Effects of Parameter Distributions and Correlations on Uncertainty Analysis of HCM Delay Model for Signalized Intersections

Transcription

1 Effects of Parameter Distributions and Correlations on Uncertainty Analysis of HCM Delay Model for Signalized Intersections Xiaojin (Jerry) Ji Ph.D. Candidate and Panos D. Prevedouros, Ph.D. * Associate Professor Department of Civil and Environmental Engineering University of Hawaii at Manoa 2540 Dole Street, 383, Honolulu, HI Telephone: (808) Fax: (808) pdp@hawaii.edu Accepted for presentation at the 2005 Annual Meeting of the TRB In consideration for publication in the Transportation Research Record Honolulu, Hawaii July 28, 2004 Revised November 3, 2004 Final Revision March 26, 2005 * Corresponding Author Paper Length 5,217 words in text 2,000 words in 8 exhibits 7,217 words total (max = 7500)

2 Ji and Prevedouros 2 ABSTRACT The uncertainty analysis of the HCM delay model often assumes parameter variances and distributions. In light of the difficulty in specifying distributions and estimating correlations, we investigated (i) whether assumed distributions have a major effect on the results, and (ii) the effect of correlations on the accuracy of delay estimates. Field data from one intersection approach in Hawaii and nine in Illinois were used. All the input variables in the delay model were considered uncertain, except for the analysis period which remained fixed at 15 minutes for consistency with the HCM. Simulation results showed that the confidence intervals of delay can be large even if the variability of each input parameter is small. The degree of saturation has a significant effect on the uncertainty of delay estimates for X 0.9. The standard deviation of input parameters is the main factor affecting the uncertainty of delay estimates. The probability distributions have a slight effect. Correlations among input parameters are often overlooked but they have a significant effect on the confidence intervals of delay estimates especially when the variability of input parameters is large and the input parameters are highly correlated. The frequency distribution of delay estimates is not normal; the shifted lognormal distribution provides a better statistical fit.

3 Ji and Prevedouros 3 INTRODUCTION The HCM 2000 provides a procedure to calculate the LOS, but it only produces a mean value as the measure of effectiveness for the facility without any statement about its accuracy. This mean value must represent a range of possible conditions due to the obviously stochastic nature of traffic and potential parameter and model errors. A discussion report of the Highway Capacity and Quality of Service (HCQS) committee (1) and an Internet survey (2) have indicated that most of the users wish to see confidence intervals around the mean value because knowledge of accuracy of the results would improve the decision-making process (2). Several methods are available for estimating the uncertainty caused by input errors. Input errors may be due to an inability to precisely quantify each input parameter or the inherent variability of each parameter. Accurate estimate of the uncertainty of output largely depends on the measurement of errors in input parameters. However, the information required to conduct an uncertainty analysis is often unavailable or incomplete. The standard deviation is an important statistic which characterizes the variability of input parameters. It can be estimated from large amounts of data or it can be estimated approximately from the lower and upper bounds of the input parameters (3). The coefficient of variation (COV) is often assumed by researchers to represent the variance if the standard deviation of the input parameters is not available. Results stemming from such simplifications are of limited usefulness. Information about distributions and correlations of input parameters is hard to derive since they can only be determined with large amounts of data. The distribution of input parameters is often assumed to be normal, lognormal or uniform (4, 5, 6). Some uncertainty analysis techniques do not require the specification of distributions such as the First Order Analysis (FOA) and the original Point Estimate Method (PEM) (7), but the underlying assumption of FOA is that the variables are all normally distributed (8, 9). This assumption may create large errors in the uncertainty analysis if the distributions of variables are significantly different from normal. A new PEM was proposed by Chang (10) to incorporate the marginal distributions of input parameters, which indicates that distributions of variables have some effect on the analysis. Another important issue for uncertainty analysis is the correlation among input parameters. Generally it is hard to approximate the correlations based on expert judgment. Empirical correlation estimates may be unstable due to small sample sizes. In practice, input parameters are often assumed to be independent to simplify the analysis. However arbitrarily assumed distributions or the disregard of correlations may produce inaccurate results. Past research on the uncertainty analysis of the HCM delay model assumed variances and distributions. Most past efforts focused on the effects of traffic volume (V) and saturation flow (s), whereas sensitivity analysis has shown that the progression factor (PF), cycle length (C) and effective green time (g) also are important factors affecting the uncertainty of delay estimates (11). In light of the difficulty in specifying distributions and estimating correlations, it is useful to investigate: whether assumed distributions of input parameters have a considerable effect on the results, and the effect of correlations on the accuracy of delay estimates. To this end, field data of uncertainty in input parameters were collected and used to perform uncertainty analysis of the delay model for signalized intersections in HCM2000. All the input parameters in the delay model,

