## Estimating turning movement volumes titlepage Estimating Turning Movement Volumes for Large Occluded formulated as an Origin-Destination matrix estimationproblem. Our method shows superiority to ALGS in One well known practical method for estimating terms of accuracy and robustness to observation error.
Turning Movement Volumes at 4-leg roundabouts is the In Section 2, we present a brief overview of cur- Algebraic Solution (ALGS). We present an extension rent methods used to estimate TMVs. In particular, to the ALGS method called the Quadratic Program- Origin-Destination based methods are discussed in Sec- ming Algebraic Solution (QPALGS). The accuracy of tion 2.1 and the ALGS method is explored in Section the two methods is compared, and the QPALGS method In Section 3, we present our extension to the is shown to have superior accuracy to ALGS. ALGS method called the Quadratic Programming Al-gebraic Solution (QPALGS). The TMV error of boththe QPALGS method and the ALGS method are char- acterized with respect to observation error in Section4.
Turning Movement Volumes (TMV) are the total sums of movements through a roundabout or inter- section during a specified period of time, typically a15 minute period called a bin. During a given bin, The utility of various field estimation techniques each vehicle’s path from entry to exit is recorded to may be characterized by their cost, complexity, inva- obtain the total numbers for each origin and destina- siveness, accuracy, and robustness to observation error.
tion. The collection of TMVs is an important factor The techniques may be categorized into direct, statisti- in the analysis of intersections and roundabouts; often cal and algebraic methods. Examples of direct methods used in determining capacity as well as ensuring safe include marking vehicles  and telephone surveys .
and efficient operation. Standard intersections are rel- Direct methods are often criticized for not accurately atively straightforward to count, but for large occluded representing true volumes due to their invasive nature.
roundabouts, the size, occlusions, and constant vehicle Statistical methods like Sequential Quadratic Pro- movement makes TMV counting very challenging. It gramming (SQP)  can be accurate provided reliable is difficult and costly to track a vehicle from entry at sampled data is obtained. They rely on observations one leg through circulation to exit at another. Fur- of approach input/output volumes and direct samples thermore, the constant entry of new vehicles from all of TMV proportions. These methods differ in terms of legs makes it difficult to record every vehicle. For these the importance placed on input/output volumes, opti- reasons, manual counting is often ruled out as a viable mization algorithms and sampling procedures. Unfor- tunately, due to the nature of large occluded round- Field estimation techniques have become an impor- abouts, directly sampling TMVs is not feasible and so tant tool for the collection of TMVs at roundabouts.
the methods are not used in practice. Most formula- One well known practical estimation technique is the tions of statistical methods may be expressed in terms ALGS method (see Section 2.2). It is non-invasive, of the general Origin-Destination problem (see Section cost effective and logistically simple, but suffers from high errors. We present an extension of this method Algebraic methods solve for TMVs given easily collected observations. The most common algebraicmethod is discussed in Section 2.2.
All methods suffer from two main sources of obser- vation error: counter error and bin assignment. Binassignment error is a natural artifact of the assignmentof vehicle movements into 15 minute bins. It arisesfrom the ambiguity in bin assignment for vehicles tak-ing the considerable amount of time required to com-plete a movement at a roundabout. Counter error maybe minimized at increased cost. Bin error is generallynot addressed but may be reduced through the use ofvideo collection and intensive analysis.
In any transportation network where there is a fixed set of zones where a vehicle may enter or exit, we maytalk about the Origin-Destination (O-D) trip matrix.
