FORECASTING DAY TO DAY VARIABILITY IN ...

FORECASTING DAY TO DAY VARIABILITY IN TRAVEL TIMES ON THE UK MOTORWAY NETWORK Sansaka Sirivadidurage, Andrew Gordon, Christopher White Mott MacDonald, UK; David Watling ITS, University of Leeds, UK 1. INTRODUCTION ‘Reliable journeys’ is one of three key objectives of the UK Highways Agency who are responsible for the motorway and trunk road network, as well as being a central theme of the Eddington report which investigated ‘Transport’s Role in Sustaining UK’s Productivity and Competitiveness’ (Department for Transport, 2009a), and a major international topic around the world. In order to measure and predict reliability, it is necessary to understand variability in travel times. Travel time variability can be divided into two components, namely variability due to incidents and day to day variability (DTDV). Specifically, DTDV refers to variations in journey time due to unpredictable changes in demand and random fluctuations in capacity. In other words DTDV is what remains after accounting for all predictable variations (time of day effects, day type effects and seasonal effects) and variability due to incidents. This paper describes how DTDV functions were calibrated for several motorway link types. These include dual three and four lane motorways with hard shoulder (D3M and D4M), mandatory variable speed limits (D3CM and D4CM) and three lane managed motorways with dynamic hard-shoulder running (D3M MM DHS). D3CM and D4CM set mandatory speed limits once flow levels reach a pre-defined threshold. 60mph and 50mph are displayed to respond to congestion and 40mph can also be set if necessary to protect traffic from queues. MM DHS on the other hand allow drivers to use the hard shoulder of the motorway as a running lane at times of high congestion. At times when hard shoulder is not in operation, variable speed limits are in force on the other lanes as per D3CM. Hard shoulder running at 60mph is under trial in the UK. The main aim of managed motorway schemes is to minimise the risk of flow breakdown and reduce accidents, thereby producing more reliable journey times. More details on managed motorway schemes can be found in Department for Transport (2009b). The paper is organised as follows. Background details are given in the next section. The source of data and data processing procedures are described in section three. Chapter four sets out statistical methods used in the analysis while chapter five reports results. The paper finishes with some brief conclusions. 2. BACKGROUND Several studies (Department for Transport, 2004; Eliasson J, 2006 and Hyder Consulting, 2007) have analysed travel time variability (TTV) (i.e. DTDV and incident variability) on urban roads where incident variability and DTDV cannot be readily separated. The first two studies produced statistical relationships to

© Association for European Transport and contributors 2009

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represent overall TTV on urban roads based on link journey times collected in Leeds and London and several streets and roads in and around central Stockholm respectively. On the other hand the TTV functions developed by Hyder Consulting (2007) are based on origin destination travel time measurements over the 10 largest urban areas in England. These functions are recommended by the UK Department for Transport (DfT) in urban variability calculations (Department for Transport, 2009c). Motorway sections, on the other hand, might be expected to be qualitatively different from streets in urban areas. Bates et al (2004) used a micro simulation approach for motorway DTDV model development using data collected on the M6. They concluded that overall TTV on a stretch of motorway is strongly related to capacity utilisation. On this particular motorway section, they noticed that DTDV appears to be quite low, meaning that incidents will be the main source of variability as long as demand is below capacity. There are no DTDV relationships for general motorway situations in the UK, hence the main focus of this paper. The DfT recognises that DTDV should be taken into account when appraising potential benefits of transport schemes or policies. Managed motorway systems such as mandatory variable speed limits and hard shoulder running can provide significant DTDV benefits. DfT commissioned Mott MacDonald to carry out a project to calibrate functions for predicting the DTDV on several road types and to incorporate these into INcident Cost benefit Assessment (INCA) software (Department for Transport, 2009d). As well as DTDV, INCA calculates delays and travel time variability costs relating to incidents, and the benefits that may arise from remedial measures to reduce their impact. In addition, the transferable DTDV functions that were estimated have the potential to be exploited in other future modelling studies and packages, where forecasts of travel time variability and its reliability impacts are required.

