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.     

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.     

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.     

Entropy in Spatial Aggregation

This paper introduces an approach to the measurement of locational phenomena in a spatial hierarchy using entropy statistics. A number of such statistics suitable for the study of spatial aggregation are derived, and each of these statistics is decomposed at different levels of the spatial hierarchy using principles of decomposition first applied by Theil. These decomposition statistics are compared with the variance analysis applied by Moellering and Tobler and with the spatial entropy measure suggested by Curry. The use of these statistics is then illustrated by data from the Reading subregion and New York City, and the paper is concluded with an analysis of a possible role for entropy and information in problems involving equal-area zoning.     

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.     

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.     

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.     

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.     

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.     

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.     

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.