Safety Impacts and Other Implications of Raised Speed Limits on High-Speed Roads (2006)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

157 C Estimating Speed Variables from Orange County Traffic Detector Data The traffic data used in the project’s analyses of Orange County, California were collected from single-loop detectors. Such detectors can only measure the traffic count (the number of vehicles passing a loop detector in a sample cycle) and lane occupancy (the fraction of total time that a loop is “occupied” by vehicles) during a given time interval. The following sections describe how average-speeds and speed variance were estimated from the 30-second count and occupancy measurements provided by Orange County’s single-loop traffic detectors. C.1 Estimation of Average Speed The average speed of an individual vehicle is the distance it travels divided by its travel time. While activating a presence-type detector, a single vehicle travels a distance equal to the vehicle length ( il ) plus the detection zone length ( dl ) during the detector’s occupancy time ( it ). The speed can thus be estimated from the following formula (May, 1990): ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ += i di i t ll v 5280 3600 (C-1) where iv = speed of individual vehicle (miles per hour), il = length of individual vehicle (feet), dl = the loop detector size (feet), and it = individual vehicle occupancy time (seconds). Thus, the average of several vehicles’ speeds during a 30-second interval can be computed in miles per hour (mi/h) as follows: ( ) ( ) ∑ ∑∑∑ +×≈⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +== i diidii t ll N tll N v v 5280 3600 5280 3600 (C-2) The final part of this equation holds only if the speeds of all measured vehicles are equal during the 30-second interval. Individual vehicle lengths are not available from single-loop detector data. Thus, in practice a single value of average vehicle length is assumed when making speed estimates, resulting in Equation C-3: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ += o dv t llv 5280 3600ˆ (C-3)

158 where ot is average occupancy time per vehicle (or %OCC * 30 sec/n, where %OCC is the percentage of time the detector is occupied during the 30-sec interval and n is the number of vehicles detected in the same interval). Equation C-3 can be modified to produce the following expression: 5280% 100 360030 ˆ , , , , dtl tl tl tl ll OCC n v + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= (C-4) where the subscripts t and l refer to the tht 30-second interval and the thl lane, respectively, and tll , represents average vehicle length during this same interval in the same lane. Note that the effective vehicle length ( vl + dl ) is unknown and not easily estimated. Many researchers have addressed the problem of computing vehicle speeds using single-loop detector data (e.g., Pushkar et al. 1994; Wang and Nihan 2000; Coifman et al. 2001; Coifman 2001; and Hellinga 2002). A number of Caltrans employees in Orange County and at Division offices were contacted in an attempt to determine a robust estimate of effective lengths for the detector sites in the project’s dataset. After several weeks of work using detection zone- and vehicle-length assumptions of 10 and 14.75 feet, respectively, it was decided to use the g-factors (the inverse of the effective lengths) from algorithms developed by the PeMS group at the University of California, Berkeley (Jia et al. 2001; PeMS 2002). These factors vary by station, lane, and every five-minute interval of every day of the week, and are automatically computed by the PeMS algorithm based on assumptions about free-flow speeds during uncongested periods (Chen et al. 2002). Use of the PeMS g-factors in the project’s speed calculations resulted in much more reasonable speed estimates than the original effective length assumption. However, it is not clear how accurate they are in any particular 30-sec interval, since they are intended to provide reasonable overall speed estimates, and suffer from a form of endogeneity bias. C.2 Estimation of Speed Standard Deviation C.2.1 Within Lanes In order to infer speed variation from estimates of 30-sec speed averages (in other words, without data on individual vehicle speeds), it was necessary to assume that the distribution of speed choices underlying any 30-sec sample remains unchanged over several successive intervals; a period of 5 intervals (2.5 minutes) was chosen for this assumption. Within any interval, the average speed over five successive 30-second intervals ( tlv ,sec150 ) is used as a central point about which to evaluate the variation in individual 30-sec averages:

159 2,1,,1,2, 2,2,1,1,,,1,1,2,2, ,sec150 ˆˆˆˆˆ ++−− ++++−−−− ++++ ++++= tltltltltl tltltltltltltltltltl tl nnnnn vnvnvnvnvn v (C-5) ( ) ∑∑ + −= + −= −= 2 2 , 2 2 2 ,sec150,,, ˆˆ t ts sl t ts slslsltl nvvnSPDNSDL (C-6) where tln , = traffic count in the tht 30-second interval ( L,2,1=t ) for the thl lane; tlv ,ˆ = the average speed estimate in this same interval and lane; tlSPDNSDL ,ˆ = the estimate of standard deviation of individual vehicle speeds in this same interval and lane. Thus, data are recognized two intervals before and two intervals after each interval for which the estimators are coded. C.2.2 Across Lanes At a given station, several detectors simultaneously produce data for each of a group of adjacent lanes: in this situation, to compute an across-lane average there is no need to average over successive time intervals. Variations in average speeds across lanes during a single 30-second interval can be used to estimate between-lane speed variation. Taken together with within-lane variation (defined above), the total section speed variance can be determined. The formulae for across-lane average speed and speed standard deviation are as follows: ∑ ∑ = l tl l tltl tesaccrosslan n vn v , ,, , ˆ (C-7) ( ) ∑∑ ⋅−= l tl l tltesaccrosslantlt nnvvNSLSD ,, 2 ,, ˆˆ (C-8) where tln , = traffic count for the tht 30-second interval in the thl lane; tlv ,ˆ = the average speed in this same interval and lane; and tNSLSD ˆ = the estimate of standard deviation in average speeds across lanes in this interval. C.2.3 Within and Across Lanes As noted, the total variation for a section consisting of multiple lanes can be estimated from its within-lane and across-lane (or between-lane, using more standard statistical terminology) variance estimates. The total is obtained from two sums of squares: the within- and between- lane sums of squares (WSS and BSS, respectively): ( )∑ −⋅= l tltltlt vvnWSS 2 ,sec150,, ˆ (C-9) ( )∑ −⋅= l tesaccrosslantltlt vvnBSS 2 ,,, ˆ (C-10)

160 ttt BSSWSSTSS += (C-11) ∑= l tl t t n TSSNSPDXSDS , ˆ (C-12) where tNSPDXSDS ˆ = the estimate of standard deviation in speeds of vehicles observed across all lanes in the section. All these estimators are of some interest, but the speed standard deviations within each lane ( tlSPDNSDL ,ˆ ) and across the entire roadway section ( tSDSXNSPD ) are probably of greatest interest. Thus, model results for these variables have been emphasized in the discussion in the body of this report.