4 Ji and Prevedouros 4 except for the duration of analysis period (T), were considered uncertain. The effects of the degree of saturation (X), standard deviations, distributions and correlations on the uncertainty of output were examined. This paper is organized as follows. A literature review is presented next followed by the methodology of analysis. The procedure of field data collection is presented next. The effects of the degree of saturation, standard deviations, distributions of input parameters and correlations among them are then investigated with numerical applications and summarized in the conclusions. LITERATURE REVIEW This section summarizes the methods used and assumptions about distributions, correlations and COV made in past research on uncertainty analysis of traffic engineering and other models. Park and Kamarajugadda (4) examined the uncertainty of delay caused by the day-to-day variation of traffic volume using an expectation method. Volume was first assumed to follow a normal distribution with a COV of 0.25 to estimate the confidence interval of delay. Various COV and different probability distributions including normal, lognormal, uniform and gamma were investigated. Since only one parameter was considered to be uncertain, no correlation information was involved. Zhao and Kockelman (5) used Monte Carlo simulation (MCS) to propagate the uncertainty in the input parameters through a travel demand model. Distributions of the input parameters were assumed to be lognormal and uncertainty in the input parameters was set to 0.30 in terms of COV. Chen et al. (6) used MCS to assess the capacity reliability of a road network. A uniform distribution with an upper and lower bound was assumed to generate the random arc capacities. The correlation among all arc capacities was assumed to be 0.5 to evaluate the effect of dependency of roadway capacities on the estimation of capacity reliability. Tarko and Tracz (12) conducted a MCS experiment to estimate the uncertainty of delay caused by the error in saturation flow with the COV assumed to be 0.1. Several papers in the field of environmental and public health present evaluations of the effects of distributions and correlations. Haan et al. (13) investigated the effect of parameter distributions on uncertainty analysis of hydrologic models. MCS was employed to propagate the uncertainty. Four commonly used distributions including normal, lognormal, uniform and triangular were assigned to the input parameters randomly. The original standard deviations were also increased by 25% and 50% to examine the effect of enlarged uncertainties. They concluded that good estimates of the means and variances of the input parameters were more important than distributions. Song and Brown (14) examined the effect of correlation among the input parameters on the output uncertainty of a water quality model. Results showed that correlations among input parameters had a large influence on the uncertainty of the output: The standard deviation of output with correlated inputs was 20% to 40% larger than that with independent inputs. Smith et al. (15) considered the more general situation of a set of correlated input variables for quantitative risk assessment. He derived a set of expressions for computing the variance of the sums of random variables with any distribution, and the mean and variance of the product of variables that are lognormally distributed. They

