A New Approach to Customization of Collision Warning Systems to Individual Drivers
This paper discusses the need for individualizing safety systems and proposes an approach including the Real-Time estimation of the distribution of brake response times for an individual driver. While maintaining high level of safety, the collision warning system should send “tailored” responses to the driver. This method could be the first step to show that safety applications would potentially benefit from customizing to individual drivers’ characteristics using VANET. Our simulation results show that, as one of the imminent and preliminary outcomes of the new improved system, the number of false alarms will be reduced by more than . We think this tactic can reach to even beyond the safety applications for designing the future innovative systems.
I Introduction111This work was supported by the NSF under CCF 0844725.
Despite the increases in safety introduced into the automobile, at latest count (2010) the number of deaths is over 30,000, the number of injuries is over two million, and the number of crashes is over ten million . Some of these accidents could have been prevented or reduced in severity if the drivers involved had been warned in time to slow down or steer away to avoid the accident. To address this problem, collision warning systems hold great promise, Fig. 1. The true potential of the various classes of warning systems to reduce crashes is seriously compromised by three interrelated factors:
The algorithms used to trigger a warning are largely ineffective when they are not adapted to the individual driver and vehicles involved directly in a crash.
Warning algorithms have relied for the most part on the behavior of threat vehicles immediately ahead and to the side.
The driver often fails to trust the warning even when it is issued in time to avoid a crash.
Radical improvement in the effectiveness of collision warning systems are now possible due to the progress that is being made in vehicular ad hoc networks (VANET). Vehicular ad hoc networks potentially allow all vehicles to communicate with each other (V2V or vehicle to vehicle communication) and with technologies embedded in the infrastructure that transmit crash relevant information (V2I or vehicle to infrastructure communication).
The effectiveness of warnings depends on how much time the driver needs to react. Therefore, to be as effective as possible, accident warning systems should be tailored to the specific characteristics of the driver. He or she could be vigilant or distracted; could perceive and react soon to an event or might have a longer perception-reaction time; could be aggressive in acceleration/deceleration or could be smoother in those. An important aspect of the specific characteristics of the driver is the distribution of brake response times (BRT) for each particular driver. The BRT is the time elapsed between a stimulus such as a lead car braking or traffic signal changing color and a braking response by the driver. Since existing collision warning algorithms don’t use the BRT distribution of individuals, drivers with different BRT in the same scenario receive the same warnings. Clearly, this approach isn’t optimal for design of safety systems. The most important contributions of this paper are:
Proposing a method for Real-Time estimation of the distribution of brake response times for an individual driver using data from a VANET system which has information about the positions, velocities, and accelerations of cars on the roads, road configurations, and the status and position of traffic signals.
Using the estimated distribution to customize warning algorithms to an individual driver’s characteristics.
The paper is organized as follows. In section II we review the relevant literature formally defining the BRT and related quantities, discussing factors that affect drivers’ BRTs, and outlining several methods that have been proposed to estimate a driver’s BRT. Section III and IV outline methods that can be used to estimate BRTs and what the distribution of a driver’s BRTs would be if he or she did not intentionally delay braking, respectively. Our concluding remarks are discussed in section V.
Ii Related Work
Ii-a Basic Ideas: Perception-Reaction Times and Brake Response Times
The time required to respond to a stimulus can be divided into several distinct phases. One such division is given by Koppa . He defines the perception time as the amount of time it takes for an individual to recognize that an event has occurred. The reaction time is then the time elapsed from detection of a stimulus to the start of a response. The response time includes the reaction time as well as the time required to complete the response. The perception-reaction time or brake reaction time is the time required to perceive and initiate a reaction to the stimulus. These divisions are illustrated in Fig. 2.
There is some ambiguity in this definition of the reaction and response times in that we must specify what is meant by the response. Commonly in driving studies, the response is operationally defined to be the act of braking . This operational definition is convenient because it is relatively easy to measure when the brakes have been applied. However, a difficulty with this definition is that a driver may intentionally delay braking, for instance if there is a large space between the driver and a traffic signal or leading car. This means that measured response times may be larger than the drivers’ “true” response times , . This delay is illustrated in the data plot reproduced in Fig. 3, which is taken from an article by Goh and Wong . We have rotated the plot to clarify that we view time headway as the independent variable and response time as the dependent variable. In this plot, the horizontal axis shows the driver’s time headway to a traffic signal at the time it turned from green to yellow and the vertical axis shows the measured brake reaction time for drivers who braked (or the actual time to pass the signal for those drivers who ran the light). We see that when the driver is a larger distance from the traffic signal, their measured brake response time is larger – likely because they chose to delay braking.
