Extremal Approaches to Estimating Spatial Interaction

Two recent theoretical approaches to the gravity model of spatial interaction are examined with emphasis on the dissimilar physical and statistical theories from which they were respectively derived. Formal relationships between the two methods are demonstrated. The two approaches jointly indicate a general method of generating new hypotheses of gravity flows. This method is discussed and evaluated, and some questions of theory comparability are explored. Comments on some of the computational features of each of the two models conclude the paper.     

Estimating Spatial Models Within A Disequilibrium Framework

Modeling demand in a spatial context requires careful handling of regional interactions. In situations where there are constraints in some markets that lead to spillovers to others it is useful to build this explicitly into the model. In this paper I present a theoretical model that is related to the disequilibrium and the spatial econometric literature. Under certain conditions the model may be estimable and an appropriate estimation technique that uses the EM algorithm, is put forward. A data set from the U.K. market for secondary school places provides a test for the procedure.     

A Note on Trip Benefits in Spatial Interaction Models

We extend the well-known transport users' benefits measure (TUB) for the doubly-constrained spatial interaction model derived by Williams (1976). The original formula expresses the TUB as composed by two terms associated with the origin and the destination zones. First, the TUB is associated here with trips instead of zones, providing a natural interpretation as a rule-of-a-half measure of benefit under inelastic demand (for the short-run case). Second, a TUB formula for the long-run case is derived, that is, when the total number of trips, trip origins, and trip destinations change. We then propose updated measures of accessibility for location behavior.     

Dynamic Models of Agricultural Location in a Spatial Interaction Framework

Recent research demonstrates that spatial interaction models may also be made to function as location models by the addition of appropriate hypotheses about structural adjustment. An appealing feature of the approach is that dynamics are explicitly incorporated. In this paper, the attempt is made to recast a problem from classical economic-geographic theory—that of agricultural location—within a spatial interaction framework. It is shown that a wide variety of models is potentially available, and the properties of a range of these models are selectively explored.     

Using Simple Genetic Algorithms to Calibrate Spatial Interaction Models

The paper investigates the use of two genetic algorithms in an attempt to obtain globally optimal parameter estimates for a mix of simple and complex spatial interaction models. The genetic algorithms work well and are strongly advocated as a more robust approach particularly for use with the more complex multiparameter models where the differences in both performance and parameter values are judged to be significant.     

A New Method of Adaptive Zoning for Spatial Interaction Models

Spatial interaction models commonly use discrete zones to represent locations. The computational requirements of the models normally arise with the square of the number of zones or worse. For computationally intensive models, such as land use–transport interaction models and activity‐based models for city regions, this dependency of zone size is a long‐standing problem that has not disappeared even with increasing computation speed in PCs—it still forces modelers to compromise on the spatial resolution and extent of model coverage as well as on the rigor and depth of model‐based analysis. This article introduces a new type of discrete zone system, with the objective of reducing the time for estimating and applying spatial interaction models while maintaining their accuracy. The premise of the new system is that the appropriate size of destination zones depends on the distance to their origin zone: at short distances, spatial accuracy is important and destination zones must be small; at longer distances, knowing the precise location becomes less important and zones can be larger. The new method defines a specific zone map for every origin zone; each origin zone becomes the focus of its own map, surrounded by small zones nearby and large zones farther away. We present the theoretical formulation of the new method and test it with a model of commuting in England. The results of the new method are equivalent to those of the conventional model, despite reducing the number of zone pairs by 96% and the computation time by 70%. Los modelos de interacción espacial suelen utilizar zonas discretas para representar áreas o puntos de interés. Los requisitos computacionales de estos modelos normalmente aumentan a razón del número de zonas elevadas al cuadrado o más. Para modelos computacionalmente intensivos como los modelos de interacción entre uso de suelo y transporte y los modelos basados en actividades para ciudades‐región, el impacto del tamaño de la zona es un problema persistente no superado aun. Esta limitación persiste a pesar de los grandes avances en la velocidad de procesamiento en computadoras, pues obliga a los modeladores a hacer concesiones entre la resolución espacial y la extensión que abarca el modelo, así como en el rigor y profundidad del análisis. En este artículo se presenta un nuevo tipo de sistema de zonas discretas que: a) tienen como objetivo reducir el tiempo de estimación de la aplicación de modelos de interacción espacial; y b) al mismo tiempo mantienen su nivel de precisión. La premisa que gobierna este nuevo sistema es que el tamaño apropiado de las zonas de destino depende de la distancia a su zona de origen: a distancias cortas, la precisión espacial es importante y las zonas de destino deben ser pequeño; a distancias mas largas, conocer la ubicación precisa es progresivamente menos importante y las zonas pueden ser mayores. El nuevo método define un mapa específico de zonas para cada zona de origen; cada zona de origen se convierte en el foco de su propio mapa, rodeada de zonas cercanas pequeñas y zonas grandes a mayor distancia. El estudio presenta la formulación teórica del nuevo método y su demostración vía un modelo de desplazamientos residencia‐trabajo en Inglaterra. Los resultados del nuevo método son equivalentes a las del modelo convencional, a pesar de reducir del número de pares de zonas en un 96% y el tiempo de cálculo en un 70%. 空间相互作用模型通常采用离散区域代表区位。模型的计算量往往与区域数量呈平方甚至更高阶增长。对于可计算的精细模型,如土地利用‐交通相互作用模型和基于行为的城市区域模型,区域尺度的依赖性是长期存在的问题,即使计算机的计算速度增加,该问题仍无法消除。因此,建模者需在模型空间分辨率和覆盖范围以及模型分析的严谨性和深度上做出权衡。本文介绍了一种新型的离散分区系统,目的在于减少空间相互作用模型估算和计算时间,同时维持其精度。新系统的前提是目标区域的适当尺度取决于与初始区域的距离：在短距离范围内,空间精确性是重要的,且目标区域必须是小的;在更远距离上,位置精度的重要性降低,目标区域可以变大。该方法为每个初始区域制定了具体的尺度地图。每个初始区域成为其自身地图的中心,被近邻的小区域和更远距离的大区域所包围。本文给出了新方法的理论公式,并以英格兰地区的通勤模型进行检验。结果显示,尽管区域对的数量减少了96%,计算时间缩短了70%,但新方法的计算结果等效于常规模型。     