3. DATA PROCESSING 3.1 HATRIS Data Extraction The source of data for this work was the Highways Agency Traffic Information System (HATRIS). The HATRIS database provides network monitoring information for the motorway and trunk road network from inductive loop sensors (Motorway Incident Detection and Automatic Signalling - MIDAS), automatic number plate recognition and matching and GPS tracking. The data stored for each link in the network includes average speed, average journey time, flow, incident flag, data quality indicator, for every 15 minute time period of every day for each link. In addition for each link HATRIS also provides the free-flow speed and the length. Data was extracted for the following links for both directions: D3M

M1 Junctions 10-15A, M4 Junctions 12-14, M6 Junctions 1626, M25 Junctions 2-5, M25 Junctions 6-7

D4M

M56 Junctions 3-4, M25 Junctions 7-10


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D3CM

M42 Junctions 3a-7

D4CM

M25 Junctions 13-15

D3M MM DHS M42 Junctions 3a-7 The reasons for this selection are included a need for data marked as ‘high’ quality in HATRIS and for links representing a range of different lengths. Data for these different road types were extracted for the time periods shown below. D3M

01/01/2006 to 31/05/2007

D4M

01/01/2006 to 31/05/2007

D3CM

29/11/2005 to 11/09/2006 (i.e. the period of the MM DHS pilot on M42 when mandatory variable speed limits were in place, but before hard shoulder running started)

D4CM

01/01/2003 to 31/12/2003 (i.e. before work began on widening this section of M25)

D3M MM DHS 12/09/2006 to 31/05/2007 (i.e. after D3CM operation on M42) Data was initially extracted from HATRIS as Microsoft Excel files. These were read into Statistical Package for the Social Sciences (SPSS), which was then used for the rest of the analysis. 3.2 Incident Identification The purpose of the DTDV curves is to estimate the remaining variability after the effect of incidents has been accounted for. Assuming that incidents occur at random, then an unbiased way of achieving this is to exclude any HATRIS data affected by incidents from the analysis. The HATRIS ‘incident flag’ which is based on an outlier detection method which identifies observations for which the journey time is more than 2 standard deviations above the mean is used for this purpose. 3.3 Calculation of Variables Variables were calculated by retaining the distinction by day type and time periods in order to ensure that seasonal, day type and within day effects are excluded from the calculation of DTDV. Each day of the year can be allocated to one of 21 day types (see Mott MacDonald, 2008a for more details). Each day type is a combination of day of the week, time of year and school term or school holiday. There were 2016 (21*4*24) data points for each link per year before filtering and most road types had data for more than a year. Therefore it was possible to conduct a more rigorous analysis than previously due to the large amount of data available. Any HATRIS data with low quality and those affected by incidents were excluded from the analysis.


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From the raw HATRIS data the next step was to calculate the mean and standard deviation of the selected variables over time periods and day types. If journey time per km is selected, then

mean(TDtl ) =

sd (TDtl ) =

∑T

dtl

d∈D

N Dtl

∑ (T

d∈D

− mean(TDtl ))

2

dtl

N Dtl − 1

where Tdtl

is the journey time per km on date d in time period t on link l

D represents day type D and d ∈ D is the set of dates which are in day type D. N Dtl is the number of days of HATRIS data for day type D, time period t and link l By retaining the distinction by day type and time period it is ensured that the standard deviation calculated above just represents DTDV and excludes seasonal, day type, and within-day effects. 3.4 Sample Size and Standard Errors The mean and standard deviation calculated above are only estimates. The uncertainty in each estimate can be quantified by the standard error (SE). The 95% confidence interval for the estimate is roughly ±1.96 x SE. The standard errors are determined by the variability in the data and the sample size: SE (mean(TDtl )) = SE (sd (TDtl )) =

sd (TDtl ) N Dtl

sd (TDtl ) 2( N Dtl − 1)

On each link the number of day type/time period combinations is potentially very large (21x96=2016) so it was decided to exclude data with small sample sizes N as this would still leave plenty of data points for function fitting. Data was only included for N>30. Even then, according to the above formulae, the 95% confidence interval for the standard deviation is still roughly ±26% of the estimate. Day type/time period/link (Dtl) combinations that operate permanently under free-flow conditions (zero mean delay and zero standard deviation) were also excluded from any subsequent analysis.


