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Uncertainty can be found at all stages of travel demand model, where the error is passing from one stage to another and propagating over the whole model. Therefore, studying the uncertainty in the last stage is more important because it represents the result of uncertainty in the travel demand model. The objective of this paper is to assist transport modellers in perceiving uncertainty in traffic assignment in the transport network, by building a new methodology to predict the traffic flow and compare predicted values to the real values or values calculated in analytical methods. This methodology was built using Monte Carlo simulation method to quantify uncertainty in traffic flows on a transport network. The values of OD matrix were considered as stochastic variables following a specific probability distribution. And, the results of the simulation process represent the predicted traffic flows in each link on the transport network. Consequently, these predicted results are classified into four cases according to variability and bias. Finally, the results are drawn into figures to visualize the uncertainty in traffic assignments. This methodology was applied to a case study using different scenarios. These scenarios are varying according to inputs parameters used in MC simulation. The simulation results for the scenarios gave different bias for each link separately according to the physical feature of the transport network and original OD matrix, but in general, there is a direct relationship between the input parameter of standard deviation with the bias and variability of the predicted traffic flow for all scenarios.

Forecasting of travel demand represents the fundamental step of planning and management of transportation facilities [

The main goal of travel demand model is traffic forecasting in different stages; generation, distribution, and assignment are to determine future values of the model output variables that are associated with a specific combination of input variables [

Although many researchers have studied uncertainty in the travel demand models, only a few studies analysed the impact of the error propagation in whole four-stage sequential transport model frameworks. For example, Zhao and Kockelman [

Future year travel demand forecasting is not an exact science, and there are complicated underlying mechanisms that inherently generate uncertainty in the forecasts. Modelling these complicated mechanisms requires numerous variables and behavioural components whose variability may be poorly determined or simply ignored. In this case, it is illogical to take a single view of the future without considering the uncertainty in travel demand modelling. Thus, to provide more efficient and reliable transport solutions for future, transport analysts and planners have to observe and predict uncertainty in transport systems [

Uncertainty becomes relevant in transportation modelling not only in case of diverging views such as if risks are very high if the policy is controversial and if there are concerns about model limitations, but also in case of certain views: several points estimated based on different scenarios are given to account for uncertainty [

This study presents a new methodology for exploring nature of uncertainty propagation deriving from input OD matrix in a four-stage transport model using Monte Carlo simulation method. MC method was used to generate data for OD matrix for three types of probability distributions (i.e., normal, lognormal, and extreme value). Then, the generated OD matrices were entered into the VISUM software to predict traffic assignment attributes. Moreover, the final step was analysing the visualizing the variability and bias of predicted for traffic flow in the transport network links.

The current study contributes to the present literature of uncertainty in transportation planning, primarily by (i) developing a methodology to predict the uncertainty in transport network depending on the variability of input OD matrix, (ii) examining the uncertainty impact on transport model by using different probability distributions in the input data, (iii) adopting a new method to visualize the uncertainty according to a probability of occurrences, and (iv) investigating the probability distributions of output traffic flow on transport network depending on the probability distributions in input data.

A new methodology was developed for quantifying and characterizing predictive uncertainty in traffic assignment models. The structure of this work directly supports a visual segmentation of uncertainty for transport network to present error and bias in traffic volumes calculated by traffic assignment models. This methodology consists of five stages: (i) input stage; (ii) MC simulation process stage; (iii) analysis of predicted traffic flow stage; (iv) predictive uncertainty stage; (v) uncertainty visualization stage. The relationships connecting these stages of the methodology are presented in Figure

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The methodology of uncertainty prediction.

The principal task in predictive modelling is to estimate the behaviour of a modelling function, in this case, traffic assignment function

Monte Carlo (MC) method or random sampling method is a division of computational mathematics. It is created from the mathematics concepts for “the frequency approximates the probability.” When the solution for a problem is the occurrence probability of a certain event or an expected value of any variants, a testing method is used to obtain the occurrence frequency of an event or the average value for these variants. MC method is based on the probability model and according to the described process by this model. The results of the simulation test are approximate solutions [

In this stage, the simulation code has been written using both Matlab software and Component Object Model (COM) of VISUM software in purpose of predicting traffic flows on the transport network. The simulation code involves two processes: the first is the generation of OD matrix (

The second process is the process of entering generated OD matrix and the transport network (TN) in the VISUM to produce the traffic flow attribute for the road network from traffic assignment function.

