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6. Model Development, Calibration, and Validation
Pages 77-109

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From page 77...
... 6.1.1 Model Selection For developing regression-based RTOR volume estimation models, flow rates were calculated using 5-minute counts of vehicles departing an intersection, as described in the previous chapter. Count variables may have many observations with values of zero and are often positively skewed.
From page 78...
... . A negative binomial mixed model considers the sample correlation by incorporating random effects into the commonly used fixed effects negative binomial model (Zhang et al.
From page 79...
... The correlation matrix shows that some independent variables are highly correlated with each other. Because highly correlated variables may contribute similar information to the model, we avoided including highly correlated variables together within the same model.
From page 80...
... 4,809 1,056 64.7 76.0 Total shadowed left-turn flow rate (veh/h/ln) 4,809 474 71.9 65.9 Total parallel pedestrian flow rate (ped/h)
From page 81...
... To account for this overdispersion, the negative binomial and quasi-Poisson regression models were fitted.
From page 82...
... Figure 37. Distribution of RTOR flow rate 82
From page 83...
... Here, y is the expected RTOR proportion, β0 , β1 , β2 ,  , βk are the regression coefficients, and X 0 , X 1 , X 2 ,  , X k are the explanatory variables. After fitting the models, the expected RTOR flow rates can be obtained by multiplying y and the total right-turn flow rate.
From page 84...
... RMSE of models for exclusive right-turn lane scenario Model form Model type Model A Model B Model C Model D Model 1 Zero-inflated negative binomial 51.3 57.9 51.0 52.3 Model 2 Negative binomial 52.3 64.9 52.5 55.4 Model 3 Logistic regression 55.9 55.7 55.6 57.9 Table 17. RMSE of models for shared right-turn lane scenario Model form Model type Model A Model B Model C Model D Model 1 Zero-inflated negative binomial 45.9 66.7 44.6 51.1 Model 2 Negative binomial 51.8 53.9 46.1 49.0 Model 3 Logistic regression 43.5 44.4 46.6 48.4 84
From page 85...
... The RMSE values for Model 1 show that zero-inflated and zero hurdle models performed comparatively better than the other models for each scenario. For Model 2 and Model 3, negative binomial and logistic regression performed better for the dual right-turn lane scenario.
From page 86...
... RMSE of models for dual right-turn lane scenario Model type Model 1 Model 2 Model 3 Poisson 43.1 69.1 36.7 Negative binomial 47.6 60.6 45.9 Quasi-Poisson 43.1 69.1 36.7 Zero-inflated negative binomial 45.6 59.1 44.3 Negative binomial hurdle 45.6 59.1 44.3 Logistic regression 88.0 89.8 83.7 Mixed effect model 45.0 26.8 23.9 Another consideration was the development of models that included useful correlations with the input data. Our objective was to develop models that would reduce the predicted RTOR flow rate as conflicting flow rate increased and that would increase the predicted RTOR flow rate with variables that would logically tend to increase it (e.g., the total right-turn flow rate, duration of red as expressed by red-to-cycle ratio, and the shadowed left-turn flow rate)
From page 87...
... For the dual right-turn lane scenario, attempts to develop an equivalent Model 1B using similar sets of variables did not produce a useful model with correlations to conflicting volumes that would yield sensible results. The inclusion of opposing and shadowed left turns yielded coefficients that had the opposite effect (i.e., the RTOR volume would increase with greater opposing left-turn flow rate and decrease with greater shadowed leftturn flow rate)
From page 88...
... Table 22. Recommended models for single exclusive right-turn lane scenario Model 1A Model 1B ZINB (All Variables)
From page 89...
... Other Models Negative Binomial Logistic Dependent Variable RTOR Flow Rate (veh/h/ln) RTOR Proportion Constant 2.013 *
From page 90...
... p < 0.01. 6.1.5 Final Model Validation As mentioned earlier, a subset of data was set aside for validation of the models, a process that was undertaken continuously during model development with the goal of arriving at a set of final models that would be both useful, well-fitted to the data, and able to provide reasonable results.
From page 91...
... Figure 39. Final model validation results.
From page 92...
... However, because the red-to-cycle ratio rarely reaches very low values, the predicted RTOR proportions rarely extend to the lower ranges and are never less than about 0.15 for any of the models, while some actual values did have RTOR proportions in that range, including some zeros. 6.2 Modeling of Right-Turn-on-Red Capacity Two different approaches were used to model RTOR capacity for different lane configurations.
From page 93...
... During some intervals (e.g., the service of the shadowed left turn) , there is no conflicting traffic, so the entire interval can be considered to be the gap subinterval.
From page 94...
... Model 1 was developed using different expressions for capacity for different lane configuration types found in the literature review. Model 2 was developed using a gap-acceptance analysis assisted by microsimulation to establish the forms of the equations and to arrive at a unified model form for RTOR capacity for all lane configuration types, with adjustments to handle specific lane configurations.