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3.5 Data Filtering It was decided to exclude data for average journey time per km over 80sec. This is the maximum that would be predicted by the UK Design Manual for Roads and Bridges volume 13 (also known as COBA curves) with a minimum speed cut-off of 45 km/h (as would be used in INCA). This ensured that the best functions are fitted for the INCA implementation range. This filtering still left a considerable number of data points for curve estimation. There are 23,173 data points for estimation of functions of which 17227 D3M, 2943 D4M, 506 for D4CM, 500 for D3CM and 1997 for D3M MM DHS. 4. METHODOLOGY 4.1 Weighted Least Squared Regression The core technique used in the function estimation was linear regression. By transforming the dependent and independent variables this can be used to fit a wide range of curves, including power and exponential functions. The main goodness of fit statistic for linear regression is the R-squared statistic. This represents the proportion of the variation in the dependent variable that can be explained by the variation in the independent variable. It takes a value between 0 and 1, with higher values indicating a better fit. However, there is a tendency to focus on R-squared to the exclusion of all else. Other properties of the fitted function also need to be considered. These focus on the properties of the residuals (or errors), which are the differences between the predicted and observed values of the dependent variable. The residuals should: •

Have no discernible trend, i.e. there should be no systematic under or over estimation at different levels of the dependent variable.

•

Be homoscedastic, i.e. the distribution of the residuals should be the same for all values of the independent variable(s).

•

Follow the assumed distribution of residuals, in our case a normal distribution.

A curve that meets the above requirements may be considered better than an alternative with a higher R-squared that does not. Real life data often exhibits heteroscedasity (the opposite of the homoscedasity discussed above), e.g. as the independent variable increases so the spread of the dependent variable also increases. This is easily seen in the scatterplot presented in Figure 1 in section 5.1. This means that the residuals will also be heteroscedastic. This is overcome by using weighted least squares (WLS) estimation. Linear regression usually finds parameters for the curve so that the sum of the squares of the residuals is minimised. WLS modifies this process by assigning weights to the residuals and minimising the weighted sum of squares. Less weight is given to residuals with higher variability.


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4.2 Weights Estimation It was noted earlier in section 4.1 that there is significant heteroscedasity in the data, which has to be dealt with by using weighted least squares (WLS) regression. SPSS includes a feature to automatically estimate weights. The weights take the form: W=

1 Xη

where X is a user-specified variable and η is a parameter to be estimated by the software. η should be allowed to vary by road type if necessary, following the results of the ANCOVA test which is described in the following section. 4.3 ANCOVA Analysis ANCOVA stands for ANalysis of COVAriances. The traditional way of thinking about ANCOVA is as an extension of ANOVA (ANalysis Of VAriances) to include continuous variables. ANOVA itself is often thought of as an extension of the t-test. The latter can be used to compare means for two groups. ANCOVA extends this principle to compare means for any number of groups. However, in the context of this project it is more useful to think of it as a way of extending linear regression to include categorical variables. Basic linear regression works with independent and dependent variables that are continuous, such as link flow or travel time. In this work we are also interested in categorical variables such as road type (e.g. D4CM or D3CM). ANCOVA was used to determine whether categorical variables could explain the dependent variable, e.g. is the level of DTDV different on different road types, even after continuous variables such as flow and journey time had been taken into account. It was also used to determine whether the regression coefficients are significantly different for different road types. 5. MODEL DEVELOPMENT AND RESULTS 5.1 Initial Curve Fitting The first stage of the initial curve fitting is to produce scatterplots for a wide range of different dependent and independent variables. An example scatterplot is given in Figure 1 below. From the scatterplot, it can be seen that heteroscedasity is clearly present in the data, i.e. the spread of the dependent variable increases with the value of the independent variable. This means that weighted least squared regression is required.


