A Regression Model for Bus Running Times in Suburban Areas of Winnipeg
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1 Journal of Advanced Transportation, Vol. 21, Winter A Regression Model for Bus Running Times in Suburban Areas of Winnipeg Attahiru Sule Alfa William B. Menzies James Purcha R. McPherson To plan new bus routes in suburban areas, expected bus running times on these routes are needed. Using most readily available relevant variables, a regression model is developed for estimating bus running times. The model is conceptually reasonable and it was tested using data other than that used for estimation. Introduction When planning to introduce new bus routes in suburban areas one major piece of information needed to assist the planner is the anticipated bus running times on the new route. In the past, system wide average speed was used to estimate bus running times. That approach is definitely based on oversimplification of the relationship between running times and independent variables which vary for different areas. Abkowitz and Engelstein (1983) pointed out that running times are a function of trip distance, number of boarding and alighting passengers (NBAP), number of signalized intersections, and some other less important variables. The aims in this paper are to identify the main variables which affect running times for fixed-route, fixed-schedule transit service in Winnipeg and to select which ones are readily available for planning purposes. Regression models that relate running time to the variables are then developed. Most of the variables ATTAHIRU ALFA is a Projessional Associate with the Transport Institute and Assistant Professor with the Dept. of Civil Engineering, University of Manitoba, Winnipeg, Canada. WILLIAM MENZIES is Superintendent of Transit Planning, Winnipeg Transit, Winnipeg, Canada. JAMES PURCHA was a Research Assistant with Dept. of Civil Engineering, University of Manitoba. Winnipeg, Canada and is now with Motor Coach Industries Ltd., Winnipeg. R. MCPHERSON was a Transit Planner with Winnipeg Transit, Winnipeg, Canada.
2 228 A. S. Alfa, W. B. Menzies, J. Purcha, andr. McPherson indicated by Abkowitz and Engelstein (1983) are readily available, especially distance and the number of signalized intersections along the route. However, variables such as the NBAP are not readily available. Seneveratne and Choo (1986) presented models that attempt to assess the effects of NBAP by relating it to total demand along the route, using data from Halifax. Data on demand is not usually very reliable even if it is available. For new subdivisions the demand data is usually based on predicted values, hence it would be even less reliable. The interest in this present study is to predict running time for planning transit services in new subdivisions using a regression model fairly similar to that of Abkowitz and Engelstein, except that the NBAP will be excluded from the model and other new variables introduced. The NBAP would not be available for a new subdivision and figures on demand would not necessarily be reliable. In its place we have decided to use variables that are readily available, reliable and can be related to demand or numbers of boarding and alighting passengers (NBAP). The NBAP is a stochastic variable but the selected variables are deterministic. The relationship between the selected variables and NBAP may therefore not be as strong as desired. The variables thought to be related to demand are as follows: i) Time ofday. It is known that demand for travel varies with time of day. Demand during the peak period is usually higher than during the off-peak period. In addition traffic flow varies also with time of day and this variable will also affect running times due to impedance caused by other vehicular traffic. This variable was considered by Abkowitz and Engelstein. It was included in their regression model as an independent variable. In this paper we did not include it as independent variable but instead obtained, for each regression model, a set of regression coefficients for each time of day. ii) Type of Street. Arterials carry more traffic than collectors and residential streets in that order. Traffic flow is a reflection of demand on that street and also affects running times due to interaction with other vehicles. iii) Number ofbus stops. Even though the number of boarding and alighting passengers (NBAP) along a route is partly reflected by demand along one route, another major variable that affects it is the number of bus stops along the route. When demand is high, the frequency for request to board or alight increases. If the number of bus stops is thus included in the model as an independent variable the resulting coefficient for this variable will reflect level of demand.
