Analysis of Naturalistic Driving Study Data: Roadway Departures on Rural Two-Lane Curves (2014)

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51 C h a p t e r 9 This chapter discusses the fourth research question: Can lane position at a particular state be predicted as a function of posi- tion in a prior state? Research Question 4 focuses more specifically on driver response to changing roadway characteristics and traffic con- ditions. Time series models were developed to incorporate the dynamic process of information acquisition and response as a driver negotiates a curve. The analysis evaluated the influ- ence of roadway geometries or traffic conditions on drivers’ lane-keeping behavior. For example, drivers on a rural two- lane roadway tend to have larger lane deviation from the center- line when there is an oncoming vehicle. Two types of dynamic linear models (DLMs) were built in this study to describe and explain the curve negotiation process: DLM with intervention analysis and DLM with autoregression and moving average (ARMA). The DLM with intervention analysis was mainly used for explanatory purposes, which related lane offset to curve characteristics and traffic conditions. The DLM with ARMA was mainly used for forecasting purposes, which could be used for roadway departure warning systems. It should be noted that Research Question 4 is similar to Research Question 2. Research Question 4 uses a different statistical method to explore whether driver/vehicle charac- teristics in one time state during curve negotiation can be modeled from previous states using a time series analysis. Research Question 4 is included as a separate research ques- tion to simplify discussion of methodology and results. The objective of Research Question 4 was to determine the feasibility of the approach because there was not sufficient time to conduct a full-scale analysis. Description of analytical approach for research Question 4 A DLM was used to analyze driver behavior. The DLM can account for the autocorrelation of the observations in time. It is a flexible model that allows the inclusion of explanatory variables and stochastic time components in the same model. The explanatory variables can evolve over time. The popu- lar Box-Jenkins autoregressive integrated moving average (ARIMA) model was not used in this study, because the underlying process of the Box-Jenkins model is assumed not to change over time; driving behavior varies in different road- way segments with different speed limits, roadway geome- tries, and weather conditions. The DLM is a better model in this case because the model can consistently update the model parameters based on the modeling errors in previous steps. In other words, the model can evolve over time based on its past observations and can be adapted to different situations. The general form of DLM can be written as follows: Observation equation Yt = Ft θt + νt, with νt ~ N1(0, Vt) State evolution equation θt = Gt θt-1 + ωt, with ωt ~ Np(0, Wt) Initial prior (θ0 | D0) ~ N(m0, C0), where (m0, C0) is fixed and Dt = {Yt, Dt-1} The model assumes that the underlying state, θt, evolves smoothly over time as an autoregressive process and that the observation at time, t, is a smooth function of the state. The state evolution equation, as formulated above, is a function of an underlying process that is unobserved. Explanatory vari- ables can be included as part of the underlying process driving the observation equation by including a linear combination of the explanatory variables in the state evolution equation. Coefficients Ft and Gt are often assumed to be constant over time, but they can also be time dependent. Data Sampling and Segmentation approach for research Question 4 One curve in North Carolina was selected that has lane posi- tion, speed, and pedal position data that were determined to be sufficiently reliable for the model. The selected trip is Analysis for Research Question 4

