 Original Paper
 Open Access
Modelling travel time distribution under various uncertainties on Hanshin expressway of Japan
 Ravi Sekhar Chalumuri^{1}Email author and
 Asakura Yasuo^{2}
https://doi.org/10.1007/s1254401301113
© The Author(s) 2013
Received: 5 October 2012
Accepted: 3 July 2013
Published: 31 July 2013
Abstract
Purpose
Studying travel time distribution or variability in travel time is very much useful in travel time reliability studies of transportation system. The properties of this distribution are described by various uncertainties which are derived from supply side, demand side and other external factors of any road network.
Method
Present study investigates the development of stochastic response surface of travel time variation under uncertain factors of traffic volume and intensity of rain fall by using Stochastic Response Surface Method (SRSM).
Analysis and Results
This model was applied to a section of Kobe Route (Nishnomiya to Awaza), Hanshin expressway, Japan. Besides hourly traffic volume data, incident data for the entire year 2006 were collected and multiple linear regression analysis for entire year data was initially performed to know the functional and significance relation between the input and output variable. Further SRSM analysis has been carried out for working days data. Results shows that travel time distribution obtained using SRSM model is better than distribution obtained by the regression model.
Conclusion
It was observed from the results that SRSM model is efficient for analyzing the stochastic relation between the response variable and uncertain explanatory variables.
Keywords
 Stochastic response surface method
 Travel time distribution
 Hanshin expressway
1 Introduction
Travel time distribution or variability in travel time is the most useful indicator to measure the performance and reliability of a transportation system. The properties of this distribution are described by various uncertainties which are derived from supply side, demand side and other external factors of a particular road network. Width in travel time distribution indicates higher uncertainties and lower travel time reliability. The measure of central tendencies of travel time distribution is unable to explain the traveler’s experience. Recently, various empirical travel time reliability studies [4, 11, 17] and Asakura [2] have extensively used travel time distribution as a tool for developing various reliability indices such as Planning Time (95 % travel time), Buffer Time Index and Planning Time Index [14]. All these reliability indices are useful to improve regional transportation planning [10].
When we intend to evaluate the effects of a transport policy on travel time reliability, it is necessary to identify the factors (source of uncertainty) that will affect travel time, and relation between the various sources of travel time. In this section various existing studies related [1] to sources of travel time variation are reviewed. Very few studies have concentrated on quantifying sources of uncertainties making travel time unreliable. Vander loop identified the main causes of unreliability of travel times for Netherlands urban roads. According to his study, 74 % of unreliability in the travel time is mainly due to internal factors of the traffic. The remaining is due to weather (8 %), road works (14 %), accidents (3to12 %) and combination factors (2 %) [18].
The US, Federal Highway Administration (FHWA) has identified seven sources of events which cause travel time variation. Further they have categorized into three main events such as traffic influence events (includes traffic incidents, work zones and weather), traffic demand events (includes fluctuations in normal traffic and special events) and physical highways features (includes traffic control devices and bottle necks) [3]. Ruimin [16], examined travel time variability under the influence of time of day, day of week, weather effect and traffic accident. In that study, the author quantified sources of travel time parameters with the help of multiple linear regressions with two way interaction models. In another study, Florida department of Transportation (Florida DOT) developed empirical travel time variability models such as function of frequency of incidents, work zones and weather conditions. For this, they have considered regression analysis on combination of different scenarios of uncertainty sources [5]. Asakura [2] further categorized the sources of travel time fluctuations in to three factors which are from demand side such as day to day traffic variation, supply side such as road closure due to accidents and external factors such as adverse weather effects and natural disaster. Most of the studies in the literature used deterministic approach to model travel time variation under the influence of various factors from supply side and demand side of the system. Travel Time variation on Hanshin expressway, Kobe route is mainly due to traffic volume, traffic accidents and amount of Rainfall.
The present study is an attempt to model the travel time distribution under various uncertainties. For this, Stochastic Response Surface Method (SRSM) has been adopted. SRSM [6] is an extension of classical Response Surface Method (RSM) to systems with stochastic inputs and outputs. The motivation for considering this model over the traditional Multiple Linear Regression (MLR) and other deterministic approaches is that both of these models fail to map stochastic behavior between response variable and explanatory variable in the system of uncertainty of travel time variation. In particular, two continuous probabilistic random factors were considered in this paper, one is traffic volume and the other is intensity of rain fall.
Archived continuous supersonic vehicle detectors data of Kobe Route on Hanshin expressway network in Japan were considered in this study. Travel time has been estimated for the study corridor by considering time slice method. Traffic incident data was collected for the same study period to model the travel time variation under various uncertainties. The same data has been considered to develop the traditional statistical model such as regression model and stochastic models. The comparative evaluation was made between these modeling approaches.
2 Study area and data collection
2.1 Study area
2.2 Data collection
3 Travel time estimation
where t_{ i } (s) denotes the travel time of section “i” at a given time “s”.
4 Modeling travel time distribution
4.1 Multiple linear regression analysis
MLR estimated coefficients for the study area
Variable  Entire year(365) data (Samples: 8364)  Working day (249) data (Samples: 5675)  

