2.5 Aggregate Models 1 Overview
2.5.4 Direct demand models:
Also known as aggregate simultaneous models, these are econometric models which forecast the number of trips between specific station pairs as a function of a vector of explanatory variables. They effectively combine the first three stages of the four stage demand model (Preston, 1987), calculating trip generation, trip distribution and modal choice (although only giving forecasts for those passengers who choose rail). Such models do have some foundation in demand theory, as their general form can be derived by using the utility maximisation principle as demonstrated by Kanafani (1983) and summarised here. An individual originating from location i has a utility function (2.6) associated with the satisfaction of a particular trip purpose, and this function depends on the trips taken by the individual to the various destinations which can satisfy this purpose. These trips have associated costs, and if it is assumed that trips to different destinations are made
independently and for each trip the traveller incurs a cost, then the total cost can be calculated using (2.7).
(
Xi Xi Xin)
U U = 1, 2,..., (2.6)∑
= j ij ij X c C (2.7) Where:Xij is the number of trips made by the individual from origin i to destination j, where there are in mutually exclusive destinations (j=1, n)
C is the total travel cost
cij is the cost of travelling from origin i to destination j
The utility maximisation principle states that the individual will select the values of Xi in order to maximise U subject to a constraint on total travel costs, which Kanafani (1983) shows leads to the general result (2.8), meaning that the marginal utility and marginal cost of additional trips to a destination should be equal. (2.8) can be seen to represent the most basic demand model of trip distribution.
ij ij X C X U ∂ ∂ = ∂ ∂ for all j (2.8)
To use this model it is necessary to make assumptions about the form of the utility
function. Beckmann & Golob (1970) suggest that U should be assumed to be the constant elasticity function (2.9), which can be substituted into (2.8) to give the result (2.10) for Xj.
∑
= j j jX U α ρ 0<ρ≤1 (2.9) ρ ρ α − = 1 / 1 ij j j c X (2.10) Where:α and ρ are parameters
Letting the attractiveness of destination j (Bj) equal (αjρ)1/1-ρ and γ equal (1/1-ρ) gives the demand function (2.11) which shows that the number of trips made to destination j decreases with the cost of travel to j and increases with Bj (Kanafani, 1983).
γ −
= j ij j B c
X (2.11)
If the total number of individuals at location i with similar utility functions to (2.9) is denoted by Ai, then the aggregate demand function (2.12) is obtained by adding up all the individual functions. This function (2.12) is the general form of the direct demand model.
γ − = i j ij ij AB c T (2.12) Where:
Tij is the number of trips from location i to location j
For rail demand analysis it is usually assumed that i and j are stations, with the terms within the function expanded to include the various components of the generating potential of station I and its catchment, of the attractiveness of station j and its surroundings, and of the generalised cost of the rail journey between the two stations and of other travel modes available over the same route.
Such direct demand models can be calibrated as time series models, with demand regressed against time series data for a single origin-destination pair, but to predict the demand for new stations it is necessary to use cross-sectional models with demand regressed against data for a single period across a matrix of O-D pairs (Lythgoe, 2004), or alternatively pooled time series and cross-sectional data. In recent years these models have become the preferred model form in the UK, with for example Lythgoe (2004) suggesting a number of possible refinements. The PDFH contains a range of direct demand models for interurban and parkway stations, but only gives a single model, based on data from 1981/2, for local stations (ATOC, 2002).
A number of possible functional forms can be used, with the final form determined during calibration depending on the level of data available. Direct demand models are usually calibrated using multiple regression to explain existing demand levels between a matrix of stations. In the past there have been severe problems obtaining suitable calibration data, with for example Preston (1987) having to combine information from a number of sources to produce a usable dataset. However, the availability of computerised ticket sales data through the CAPRI system and its more sophisticated successor, LENNON, mean that obtaining data on rail use is now less of a problem (Preston 2001).
Direct demand models have been widely used to predict rail demand overseas, but these applications tend to be for inter-urban rather than local travel and use time-series rather than cross-sectional or pooled data, as illustrated by the following examples.
• McDonough (1973) used a regression model to predict the short-run demand for
commuter rail services in Boston, USA. No serious attempt was made to specify catchments for the stations involved.
• Talvitie (1973) used constrained linear and logarithmic cross-sectional regression
models to predict the number of work trips by public transport between zones in the Chicago area. This was based on a small sample of data from a disaggregate travel survey.
• McGeehan (1984) used a simple aggregate model to forecast short-run demand
changes on Coras Iomparr Eireann (CIE) interurban services in the Republic of Ireland.
• Doi & Allen (1986) used linear and logarithmic time series regression models to
Philadelphia, USA, and found significant seasonal variations in demand.
• Hadj-Chikh & Thompson (1998) estimated ridership levels between commuter rail
stations in South Florida using a direct demand model based on catchment populations and the distance between stations.
• Walters & Cervero (2003) developed a logarithmic regression model to forecast
ridership on potential extensions to the BART network in San Francisco, USA. This was used together with a conventional 4-stage travel model to account for macro-level travel patterns and micro-area sensitivities resulting from alternative project options.
• Kuby et al. (2004) produced a multiple regression model of light rail boardings
calibrated on data from 268 stations in nine US cities, using a raster-based GIS algorithm to define catchments (see Upchurch et al., 2004).