Analytical Error Propagation in Four-Step Transportation Demand Models

Transportation demand models currently lack a rigorous and analytic treatment to quantifythe error propagation from different sources through the models. The error of trafficforecasts is attributed to two main sources: the model specification error and the inputvariable measurement error. Since Four-Step Transportation Demand Model (FSTDM) iscommonly used in practice but its error is not well-studied, the first part of the current studyillustrates how the errors of the input variables as well as of the model specification arepropagated analytically step by step and how these errors interact to result in inaccuratetraffic forecasts. The proposed approach is able to quantify separately and collectively the share of differentsources of error in the traffic forecast error. The proposed procedure is an efficient methodthat is less time consuming than existing simulation-based methods. This enables theproposed procedure to analyse the sensitivity of the traffic forecast to the inputmeasurement error and the quality of modelling in large scale networks. Moreover,comparing the output errors using the proposed approach with the acceptable ranges oferror specified in transportation guidelines, decision makers will have a clear opportunity torealise the credibility of a point traffic forecast and its associated variance.The proposed approach derives the variance from calibrated models in each of the foursteps, to obtain the variance of the output based on the variance of inputs. The resultingvariance formula provides an analytical expression to estimate the forecast errors from theinput errors. In addition, the model specification error of each step of the FSTDM is addedto the propagated input measurement errors. The proposed approach is applied to the cityof Brisbane as a case study spanning the four-step models for eight different trip purposes.As an example, a measurement error of 10 percent for the input variables of the BrisbaneFSTDM (BFSTDM) as well as the specification errors of models calibrated for the HomeBased Work - Blue collar (HBWB) trip purpose were explored. The model specificationerror produces variances of 1760.77 (trip/h)2, 976.72 (trip/h)2, 0.01082 (trip/h)2 and0.001327 respectively for trip production, trip attraction, trip distribution and modal splitsteps. Subsequently, the variance of output errors for the same steps are respectively, onaverage, 2885.50 (trip/h)2, 7218.70 (trip/h)2, 0.25 (trip/h)2 and 0.18. The variance of outputerror in the traffic assignment step is calculated to be 2097.20 (veh/h)2 for all trip purposes,while the model specification error of the same step is 1056 (veh/h)2. Having the existing 868 traffic zones, from the first to the third step, a reduction in the variance of trips perorigin-destination (O-D) pair is observed. At the same time, in the traffic assignment step,considering all trip purposes, the size of the forecast error variance per link increases.In the second part of the present study, the specification error of a user equilibrium trafficassignment (UETA) is measured using validation techniques. Moreover, the propagation ofO-D demand measurement errors to the UETA output is investigated using two differentmethods: the proposed analytical sensitivity-based method and a simulation-basedmethod. The analytical method uses the results of a sensitivity analysis (SA) on the UETAmathematical program, while the simulation-based method runs a Monte Carlo Simulation(MCS). The proposed method for error propagation is applied to an illustrative example to addressthree main questions: the number of samples that ensure a reasonably accurate result forthe MCS method; the size of the O-D demand measurement error for which the analyticalmethod is valid; and, the share of the path flow rate variance and covariance from thevariance of the O-D demand measurement error.