3 Bus Running Time 229 While number of signalized intersection is recognized as one of the major variables affecting running times, the number of stop signs is usually not considered in most of the literature. This variable we feel should be considered because it contributes to running times. The following variables were therefore considered as candidate variables to be investigated in this analysis: - number of bus stops - number of stop signs - number of traffic lights - length of route segment - time of daylweek - type of street (arterial, collector, residential) - speed limit. Speed limit was included for the obvious reason that a vehicle cannot exceed that specified limit. Other variables considered but later eliminated during the analysis were: number of yield signs, number of bad turns, number of lanes in a segment, land use, whether the street is divided or undivided and number of turns in a segment. Bad turns are defined as turns against traffic. The NBAP was not included as it cannot be reliably predicted for new bus routes. Data was collected for a number of route segments in The sample route segments were selected to reflect a large observable range for each of the independent variables. Three multiple regression models were developed for the running times using the data collected. Using new data collected for 1987 on some of the routes the models were tested and compared. The data collection will be described in the second section and the three models will be presented in the third section. In the fourth section model validation tests will be described. In the final section the three models will be discussed and conclusions will be drawn. Data Collection Forty-six segments were chosen throughout the city of Winnipeg. Each segment had a unique set of characteristics which distinguishes it from the other segments. These characteristics included: a) number of bus stops (XI) b) number of stop signs (X,)
4 230 A. S. Alfa, W. B. Menzies, J. Purcha, andr. McPherson Table 1. Segment Type Characteristics Characteristics Segment No. of Divided VPe Speed Limit Lanes Land Use Street? 1 50 km/h 2 Residential no 2 50 km/h 4+ Non-residential n km/h 4+ Non-residential Yes 4 60 km/h 4+ Non-residential Yes 5 60 km/h 4+ Non-residential no 6 70 km/h 4+ Highway Yes 7 80 km/h 4+ Highway Yes number of traffic lights (X3) number of turns against traffic (X,) number of yield signs (X,) number of turns in the segment (x6) distance from start to finish (X7) speed limit on road (Xg) number of lanes in the segment (X,) land use (residential, non-residential, or highway) (X~O) divided or undivided street (X11) Segments were further broken down by distinguishing directions of travel and times of day. Direction of travel was defined as inbound, outbound, clockwise, or anti-clockwise. For this study, five time periods were defined as follows: AM Peak Midday PM Peak Evening Night 2100-End of Service The 46 segments were divided into 7 different segment types according to their characteristics, as shown in Table 1. Seventeen of the segments were of the residential - 50km/hr - non divided - two lanes type. With this heavy weighting in mind, a sample size was selected. It was decided that 5 observations per direction per time period were to be gathered for segment types 2 through 7, while 3 observations per period per direction were gathered for
5 Bus Running Time 23 1 segment type 1. Observations were made such that a representative sample of each time period was recorded. In other words, the individual observations were spread out throughout the time periods so that, for example, all 5 observations for a midday period would not be recorded between 1O:OO-11:OO. The following definition of a segment was used by the checkers when recording the data. If a bus made a stop immediately before or after an intersection which marked the start point of the segment, then the timing would begin at that moment. A stop might occur for any of the following reasons: bus stop, stop sign, traffic light, or left turn against traffic. If the bus did not stop at the start of the segment, then the start time would be the time the bus reached the middle of the intersection. Similar definitions were used for the end point of the segment. The type of data collected for the study consisted of time intervals as a bus travelled from one predetermined point to another. Times were collected to the nearest second on segments of bus routes which varied over a number of characteristics. Regression Models Initially, a multiple linear regression model was considered. Based on the correlation matrix obtained, only four variables X I, X2, X3 and X7 were candidates for inclusion in the model. A regression model was calibrated for each time period using the following specification: Model 1 where Y = running time X1 = no. of bus stops X2 = no. of stop signs X3 = no. of traffic lights X7 = length of the segment (km). Y = a0 + U lxl + a2x2 + a3x3 + a7x7 The multiple correlation, F-value and the standard error of estimate for each time period for this model are shown in Table 2. The resulting coefficients
6 232 A. S. Alfa, W. B. Menzies, J. Purcha, andr. McPherson Table 2. Multiple correlations, F-values and standard errors of estimate by time period for Model 1 Degrees Standard of Error of Multiple Freedom Estimate Time Period Correlation F-value (VI, v2) Prob.* based on Y 1 - AM peak (4,403) O.OOO Midday (4,443) o.ooo PM Peak (4,455) o.ooo Evening (4,367) O.OOO Night (4,322) O.OOO *Probability that all coefficients are zero. Table 3. Regression Coefficients for Model 1 a0 a1 a2 a3 a7 Time Period Intercept Stop Signs Bus Stops Traffic Lights Distance 1 - AM Peak Midday PM Peak.I Evening Night.I for each time period are shown in Table 3. Demands vary for each of the 5 time periods, therefore the set of coefficients for each time period is a reflection of the demands during that period. Except for the night time, the set of coefficients for the other four time periods are not significantly different from each other. Demand at night is very low. The coefficient a2 for bus stops drops significantly at this period while a1 and a7 increase significantly. One may thus comment that even though the number of bus stops affects running time, the effect is more significant when demand is high. This is a reasonable observation because higher demand often increases the frequency of boarding and alighting at bus stops. This multiple linear regression model is similar to that by Abkowitz and Engelstein (1983). One main weakness of this model, as we see it, is that of the boundary condition when the length of the segment is zero. When X7 = 0, Y should be zero. This linear model violates that condition. Even though this is only of theoretical value as one may impose the condition of setting Y = 0 ifx7 = 0, it is important that the model makes conceptual sense. This led to