The entries of the O-D matrix, tij are the total numberof vehicles entering at zone i and exiting at j. Exten-sive research exists on the general problem of estimat-ing the unknown O-D matrix given point observationson links within the network . Indeed, roundaboutshave been modelled in this way. Statistical methods forestimating TMVs at roundabouts, including SQP, haveincorporated techniques from this area. In all cases, theobservations used do not provide enough informationto completely determine the TMVs. Effective sampling Figure 1. Circulating, Out, In, and Next Vol-
is required to provide enough reliable prior information umes algebraically expressed in terms of the
turning movement volumes
origin to their destination and that no u-turns are per- The ALGS  method approaches the problem of formed. Figure 1 shows the relationships between turn- finding the TMVs as an algebraic problem relating the ing movements and the exact COIN volumes as de- TMVs to measurable observations and a number of as- scribed in . Traditionally the ALGS method has only been formulated for 4-leg roundabouts. We denote For each bin, simple point observations are collected the 16 unknown origin-destination TMVs as t at each leg using the following guidelines: i ∈ {0, 1, 2, 3} is the leg of origin, and j ∈ {0, 1, 2, 3} is In The number of vehicles entering the roundabout at the destination. The relations for leg 0 can be written Out The number of vehicles exiting the roundabout Circulating The number of vehicles entering or pass- ing a given leg without exiting at the next leg Next The number of right turns for North American Where C0 is the Circulating volume at leg 0; O0 is We refer to these standard observations as the Cir- the volume of Outs; I0 is the volume of Ins at leg 0; and culating, Out, In, and Next volumes (COIN). The as- N0 is the volume of Next turns. The expressions for sumptions are that vehicles navigate directly from their legs 1,2, and 3 are similar. Eq. (1b) - (1d) contain all of the COIN observations, while Eq. (1a) and Eq. (1e) A is a 20 × 16 matrix populated with coefficients from encode the assumptions made in . Extending the 5 equations in Eq. (1) to all 4 legs gives 20 equationsfor 16 unknowns. Provided the COIN observations are t is the column vector of unknown TMVs tij for all exact and the assumptions are met, Eqns. (1) are con- sistent and admit a unique solution as given in , and Notice that this system cannot, in general, be satisfied with equality. Instead, the idea from GLS is to definea cost function: Here, the cost function represents the L2 distance between the COIN volumes for a proposed set of TMVs However, even when small observation errors are mize the distance in Eq. (4) by solving the following present, the solutions from Eq. (2) often result in large errors in the TMVs  including negative volumes (seeSection 4). Unfortunately the ALGS method is par- ticularly susceptible to binning error since a vehicle executing a movement at a bin boundary may be split across the boundary resulting in errors in the COIN ob- servations in both bins. This may be resolved through We would like our estimate to be in the domain of the use of video collection units and intensive video TMVs, which requires each component to be a non- processing for proper time bin assignment which is be- negative integer. Denote the unknown optimal integer yond the scope of the current discussion.
solution by t∗Z ∈ Z16. The convexity of the objectivefunction provides that: • It is practical to find the relaxed solution t∗ ∈ R16 The traditional ALGS method reduces the 20 equa- tions to 16 by eliminating redundant information. Un- • t∗Z may be found by searching the neighbours in fortunately, this process is only valid for perfect COINobservations. When subject to observation error, the Z16 around t∗ requiring 216 evaluations of the ob-jective function.
(possibly inconsistent) equations in Eqns. (1) actu-ally represent multiple samples rather than redundant • Alternatively, rounding to integers has a guaran- information. Simply using Eqns. (2) amounts to se- teed error bound round(t∗ ) − t∗Z ≤ 1 for all i, j.
Instead, we treat the TMV estimation as a special For expedience, we use rounding. The result is a set of case of the O-D trip matrix estimation problem , non-negative integer TMV estimates which best agree . We employ the Generalized Least Squares (GLS) with the COIN observations and our assumptions si- solution as described in , , and . The GLS is capable of incorporating a known prior distribution of The objective function in Eq. (4) is not necessar- TMVs. For large occluded roundabouts it is not fea- ily exclusive to COIN observations. Other observa- sible to sample TMVs. We use the u-turn assumption tions or prior knowledge may easily be incorporated as our prior distribution and consider any value for the as additional linear equations. Indeed, with sufficient other turning movements equally likely. In using this additional observations it is practical to make similar method, we keep all of the original information from estimations for TMVs at 5-leg roundabouts.
the 20 equations represented by Eq. (1). We denotethis by the over-determined linear system f is the column vector of our observations of COIN Given a known error in COIN observations, we would like to know the accuracy of our TMV estimates.
Figure 2. RRMSE (see Eq. 6) vs. Observation Error for the ALGS method (×) and the QPALGS method
(
). Left, low volume: µV = 10 vehicles per movement per 15 minutes. Right, high volume: µV = 47.5
vehicles per movement per 15 minutes.
Table 1. RRMSE for QPALGS and ALGS methods for observation error of 3% to 8% for high volume
traffic (47.5 vehicles per movement per 15 minutes)

Thus the relative error in the estimated TMVs, de- the ALGS method was performed on the same data set noted ∆t/t, as a function of the relative error in the as QPALGS. The simulation was intended to present COIN observations, denoted ∆f /f , is critically impor- the typical error for 15 minute binned data in the case tant. Case studies  have previously been conducted of low, but non-zero u-turns. For each given obser- to establish practical estimates on the error of ALGS.
vation error e = ∆f /f in the range 0% to 15%, and For O-D based methods, Cascetta  formulated an average movement volume µV , we simulated 10, 000 15 analytical expression for the covariance of the Aitken minute bins of complete roundabout data by the fol- estimator. Unfortunately, the expression is only ap- plicable when the solution domain is extended to real 1. For each non-uturn movement from approach i to numbers and the non-negativity constraint from Eq.
j, i = j, generate a count by drawing from a nor- mal distribution with mean µV and standard de- In , Dixon reported the Mean Absolute Percent viation 10% × µV and round to the nearest non- Error (MAPE) as a measure for comparing estima- tion procedures. Observation error was not discussed.