Spatial Characteristics and Comparisons of Interaction and Median Clustering Models

Cluster analysis has been pursued from a number of directions for identifying interesting relationships and patterns in spatial information. A major emphasis is currently on the development and refinement of optimization‐based clustering models for the purpose of exploring spatially referenced data. Within this context, two basic methods exist for identifying clusters that are most similar. An interesting feature of these two approaches is that one method approximates the relationships inherent in the other method. This is significant given that the approximation approach is invariably utilized for cluster detection in spatial and aspatial analysis. A number of spatial applications are investigated which highlight the differences in clusters produced by each model. This is an important contribution because the differences are in fact quite significant, yet these contrasts are not widely known or acknowledged.     

Analysis of Dynamic Spatial Interaction Models by Means of Optimal Control

This paper deals with the design of general classes of dynamic spatial interaction models. On the basis of a general (well-behaved) multiperiod objective function and of a dynamic model representing the evolution of a spatial interaction system, an optimal control model is constructed. Particular attention is given to the equilibrium and stability conditions. It turns out that it is possible to identify steady-state solutions for a dynamic spatial interaction model. Furthermore, it can also be demonstrated that the entropy model is a specific case of this spatial interaction system. A simple illustration for urban dynamics is given as well.     

Explorations and Syntheses of Linear Programming And Spatial Interaction Models of Residential Location

Various “linear programming” models of residential location are explored in some detail. It is then shown that entropy maximizing “sub-optimal” versions of these can be developed, which may be more realistic at typical levels of resolution used for analysis but can still be given some of the interpretations of the programming models. Further, any programming model can be seen as a limiting case of some entropy maximizing model in which some of the parameters tend to infinity. Some empirical tests of both kinds of models are presented, and the limiting relationships are discussed.     

Building an Automated Modeling System to Explore a Universe of Spatial Interaction Models

A methodology is described for harnessing the power of supercomputer hardware to build an automated modeling system. The problems of applying this new approach to mathematical modeling in geography are discussed by reference to the design of better performing spatial interaction models. It is also possible that the results may provide clues about new theories and knowledge.     

Learning in Single Hidden-Layer Feedforward Network Models: Backpropagation in a Spatial Interaction Modeling Context

Learning in neural networks has attracted considerable interest in recent years. Our focus is on learning in single hidden-layer feedforward networks which is posed as a search in the network parameter space for a network that minimizes an additive error function of statistically independent examples. We review first the class of single hidden-layer feedforward networks and characterize the learning process in such networks from a statistical point of view. Then we describe the backpropagation procedure, the leading case of gradient descent learning algorithms for the class of networks considered here, as well as an efficient heuristic modification. Finally, we analyze the applicability of these learning methods to the problem of predicting interregional telecommunication flows. Particular emphasis is laid on the engineering judgment, first, in choosing appropriate values for the tunable parameters, second, on the decision whether to train the network by epoch or by pattern (random approximation), and, third, on the overfitting problem. In addition, the analysis shows that the neural network model whether using either epoch-based or pattern-based stochastic approximation outperforms the classical regression approach to modeling telecommunication flows.     