5 Ji and Prevedouros 5 concluded that substantial errors result if correlations are neglected, but there are circumstances when the dependencies could be ignored such as when weak correlations or correlations among variables with small variance are present. Bukowski et al. (16) extended the research by Smith et al. (15) to a more general condition by investigating the sum, product and division functions of two variables with a set of possible distributions including normal, lognormal, uniform and triangular and four levels of correlation: 0.0, 0.25, 0.75 and METHODOLOGY Monte Carlo Simulation (MCS) was selected for conducting the uncertainty analysis of the delay model for signalized intersections in HCM2000. MCS is the most widely used method for uncertainty analysis. It involves random sampling from the distribution of inputs and successive model runs until the desired accuracy of the outputs is reached. Not only the means and variances but also the distributions of input parameters are required to perform the MCS. Furthermore the correlation among the input parameters should also be considered in the simulation if they are correlated. The main advantage of the MCS is its general applicability to any form of functions. MCS has been criticized for its intensive computational requirement since thousands of repetitive runs of a model are needed to reach convergence and the evaluation of a complex model with a large number of sample sizes may take a long time. The computational challenges have been relaxed substantially with fast desktop computers. Another limitation of MCS is that rules determining the necessary number of repetitions for reaching convergence are not available. Tyagi (9) stated that the error in the estimation of the population mean from MCS is inversely proportional to the square root of the number of runs, which indicates that the accuracy of MCS is dependent on the sample size. Morgan and Henrion (17) described a way to estimate the runs needed based on the approximated variance and desired precision of the output. Robert and Casella (18) presented several theories to determine the sample size, one of which was suggested by Raftery and Lewis (19). A crude estimation from it implied that for a precision of 99.5%, the minimum size is about 15,000. Various sample sizes were used in the literature: Chen et al. (6) used 5,000 runs; Tarko and Tracz (12) used 1,000; Haan et al. (13) used 1,500 and Bukowski et al. (16) used 5,000 repetitions. DATA DESCRIPTION The westbound approach of Vineyard Blvd. at the actuated signalized intersection with Punchbowl St. in downtown Honolulu, Hawaii, was selected for the data collection site. It is the same site as used in (20), which includes a detailed description and geometric and signal characteristics for this approach. Two through lanes were selected as the subject lane group. Data from eight days, Monday to Thursday, in two consecutive weeks, were collected from traffic surveillance videotapes. Traffic volume was counted with the Autoscope using a one-minute interval from 6:00 to 10:00 AM to trace the peak hour on different days. The peak hour of traffic volume was found to be between 7:30 and 8:30 AM with only a five-minute observed shift among the days examined. The main reason of this is that the traffic is composed almost exclusively of daily commuters and this approach stems from a two-lane freeway off-ramp. Saturation headways, cycle lengths and greens were collected manually for 15 cycles between 7:30 and 8:30 AM.

6 Ji and Prevedouros 6 Correlations among input parameters were derived by combining these data with similar data from nine intersection approaches obtained in the mid-1980s in the suburbs of Chicago, Illinois. All ten approaches in the sample have 3.6 m (12 ft.) wide lanes, exclusive left turn bays, at least two exclusive lanes for the through movement, and actuated controls. The data from suburban Chicago, Illinois were collected in the field; about 20 cycles for each approach. The data from Honolulu were collected from surveillance camera tapes using an identical data extraction process. All data herein are from through lanes. Overall, data from all sites were consistently collected at similar high-design intersection approaches of intersections with traffic-actuated signalization. Mean and Standard Deviation All the input parameters in the HCM delay model, except for the duration of analysis period (T), are considered uncertain. They are saturation flow (s), traffic volume (V), peak hour factor (PHF), cycle length (C), green (g), progression factor (PF), incremental delay factor (k) and upstream metering factor (I). The values for s, C and g for each day were averaged from data from 15 cycles. PHF was calculated from 15-minute interval volumes. The final mean and variance of these variables were calculated from eight values (one per day). Daily means and variances were used instead of per-cycle data because only day-to-day variability was considered as the error in the input parameters. Values for the PF were estimated from HCM2000 and included in the analysis for illustration purposes given that arrivals on the examined approach are random (AT=3), e.g. traffic exiting the freeway. Arrival types were assumed to be AT=2 and AT=3 to model a potentially inaccurate determination of PF. Given that the mean green ratio (g/c) of the subject lane group is 0.47, a range of 1.00 PF was obtained from HCM. The mean value of PF was ( ) ( ) / 2 = /12 = and the variance was calculated from the formula assuming that the PF follows a uniform distribution. The subject lane group is controlled by an actuated traffic signal. The degree of saturation (X) for eight days ranged from 0.81 to 0.91 and the unit extension of the actuated signal is 4.0 sec. With these inputs, a range from 0.37 to 0.44 was obtained for parameter k from the HCM. The mean and standard deviation for k are 0.41 and 0.02, respectively. They were calculated in the same way as for PF. Since the subject approach stems from a two-lane freeway off-ramp, the traffic flow is not filtered by upstream signals. However the value for I was assumed to be less than 1.0 for illustration purposes. Given that the degree of saturation ranges from 0.81 to 0.91, a range of I was obtained from HCM. The mean and standard deviation for I are and 0.06, respectively. They were calculated in the same way as for PF. Distribution Since data from only eight days were available, it was not possible to derive accurate distributions for the eight input parameters. Little information about the distributions of these parameters is available in the literature. The Lilliefors test in (20) suggested that saturation headways are normally or lognormally distributed. Yi and