Coefficients  t Stat  Coefficients  t Stat  
Intercept  421.38  52.53  439.44  43.61 
Traffic Volume (veh/hr)  0.27  44.93  0.28  38.58 
Traffic Accident (Y = 1/No = 0)  241.75  15.37  244.94  13.04 
Road Works (Y = 1/No = 0)  7.09  0.62  12.66  0.97 
Breakdown (Y = 1/No = 0)  33.87  2.35  5.47  0.31 
Cleaning (Y = 1/No = 0)  25.17  1.74  16.37  0.98 
Other Incidents (Y = 1/No = 0)  115.68  3.30  134.15  3.33 
Rain Fall (mm/hr)  20.77  5.65  22.58  5.31 
Similarly MLR analysis carried out for working days data (249 days) to understand the effect of incidents on travel time variation during these days. From observation of t stat values of traffic volume, traffic accident, rain fall and other incidents on working days have high magnitude of significance in travel time variation (t stat value is greater than the critical value of 1.64 at 5 % level of significance). Road works and road cleaning incidents generally taken place on non working days except during emergency cases on Hanshin Expressway. Analysis of Variance (ANOVA) of this model having high F value (275.20) and with very low probability value (p < 0.005) demonstrates a very high significance for the regression model. The goodness of fit of the model R^{2} value indicates that 25 % of the total variation is explained by this model. From Table 1, it was concluded that the traffic volume, traffic accidents and rain fall incidents are highly significant for travel time variation on this section of Kobe Route.
Model coefficient estimated by multiple regression analysis
Model  Coefficients  t Stat 

Intercept  440.10  39.71 
Traffic Volume (veh/hr)  0.28  34.10 
Rain fall intensity (mm/hr)  24.40  5.06 
Model coefficient estimated by non linear analysis
Model  Coefficients  t Stat 

Intercept  473.543  22.52 
Traffic Volume (TV) (veh/hr)  0.203  5.024 
Rain fall intensity (RF) (mm/hr)  28.081  1.993 
TV^{2}  2.94E05  1.779 
RF^{2}  −3.922  −4.732 
TV*RF  0.029  3.174 
Where TV Traffic Volume (veh/hr) and RF is Rain fall intensity (mm/hr)and β_{0} to a β_{5}
Further these model coefficients were considered for estimating travel time for the collocation points (Table 6) generated for 2nd order polynomial equations. The following sections discuss the SRSM analysis for working days data.
4.2 Stochastic response surface method
4.2.1 Methodology
Stochastic Response Surface Method (SRSM) [6, 7] is an extension to the classical deterministic response surface method (RSM). RSM is a collection of mathematical and statistical techniques that are useful for the modeling and analysis of problems in which response of interest is influenced by several variables [13]. RSM also quantifies relationship among the measured responses and the input factor. The main difference between RSM and SRSM is the way the input parameter are supplied. SRSM is one of the ideal conventional sampling based method for uncertainty analysis and this is accomplished by approximating both inputs and outputs of the uncertain system through stochastic series of wellbehaved standard random variable (srv). The series expansion of the outputs contains coefficients that can be calculated from the results of limited number of model simulations.