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SD of mean journey time per km SD of delay per km (seconds)

Mean journey time per km Delay per km (seconds)

Figure 1: Example scatterplot of dependent variable vs. independent variable Based on the scatterplots the pairs of variables shown in Table 1 were taken forward for initial curve fitting. Table 1: Pairs of variables for initial curve fitting Dependent variable

Independent variable

Standard deviation of delay

Mean delay

Variance of delay

Mean delay

Standard deviation of delay per km Mean delay per km Variance of delay per km

Mean delay per km

Coefficient of variation of link travel Link congestion index time

The Curve Estimation function in SPSS was used to estimate standard Linear, Quadratic, Cubic, Power, Logarithmic and Exponential functions. In addition to the standard curves list above, the following were also estimated, as they have been used in previous work on urban DTDV: CV = aCI b D c

(

CV = exp a + b(CI − 1) + d (CI − 1)

3

)


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where CV is the coefficient of variation of journey time (i.e. standard deviation of journey time over mean journey time), CI is the congestion index (i.e. mean journey time over free flow journey time), D is the link length and a, b, c and d are coefficients to be estimated. The former is recommended in WebTAG Unit 3.5.7 (Department for Transport, 2009c) on reliability. We refer to this as the ‘WebTAG’ curve. The latter is used by Eliasson J (2006). At this stage we were interested in identifying a number of candidate curves and variables to take forward for more detailed analysis hence Ordinary Least Squares (OLS) regression was used. The results confirmed the relative lack of fit when modelling CV as a function of CI. For all pairs of variables the linear/quadratic/cubic family of curves gave the best fit. On this basis the CV versus CI curves were discarded for subsequent analysis as were all curves apart from linear, quadratic and cubic. More details can be found in Mott MacDonald (2008a). For each of the four pairs of variables ANCOVA was run using the cubic function of the independent variable. The results showed that, in each case, fitting a separate function for each link type gives a significant improvement in the overall fit. This result was taken through all subsequent work, with the data segmented by link type at each step.

Type

WLS regression was then used to fit cubic functions for the four pairs of variables listed above, separately for each link type. At this stage we had data only for three motorway link types. Adjusted R-squared statistics are presented in Table 2 below. Since the analysis of residuals did not favour any one functional form these results were the main determinant of the preferred model form. Table 2: Results of WLS regression – adjusted R-squared values Dependent SD(delay Var(delay SD(delay) Var(delay) variable per km) per km) Independent Mean(delay Mean(delay Mean(delay) Mean(delay) variable per km) per km) D3M MM 0.70 0.52 0.71 0.50 DHS 0.66 0.57 0.71 0.51 D3M D4M

0.73

0.57

0.79

0.73

Two main conclusions can be derived from above: 1. It is desirable to have normalised or dimensionless variables, to make it easier to transfer the curves to other links. 2. The curves of standard deviation of delay per km as a cubic function of mean delay per km gave a good fit.


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Working with delay per km gives a marginally better fit than just delay. However there was a difficulty in determining delay. Delay is calculated using the free flow time which was based on the free flow speed from HATRIS. This is defined as 107.82 km/h (equal to 67.00 mph) on all links. That this is less than the speed limit is a consequence of the HATRIS data being an average over all vehicle types, including heavy goods vehicles. The inaccuracy of the delay calculation forced us to find alternative variables for the analysis. It was then decided to look at functions based on journey time per km, rather than delay per km. This is because: 1. Journey time per km is simply delay per km plus free flow journey time per km 2. The standard deviation of journey time per km is equal to the standard deviation of delay per km. 5.3 Detailed Curve Fitting The analysis presented in this section concentrated on fitting standard deviation of journey time per km as a cubic function of mean journey time per km. 5.3.1 ANCOVA Results and Weights ANCOVA analysis was used to investigate the significance of road type (All five motorway link types are used for this analysis) in explaining the level of DTDV. Hence ANCOVA was run using the cubic function of the independent variable. Table 3 show that, in each case, fitting a separate function for each road type gives a significant improvement in the overall fit. This result was taken through all subsequent work, with the data segmented by road type at each step. For the estimation of weights we took X to be the mean delay per km. The range of estimated η values varies from 0.6 to 2.0 for different road types. The significant variations in the range of weights indicate that there are different levels of heteroscedasity in the data for different road types.

