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entropy-maximizing models

  Statistical models for identifying the \'most likely\' spatial allocation pattern in a system subject to constraints. The approach was introduced into geographical modelling by A.G. Wilson in 1967 as the basis for a more rigorous interpretation of the gravity model, and has been extensively used since for spatial interaction modelling in urban regions and for modelling inter-regional flows of traffic and commodities. It is based on the concept of entropy, a measure of the uncertainty or \'likelihood\' in a probability distribution.

A journey-to-work model illustrates the method. For a city divided into k zones, we wish to calculate the best estimate of interzonal commuting flows Tij without knowing the detailed information of each individual movement. Assume that there are N total commuters. Any specific trip distribution pattern Tij, known as a \'macrostate\' (see entropy), can arise from many different sets of individual commuting movements or \'microstates\'. Entropy measures the number of different microstates that can give rise to a particular macrostate:

{img src=show_image.php?name=bkhumgeofm7.gif }

In the absence of detailed microstate data, we assume that each microstate is equally probable, and that the macrostate {Tij} with the maximum entropy value is the most probable or most likely overall pattern.

Additional information is also normally available, notably the number of commuters originating from each zone Oi, the total number of jobs available in each zone Dj, and estimates of the average or total travel expenditure for the city, C (usually based on survey data). The entropy-maximizing method then consists of maximizing W({Tij}) subject to the constraints

{img src=show_image.php?name=bkhumgeofm6.gif }

where cij is the travel cost from zone i to zone j. This maximization is a non-linear optimization problem, and must be solved by iterative search methods, systematically trying out different sets of values until the maximum is found.

Entropy-maximizing models not only fit empirical trip-distributions well, but also facilitate easy calculation of the effects of new housing or jobs (altering the Oi and Dj terms), and so have been widely used in more general urban models. Wilson and his Leeds colleagues have extended the model in many ways, making it dynamic, linking it to industrial and urban location theory, and including several types of disaggregation (e.g. by mode of travel). Recent work links the method to geographical information systems (GIS) to provide corporate and public sector location and strategic planning (Birkin et al., 1996). Other (non-transport) applications of the entropy-maximizing approach include its use to predict the most likely \'flows\' or changes in votes between parties in English parliamentary constituencies at general elections (Johnston, 1985).

The entropy-maximizing trip distribution, based on given total cost C, can be related to the optimizing minimum-cost distribution generated by the transportation problem: as C is reduced to its minimum feasible value the entropy-maximizing pattern converges to the linear-programming transportation problem pattern of flows. (LWH)

References Birkin, M., Clarke, G., Clark, M. and Wilson, A.G. 1996: Intelligent GIS. Location decisions and strategic planning. Cambridge: GeoInformation International. Johnston, R.J. 1985: The geography of English politics: the 1983 general election. London: Croom-Helm. Wilson, A.G. 1967: A statistical theory of spatial distribution models. Transport Research 1: 253-69.

Suggested Reading Wilson, A.G. and Bennett, R.J. 1986: Mathematical methods in human geography and planning. Chichester: John Wiley.



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