7 Ji and Prevedouros 7 Wolfenbarger (21) suggested that PHF is a normal random variable for 0.85<PHF<1.0. Traffic volume was assumed to follow a normal distribution in (4). Based on these findings, s, V and PHF were all set to follow a normal distribution for the base analysis. C and g were assumed to follow a uniform distribution. Only ranges of the values were available for parameters of PF, k and I, thus a uniform distribution was assumed (13). The final input data are shown in Table 1. Correlation Correlations among the input parameters were calculated from the eight-day hourly data, which are shown in Table 2a. Similar data from nine approaches of high design intersections with traffic actuated control were available from Illinois (22). The database from Illinois contains cycle-to-cycle data of s, V, C, g and PF. Each approach was investigated for about 20 cycles, which resulted in a total of 206 data points available for each input variable. The correlations calculated from the Illinois data are presented in Table 2b. Tables 2a and 2b indicate that most of the values are similar between the two sources. However, the signs of the correlation between V and C, and V and g are different. They are negative for the data from Honolulu while they are positive for the data from suburban Chicago. Intuitively the sign should be positive because it is expected that larger volumes should be served with longer cycle lengths and greens. The one intersection sample from Honolulu is likely misleading in this respect. A simple weighted average calculation was used to combine the Illinois and Hawaii correlations. Certain assumptions were made. The correlation between C and s was set to zero because at heavily loaded intersections during peak periods the signal is max out, therefore a correlation between C and s cannot be established. A positive correlation was set between C and V because a higher rate of arrivals (volume) forces the signal controller to increase affected green phases and thus increase cycle length. Li and Prevedouros (20) indicated that saturation headway tends to decrease at heavily congested conditions. So the strong positive dependency between s and V is expected. Since there is no information about the correlations of k and I, these were set to zero. The final correlations are shown in Table 2c. UNCERTAINTY ANALYSIS The base uncertainty analysis was performed using MCS with the data of Table 1, assuming that the input parameters are independent. A mean delay of 43.2 sec/veh resulted from 5,000 runs. The standard deviation is 4.26 and the 95% confidence interval is ±16.7 resulting in a lower value of 36.1 sec/veh and an upper value of 52.8 sec/veh both of which fall within LOS D. In order to check the convergence of delay with the 5,000 runs, the MCS was performed three times with 5,000 runs. The difference in the mean value of delay was always less than 1.0 sec/veh and less than 0.50 for the standard deviation. Thus 5,000 runs can provide stable and precise results for the uncertainty analysis of delay in this example. Of the 5,000 delay samples, 97.8% fell within LOS D with only 1.22% in LOS C and 0.94% in LOS E. The frequency histogram and cumulative density function are shown in Figure 1. The histogram of delay is bell-shaped with a positive skewness of 0.50, which implies that the lognormal distribution may provide a good fit.