Table 3: ANCOVA results Dependent Variable: SD(mean JT per km ) Type I Sum of Mean Source df F Squares Square Corrected Model 690025.9(a) 19 36317.2 5365.4 Intercept 236999.2 1 236999.2 35013.9 standard 3244.0 4 811.0 119.8 Mean JT per km 635350.9 1 635350.9 93865.9 Mean JT per km 23370.3 1 23370.3 3452.7 _squared Mean JT per km 113.0 1 113.0 16.7 _cubed


Sig. .000 .000 .000 .000 .000 .000

9

standard * Mean JT per km standard * Mean JT per km _squared standard * Mean JT per km _cubed

25787.2

4

6446.8

952.4

.000

1884.8

4

471.2

69.6

.000

275.7

4

68.9

10.2

.000

Error 156715.8 23153 Total 1083740.9 23173 Corrected Total 846741.8 23172 a R Squared = .815 (Adjusted R Squared = .815)

6.8

5.3.2 Function Fitting WLS regression was then used to fit cubic functions separately for each road type. 5.3.3 Analysis of Residuals For the fitted functions the weighted residuals were calculated and then standardised. This means that they are transformed by subtracting the mean residual value and dividing by the standard deviation. The standardised values have a mean of zero and standard deviation of one. Plots of the standardised weighted residuals against the independent variable are given in Figure 2. Ideally these should show no discernable trend, i.e. the mean value of the residual should not increase or decrease with the value of the independent variable. They should also show homoscedasity, i.e. the spread of residual values around zero should be the same for all values of the independent variable.


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D3M MM DHS standard: ATM HSR

WResStd_1

Weighted standard residual

30.00

20.00

10.00

0.00

-10.00

-20.00 30.00

40.00

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70.00

80.00

avgjourneytimekm_mean

Mean journey time per km (seconds)

D3M standard: D3M 30.00

WResStd_1

Weighted standard residual

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0.00

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-20.00 30.00

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avgjourneytimekm_mean Mean journey time per km (seconds)


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D4M standard: D4M

WeightedWResStd_1 standard residual

10.00

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-5.00

-10.00 30.00

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D4CM standard: D4CM

Weighted standard residual WResStd_1

4.00

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D3CM standard: D3CM

WeightedWResStd_1 standard residual

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Figure 2: Weighted Standardised Residuals


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Figure 2 indicates that residuals from weighted regression have no discernible trend and show homoscedasity for most of the road types. Overall, the residuals show the desired attributes. 5.3.4 R-squared Statistics The analysis presented in this paper concentrated on fitting standard deviation of journey time per km as a cubic function of mean journey time per km. While working with average journey time per km it proved necessary to ‘centre’ the data to avoid collinearity between the different powers. Centring involves subtracting the mean value of the independent from each data point. Adjusted R-squared statistics are presented in Table 4 below.

Road type

Table 4: Results of WLS regression for individual link types – adjusted Rsquared values Dependent variable Independent variable D4CM D3CM D3M MM DHS D3M D4M

SD(journey time per km) Mean(journey time per km) 0.61 0.42 0.68 0.80 0.94

The adjusted R-squared values are low for some link types. As mentioned before in sections 3.1 and 4.4 a curve that has well behaved residuals may be considered better than an alternative with a higher R-squared that does not. Therefore the above results are assumed appropriate. However in the long run it would be appropriate to update models for managed motorway roads (D3CM, D4CM and D3M MM DHS) once more data is readily available. 5.3.5 Function Coefficients and Graphs The estimated coefficients for each road type are shown in Table 5 below. The coefficients were estimated using the centred data then adjusted so that they can be applied to uncentred data. It is the latter set that is shown here. The t-statistic and corresponding significance level are also shown. A tstatistic of >1.96 or