Aura Reggiani

Entropy, spatial interaction models and discrete choice analysis: Static and dynamic analogies

Abstract This paper aims at analysing the existence of a formal correspondence between spatial interaction models emanating from entropy theory and micro-economic discrete choice theory (in particular, multinomial logit models.). After a concise review of the literature on this issue, the emphasis is placed on an interpretation of formal analogies between both classes of models in a dynamic context. A simple dynamic spatial interaction model—based on optimal control theory—is proposed, and it is shown that the results confirm also the existence of a formal analogy between (macro) dynamic interaction models and (micro) choice models. Similar results are also derived for Alonsos general theory of movement in a spatial system.

Dynamic Spatial Interaction Models: New Directions

Spatial interaction models have received a great deal of attention in the past decade. In recent years, various approaches have also been developed to take into account dynamic aspects of spatial interaction models, by means of, for instance, optimal control theory, bifurcation theory, or catastrophe theory. The present paper deals with new directions in dynamic spatial interaction research. The focus is on a general dynamic interaction model analyzed in the framework of optimal control theory. The objective function used is a bicriterion utility model, to be maximized subject to a set of differential equations which bear some resemblance to those used by Wilson in a shopping-centre context. The link between the model presented and a catastrophe type of model is investigated. It is demonstrated that catastrophe behaviour may emerge as a particular case of this optimal control model. Finally, it is shown how external influences (for example, stochastic impacts of the Brownian motion type) affect the optimal trajectory.

Logit Models and Chaotic Behaviour: A New Perspective

Chaos theory and discrete choice theory have been developed as two separate analytical tools from various disciplinary backgrounds. In this paper the aim is to link chaos theory (emerging mainly from physics) to discrete choice theory (emerging mainly from geography and economics) by showing the formal conditions under which a dynamic logit model can exhibit chaotic behaviour. It will be shown that under certain conditions a generalized predator-prey model arises from a dynamic logit model. Furthermore, the analysis will be extended by developing a time-delayed logit model related to a modal (or route) choice problem in which congestion effects are incorporated in a dynamic framework. The analysis will be illustrated by means of simulation experiments, in which it is shown that different types of behaviour (including chaotic movements) can emerge depending on critical values of the utility function.

A new approach to modal split analysis: some empirical results

This paper describes a new approach (SD-MNL) to analyze modal choice. The SD-MNL method allows one to consider travelling modes under uncertainty. In particular, different states of nature for the alternatives are taken into account. In this scenario, Stochastic Dominance rules (SD) will be applied together with Multinomial Logit models (MNL) to describe a flow pattern in Regione Lombardia, Italy. The results obtained fit statistically the observed data and give rise to interesting considerations about travelling modes.

Aggregation in decisionmaking: a unifying approach

The aim of this paper is to describe a new approach in decisionmaking under uncertainty related to discrete choice models (DCM) and stochastic dominance (SD). A common basic structure is recognized for DCM and SD and both are embedded in the new approach.

Retrospect and Prospect

In this study various research directions and models centering around SIMs have been analyzed. In the first part of the book the attention has been focused on static developments in SIMs, while the second part has been devoted to an analysis of dynamic aspects of SIMs with particular emphasis on chaos theory.

Spatial Modelling and Chaos Theory

In the previous chapter we have shown that a SIM which takes into account external stochastic influences is highly dependent on the diffusion component (i.e. the amplitude) of these exogenous forces. However, as mentioned before, various types of fluctuations may influence spatial-economic behaviour, and some of them may be ‘governed’ by deterministic impact mechanisms. In other words, such irregular oscillations may be created endogenously through a non-linear lag structure in the behaviour of a system and described by relatively simple differential equations. This second way of explaining — seemingly stochastic — endogenous fluctuations will be treated in the present chapter by means of concepts stemming mainly from chaos theory.

Spatial Interaction Models and Chaos Theory

In the previous chapter the relevance of chaos theory, with special emphasis on spatial systems, has been demonstrated. In this context various important issues still deserve further attention. Firstly, a critical issue — according to Sterman (1988) — is that “the significance of the results hinges in large measure on whether the chaotic regimes lie in the realistic region of parameter space or whether they are mathematical curiosities never observed in a real system” (p. 148). Secondly, there are evidently major questions concerning the validity of both model specifications (notably, are model specifications compatible with plausible economic-spatial hypotheses) and of testability of model results (notably, are model results qualitatively or quantitatively justifiable from possible nonlinear patterns in the underlying data set).

Dynamic and Stochastic Spatial Interaction Models

In the first part of this study the attention has been focussed on static SIMs. The second part of the present study will be devoted to the analysis of dynamic modelling, with particular reference to dynamic SIMs.

Spatial Interaction Models and Utility Maximizing Behaviour at the Micro Level

In Chapter 3 it has been demonstrated that the family of SIMs can be derived from different formulations of an entropy (or utility) maximizing macro approach and hence viewed as an optimum system’s solution. In the present chapter the attention will be devoted to a further investigation into the connections between SIMs and micro-economic theory.