This simulation is repeated for a finite number of iterations

One of the most important aspects of a simulation study is an analysis of simulation experiments. In this stage, the results of the simulation process are analysed for both statistical characterization and examining the fitting for the distribution (Goodness-of-Fit Test). The outcomes of the simulation process are traffic flow attributes for all links on the transport network. The number of obtained attributes is equal to the number of iterations

The statistical characteristics of predictive traffic flow are defined by two parameters: the average value results (

Meaningful quantification of data and structural uncertainties in conceptual travel demand modelling is a significant challenge in the transportation system. Heterogeneous, insufficient, unstable, and unsteady characterize data used to build OD matrix because of the difficulty in measurements, and the nature of the individual behaviour leads to variation of the parameter of OD matrix in one side. Besides limitations and lack of information in mathematical concepts and the structure in four-stage sequential transport model frameworks lead to variation in traffic flow in transport networks.

The predictive uncertainty is defined by joint consideration of the mean predictive error (i.e., statistical bias) and the predictive variability (i.e., statistical standard deviation). In this methodology, the uncertainty has been predicted by joining the traffic flow attribute from input stage (first stage) and the traffic flow attributes obtained by simulation processes (fourth stage).

The mathematical operations in this stage include both calculating bias in traffic flow as (

In any uncertainty quantification process, setting limits for the predictive uncertainty is required to increase understanding of the researchers to models behaviour in both bias and predictive variability. The GEH statistic has been used as a limitation of the bias in this study. The GEH statistic is a form of Chi-squared statistic that can be used to compare observed and modelled counts as (

GEH statistic bands less than 5 [

The allowed accuracy and variability are presented in (

The relationship between precision and accuracy.

The last stage of this methodology is uncertainty visualization. Uncertainty visualization is endeavouring to display data together with additional uncertainty information. These visualizations present a more complete and accurate interpretation of data for researchers to analyse [

Applications that use visual graphs and comparative figures to indicate information variability or draw levels of confidence in data values help analysts better understand and cope with uncertain information better than using digital tables and metadata [

There are different methods used to visualize uncertainty: statistical and probability-based visualization, point and global visualization, used colours, financial visualization, icons, ontology, lexicon, etc. [

The first step classifies the predicted traffic flow values into four cases according to bias and variability.

This case occurred when the predicted traffic flows (

This case occurred when the predicted traffic flows (

This case occurred when the predicted traffic flows (

This case occurred when the predicted traffic flows (

The second step of the visualization process is giving specific colour for each case of uncertainty. Table

This table showsthe classification of the four cases of uncertainties.

Variability limitations (Precision) | |||
---|---|---|---|

Low | High | ||

Bias limitations (Accuracy) | Low | Case I | Case III |

High | Case II | Case I |

The probability density curves of the predicted traffic flow for the four cases.

The probability density of predicted traffic flows addressed with the uncertainty cases.

The major role of this visualization of uncertainty is to give information about the level of uncertainty to the decision-makers. Based on these cases they can see how reliable the predictions of the model are and if they can accept the risks relating to the given predictive uncertainty. If predictions that are more precise are needed, then sensitivity analysis helps to identify the input data most dominantly influencing the predictive uncertainty. Then these input data should be measured more precisely.

The developed methodology has been applied in a small city, Ajka, located in Hungary. The results of case study were presented earlier in [

Study area (Ajka, Hungary), showing observed traffic flows on the transport network [

The required data include information about the physical feature of the study area, where the number of TAZ=25 and the number of links (

In this case, three types of probability distributions (PD) have been used:

In this simulation, 30 scenarios have been experimented. These scenarios have been grouped into three groups according to probability distribution (PD) type. And for each group, the parameter of standard deviation (

The results of the simulation process are represented by attributes of traffic flow for the links of the transport network. The number of these attributes is equal to the number of simulation iterations. Consequently, the outcomes for each link have been studied separately, by finding the statistical parameters for predictive traffic flows: both of the average value (

The relationship between the standard deviation for OD matrices (input) and the bias in the predicted traffic flows for link No. 6.