7 Bus Running Time 233 further investigations to try and consider a more theoretically sound but simple enough model. A non-linear multiple regression model was thus considered. Based on the simple relationship that running time is given by distance divided by speed and the assumption that speed is a function of number of lights, number of stop signs and number of bus stops, the following relationship was selected: Model 2 Y = b& exp(blxi + b2xi + b3xi) where exp(blxi + b2xi + b3xi) =fl(l/speed) b&p = fz(distance). and Eq. (2) was linearized to read where? = lny bo = lnbo and Xf = Xi/X7; i = 2,3,4 Xi = number of stop signs per km Xi = number of bus stops per km XI = number of stop signs per km X$ = distance. The multiple correlation, F-value and the standard error of estimate for each time period for Model 2 are shown in Table 4. The resulting coefficients for each time period are shown in Table 5. The third model considered is another non-linear model similar to the second model except that the three independent variables are not on a per unit distance basis. The model is of the form Model 3
8 234 A. S. Alfa, W. B. Menzies, J. Purcha, and R. McPherson Table 4. Multiple Correlations, F-values and Standard errors of estimate by time period for Model 2 Degrees Standard of Error of Multiple Freedom Estimate- Time Period correlation F-value (VI, v2) Prob. * based on Y 1 - AM Peak (4,403) O.OOO Midday (4,443) o.ooo PM Peak (4,445) o.ooo Evening (4,367) O.OOO Night (4,322) O.OOO *Probability that all coefficients are zero. Table 5. Regression Coefficients for Model 2 b0 bl b2 b3 b7 Stop Signs Bus Stops Traffic Lights Time Period Intercept per km per km per km Distance 1 - AM Peak.38.I Midday.24.I0.I0.I PM Peak lI Evening.I8.I1.I Night.I which is linearized to read The multiple correlation, F-value and the standard error of estimate for each period for this model are shown in Table 6 and the resulting coefficients are reported in Table 7. In general, the three models produced multiple correlation coefficients that are fairly similar, ranging from 0.72 to The resulting F-values are also similar and the coefficients have correct signs. The standard errors of estimate for models 2 and 3 are about the same but cannot be compared to those of model 1. The standard errors of estimate for model 1 are based on Y while for the other two models they are based on?(in Y). Except for very marginal differences, one can say that the three models are identical, based on the
9 ~ ~ Bus Running Time 235 Table 6. Multiple Correlations, F-values and Standard errors of estimate by time period for Model 3 Degree Standard of Error of Multiple Freedom EstimateA Time Period Correlation F-value (vl, v2) Prob.* based on Y 1 - AM Peak (4,403) O.OOO Midday (4,443) o.ooo PM Peak (4,445) o.ooo Evening (4,367) O.OOO Night (4,322) O.Ooo *Probability that all coefficients are zero. Table 7. Regression Coefficients for Model 3 i.0 c1 c2 c3 c7 Time Period Intercept Stop Signs Bus Stops Traffic Lights Distance 1 - AM Peak Midday PM Peak Evening Night numerical results. One may argue that the second model is more conceptually acceptable than the third model and the third model more acceptable than the first one. Validation Tests To assess how well each of the three models predicts running times of buses, new data was collected in 1987 for some of the segments used in estimating the models. The running times for these segments were then predicted using the three models. The Pearson Correlation coefficient (PCC) between the predicted and observed running times for each model are given in Table 8. Model 1 performed slightly better than Model 3 which in turn performed better than Model 2, based on the PCC values. On the other hand, Model 2 seems more reasonable on the basis of the concept by which it was developed.
10 236 A. S. Alfa, W. B. Menzies, J. Purcha, andr. McPherson Table 8. Pearson Correlation Coefficient (PCC) Time Period Model 1 Model 2 Model AM Peak Midday.I PM Peak Evening Moreover, one may also argue that the difference in the PCC values are not convincingly large enough to reject the basis for preferring Model 2. Discussion and Conclusion A regression analysis simply produces, for a predetermined functional relationship, the most appropriate coefficients that fit a given set of data. The statistics that assess the fitness of the data together with the magnitude and signs of the regression coefficients are then used for accepting a specific functional relationship as appropriate. The same statistics are also used for selecting the most appropriate functional relationship from among the candidate models considered. This procedure is reasonably adequate if the regression model is to be used only for interpolation purposes. If there is a slight chance that the model might be used for extrapolation then the functional form of the regression equation should make conceptual sense in addition to the other conditions stated earlier. Moreover, even if the statistics that assess the goodnessof-fit of the model, and the magnitude and signs of the regression coefficients are reasonable, the model still needs to be tested using new data. These factors overall point to Model 2 as the most appropriate regression model for our problem if extrapolation is likely to be undertaken in the future. If the use of the model is limited to interpolation then model 1 is the most appropriate one. Acknowledgements Dr. Alfa s research was supported in part by grant No. A 6584 from the National Science and Engineering Research Council of Canada.
11 Bus Running Time 237 References Abkowitz, M. and Engelstein, I. (1983). Factors affecting running time on transit routes. Trunsportation Research -A. 17A, Seneveratne P. and Choo C. (1986). Bus journey times in medium size urban areas. Journal of Advanced Transportation. 20,
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