Optimal Zoning Systems for Spatial Interaction Models

The design of zoning systems for spatial interaction models is a major problem which affects both the interpretation and acceptability of these models. This paper demonstrates that zoning-system effects on parameter values and model performance are nontrivial, and that their magnitude is far larger than was previously thought likely. An approach which is most appropriate in an applied context, where there is also the problem of poor model performance, is to identify a zoning system which will approximately optimise model performance. The paper gives details of how this may be achieved. This method is demonstrated by a series of empirical studies. Finally, there is a brief discussion of the general implications for spatial model building.

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Empirically Derived Deterrence Functions for Maximum Performance Spatial Interaction Models

Environment and Planning A Economy and Space ◽

10.1068/a091067 ◽

1977 ◽

Vol 9 (9) ◽

pp. 1067-1079 ◽

Cited By ~ 18

Author(s):

S Openshaw ◽

C J Connolly

Keyword(s):

Goodness Of Fit ◽

Spatial Interaction ◽

Model Performance ◽

Interaction Model ◽

Theoretical Research ◽

Maximum Performance ◽

Interaction Models ◽

Spatial Interaction Models ◽

The Relationship ◽

Descriptive Models

The relationship between the choice of deterrence function and the goodness of fit of a singly constrained spatial interaction model is examined as a basis for improving model performance. The results show that there is no significant improvement in model goodness of fit until a deterrence-function characterisation is used which is based on a family of functions, with the spatial domain of each function being determined in an approximately optimal manner. These findings are consistent with theoretical research on microlevel trip behaviour and can be used to identify descriptive models which possess maximum levels of performance.

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

Environment and Planning A Economy and Space ◽

10.1068/a040205 ◽

1972 ◽

Vol 4 (2) ◽

pp. 205-233 ◽

Cited By ~ 104

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.

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

Environment and Planning A Economy and Space ◽

10.1068/a140629 ◽

1982 ◽

Vol 14 (5) ◽

pp. 629-658 ◽

Cited By ~ 28

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.

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Proximate Aggregation-Estimation of Spatial Interaction Models

Environment and Planning A Economy and Space ◽

10.1068/a160467 ◽

1984 ◽

Vol 16 (4) ◽

pp. 467-486 ◽

Cited By ~ 10

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.

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

Environment and Planning A Economy and Space ◽

10.1068/a101151 ◽

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.

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

Environment and Planning A Economy and Space ◽

10.1068/a130217 ◽

1981 ◽

Vol 13 (2) ◽

pp. 217-224 ◽

Cited By ~ 15

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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