Networks and Application to the Seoul Bus System

This study presents an algorithm for measuring the time‐distance accessibility based on the flow data of actual transportation networks and examines resulting accessibility distributions. Specifically, we construct an extended graph of the transportation network to take into account not only the average speed and transfer time extracted from the data set but also various possible trajectories taken via transfers. Employing the modified Floyd algorithm which finds the shortest time distance, we compute the time‐distance accessibility of every bus stop in the Seoul bus system, which yields distinctive skew distributions. We then introduce a time‐distance cutoff to focus on effective connections and probe the emergent spatial distributions as the cutoff is varied. Revealed are the characteristic scales as well as spatial structures of the system. It is suggested that the time‐distance accessibility can serve as a significant measure to describe and predict the urban land use pattern.     

Gibrat Revisited: An Urban Growth Model Incorporating Spatial Interaction and Innovation Cycles. 再评吉尔布瑞特：顾及空间相互作用和创新周期的城市增长模型

We designed a geographical model for simulating the distribution of urban growth in systems of cities. The model incorporates the hierarchical and spatial diffusion of innovation cycles through gravitational interactions within a set of cities. Using theoretical simulations, we demonstrate that this model is able to reproduce the observed properties of urban systems for the log‐normal distribution of city sizes as well as the observed distribution of growth rates. Our experimentation was performed on a large harmonized historical database that includes a few hundred French urban agglomerations between 1831 and 1999 (Pumain‐INED database). Both spatial interaction and innovation cycles are necessary ingredients to explain the evolution of urban hierarchies. We suggest that Gibrat's generic stochastic growth model based on independent entities should be replaced by a more relevant model of spatially and temporally interdependent geographical entities.     

Spatial Autoregressive Models for Geographically Hierarchical Data Structures

This article discusses how standard spatial autoregressive models and their estimation can be extended to accommodate geographically hierarchical data structures. Whereas standard spatial econometric models normally operate at a single geographical scale, many geographical data sets are hierarchical in nature—for example, information about houses nested into data about the census tracts in which those houses are found. Here we outline four model specifications by combining different formulations of the spatial weight matrix W and of ways of modeling regional effects. These are (1) groupwise W and fixed regional effects; (2) groupwise W and random regional effects; (3) proximity‐based W and fixed regional effects; and (4) proximity‐based W and random regional effects. We discuss each of these model specifications and their associated estimation methods, giving particular attention to the fourth. We describe this as a hierarchical spatial autoregressive model. We view it as having the most potential to extend spatial econometrics to accommodate geographically hierarchical data structures and as offering the greatest coming together of spatial econometric and multilevel modeling approaches. Subsequently, we provide Bayesian Markov Chain Monte Carlo algorithms for implementing the model. We demonstrate its application using a two‐level land price data set where land parcels nest into districts in Beijing, China, finding significant spatial dependence at both the land parcel level and the district level.     

Daily urban systems explained from the household perspective: an exploratory research on how the spatial interactions of the individual households shape the structure of daily urban systems

The concept of the Daily Urban System (DUS) has gained relevance over the past decades as the entity to examine and explain the functionality of the urban landscape. Daily Urban Systems are usually defined and measured by the strength of commuter or shopper flows between the nodes of the system. It is important to realize that these Daily Urban Systems are the accumulated pattern of individuals making frequent, recurring trips to other localities than their own. Understanding the microeconomic decisions behind these spatial interactions will help in assessing the functional and spatial structure of DUS. In this paper is explored how, based on Dutch empirical data, the individual household’s spatial interactions shape the daily urban system and how the destination of these interactions correlates with personal and spatial variables and motives for interaction. The results show that the occurrence of non-local spatial interactions can be explained by the size-based Christallerian hierarchy of the localities of residence, but that it is the regional population – or market potential – that explains and moderates the sorting of households and the intensity and direction of their spatial interactions in the DUS, matching agglomeration theory.     

p‐Functional Clusters Location Problem for Detecting Spatial Clusters with Covering Approach

Regionalization or districting problems commonly require each individual spatial unit to participate exclusively in a single region or district. Although this assumption is appropriate for some regionalization problems, it is less realistic for delineating functional clusters, such as metropolitan areas and trade areas where a region does not necessarily have exclusive coverage with other regions. This paper develops a spatial optimization model for detecting functional spatial clusters, named the p‐functional clusters location problem (p‐FCLP), which has been developed based on the Covering Location Problem. By relaxing the complete and exhaustive assignment requirement, a functional cluster is delineated with the selective spatial units that have substantial spatial interaction. This model is demonstrated with applications for a functional regionalization problem using three journey‐to‐work flow datasets: (1) among the 46 counties in South Carolina, (2) the counties in the East North Central division of the US Census, and (3) all counties in the US. The computational efficiency of p‐FCLP is compared with other regionalization problems. The computational results show that detecting functional spatial clusters with contiguity constraints effectively solves problems with optimality in a mixed integer programming (MIP) approach, suggesting the ability to solve large instance applications of regionalization problems.     