8 Ji and Prevedouros 8 The Kolmogorov-Smirnov test was used to test the distribution of simulated delays. The maximum difference from the normal test is 0.037, which is greater than the critical value of at 95% level of significance (23). Thus the normality hypothesis is rejected. The shifted lognormal distribution was tested and it was not rejected at the 5% significance level. It should be noted that the subject approach of the study intersection is affected by upstream bottlenecks on the freeway and, as a result, it has stable traffic characteristics. Table 1 shows that all the COV, except for parameter I, are less than Sensitivity analysis indicated that parameter I did not have a significant effect on the uncertainty of delay (11). The small variability of input parameters still produces a wide confidence interval of ±16.7 sec/veh. This indicates that the combined effect on delay of the variability in the input parameters can be large. In order to do a more general uncertainty analysis of the delay model in HCM, the effects of saturation level, augmented standard deviations, randomly selected distributions and correlations among input parameters were evaluated. Effect of Saturation Level Generally delay increases dramatically when demand approaches or exceeds capacity. In order to vary X=V/s, the traffic volume was changed and the COV was kept unchanged. Uncertainty analysis was performed for each level of X ranging from 0.4 to 1.1, as shown in Table 3. Numbers in brackets represent the lower and upper 95% C.I. Table 3 shows that the difference of the mean delay estimates is small for X < 0.9. The standard deviation of delays estimates increases sharply for X 0.9. The corresponding confidence intervals cover LOS D, E, F when 1.0 X 1.1 uncertainty of delay for X The simulation results in Table 3 indicate that the level of saturation has a significant effect on the Effect of Standard Deviation Simulations were performed with augmented standard deviations. The base standard deviations of all variable input parameters, except for k and I, were increased by 20%, 50%, 100% and 150%. Parallel research on sensitivity analysis and variance decomposition showed that the effects of errors in k and I on the uncertainty of delay were not significant (11). The range of PF was determined based on the g/c ratio and assumed arrival type (AT). Simulations were performed for each standard deviation augmentation as shown in Table 4. The results show that the mean value of delay does not change appreciably until the amplification factor reaches 2.5. On the other hand, standard deviation, COV and confidence intervals of delay display significant increases for all levels of amplification. The resultant standard deviation for the delay increases fivefold with variance amplification at 2.5. The 95% confidence intervals cover three or four LOS. It is noted that the COV for all input parameters, except for PF, are small at less than 0.1 even when they are amplified 2.5 times. These results suggest that large standard deviations for the input parameters have a small effect on the mean value of delay and a large effect on the standard deviation and confidence interval of delay.

9 Ji and Prevedouros 9 Effect of Distribution The distributions of the input parameters must be specified to conduct uncertainty analysis with MCS. Due to the difficulty in determining the correct probability density functions of input parameters, assumptions relative to the distributions are often made (as described in the literature review section), which, in turn, raise concerns about the reliability of outputs. Four commonly used distributions were investigated: Normal, lognormal, uniform and triangular. A total of seven combinations, including the base, were investigated as shown in Table 5. The mean values and standard deviations of distributions were kept the same so that results from different combinations of distributions are comparable. Data for the lognormal distribution were generated directly from it with the base mean and standard deviation rather than generating them from a normal distribution and then converting them to a lognormal. The variance of triangular distribution is described mathematically in Ayyub (23) as follows. a,b, c a + b + c ab ac bc = 18 σ (1) Where and are lower limit, upper limit, and mode respectively. The triangular distribution was assumed to c a b be symmetric about its mean, so the mean value equals its mode,, and has the same difference to and. Thus the lower and upper limits of the triangular distribution can be determined from Equation (1), which are The lower and upper limits of the uniform distribution are similarly determined by µ m 3σ. µ m 6σ. Monte Carlo simulations were performed with mean values and standard deviations of input parameters as shown in Table 1 and distribution combinations as shown in Table 5 and no correlations among input parameters. The results are shown on the left part of Table 6. They show that mean values, standard deviations and the 95% CI of delay obtained from different combinations of distribution are almost the same. It appears that the distributions have a minimal effect on the uncertainty of delay, but this result may be influenced by the small standard deviations of input parameters. The same six groups of simulations were performed again after doubling the standard deviations. The results are shown on the right part of Table 6. Again, the different combinations of distributions did not cause much difference in the uncertainty of delay. Thus, assuming a normal, lognormal, uniform or triangular distribution for the input parameters does not appear to have a significant effect on the uncertainty of delay estimates. Effects of Correlation In past research, correlations were often assumed to be equal to zero to simplify the analysis because of the difficulty in determining them. However this assumption may produce inaccurate results. In this section correlations among input parameters are considered in the evaluation of the uncertainty of delay. The correlations shown in Table 2c were used. The simulation was performed with the base standard deviation and distributions listed in Table 1. To further examine the effect of dependencies, MCS was performed considering not only base correlations shown in