Step1 Representation of stochastic model inputs: For each uncertain input, corresponding srv is assigned and the input random variable is expressed in terms of the srv. If the input random variables are mutually independent, the uncertainty in the ith input variable X_{i} is expressed as a function of the srv.${\mathrm{X}}_{\mathrm{i}}={\mathrm{f}}_{\mathrm{i}}\left({\xi}_{\mathrm{i}}\right)$(4)

Step2 Functional approximation of model output: Each model output is expressed as a series of expansion in terms of srv as a multidimensional hermit polynomial with unknown coefficients. A Second order polynomial approximation is generally recommended in the literature. Also this approximation can be refined further using higher order terms depending on the accuracy needs. In this study second order polynomial function with two independent variable ξ_{1} andξ_{2} were considered and the mathematical expression was presented at Eq. (5)$\mathrm{y}={\mathrm{a}}_{0}+{\mathrm{a}}_{1}{\xi}_{1}+{\mathrm{a}}_{2}{\xi}_{2}+{\mathrm{a}}_{3}\left({\xi}_{1}^{2}1\right)+{\mathrm{a}}_{4}\left({\xi}_{2}^{2}1\right)+{\mathrm{a}}_{5}{\xi}_{1}{\xi}_{2}\phantom{\rule{2.75em}{0ex}}$(5)

Step 3: Estimation of unknown coefficients in functional approximation: The unknown coefficients in Eq. (2) are estimated by equating model outputs with the corresponding polynomial expansions at a set of possible collocation points. Preferably next higher order of functional approximation routes to be considered for the generation of collocation points [15].

Step 4: Calculation of the statistical properties of model outputs: The model outputs are estimated followed by the estimation coefficients. The statistical properties of the outputs such as probability density function, moments of “y” can be readily calculated. This can be accomplished by generating large number of the srvs and the calculation of the values of inputs and the outputs from the transformation of Eqs. (4) and (5)
4.2.2 SRSM analysis
Uncertainty ranges of model parameter and response variable
Parameter  Traffic volume (veh/hr)  Rain fall (mm/hr)  Travel time (seconds) 

Minimum value  190  0  503 
Maximum value  2429  19  4383 
Average value  1222  0.20  784 
Standard deviation  589  0.97  350 
Distribution type  Lognormal  Exponential  
Distribution parameter  μ = 6.95; σ = 0.61  λ = 5.23 
Second order SRSM model was considered to approximate the response of travel time (Eq. 5). In order to solve for the second order polynomial expansion, the roots of the third order hermit polynomial, $+\sqrt{3},\surd 3$ and zero are used. The points are selected such that each srv takes the value of either 0 or one of the roots of the polynomial. Therefore there are nine possible collocation points they are $\phantom{\rule{0.25em}{0ex}}\left(0,0\right),\left(\surd 3,0\right),\left(0,\phantom{\rule{1em}{0ex}}\surd 3\right)\phantom{\rule{0.5em}{0ex}},\left(\surd 3,0\right),\left(0,\surd 3\right),\phantom{\rule{0.5em}{0ex}}\left(\phantom{\rule{0.25em}{0ex}}\sqrt{3},\surd 3\right),\left(\surd 3,\surd 3\right),\left(\surd 3,\surd 3\right)\mathit{and}\left(\sqrt{3},\surd 3\right)$.
Set of model input points for traffic volume and rainfall at the points were generated by using transformation technique and presented at Table 5. For lognormal distribution exp(μ + σξ_{1}) and for exponential distribution $\frac{1}{\lambda}$ log $\left(\frac{1}{2}+\frac{1}{2}\mathit{erf}\left(\frac{{\xi}_{2}}{\sqrt{2}}\right)\right.$ was considered [6] where erf is a error function.
Collocation points for 2nd order polynomial equations and corresponding travel time
Collocation points  Traffic volume (veh/hr)  Rain Fall (mm/hr)  Travel Time (seconds)  