52 shown in Figure 9.1. The sample curve is a single curve with a radius of 1,128 m. The driver was driving from northwest to southeast on the outside lane on a rural two-lane highway. The speed limit is 55 mph. The trip occurred at night, and there was only one oncoming vehicle in this trip. Variables Used for research Question 4 Time series data output by the DAS with additional variables added were used for this model. The distance from the left wheel to the left lane marking was used as a dependent vari- able. A positive value for the left distance indicates that the left wheel is within the lane; a negative value for the left distance indicates that the left wheel crossed the centerline marking. The raw data were collected at every 0.1 s but were aggregated to the 1-s level for this analysis. The time series plot of the raw data for the left distance is shown in Figure 9.2. The plot also labels the area of influence under the oncoming vehicle and the curve. The figure shows that the vehicle had a larger left distance, so it moved away from the centerline when there was an oncoming vehicle. In contrast, the vehicle moved closer to the centerline when the driver was driving inside the curve. results for research Question 4 This analysis focused on the use of DLM for intervention analysis and forecasting. The next section introduces the intervention analysis. The subsequent section focuses on the use of DLM for forecasting. Intervention Analysis with Dynamic Linear Model The objective of DLM is to describe and explain the lane posi- tion of the vehicle in the curve negotiating process. The pro- posed model assumes an additive model in which the lateral position is the sum of normal driving positions, the lane deviation due to the oncoming vehicle, and the lane deviation due to the curve. The influence of the oncoming vehicle and the curve are introduced into this model as intervention vari- ables Wv,t and Wc,t, respectively, as follows: ( ) ( ) ( ) ( ) = µ + β + β + ε ε σ µ = µ + ξ ξ σ β = β + τ τ σ β = β + ρ τ σ ε + ξ + τ + ρ NID 0, NID 0, NID 0, NID 0, , , , 2 1 2 , 1 , 2 , 1 , 2 y W Wt t v t v t t c t t t t t t t v t v t t t c t c t t t ∼ ∼ ∼ ∼ For t = 1 . . . n, where µt is a stochastic-level variable at time t, the model represents the normal driving positions without the influence of the oncoming vehicle and the curve. Wv,t is a dummy variable with 0 for the absence and 1 for the existence of an oncoming vehicle. Wc,t is a dummy variable with 0 for outside the curve and 1 for inside the curve. The coefficients bv,t and bc,t are the coefficient matrices of these intervention effects. The variable et is the random noise in the observa- tion equation. The model parameters xt, tt, and rt are the random noise in the state equations. Source: World_Street_Map (Esri, DeLorme, NAVTEQ, USGS, Intermap, iPC, NRCAN, Esri Japan, METI, Esri China [Hong Kong], Esri [Thailand], and TomTom, 2013). Figure 9.1. Sample trip highlighted in ArcGIS. Figure 9.2. Time series plot of distance of left wheel to left lane marking.

53 The model was estimated using maximum likelihood esti- mation based on the DLM package in R. The predicted mean level of the model is plotted in Figure 9.3. This model assumes the variance for the state equations (xt, tt, rt) to be zero. In this way, the model is forced to fit a straight line for the mean effect. The influence of the oncoming vehicle is treated as an intervention effect and causes a level shift to a larger left dis- tance, whereas the curve causes a level shift to a smaller left distance, as illustrated in Figure 9.3. After the model was fitted with the DLM, the model was further decomposed into three components: the mean level at normal driving, the intervention effect due to an oncoming vehicle, and the intervention effect due to the curve. The decomposition of each effect allows the evaluation of the three components separately, as shown in Figure 9.4. The top panel in Figure 9.4 illustrates the mean level of lane deviation as if there were no intervention effect from the oncoming vehicle and the curve. It also represents the mean lane position for normal driving conditions, in which the left edge of the vehicle is approximately 0.44 m (1.44 ft) from the centerline (to the right of the centerline). The middle panel shows the influence of an oncoming vehicle on the lane devi- ation. The positive sign means the vehicle was moving away from the centerline for an additional 0.43 m (1.41 ft) (a total Figure 9.3. Actual lane deviation versus predicted lane deviation based on DLM. Figure 9.4. Decomposition of lane deviation data into three components: Stochastic mean effect, lane deviation due to oncoming vehicle, and lane deviation due to curve.