10 Ji and Prevedouros 10 Table 2c but also with these correlations multiplied by 0.5 and 2.0. If after multiplication the resultant value exceeded 1.0, then 0.99 was used. Results with and without accounting for correlations are presented on the left part of Table 7, which shows that the uncertainty of delay estimates decreases when correlations are accounted for and that the uncertainty in delay estimates decreases with increasing correlations. The mean values of delay are almost the same whether correlations are accounted or not, and they are not affected by the strength of the correlations. However standard deviation and 95% CI are affected. In the example herein, the confidence interval when the base correlations (Corr*1.0 in Table 7) are accounted for is narrowed by almost 30%, compared to the estimates without considering the correlations. To further evaluate the effect of correlations, similar simulations were conducted again with doubled standard deviations. The results are shown on the right part of Table 7. Results with doubled standard deviations indicate that the mean delay estimates remain largely unaffected. However, the standard deviation and 95% CI display a significant decrease when correlations are accounted for. The standard deviation of delay resulting when the base correlations are accounted for is almost half of the standard deviation of delay resulting when correlations are omitted. The standard deviation decreases with increasing correlations. The results in Table 7 suggest that correlations only have a slight effect on the mean value of delay, but they can decrease the standard deviation and the CI considerably, especially when the variability of input parameters is large and the input parameters are highly correlated. CONCLUSION Uncertainty analysis of the delay model for signalized intersections in HCM2000 was conducted using Monte Carlo simulation. All the input parameters in the delay model, except for the duration of analysis period, were considered uncertain. Field data from an intersection approach in Honolulu, Hawaii and nine intersection approaches in suburban Chicago, Illinois were used. All data examined are for the movement only on exclusive through lanes of traffic-actuated, high-design signalized intersections. Only the day-to-day variation of the input parameters was considered as the source of input errors. Simulation results from the base values show that the 95% confidence intervals of delay estimates can be large even when the variability of each input parameter is small. This highlights the importance of adding a measure of statistical confidence to the mean estimate of delay used in the HCM. Conclusions based on limited samples are listed below: The level of saturation has a significant effect on the uncertainty of delay estimates when the degree of saturation (X) is at or above 90%. This finding also appears in many a past research and in HCM The frequency distribution of delay estimates is not normal. The shifted lognormal distribution provides a better statistical fit: It was not rejected at the 5% significance level (K-S test). Large standard deviations for the input parameters have a small effect on the mean value of delay and a large effect on the standard deviation and confidence interval of delay.