Nonparametric Significance Test for Discovery of Network‐Constrained Spatial Co‐Location Patterns

Spatial co‐location patterns are useful for understanding positive spatial interactions among different geographical phenomena. Existing methods for detecting spatial co‐location patterns are mostly developed based on planar space assumption; however, geographical phenomena related to human activities are strongly constrained by road networks. Although these methods can be simply modified to consider the constraints of networks by using the network distance or network partitioning scheme, user‐specified parameters or priori assumptions for determining prevalent co‐location patterns are still subjective. As a result, some co‐location patterns may be wrongly reported or omitted. Therefore, a nonparametric significance test without priori assumptions about the distributions of the spatial features is proposed in this article. Both point‐dependent and location‐dependent network‐constrained summary statistics are first utilized to model the distribution characteristics of the spatial features. Then, by using these summary statistics, a network‐constrained pattern reconstruction method is developed to construct the null model of the test, and the prevalence degree of co‐location patterns is modeled as the significance level. The significance test is evaluated using the facility points‐of‐interest data sets. Experiments and comparisons show that the significance test can effectively detect network‐constrained spatial co‐location patterns with less priori knowledge and outperforms two state‐of‐the‐art methods in excluding spurious patterns.     

A Spatial Mixture Model of Innovation Diffusion

The diffusion of new product or technical innovation over space is here modeled as an event‐based process in which the likelihood of the next adopter being in region r is influenced by two factors: (i) the potential interactions of individuals in r with current adopters in neighboring regions, and (ii) all other attributes of individuals in r that may influence their adoption propensity. The first factor is characterized by a logit model reflecting the likelihood of adoption due to spatial contacts with previous adopters, and the second by a logit model reflecting the likelihood of adoption due to other intrinsic effects. The resulting spatial diffusion process is then assumed to be driven by a probabilistic mixture of the two. A number of formal properties of this model are analyzed, including its asymptotic behavior. But the main analytical focus is on statistical estimation of parameters. Here it is shown that standard maximum‐likelihood estimates require large sample sizes to achieve reasonable results. Two estimation approaches are developed which yield more sensible results for small sample sizes. These results are applied to a small data set involving the adoption of a new Internet grocery‐shopping service by consumers in the Philadelphia metropolitan area.     

Adaptive Cell Tower Location Using Geostatistics. 基于地统计学的适应性基站发射塔选址研究

In this article, we address the problem of allocating an additional cell tower (or a set of towers) to an existing cellular network, maximizing the call completion probability. Our approach is derived from the adaptive spatial sampling problem using kriging, capitalizing on spatial correlation between cell phone signal strength data points and accounting for terrain morphology. Cell phone demand is reflected by population counts in the form of weights. The objective function, which is the weighted call completion probability, is highly nonlinear and complex (nondifferentiable and discontinuous). Sequential and simultaneous discrete optimization techniques are presented, and heuristics such as simulated annealing and Nelder–Mead are suggested to solve our problem. The adaptive spatial sampling problem is defined and related to the additional facility location problem. The approach is illustrated using data on cell phone call completion probability in a rural region of Erie County in western New York, and accounts for terrain variation using a line‐of‐sight approach. Finally, the computational results of sequential and simultaneous approaches are compared. Our model is also applicable to other facility location problems that aim to minimize the uncertainty associated with a customer visiting a new facility that has been added to an existing set of facilities.     

An Introduction to the Network Weight Matrix

This study introduces the network weight matrix as a replacement for the spatial weight matrix to measure the spatial dependence between links of a network. This matrix stems from the concepts of betweenness centrality and vulnerability in network science. The elements of the matrix are a function not simply of proximity, but of network topology, network structure, and demand configuration. The network weight matrix has distinctive characteristics, which are capable of reflecting spatial dependence between traffic links: (1) elements are allowed to have negative and positive values capturing the competitive and complementary nature of links, (2) diagonal elements are not fixed to zero, which takes the self‐dependence of a link upon itself into consideration, and (3) elements not only reflect the spatial dependence based on the network structure, but they acknowledge the demand configuration as well. We verify the network weight matrix by modeling traffic flows in a 3 × 3 grid test network with 9 nodes and 24 directed links connecting 72 origin‐destination (OD) pairs. Models encompassing the network weight matrix outperform both models without spatial components and models with the spatial weight matrix. The network weight matrix represents a more accurate and defensible spatial dependency between traffic links, and offers the potential to augment traffic flow prediction.