From page 95...
... Regime A can be defined as the occurrence of RTOR under acceptable gaps in the conflicting through or left-turn traffic, while regime B represents RTOR movements during the shadowed left turn. For regime A, capacity models consider three possible gap acceptance patterns.
From page 96...
... Equation 53 Equation 56 + Equation 57 Equation 55 6.2.3 Use of Simulation Results to Determine New Model Forms To develop a new mathematical model of RTOR capacity, simulation networks with different lane configurations were developed in microsimulation to establish a dataset in which the intersections operate at capacity. This enabled the impact of one variable on the capacity to be identified independently.
From page 97...
... Equations for the RTOR capacity during the unsaturated green period were obtained by using a curve fitting procedure with the R software package. Additional details are provided in the following discussion for each of the three lane configurations.
From page 98...
... From the simulation data, we found that the RTOR capacity follows a similar pattern to an exponential decay curve, as shown in Figure 40. Therefore, an equation similar to an exponential decay equation was used to model RTOR capacity c′ during the unsaturated green period of the conflicting traffic: =c′ sRTOR exp [ − λ ⋅ VC ]
From page 99...
... Forms of equations for the RTOR capacity of each lane were found that were similar to those obtained for the single right-turn lane.
From page 100...
... Each lane may have different gap acceptance behavior, so the calculations are done independently for each lane. For the shadowed left-turn interval, Equation 62 is used with g s = 0.
From page 101...
... Figure 41. RTOR capacity during the unsaturated green period of the conflicting traffic for dual right-turn lanes: (a)
From page 102...
... Therefore, the RTOR flow rate in the absence of conflicting traffic is adjusted by a factor fT : 3600 vRTOR =fT ⋅ sRTOR =fT ⋅ Equation 64 tf From an analysis of the simulation data, the following expression was obtained for this adjustment factor: fT = 0.01 ⋅ e 4.3 p Equation 65 The following expression gives the potential RTOR capacity: =c′ vRTOR exp [ − λ ⋅ CV ] Equation 66 Using the simulation data for curve fitting, the following equation form was obtained for λ: 4 ⋅ p + 0.3 ⋅ tc − 1 λ= Equation 67 1000 The following expression was obtained for the RTOR capacity: g − gs c = c′ Equation 68 C Similar to the analysis of the exclusive right-turn lane, Equation 68 is used for all three intervals.
From page 103...
... Equation 62 lane separately) Equation 68 6.2.4 Adjustment Factors for Pedestrians Because pedestrians may influence the RTOR movement at an intersection, appropriate adjustment factors should be applied to the models to obtain a more accurate prediction of RTOR capacity.
From page 104...
... e −5 OCCr = 3600 Equation 71 g p − gq where: OCCr = relevant conflict zone occupancy g p = effective green time for the conflicting through movement v0 = opposing demand flow rate (veh/h) Finally, the pedestrian adjustment factor f p can be calculated using the time the conflict zone is unoccupied: f p = 1 − OCCr Equation 72 104
From page 105...
... (veh/h/ln) Location lane thru lanes lanes Mean Max Mean Max Mean Max Mean Max NE 112th Ave & NE 18th St Exclusive 2 2 151 171 200 284 197 258 4 8 N Hoagland Blvd & US 192 Exclusive 1 1 400 414 264 380 1,395 1,540 178 236 I-10 & S Acadian Thruway Shared 0 1 626 649 392 464 0 0 391 464 Naperville Wheaton Rd &Ogden Ave Shared 2 1 696 696 272 312 1,037 1,136 42 56 Civic Dr & Ygnacio Valley Rd Dual 3 2 586 622 48 70 1,577 1,800 122 152 I-680 NB OffRamp & Ygnacio Valley Rd Dual 1 1 508 559 358 424 68 124 2 8 HCM-recommended default values of critical gap and follow-up time were used for model validation.
From page 106...
... Figure 43. Predicted versus observed RTOR capacity for different lane configurations: (a)
From page 107...
... The existing delay equation is based on an analysis of random arrivals at the intersection served by a single green interval, but RTOR introduces multiple new intervals in which the right-turning traffic may be served. Figure 44 shows queue accumulation polygons for potential delay models: • No modeling of RTOR (Figure 44a)
From page 108...
... Queued Vehicles Time Interval 1 Interval 2 Interval 3 Effective Green (e) Queued Vehicles Time Interval 1 Interval 2 Interval 3 Effective Green Figure 44.
From page 109...
... The models presented here could be used to augment the signalized intersection methodology of the HCM, which in its current form is known to produce inaccurate estimates of delay and LOS for right-turn movements by largely ignoring the additional capacity that may result from the RTOR maneuver. In the absence of field data, the analyst is currently asked to assume an RTOR flow rate of zero.


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