11 Ji and Prevedouros 11 The distributions of input parameters does not have a significant effect on the uncertainty of delay and normal, lognormal, triangular or uniform distributions can be assumed with little impact on delay estimation accuracy. Correlations among input parameters are often overlooked by researchers but simulation results show that correlations have a significant effect on the accuracy of delay estimates. Correlations only have a slight effect on the mean value of delay, but they reduce standard deviation and confidence intervals especially when the variability of input parameters is large and the input parameters are highly correlated. Future research will address the effect of individual parameters in the saturation flow estimation and enrich these analyses with data from other sites. The development of a simplified methodology for adding approximate confidence intervals to HCM-based delay estimates for signalized intersections with limited sampling requirements of prevailing conditions and reasonably safe assumptions for parameter distributions and correlations is under way. REFERENCES 1. TRB committee AHB40 on Highway Capacity and Quality of Service, A Research Program for Improvement of The Highway Capacity Manual. December Tarko, A. and P. Songchitruksa. Reporting uncertainty in the highway capacity manual-survey results. Transportation Research Board, CD-ROM, National Research Council, Washington, D.C., Prasher, S. and S. Russell. First and Second Order Analyses of Parameter Uncertainty in Drainage Design. Transactions of ASAE, Park, B. and A. Kamarajugadda. Estimating Confidence Interval for Highway Manual Delay Equation at Signalized Intersections. Transportation Research Board, CD-ROM, National Research Council, Washington, D.C., Zhao, Y. and K. M. Kockelman. The Propagation of Uncertainty through Travel Demand Models Chen, A., Yang, H., Lo, H. K., and Tang, W. H. Capacity Reliability of a Road Network: an Assessment Methodology and Numerical Results. Transportation Research Part B 36, 2002, pp Rosenblueth, E. Two-point Estimates in Probabilities. Appl. Math Modeling, Vol. 5, 1981, pp Johnson, P. Uncertainty of Hydraulic Parameters. Journal of Hydraulic Engineering, Vol. 112, No. 2, Tyagi, A. A Simple Approach to Reliability, Risk, and Uncertainty analysis of Hydrologic, Hydraulic, and Environmental Engineering Systems. PhD Dissertation, Oklahoma State University, Chang, C., J. Yang, et al. Uncertainty Analysis by Point Estimate Methods Incorporating Marginal Distributions. Journal of Hydraulic Engineering, Vol. 123, No. 3, 1997, pp Ji, X., and P. Prevedouros. Comparison of Methods for Sensitivity and Uncertainty Analysis of Signalized Intersections Analyzed with HCM Paper Annual Meeting of the TRB, National Research Council, Washington, D.C., January Tarko, A. and M. Tracz. Uncertainty in Saturation Flow Predictions. Proceedings of 4th International Symposium on Highway Capacity, 2000.

12 Ji and Prevedouros Haan, C. T., T. AI-Issa, S. Prabhu, G. J. Sabbagh and D. R. Edwads. Effect of Parameter Distributions on Uncertainty Analysis of Hydrologic Models. American Society of Agricultural Engineers, Vol. 41, No. 1, 1998, pp Song, Q. and L. C. Brown. Do Model Uncertainty with Correlated Inputs. Journal of Environmental Engineering, Vol. 116, No. 6, 1990, pp Smith, A. E., P. B. Ryan and J. S. Evans. The Effect of Neglecting Correlations when Propagating Uncertainty and Estimating the Population Distribution of Risk. Risk Analysis, Vol. 12, No. 4, 1992, pp Bukowski, J., L. Korn and D. Wartenberg. Correlated Inputs in Quantitative Risk Assessment: The Effects of Distributional Shape. Risk Analysis, Vol. 15, No. 2, 1995, pp Morgan, M. G. and M. Henrion. A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis. Cambridge University Press, Cambridge, New York, Robert, C. P. and G. Casella. Monte Carlo Statistical Methods. Springer-Verlag New York, Inc., Raftery, A. E. and S. Lewis. How Many Iterations in the Gibbs Sampler? In Bayesian Statistics 4. J. O. Berger, J. M. Bernardo, J. M. Bernardo, A. P. Dawid and A. F. M. Smith (Eds). Oxford University Press, Oxford. 1992, pp Li, H. and P. Prevedouros. Detailed Observations of Saturation Headways and Start-up Lost Times. In Transportation Research Record: Journal of the Transportation Research Board, No. 1802, TRB, National Research Council, Washington, D.C., pp , Yi, P. and J. Wolfenbarger. Peak Flow Factor Characteristic. Transportation Research Board, CD-ROM, National Research Council, Washington, D.C., Jovanis, P., N. Rouphail, and P. Prevedouros. Design and Operation of Signalized Intersections in Illinois. Report for Illinois DOT. The Transportation Center, Northwestern University, Evanston, IL, Ayyub, B. M., and McCuen, R. H. Probability, Statistics, and Reliability for Engineers and Scientists, 2 nd Edition. Chapman & Hall/CRC, 2003.