54 of 0.77 m to the right of the centerline) to avoid the conflict with the oncoming vehicle. The bottom panel illustrates the intervention effects of the curve on the lane position. The effect of the curve on the lane position is -0.53 m (1.73 ft) to the left of the normal driving position, which places the vehi- cle about 0.09 m (0.30 ft) beyond the centerline (i.e., the left edge of the vehicle crosses the centerline by 0.1 m). Forecasting with Dynamic Linear Model The second model is a DLM representation of ARMA model. The classical ARMA (p, q) process model can be defined as the following formula: ∑ ∑= φ + ψ ε + ε− = − =1 1 Y Yt j t j j p j t j j q t where Yt-j = the past observation at time t - j; et-j = the error of the model at time t - j; and et = the residual of the model at time t. This model predicts the future position of the vehicle based on the past p number of observations and q number of mod- eling errors. The best model for this time series data is the ARMA (2, 1) model, as follows: = + + ε + ε − − − 0.89 0.06 0.981 2 1Y Y Yt t t t t The variance of the model residuals is 0.015. The one-step- ahead prediction with 95% confidence interval is plotted in Figure 9.5. The figure shows that 95% of the one-step-ahead predictions will fall into this interval. The width of the confi- dence interval is approximately 0.4 m. The plot shows that the model predicted the future observations accurately. The model fits and residuals were checked with a variety of tests. The Q-Q plot and Shapiro-Wilk normality test were used to check the normality of the residuals. The Q-Q plot followed a straight line, and the p-value for the Shapiro-Wilk normality test is 0.56, which is higher than the 0.05 confidence level. Therefore, the model satisfies the normality assumption. The standardized residuals are plotted in Figure 9.6. Most of the standardized residuals in the plot are lower than 3, which means no significant outlier was detected. Both an autocovariance function (ACF) and Ljung-Box statistics were used to check the correlations in the residuals. The ACF values for all lags are within the limits. The p-values of the Ljung-Box statistics are all above 0.05, which indicates no sig- nificant autocorrelation in the residuals. The forecast of the model is also reasonably similar to the actual values. There- fore, it was concluded that the DLM fit the data well. Again, the limitation is that the model can only be used to treat one time series data row at a time. The model is also not appropri- ate to generalize this result to other drivers on other curves. Summary and Implications This chapter has described a time series analysis of driver behavior on rural two-lane curves using SHRP 2 NDS data. DLM was used to analyze driver behavior as a dynamic pro- cess. Distance from the centerline was chosen as a surrogate to represent roadway departure risks. Two types of DLM were used to analyze the time series data: DLM with intervention analysis and DLM with ARMA for forecasting. The intervention analysis evaluated the influence of the oncoming vehicle and the curve on lane position. The two effects were included in the DLM as intervention effects. The model was decomposed into three components: the mean level of lane position, the lane deviation due to the oncoming vehicle, and the lane deviation due to the curve. This analysis found that the average distance from the centerline to the left edge of the vehicle under normal driving conditions is approximately 0.44 m (1.44 ft) to the right of the centerline. The vehicle moved away from the centerline by an additional 0.43 m (1.41 ft) when there was an oncoming Figure 9.5. Predicted left distance with 95% confidence interval.

55 vehicle (a total of 0.77 m to the right of the centerline). At the inside of the curve, the vehicle moved closer to the centerline by about 0.53 m (1.73 ft), which placed the left vehicle edge across the centerline by 0.09 m (0.30 ft). However, these findings are based on one sample trip only and cannot be generalized to other drivers and curves. The overall safety benefit of this finding is arguable. On the one hand, larger lane deviation from the centerline increased the probability of a roadway departure crash. On the other hand, it decreased the likelihood of a head-on collision with an oncoming vehicle. Therefore, the overall safety benefits of the lane position should be further evaluated and discussed in a future study. The second model, DLM with ARMA, successfully fitted the sample trip for forecasting purposes. The diagnostics of the model residuals indicated that the fitting is adequate for the time series data. The model predicted the future position of the vehicle based on the past observations and errors. One- step-ahead prediction showed that the predicted values are very close to the actual observations. The 95% confidence interval was also plotted with the forecast values. Therefore, the research team concludes that the model fit the data well Figure 9.6. Diagnostics of residuals: Standardized residuals, autocovariance function of residuals, and p-values for Ljung-Box statistics. and could potentially be used for roadway departure crash warning systems. The model results are only applicable to the scenarios tested and cannot be extrapolated to other drivers or other curves. The intent of Research Question 4 was to show proof of con- cept. However, results suggest that the time series data can be used to model driver lateral control as a function of external variables (oncoming vehicles, position within the curve). This indicates that lane position data from the SHRP 2 NDS that are sufficiently reliable could be used in development of colli- sion warning system algorithms. It should be noted, however, that the models predicted lane position at a high resolution (i.e., 0.09 m). The accuracy of offset and other lane position variables has not yet been pub- lished. The accuracy also depends on the quality of the for- ward view, quality of lane lines, and other factors. As a result, 0.1 m (0.3 ft) is likely smaller than the accuracy of the vari- ables used to calculate position from the lane edge. Therefore, whether a vehicle crosses a lane line and the magnitude of the crossing should be evaluated cautiously. The results do show that the model can be used to predict position and shifting position due to external variables.