A New Set of Spatial-Interaction Models: The Theory of Competing Destinations

1983 ◽  
Vol 15 (1) ◽  
pp. 15-36 ◽  
Author(s):  
A S Fotheringham
Keyword(s):  
Spatial Interaction ◽  
Model Misspecification ◽  
Distance Decay ◽  
Gravity Models ◽  
Parameter Estimates ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
The Family ◽  
Decay Parameters ◽  
Alternative Set

Members of the family of spatial-interaction models commonly referred to as gravity models are shown to be misspecified. One result of this misspecification is the occurrence of an undesirable ‘spatial-structure effect’ in estimated distance-decay parameters and this effect is examined in detail. An alternative set of spatial-interaction models is formulated from which more accurate predictions of interactions and more accurate parameter estimates can be obtained. These new interaction models are termed competing destinations models, and estimated distance-decay parameters obtained in their calibration are shown to have a purely behavioural interpretation. The implications of gravity-model misspecification are discussed.

Model Misspecification and Spatial Structure in Spatial-Interaction Models

1983 ◽  
Vol 15 (3) ◽  
pp. 319-327 ◽  
Author(s):  
M Baxter
Keyword(s):  
Spatial Structure ◽  
Spatial Interaction ◽  
Model Misspecification ◽  
Parameter Estimates ◽  
Interaction Models ◽  
Misspecified Models ◽  
Spatial Interaction Models ◽  
Misspecified Model ◽  
Special Cases ◽  
Explicit Expressions

Concern has been expressed about the effects of spatial structure on parameter estimates from spatial-interaction models. The problem is essentially one of model misspecification. With a correctly specified model assumed, in which destination attraction depends on whether it is near to an origin or not, the consequences of using a misspecified model are examined. Explicit expressions for bias in the parameter estimates are derived; these are complex, but depend on terms that can be clearly interpreted in terms of aspects of spatial structure, such as scale, compactness, shape, remoteness of destinations, etc. Some simple special cases show how, with misspecified models, estimates from different systems will almost certainly differ. Extensions of the analysis and problems of estimation and interpretation are discussed.

Spatial Flows and Spatial Patterns

1984 ◽  
Vol 16 (4) ◽  
pp. 529-543 ◽  
Author(s):  
A S Fotheringham
Keyword(s):  
Spatial Structure ◽  
Spatial Patterns ◽  
Spatial Interaction ◽  
A Priori ◽  
Parameter Estimates ◽  
Spatial Interaction Models ◽  
Significant Bias ◽  
Spatial System ◽  
Decay Parameters ◽  
Spatial Flows

The misspecification of gravity spatial interaction models has recently been described by the author. The bias in parameter estimates that results from such misspecification appears to produce the ‘map pattern effect’ or ‘spatial structure bias’ in estimated distance-decay parameters. A further aspect of the misspecification bias in gravity parameter estimates is explored here. The severity of the bias is shown to vary in a predictable manner with variations in spatial structure. In particular, the bias is shown to be dependent upon the pattern of accessibility that exists within a spatial system. The relevant aspects of this pattern are discussed for intraurban and interurban (or interregional) flow matrices. It is shown that from an examination of the spatial structure of centres in a spatial system it is possible, a priori, to identify whether significant bias will arise in the calibration of a gravity model. Certain configurations of centres are shown theoretically to produce maximal bias. The author thus answers the question, “why do gravity parameter estimates appear to be biased in some systems but not in others?”

Misspecification in Spatial Interaction Models: Further Results

1985 ◽  
Vol 17 (5) ◽  
pp. 673-678 ◽  
Author(s):  
M J Baxter
Keyword(s):  
Spatial Structure ◽  
Spatial Interaction ◽  
Parameter Estimates ◽  
Interaction Models ◽  
Spatial Interaction Models

Commonly used models for data on flows between sets of origins and destinations may often be misspecified because of a failure to account for the influence of spatial structure and other factors that affect flows. General expressions for the bias in parameter estimates that arises from such misspecification are derived for the most usual methods of estimation.

The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

1972 ◽  
Vol 4 (2) ◽  
pp. 205-233 ◽  
Author(s):  
M Batty ◽  
S Mackie
Keyword(s):  
Maximum Likelihood ◽  
Spatial Interaction ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Calibration Methods ◽  
The Family ◽  
Newton Raphson ◽  
Best Parameter ◽  
Parameter Values ◽  
Raphson Method

This paper presents a methodology for deriving best statistics for the calibration of spatial interaction models, and several procedures for finding best parameter values are described. The family of spatial interaction models due to Wilson is first outlined, and then some existing calibration methods are briefly reviewed. A procedure for deriving best statistics based on the principle of maximum-likelihood is then developed from the work of Hyman and Evans, and the methodology is illustrated using the example of a retail gravity model. Five methods for solving the maximum-likelihood equations are outlined: procedures based on a simple first-order iterative process, the Newton—Raphson method for several variables, multivariate Fibonacci search, search using the Simplex method, and search based on quadratic convergence, are all tested and compared. It appears that the Newton—Raphson method is the most efficient, and this is further tested in the calibration of disaggregated residential location models.