The expression for the covariance provided in  does 2. For each u-turn movement, generate a count of 0, provide a relationship between ∆f and ∆t. Unfortu- 1, 2, 3, or 4 with probability 85.5%, 7.70%, 3.85%, nately, the form expressed is not a relative error and 1.93%, or 0.97%. Note, the expected number of u- does not generalize. Thus  and  fail to provide a turns is 1, slightly violating the u-turn assumption.
relationship between ∆f /f and ∆t/t.
3. Let ttrue be the TMVs generated in 1 and 2. Gen- To determine the relationship between ∆t/t and erate the corresponding COIN volumes f true using ∆f /f , a simulation was performed. For comparison, 4. Generate noisy COIN observations based on the ideal measurement f true. The kth observation f obsis generated from a normal distribution with mean This paper has outlined a new method for estimating f true and standard deviation e% × f true.
Turning Movement Volumes at large occluded round-abouts. The new method, QPALGS is an extension 5. repeat steps 1-4 for N = 10, 000 15 minute bins.
to the ALGS method using Origin-Destination matrixestimation theory. The new method is non-invasive, The noisy COIN observations f obss are representative accurate and robust to observation error. In practice, of counts collected by counters with known relative the TMVs can be expected to be accurate ±11.3% for error e. The simulation was performed for both low observation error ±4%. The method has the potential and high volume scenarios corresponding to means of to be extended to roundabouts with more legs.
µV = 10 vehicles per 15 minutes and µV = 47.5 vehi-cles per 15 minutes. For each f obs, the ALGS andQPALGS TMV estimates were computed and com- pared to the known ttrue. Note, often the resultingALGS volumes were negative. For high observation error, negative volumes became prevalent occurring in destination matrices using traffic counts- a litera- 63% of the bins. The relative error in the TMV es- ture survey. Working paper, International Institute timates was measured using the Relative Root Mean  AC Aitken. On least squares and linear combina- tion of observations. Proc. R. Soc. Edinb, 55:42–48, destination matrices by constrained generalised where m is the method (ALGS or QPALGS). The RRMSE was preferred over the MAPE  or the es- Methodological, 25(1):13 – 22, 1991.
timator’s covariance matrix  because the RRMSEcharacterizes both the bias and precision of the esti-  Ennio Cascetta. Estimation of trip matrices from mated TMV (the MAPE does not), and is a relative traffic counts and survey data: A generalized least squares estimator. Transportation Research Part B: Figure 2 presents the relationship between the ob- Methodological, 18(4-5):289 – 299, 1984.
servation error and the error in the estimated TMVsfor both QPALGS (⊙) and ALGS (×). The value of  Michael P. Dixon, Ahmed Abdel-Rahim, Michael N = 10, 000 was chosen to obtain a precision of ±.05% for the RRMSE. Table 1 shows a subset of data ex- Rodegerdts. Field evaluation of roundabout turn- tracted from Figure 2. Figure 2 and table 1 are in- ing movement estimation procedures. Journal of tended to be used to establish the accuracy of a par- Transportation Engineering, 133(2):138–146, 2007.
ticular study given a known or estimated observation error. For example, if a roundabout study is performed destination demands from links and probe counts.
with high average movement volume and the counters Depertment of Civil Engineering, Queens Univer- are known to be accurate ±4% then the expected error in the TMVs is ±11.3% for QPALGS (⊙) or ±15.3%for ALGS (×). Figure 2 clearly shows that QPALGS  Bruce W. Robinson, Joe Georges. Bared, United outperforms the ALGS in all cases. In addition, the States., Turner-Fairbank Highway Research Cen- QPALGS exhibits a robustness to observation error far ter., Kittelson, and Associates. Roundabouts : an superior to the ALGS method. Table 1 clearly shows informational guide / [principal investigator, Bruce the importance of knowing and minimizing observation W. Robinson]. U.S. Dept. of Transportation, Fed- error. With the use of Automated Traffic Data Collec- eral Highway Administration, [Washington, D.C.] tion solutions such as Miovision Technologies Inc., ob- servation error can be minimized guaranteeing TMVaccuracy in excess of 88%. Without such guaranteesand using traditional methods, TMV accuracy can eas-ily be less than 70%.