ξ _{1}  ξ _{2}  
0.000  0.000  1045  0.133  736.32 
1.732  0.000  2996  0.133  1282.57 
0.000  1.732  1045  0.008  733.30 
−1.732  0.000  365  0.133  545.70 
0.000  −1.732  1045  0.608  747.83 
1.732  −1.732  2996  0.608  1294.08 
−1.732  1.732  365  0.008  542.73 
1.732  1.732  2996  0.008  1279.56 
−1.732  −1.732  365  0.608  557.26 
Model coefficients estimated by SRSM model
a_{0}  a_{1}  a_{2}  a_{3}  a_{4}  a_{5}  

Coefficients  797.015  212.709  −4.193  59.282  1.416  0.0001 
Once the coefficients are estimated the travel time distribution can be fully described by random generation of a large number of samples. In this study the 4180 random samples (same size original data) are generated for SRSM analysis. All this procedure was implemented in MATLAB environment [12]. Travel Time estimated by SRSM model and MLR models are compared against with actual travel time is presented at Fig. 6. From this figure it can be observed that SRSM probability distribution is unimodal (having one maximum at 625 sec), asymmetrical and similarly follows the actual travel time distribution. Whereas travel time distribution obtained by MLR models are bimodal frequency curves having two peaks, one maximum at 625 s and the other maximum at 925 s. Even travel time distribution estimated by MLR model by considering all the uncertainty parameters (Table 1) also follows bimodal frequency. From this it can be conclude that the MLR models are overestimating beyond the average actual travel time (783 s). It can also be concluded from the Fig. 5 that even if more uncertainty parameters are considered for modeling travel time MLR models are unable to follow the actual travel time distribution. Further, from travel time distribution it can be observed that travel time obtained by SRSM model is well distributed between travel time, 542 s to 2,302 s. Whereas MLR models estimated travel time distribution varies between 493 to 1,363 s. From this it can be concluded that MLR models are unable to map the worst case scenarios, this we can observe from tails of the probability distribution of travel time (Fig. 5).
From the above discussion of results, it was observed that SRSM models are capable to analyze the stochastic behavior of uncertain variable and also these models are performs better than the conventional regression model to model travel time distribution. The algebraic expressions in terms of standard random variable (srv) are smooth and continuous could efficiently model the tails of the probability distributions of the outputs (Fig. 5). This explains that the SRSM models are capable to model the worst case scenarios. Further the observable difference between the estimated distribution of SRSM model and actual distribution can be improved by increasing the number of uncertainty parameters in the model.
To validate the distributions obtained by both the models, chisquare nonparametric statistical goodness of fit have been carried out between actual travel time considered as observed frequency and travel time estimated by SRSM and MLR model considered as expected frequency and 30 s travel time intervals have been considered for frequency estimation. From the results it can be concluded that MLR models have higher estimated chisquare value (6415) than the SRSM models (2182). This emphasizes that MLR models have grater discrepancy between actual distribution and estimated distribution than SRSM model.
5 Conclusions
Travel time distribution is the most useful indicator to measure performance of any transportation system and properties of this distribution was influenced by various uncertainties which are derived from supply side, demand side and other external factors of any transportation system. Regression analysis between travel time and various uncertain parameters were considered to develop the functional relationship among them. From the tstatistic value it was observed that the effect of traffic volume, traffic accidents and amount of rain fall influence is quite significant on Hanshin Expressway study area. Further, SRSM models were applied in this study to resolve a probabilistic analysis. The uncertain parameters considered in this analysis are traffic volume and rain fall intensity for modeling travel time distribution. The travel time distribution obtained by SRSM model was compared with regression models and observed that SRSM model is better than the regression model and also following the actual travel time distribution. Further the difference between the estimated distribution by SRSM model and actual distribution may be improved by increasing the number of uncertainty parameters in the model.
Declarations
Acknowledgments
The authors would like to express their great appreciation to the HanShin Expressway Public Corporation for providing the data and other required information. Authors are also thankful to the Dr. Purnima Parida, Head, Transportation Planning division and S. Gangopadhyay, Director of CRRI to publish this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
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