Spatial Aggregation in Gravity Models: 3. Two-Dimensional Trip Distribution and Location Models

1982 ◽  
Vol 14 (5) ◽  
pp. 629-658 ◽  
Author(s):  
M Batty ◽  
P K Sikdar
Keyword(s):  
Spatial Interaction ◽  
Model Performance ◽  
Spatial Aggregation ◽  
Gravity Models ◽  
Canonical Forms ◽  
Two Dimensional ◽  
Variable Model ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Spatial Entropy

This is the third of four papers and in it the methodology for analysing spatial aggregation in gravity models outlined in the first paper is further elaborated. In the second paper, the methodology was applied to one-dimensional spatial interaction models of the population density type, with some success; and here it is proposed to apply the methodology to two-dimensional spatial interaction models using the same data base, the Reading (UK) region. Accordingly, the methodology is first stated for linking information in data measured by spatial entropy to the parameters of models generated from spatial entropy. The family of four spatial interaction models due to Cordey-Hayes and Wilson is then derived, the canonical forms of their associated spatial entropy functions presented, and the analytic properties of such models explored. These four models are then fitted to spatial aggregations of the Reading region, and various empirical relationships between their entropies and parameters described. The results are not as regular as those of the models in the second paper because of more variable model performance, but nevertheless a means of approximating scale parameters from data based on the work of Kirby is outlined. This enables estimates of the dispersion parameters to be made through the canonical forms. Although the results are poor because of model performance, the methodology outlined here serves as a basis for the more fully fledged application to be discussed in the final paper.

Interzonal Migration: Some Historical Tests of Spatial-Interaction Models

1978 ◽  
Vol 10 (10) ◽  
pp. 1187-1200 ◽  
Author(s):  
J C H Stillwell
Keyword(s):  
Spatial Variation ◽  
Spatial Interaction ◽  
Interaction Model ◽  
Regional Variations ◽  
Interaction Models ◽  
Spatial Interaction Model ◽  
Spatial Interaction Models ◽  
Disaggregated Data ◽  
Migration Flows ◽  
Decay Parameters

Observed migration and survival flows between counties and between standard regions are used to test alternative calibrations of a doubly constrained spatial-interaction model. Spatial variation in the propensity to migrate over distance is examined in an analysis of zone-specific decay parameters, and two methods of splitting aggregate migration flows according to reason for move are investigated. The results of the model tests for age/sex-disaggregated data underline regional variations in propensities to migrate and in mean distances migrated.

Proximate Aggregation-Estimation of Spatial Interaction Models

1984 ◽  
Vol 16 (4) ◽  
pp. 467-486 ◽  
Author(s):  
M Batty ◽  
P K Sikdar
Keyword(s):  
Spatial Interaction ◽  
Gravity Models ◽  
New Method ◽  
Entropy Measure ◽  
Interaction Models ◽  
Information Measures ◽  
Spatial Interaction Models ◽  
Entropy Measures ◽  
Log Linear ◽  
Parameter Values

In this paper the authors introduce a method of approximating the parameter values of gravity models from measures of information or entropy associated with the observed pattern of spatial interaction. The method builds on the previous work of the authors in which parameter values were estimated in a two-stage process which involved utilising the log-linear properties of entropy models through the canonical form of entropy, together with other approximations based on Kirby's method. Here the method is elaborated by adopting a consistent set of information measures to which the parameters of the model are related and this negates the need for other approximations. The original framework is first reviewed and then elaborated through the introduction of a weighted entropy measure. The traditional family of spatial interaction models is sketched and the new method developed for each of these models. The models are then applied to various aggregations of trip data in the Reading (United Kingdom) subregion, and estimates of parameter values based on the old, new, and conventional methods are compared. The new method is demonstrably superior to the old method and various extensions through the spatial disaggregation of entropy measures are noted in conclusion.

A Note on the Calibration of Maximum-Performance Spatial-Interaction Models

1978 ◽  
Vol 10 (10) ◽  
pp. 1151-1154
Author(s):  
M J Baxter
Keyword(s):  
Spatial Interaction ◽  
Model Performance ◽  
Maximum Performance ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
The Family

Evidence is presented to show that the improvement in model performance achieved by the family of maximum-performance spatial-interaction models developed by Openshaw and Connolly (1977) may be explicable solely for statistical reasons and that there is no need for a geographical or behavioural explanation, as was suggested.

A Note on the Treatment of Flows across System Boundaries in Spatial-Interaction Models

1979 ◽  
Vol 11 (4) ◽  
pp. 447-454 ◽  
Author(s):  
I Masser
Keyword(s):  
Spatial Interaction ◽  
System Boundaries ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
The Family ◽  
Constrained Model

Some problems that must be resolved by the analyst in connection with the treatment of flows across system boundaries to and from external zones are discussed in this paper. These problems make it necessary to make modifications to the conventional formulations both of the doubly and of the singly constrained members of the family of spatial-interaction models. None of the possible modifications wholly satisfies theoretical requirements in terms of the doubly constrained model, and the advantages and limitations of various approaches can only be assessed in an operational situation. For this reason some of the findings from a study of the Amersfoort region are presented which help to throw light on this problem.

On the Relationship between Alonso's Theory of Movement and Wilson's Family of Spatial-Interaction Models

1981 ◽  
Vol 13 (2) ◽  
pp. 217-224 ◽  
Author(s):  
J Ledent
Keyword(s):  
Input Data ◽  
Spatial Interaction ◽  
System Of Equations ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Standard Family ◽  
Original Contribution ◽  
Standard Models ◽  
The Relationship

This paper compares the system of equations underlying Alonso's theory of movement with that of Wilson's standard family of spatial-interaction models. It is shown that the Alonso model is equivalent to one of Wilson's four standard models depending on the assumption at the outset about which of the total outflows and/or inflows are known. This result turns out to supersede earlier findings—inconsistent only in appearance—which were derived independently by Wilson and Ledent. In addition to this, an original contribution of this paper—obtained as a byproduct of the process leading to the aforementioned result—is to provide an exact methodology permitting one to solve the Alonso model for each possible choice of the input data.

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