John R. Roy

ON APPROPRIATE MICROSTATE DESCRIPTIONS IN ENTROPY MODELLING

Using an information-theoretic approach to entropy modelling, expressions for the possible number of microstates are derived from the sets of choices available to households and firms in given location/transport planning situations. In planning applications, the urban space is usually subdivided into discrete zones, and various classes of householders and firms are grouped into separate activities. Nevertheless, the location sites within zones may vary considerably in accessibility, price and quality, and amenity v. cost trade-offs between location and travel many differ markedly between individuals within the same activity group. The inclusion of such variations within a random utility framework is demonstrated to be equivalent to maximizing entropy with the microstates disaggregated to the level where these variations occur. Using constraints based on spatial conditions, observed behaviour and planning policy, entropy maximization is used to determine the most probable macrostate according to each choice set specification. The resulting distributions are compared and their revelance discussed.

Minimizing information loss in simple aggregation

A criterion is presented to indicate a strategy for additively combining corresponding discrete rows and columns of a square array to form an array of reduced size such that the loss of information thereby incurred is minimized. The objective function is expressed in terms of an information loss criterion between the original array and an array of the same size acting as a surrogate for the reduced array. In contrast to the usual application of information theory in planning, where arrays are filled out based on incomplete information, the converse case is implied here, in which a particular partial representation of the array is sought (that is, the aggregated array) which most closely approaches the information content of the original full array. The method is expected to be useful for situations where available large arrays of data need to be appropriately reduced in size so that they can be efficiently used in further computations. Alternatively, the original array may contain current observed data, and the primary aim may be to determine from this array the aggregation scheme to use in future forecasting calculations. Typical applications may include the aggregation of spatial zones, the aggregation of economic sectors in input-output analysis, as well as various clustering problems.

An Alternative Information Theory Approach for Modelling Spatial Interaction

In the use of information theory for the development of forecasting models, two alternative approaches can be used, based either on Shannon entropy or on Kullback information gain. In this paper, a new approach is presented, which combines the usually superior statistical inference powers of the Kullback procedure with the advantages of the availability of calibrated ‘elasticity’ parameters in the Shannon approach. Situations are discussed where the combined approach is preferable to either of the two existing procedures, and the principles are illustrated with the help of a small numerical example.

A model of trade flows in differentiated goods

In modelling trade competition for final products within a sector, the assumption of product homogeneity is often quite unrealistic. Even where the demand for different versions of the product can be attributed to relatively independent submarkets, considerable heterogeneity may still exist within each submarket. For each submarket, such as that for a medium-priced family car, it is assumed that more product heterogeneity exists between countries or regions than within them. Heterogeneity between regions is related to the existence of importers, acting as intermediate agents, who make trade-offs between cif import prices of goods and the willingness-to-pay by consumers. Aggregate spatial supply and demand functions based on plausible distributional assumptions are presented and estimated using a unified theoretical basis for each regional grouping of producers, importers and consumers. Conditions for simultaneous equilibrium for these three sets of agents are established, and illustrated via a small numerical example.

A Dispersed Equilibrium Commodity Trade Model.

For many bulk commodities, such as mineral ores, crude oil, building materials and food grains, the suppliers are increasingly in the position of being price-takers. This means that, in the short run, their main decisions relate to spatial choice of markets and setting of production levels within the currently available capacity. In this paper, an entropy maximisation framework is introduced to handle dispersion about the profit-maximising choice of markets and production levels by the suppliers. The model also uses information theory to implicitly account for certain rigidities in trading relationships resulting from non-price factors. Although demand functions must be provided exogenously, cost functions can be inferred from regional vintage production data, which in turn allow profit functions to be defined for each producing region or country. A unique and stable dispersed price equilibrium of the Walrasian type is established for this spatial system under quite general conditions.

On planning and forecasting the location of retail and service activity

In countries with mixed economies, planning authorities cannot usually enforce micro location and travel decisions of urban activities, but are confined to forming macro location and transport policy, within which agents can operate according to the market. An attempt is made here to describe such multilevel behaviour of the planning authority, landlords, firms and customers, and to obtain non-cooperative equilibria in terms of the controls exerted by each group. The behaviour of the market groups is described via entropy maximization, while alternative modes of behaviour are ascribed to the authority. In addition, a Pareto-type solution is examined.

Comment on “The Development of a New Gravity—Opportunity Model for Trip Distribution”

The integration of the gravity and intervening-opportunities model in the recent paper by Gonçalves and Ulysséa-Neto provokes interesting questions on the role of the number of intervening opportunities on either decreasing or increasing travel. In this note I discuss such issues, as well as making comparisons with recent innovations in spatial interaction theory.

Entropy Models with Entropy Constraints on Aggregated Events

Most spatial models deal with a number of objects, such as segments of demand, as well as supply choices including houses, jobs, transport modes or routes, etc. A choice of or by any of these objects represents a probabilistic event (decision step). The problem is to determine the joint probability density for the collection of events. In this paper, the joint probability function is estimated using an extended entropy model, constrained by additional information on the entropy of aggregated events. The resulting models are compared with existing formulations, based either on entropy maximization or on utility maximization. It is also shown that models which are used for predicting demand can be generalized to allow for the dependence of demand on supply, allowing for nonhomogeneous zonal supply and supply – demand imbalances.

Planning models for non cooperative situations: a two player game approach

Abstract In the analysis of urban systems, attempts are usually made to subdivide the system into a set of interest groups, according to the.pursuit of certain common purposes. The behaviour of each group is described in terms on an objective and a set of variables under its control. Groups may be linked through the occurrence of variables controlled by other groups either in their objectives, in their constraints, or both. In this paper, the problem is addressed, whereby equilibrium states are sought for situations where one group is the planning authority, exercising policy instruments to influence system outcome at a level different to that required to reasonably describe the macro behaviour or objectives of all groups in the system, including the authority itself. The principles are illustrated on some specific models, and alternative means of handling various classes of intergroup linkages are investigated.

A note on consistency of choice units in entropy models

In using entropy maximization models to forecast locational and travel behaviour, one is confronted with the problem of delineating the choice process as precisely as possible. In addition to defining a fine-grain choice structure implying individuals seeking distinct location sites within residential zones and travelling to distinct jobs or shops within destination zones, this note also accounts for the fact that the location choice is of a site for a household or firm, but the corresponding travel choices are by individual members of a household. In conjunction with disaggregation across quantities with large variance, the above principles are applied to formulate improved versions of residential and shopping location models.