The relationship between the standard deviation for OD matrices (input) and the variability of the predicted traffic flows (output) for link No. 6.

The predicted uncertainty for the link No. 6, applying normal distribution scenarios [

The obtained data from the previous stage has been processed by Excel-sheet to find the uncertainty in traffic flows. The mathematical calculations determine the bias and variability for the predicted traffic flows, as well as the accurate limits for each link separately. For the link No. 6, the lower accurate limit (

Finally, the probability of the uncertainty cases for the predicted traffic flows in transport network has been visualized into bar-charts by merging Figures

The coloured bar-charts give a whole idea about the probability ratios of the uncertainty cases. In general, the percentage of green colour (Case I) means the probability of predicted traffic flow will lie within the allowed limits of accuracy and variability. And the percentage of yellow colour (Case III) means the probability of predicted traffic flow will lie still within the allowed accuracy but outside the allowed variability. Both percentages of green colour (Case I) and yellow colour (Case III) are located within the accuracy range, while the percentage of blue colour (Case II) means the probability of predicted traffic flow will lie outside the allowed accuracy but still within the allowed variability. And, the percentage of red colour (Case IV) means the probability of predicted traffic flow will lie outside the allowed limits of accuracy and variability. Both percentages of blue colour (Case II) and red colour (Case IV) are located outside the accuracy range.

Although the purpose of the research is visualization of the uncertainty, it is good to give more details by text. In the same example, for the link No. 6, concerning normal distribution scenarios as shown in Figure

Similarly, we can interpret Figure

The predicted uncertainty for the link No. 6, applying lognormal distribution scenarios [

Likewise, we can interpret Figure

The predicted uncertainty for the link No. 6, applying extreme value distribution scenarios [

Figures

Visualization of predictive uncertainties helps to understand the stochastic nature of predictions. To be able to make decisions based on model predictions, decision-makers should have information about the accuracy and precision of the predictions. If the predictive uncertainty of the model is not acceptable, then sensitivity analysis helps to identify the input data most dominantly influencing the predictive uncertainty.

In this paper, a new methodology has been presented to predict traffic flow and visualize the uncertainty in those predicted values. This methodology enables applying various scenarios showing the variation in traffic flow on transport network by supposing that the input values of OD matrix are varying according to a specific probability distribution. The importance of this methodology is permitting transport planners and decision-makers to monitor and identify which of the links suffers from bias and unexpected change in traffic volumes in the event of a change in the conditions of inputs OD matrix.

The algorithm of this methodology consists of two parts: the first part has been built on Monte Carlo simulation method to generate numerous OD matrices, and VISUM software for getting the traffic assignment on a transport network. The results of this part represent predicted traffic flows on each link of the transport network. These predicted traffic flows suffer from uncertainty in both a bias from the observed value and variability from the average predicted value, while the second part of the algorithm was designed to categorize the uncertainty of the predicted traffic flows into four cases according to variability and bias: Case I (low variability, low bias), Case II (low variability, high bias), Case III (high variability, low bias), and Case IV (high variability, high bias). Finally, the percentages of these cases have been visualized in coloured bar-charts. The percentage of each case represents the likelihood of occurring (i.e., the likelihood of the predicted traffic flow to biasing or varying depends on the percentages of these cases).

Finally, the methodology has been tested in a small study area using three main-scenarios; each of them has (i) different probability distribution (normal, lognormal, and extreme value) and (ii) 10 subscenarios different according to standard deviation parameter graded from 0.05 to 0.50. The obtained results of this study area showed that uncertainty in traffic flow is found on all links of the transport network but in different degrees, depending on the scenario’s parameters and the observed traffic flow.

The current case study shows that the effect of applying scenarios had the same simulation parameters for all zones. Future research will consider applying different simulation parameters in the same scenario according to land use characteristic of each zone and how the accuracy and precision of the predicted traffic flows can be improved once the case of uncertainty of the predicted traffic flow is known.

The data used to support the findings of this study are available from the corresponding author upon request. The software used in this study can be requested from the corresponding author.

The authors declare that there are no conflicts of interest.