Causes of Spatial Non-Stationarity

Fotheringham et al. (2002) identified three main reasons for spatial variations in model parameters, which are sampling variation, intrinsic differences across space in relationships between variables, and poor specification of model form. The latter two illustrate a

fundamental schism in social scientific theory, which merits brief consideration here as it can be argued that the reasons are in fact related.

3.2.2.1 Sampling variation

The parameters vary depending on the sample of data used. For example, a rail demand model calibrated using LENNON data might have different parameter values from one calibrated using travel survey data. Such variation is likely to result either from

deficiencies in data collection or from inconsistent model implementation, and while not related to any underlying spatial processes can still complicate the identification of these processes (Fotheringham, 1999).

3.2.2.2 Relationships between variables intrinsically different across space

attitudes or preferences across space. For example, the provision of wireless internet facilities on local trains might increase patronage more in one area than in another because travellers in the first area valued this provision more highly than those in the second area, perhaps because of a higher propensity to work whilst in transit. While place is an important factor in understanding behaviour, all this factor provides on its own is a

description of the spatial variation that exists. It does not give any explanation of why this variation exists, and it has been argued “that the magnitude of contextual variations will be inversely proportional to the adequacy and completeness of the underlying model of relationships between individual attributes” (Hauser, 1970). There are obviously spatial variations in the relationships between the variables affecting rail demand, but ‘space’ is not a causal factor in itself, merely a proxy for societal factors which are not captured by the model. To state that spatial variations in parameters occur because they are affected by indefinable differences in ‘space’ is to take a backward step from global models which hide these variations, as they at least attempt to explain relationships between variables at some level. This means that it is important that local analysis methods are used as a tool to identify further model variables, to explain spatial variations in parameters and to enhance model performance rather than just as a way of identifying variability. While it is

impossible to identify and quantify every causal factor behind variations in rail demand, virtually all this variation does still result from a definable cause, however micro-scale this cause may be. Fotheringham et al. (2002) suggest that, within a postmodern framework, the identification of local variations in relationships would be a useful precursor to more intensive studies that highlight why such differences occur, but this makes it difficult to see how this cause of variation differs from the third reason (poor model specification).

3.2.2.3 Poor specification of model form

Variations in parameter relationships may occur because the model from which these relationships are estimated is a misspecification of reality with one or more of the

independent variables affecting rail demand being omitted from the model or represented by an inappropriate functional form. This is the most likely cause of parameter variation in local rail demand models as several important causal factors are difficult to model

effectively. For example, bus competition may mean that models over-predict rail demand from inner city areas where bus use is high, and under-predict demand from outer suburban areas where bus services are less attractive, but modelling this is not straightforward. Another example is micro-level access to stations, where a station might have a large

catchment population and a good level of service but because the only means of access was along a poorly lit footpath or through a run-down industrial estate passenger usage would be much lower than might be expected. The fear of crime can be a significant factor in dissuading potential rail users (Cozens et al., 2003), and while such micro-level variation is almost impossible to quantify it has the potential to distort global model calibration if it is not taken into account.

While this is a positivist point of view, it is actually only an extension of the approach outlined in Section 3.2.2.2, which aims to include the factors responsible for the spatial non-stationarity of the previously considered variables in the model. If local analysis shows that there is after all some intrinsic variation in levels of rail demand over space which cannot be explained in terms of any other variable then this analysis will still increase our understanding of its precise nature. Local modelling can be seen as the statistical equivalent of a microscope, which reveals much additional detail (Fotheringham et al., 2002).

It is indeed questionable whether any rail forecasting model is truly global, as no models treat the area served by the railway network as an unvarying plane where a railway station provided with certain facilities and service frequencies would generate the same number of trips wherever it was located. Even the most basic models include the size of the

population inhabiting an area, which is just as much a local feature as the area’s socio- economic or cultural composition. Therefore all ‘local’ models actually do is include more detailed local variation than is considered by so called ‘global’ models.

It is important to note that the ‘local modelling’ discussed here refers to local adaptations of aggregate models, as opposed to disaggregate approaches which it could be argued are even more ‘local’. Fairly extensive disaggregate analysis of rail demand has been

undertaken (see Section 2.4), and in some cases this has included spatial stratification, with for example Sheskin & Stopher (1988) examining the difference in attitudes to transit services between rural and urban areas.

3.2.3 Types of local multivariate analysis