13 Ji and Prevedouros 13 LIST OF TABLES Table 1. Input data for Uncertainty Analysis Table 2. Correlations Used for Uncertainty Analysis Table 3. Simulation Results with the Varied Demand Level Table 4. Simulation Results with Varied Standard Deviations of input parameters Table 5. Combinations of Probability Distributions Used Table 6. Simulation Results from Different Combinations of Probability Distributions Table 7. Simulation Results with Varied Level of Correlations LIST OF FIGURES Figure 1. Frequency histogram and cumulative density function of delay

14 Ji and Prevedouros 14 Table 1. Input Data for Uncertainty Analysis Input Mean Std. Dev. COV Distribution s Normal V Normal PHF Normal C Uniform g Uniform PF Uniform k Uniform I Uniform T 0.25 N/A N/A Fixed Value

15 Ji and Prevedouros 15 Table 2. Correlations Used for Uncertainty Analysis (a) Correlations Calculated from Hawaii data (1 approach) s V C g PHF s V C g PHF 1.00 (b) Correlations Calculated from Illinois data (9 approaches) s V C g PF s V C g PF 1.00 (c) Combined Correlations from Hawaii and Illinois s V PHF C g PF k I s 1 V PHF C g PF k I

16 Ji and Prevedouros 16 Table 3. Simulation Results with Varied Saturation Level X HCM delay Sim. Mean Std. Dev. 95% C.I. LOS [26.0, 35.6] C-D [27.6, 38.0] C-D [29.6, 40.6] C-D [31.8, 44.2] C-D [34.5, 48.9] C-D [37.6, 61.7] D-E [42.5, 105.5] D-E-F [54.1, 148.9] D-E-F

17 Ji and Prevedouros 17 Table 4. Simulation Results with Varied Standard Deviations of Input Parameters Amplification factor Sim. Mean Std. Dev. COV 95% CI LOS span based on CI [36.1, 52.8] D [34.6, 55.4] C-D-E [32.6, 60.8] C-D-E [29.9, 82.7] C-D-E-F [27.4, 113.6] C-D-E-F

18 Ji and Prevedouros 18 Table 5. Combinations of Probability Distributions Used Combinations Input Base s N U N Ln N Tri Ln V N U N Ln N Tri Ln PHF N U N Ln N Tri Ln C U U N N Ln U Tri g U U N N Ln U Tri PF U U N N Ln U Tri k U U N N Ln U Tri I U U N N Ln U Tri Note: N Normal; Ln Lognormal; U Uniform; Tri Triangular

19 Ji and Prevedouros 19 Table 6. Simulation Results from Different Combinations of Probability Distributions Base Std. Dev. Doubled Std. Dev. Combinations Mean Std. Dev. 95% CI Mean Std. Dev. 95% CI Base [36.1, 52.8] [30.0, 83.0] [35.9, 51.9] [29.9, 79.7] [35.4, 52.0] [29.7, 79.7] [35.9, 52.6] [29.8, 82.0] [35.8, 52.4] [30.3, 80.5] [35.9, 52.4] [29.7, 78.9] [35.9, 52.8] [29.8, 82.0]

20 Ji and Prevedouros 20 Table 7. Simulation Results with Varied Level of Correlations Base Std. Dev. Doubled Std. Dev. Correlation Mean Std. Dev. 95% CI Mean Std. Dev. 95% CI No Corr [36.1, 52.8] [29.9, 82.7] Corr* [36.7, 50.6] [31.3, 66.9] Corr * [37.2, 49.3] [32.4, 58.5] Corr * [37.8, 48.4] [33.2, 54.9]

21 Ji and Prevedouros 21 Figure 1. Frequency histogram and cumulative density function of delay.