Combining Geostatistical Methods and Hierarchical Scene Models for Analysis of Multiscale Variation in Spatial Data

Concepts from Hierarchical Analysis of Variance (ANOVA) can be combined with ideas from geostatistics to describe the multiscale structure of spatial data. Hierarchical ANOVA involves modeling spatial data as the sum of effects associated with processes acting at different spatial scales. These effects can be modeled as stationary regionalized variables, whose spatial structure can be described using the variogram. According to this model, the variogram of the spatial data is the sum of variograms and cross‐variograms of the effects. Whereas hierarchical ANOVA reveals the relationship between scale and variability, the hierarchical decomposition of the variogram relates scale with spatial structure. This analysis method can reveal otherwise undetected features of spatial data, and can guide further analysis.     

Computing the Jacobian in Gaussian Spatial Autoregressive Models: An Illustrated Comparison of Available Methods

When estimating spatial regression models by maximum likelihood using spatial weights matrices to represent spatial processes, computing the Jacobian, ln(|I ? λW|), remains a central problem. In principle, and for smaller data sets, the use of the eigenvalues of the spatial weights matrix provides a very rapid resolution. Analytical eigenvalues are available for large regular grids. For larger problems not on regular grids, including those induced in spatial panel and dyadic (network) problems, solving the eigenproblem may not be feasible, and a number of alternatives have been proposed. This article surveys selected alternatives, and comments on their relative usefulness, covering sparse Cholesky and sparse LU factorizations, and approximations such as Monte Carlo, Chebyshev, and using lower‐order moments with interpolation. The results are presented in terms of component‐wise differences between sets of Jacobians for selected data sets. In conclusion, recommendations are made for a number of analytical settings. Al estimar modelos de regresión espacial con el método del máxima verosimilitud (máximum likelihood) y usando matrices de pesos espaciales para representar procesos espaciales, cálculo del término jacobiano (jabobian)—ln(| I ?λ W |)‐ sigue siendo un problema central. En principio, y para bases de datos más pequeñas, el uso de los valores propios (eigenvalues) de la matriz de pesos espaciales proporciona una solución muy rápida. Los eigenvalues analíticos para retículas o grillas grandes y regulares son ya conocidos. Para problemas más grandes, que no se presentan en mallas regulares ‐incluyendo aquellos que se inducen en problemas de paneles espaciales y en problemas de (redes) diádicas‐, es posible que resolver el eigenproblem no sea posible. Este artículo estudia una selección de alternativas y comenta acerca de su relativa utilidad. Se cubren las facorizaciones de tipo Cholesky disperso (sparse Cholesky) y de tipo LU dispersas (sparse LU), las aproximaciones Monte Carlo, y Chebyshev, así mismo se utiliza momentos de bajo‐orden (lower‐order) con interpolación. Los resultados se presentan en términos de diferencias de componentes entre sets de términos jacobianos para bases de datos seleccionadas. En conclusión, se hacen recomendaciones para una serie de contextos analíticos. 当采用表征空间过程的空间权重矩阵对空间回归模型进行最大似然估计时,雅可比矩阵ln(|I?λW|)的计算仍是核心问题。对于小数据集,原则上可利用空间权重矩阵的特征值提供一种快速的解决方案,对于大型规则格网数据特征值分析同样有效。但对于不规则格网大型问题,包括从空间面板和二元（网络）问题中引伸的问题,利用特征值的解决方案可能不适用,对此学术界提出了多种可选替代方案。本文选取已有的几种替代方案并评论各自的相对有效性,其中包括稀疏Cholesky分解和稀疏LU分解法,Monte Carlo和 Chebyshev近似模拟法以及低阶矩插值法。结果以所选数据集雅可比矩阵间特定组份的差异方式显示。最后,推荐了一些分析设定。     

Spatial Models of Spread-Backwash Processes

Spread-backwash processes are central elements in regional development thought. This article argues that the main spread-backwash processes produce developmental change that has a general spatial form. Straightforward models of the processes of government incomes and expenditure flows, private capital flows, intraregional trade, migration-commuting, and the diffusion of innovations provide the rationale for the general distance decay form of developmental impact. Each model is supported by evidence from the relevant literature. It is argued that there is not a dichotomous set of spread versus backwash processes, but rather that the same set of processes yields differential spatial impacts.     

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.