SPATIAL AGGREGATION EFFECTS IN GRAVITY MODEL FOR TRIP ANALYSIS

Abstract

In urbrn regions, the importance of the transportation planning for ? "better environment has been long accepted. The role played by the trip-analysis for the better understanding of the transport phenomenon has been acknowledged by the analysts. An appropriate transport analysis is very much necessary for decision-making in the transport planning. The most favoured modelling technique adopted for the predicting the trip-interchange is based on the 'gravitational hypothesis'. The effects of the spatial aggregation of date have not been given due attention in the analysis of these gravity models. This thesis presents a new method of accounting for the spatial effects and modelling the spatial interactions in the trip-analysis based on gravitational hypothesis.lt deals with the spatial aggregation of spatial data. The information theory has been successfully used^ JJ in analysing effects of the spatial aggregation comprising of the scale effect, and the pure aggregation effect, resulted from the spatial aggregation of data, required for the scientific enquiry. The established 'Entropy Maximizing methodology' has been shown to be unsuitable in exploring the aggregation effects in the data and the model as well* The information theoretic •Minimum Information Principle', (MIP)^ >' hrs been shown to be a powerful tool in predicting the "most probable state"(MPS)** occurring, in a given system and has been used in search of a suitable information measure which appropriately deals with the spatial aggregation effects enabling the modelling of the ii iii spatial interactions. The suitable measure thus derived has been called as 'minimum cross-entropy measure'. The cross-entropy mepsure consists of two trip- distri bution probabilities;ofle.ut data level and another at the aggregated level. This expression for the cross-entropy ha-? en inconsistency of dimensions for these trip-distributions as the two.different levels are referred to in the aggregation process. The dimensional inconsistency in the cross-entropy expression has been removed by developing a methodology of 'information equivalence'. The methodology is b?sed on the minimum information loss while transferring the information M 29") from the aggregate level to the model levelv y , The methodology finds a surrogate prior distribution which h?s the needed dimensional consistency and has its information equivalence with that of the aggregated prior distribution of probabilities. By Incorporating the surrogate prior distri bution at the model-level in place of the aggregated prior probability distribution in the cross-entropy met" sure, the spatial aggregation process has been accounted for in the spatial interaction-modelling. In the spatial analysis, it has been observed^ ' that a good theory - or any theory for that matter- yields the best results only at a paiticular level of analysis, hence, the same theory if applied at other levels of analysis may -yield inferior results and Interpretations. As a result of the spatial aggregation these interpretations change with iv the level of analysis, and many a time , might be even contradictory to previous interpretations. Hence, a search for the optimum level is necessary even after modifying the theory to explore the aggregation effects. The optimum level of analysis should,first,be investigated. Hence the metho dology of finding the optimum level of analysis is developed. The exploretions are made with the help of :;the information loss occurring as a result of the aggregation and the frame work for computing such a level has been developed. The role of the modelling technique in this frame-work has been presented, finally, the technique for the search for the optimum level has been elaborated with reference to the modelling techniques,newly developed. In the next part of the thesis, a family of 'Minimum Cross-entropy' models is derived. The structure of the models is retained which is similar to that of 'Maximum Entropy' models. An addition of a prior probability distribution is made in the form. The cross-entropy has been shown to decom pose into Shannon's entropy and Kerridge correction fac tor .The role played by the prior probability distribution has been fully explored. This displayed the necessity of choosing a prior probability distribution from a proper level. This level of the prior need not be related to the optimality of the level. The prior from the concerned level provides the missing information regarding the aggregation phenomenon and the scale effect which otherwise w?s found to be very difficult to be expressed as the information through const raints equations, in terms of level at which the model was stated. To each level of the analysis, there are other levels in the process of aggregation or disaggregation which contribute priors, impregnated with missing information. Depending on the way of choosing these probabilities, two new modelling technioues have been developed. The first techni que is a 'constant level technique', in which, the model has been kept at the constant level and other is a 'moving level technique' in which, as the name suggests, the model-level is driven from one level to the next aggregated level. When fcbe moving level technique chooses the observed distribution as the prior distribution, the estimated distribution predicts with reference to the individual trip having theoretically maximum dimensions equal to the total number of trips. This accounts for the pggregation-effect accrued to the observed, trips which otherwise is impossible to account for»as there are no means to trace them. The predictions obtained by using moving level technique are, thus, nearest to the absolutely true distribution^ ° . It has been observed that the spatial aggregation has also an effect on the working of the model itself. It was observed that in using these techniques, the modelling has not been indifferent to the aggregation phenomenon. The effects of the spatial aggregation and the influe nce of the scale ha*ebeen explored. It is attempted to under stand the same through analysing the differences in the VI information loss between consecutive levels of aggregation. By the thorough analysis of these differences in the information loss, obtained by using both the techniques of modelling, for a particular pair of aggregation level and the model level;; the pure aggregation loss and the pure scale effects hcve been isolated and quantified. It is an accepted fact that howsoever perfect a modal may be, it is ought to introduce some effect (error) in the predictions. As such, the model effect also can be isolated from the total aggre gated effect. The concept of isolation of the model-effect achieved as a result of these explorations, may open a new field of investigations, regarding the design of modelling technique; in spite of the fact that a lot of research work has been conducted in this direction. The strategy of iso lation and '• quantification is expected to give much insight in understanding the aggregation phenomenon and the spatial interactions. It is the pattern of the loss of information that suggests the appropriateness of the level of aggregation. The qualitative and the quantitative variational pattern of the information loss can give inferring insights regarding the pattern of variation in data, due to the aggregation and regarding the influence of the scale on the inference which is likely to be drawn from the observations of such a pattern. It can be observed that implicit in the information loss is the modelling technique utilised. Thus, it can be seen that Vll in understanding the problem of the aggregation, the role of the theory defining the phenomenon, the quantifying measures, and the modelling technique or methodology are ground together in yielding the inferences regarding the optimality of the level, scfle effect and aggregation and modellingtechnique effect. Untill now,the canonical form of the measure formu lating a model is used for only establishing the link bet- (1 h) ween the data and model parametersv . Such a link has been established and the spatial variation in the data as well as the shift of the model parameters from level to level hcve been observed as a part of the investigations of the effects of spatial aggregation in the model parameter, JE-ut the un-noticec suggestive power of the canonic pi form discovered is that the canonical form of the minimum crossentropy measure suggests that the adopted measure was missing some form of prior information regarding the marginal probabilities at origins and destinations and should have been included in it as a part of prior information. Inter preting the insights obtained from the canonical form, the cross-entropy model was modified and a modified family of spatial interaction models was derived and further investi gated and the performances of both families of models have been compared. The very strength of the canonical form which - in the manner of a feed b?ck - investigates into the appropriateness of the measure for the given prior informa tion and if needed proposes the nature of the modifications, VI11 if any,available information was not initially accounted for, may become another area of further investigations in the field of spatial modelling. In the final part of the thesis, the modified methodology, for modelling the spatial interactions, in the context of the effects of spatial aggregation, has been investigated empirically using travel-data from Reading Region, (U.K.), and Pune Metropolitan Are&£India). These results have been presented and the observations have been discussed and interpreted in the light of the theory developed in this thesis. The conclusions regarding these observations have been drawn and the scope for the further research haSc been presented.