Spatial Interaction Models and Fisher Information: a new calibration algorithm

The theoretical background of spatial interaction models is reviewed and used as a basis for the derivation of a novel approach for directly calibrating spatial interaction models concurrently with the main solution procedure. The analysis was prompted by a link with Fisher Information. The new approach is compared with a number of earlier approaches, particularly that of Sen and Smith. 1. Three approaches to spatial interaction modelling. The intention in this paper is to look afresh at the work of one of the present authors see Wilson (1967, 1970) in the context of the theoretical developments in spatial interaction modelling of Sen and Smith (1995) and the calibration issues relating to iterative numerical methods. The new developments reported here relate to the concept of Fisher Information. These explorations arise from Frieden's (1998) claim that much of physics can be derived from Fisher information a statistical inferencing concept; and that in the case of statistical mechanics, Fisher information and entropy are related. Intuitively, therefore, this implies a connection between entropy-based approaches to spatial interaction modelling and Fisher information. It has been found in the past that there are considerable benefits from exploring how different mathematical approaches can address the same modelling task. In particular, it is often the case that techniques in one approach complement those in another so that the power of the set of techniques available from a number of approaches is greater than the sum of the parts. There is a history of this in urban modellingdiscussed for example in Wilson (2000). In this study, the outcome is a deeper understanding of how a number of approaches can be linked and, specifically, a new algorithm for calibrating spatial interaction models can be constructed. In what follows, it is assumed that the model being considered is that described in summary form in pages 62-63 of Wilson (2000), or pages 15-23 of Wilson (1970)or pages 243-253 of Robinson (1998). Consider n spatial zones and suppose that the random variable (or alternatively the observed value ) describing the number of trips from any zone i to any zone j is ij N and the estimate of this value to be obtained from a model is ij T . In what follows, it is assumed that the expected value based on the observed values is given by E ( ij N )= ij T , (1) see Sen and Smith (1995). Let i O be the number of work trips originating in zone i and let j D be the total number of work trip destinations in zone j. (These are taken as given and can be actual values.)There are then two sets of constraints , 0 = − ∑ i j ij O T (2) . 0 = − ∑ j i ij D T (3) The total number of elements in the trip interaction matrix is given by T n where ∑ = ij T n 1 There is also a cost constraint equation involving weighted summation over the T n terms of ij T : 0 = − ∑ C T c ij ij ij (4) where ij c is the generalised cost of travelling between i and j. Let T be the total number of trips i.e. ∑ = ij ij T T (5) From elementary combinatorial theory the number of ways in which individuals can be arranged to get, say, the kth particular pattern of trips is given by Wilson (1967) as the quantity }) ({ ij k T w where ) ! ( ! }) ({ ij ij ij k T T T w Π = (6) The approach is then to find the matrix } { ij T which maximises the quantity defined by L T w M ij k + = })) ({ log( (7) where L consists of the Lagrange multiplier terms given by ) ( ) ( ) ( ) 2 ( ) 1 ( ij ij ij i ij j i j j ij i i i T c C T D T O L ∑ ∑ ∑ ∑ ∑ − + − + − = β λ λ (8) where ) 2 ( ) 1 ( , j j λ λ and βare the Lagrange multipliers to ensure that the constraints (2) to (4) are satisfied. The maximum value of M is found by solving the equations , 0 = ∂ ∂