In this paper we define the potential brake response time (PBRT) as the time that a driver could have braked in if he or she did not choose to delay braking, which is the relevant quantity for the purposes of an accident warning system. We will use the term “brake response time” (BRT) to refer to the observed quantity, the time elapsed between a stimulus such as a traffic signal color change and when the driver applies pressure to the brake pedal. These definitions are illustrated in Fig. 4. The estimation of BRT and PBRT both present technical difficulties. We review methods that have been proposed to estimate these quantities by previous researchers in the next two subsections.
Virtually every study to examine reaction times has found that the population distribution of reaction times is skewed right and several have shown that it is well approximated by a lognormal distribution , , , , , , . A close examination of the plot in Fig. 3 above indicates that the distribution of BRTs is also skewed right at a fixed value of time headway. We will make use of this observation later in our data analysis.
Ii-B Estimation of Brake Response Time from Car-Following Data
Several previous studies have examined how BRT can be estimated from car-following data automatically. Here we review several of these methods, focusing in particular on the effectiveness of these algorithms for obtaining an accurate estimate of the BRTs to several distinct events and on the feasibility of implementing them with the limited computational resources available in an on-board computer system in a car.
The main ideas we build on in this paper were proposed by Zhang and Bham . Their method is based on intuitive reasoning about the relationships between the distances, speeds, and accelerations of two cars when the following car reacts to an action taken by the lead car. The starting point in their algorithm is to identify two cars which go for a period of at least 4 seconds in which they are separated by less than or equal to 250 feet and their speeds are within 5 ft/s, or 1.52 m/s. These cars are said to be in a steady state. They then observe a time A when the the distance between the cars decreases or increases while the follower has an acceleration rate of . This change in distance between the cars is caused by acceleration or deceleration of the leader. Next they find the time B when the follower decelerates or accelerates at a rate . The difference between times A and B is then an estimate of the follower’s BRT. The advantages of this method are that it is intuitively reasonable, relatively easy to implement, and it yields reasonable reaction time estimates. However, the requirement that the cars be in steady state is restrictive. To obtain more information about drivers’ reaction times, it would be helpful to extend this approach to estimate reaction times in other situations than the steady state.
Another method for BRT estimation was proposed by Ma and Andréasson and is based on techniques designed to find the lag between two linearly related time series . The inputs to this technique are two time series and – one which represents a stimulus, such as the difference in speed between the two cars, and another which captures the response of the follower, such as that car’s acceleration. The basic idea of the method is to examine the covariance between the time series in the frequency domain, as measured by the coherency. There are two major limitations to this approach that make it inappropriate for our purposes. First, the coherency measures linear association between the two time series. If acceleration is not a linear function of the difference in speeds between the cars, this approach may not measure the lag between the time series appropriately. Second, this method does not allow us to estimate separate BRTs to separate events in a natural way. Instead it views the entire output time series as being a linear function of the entire input time series and estimates an overall time delay. To estimate separate BRTs in order to build up a distribution of response times, we would have to manually divide the time series into shorter pieces corresponding to each reaction event.
A third approach was taken by Ahmed, who specified a reaction time distribution as part of a larger model of car-following behavior, and estimated all parameters of this model jointly through maximum likelihood techniques . However, the maximum likelihood estimates had to be obtained numerically, which is computationally intensive due to the complexity of the model. Therefore, this method would not be practical to implement in an accident warning system where the BRT distribution must be obtained with limited computing resources. Furthermore, one of the desired requirements for the warning systems is to use the individual perception reaction time data online. In other words, the model needs to become more accurate as more information becomes available from VANET system. However, based on most of the current methods we cannot update the algorithm in Real-Time.
Ii-C Estimation of Potential Brake Response Times from Observed Brake Response Times
Three previous studies have addressed the problem of estimating the distribution of “true” reaction times based on observed brake response times. All of these studies examined this problem in the context of traffic signals, and focused on estimation of population distributions, rather than distributions of response times for a particular individual.
Maxwell and Wood simply used the mode response time as a point estimate for the average brake response time in a population, arguing that this measure would be less sensitive to large reaction times that include a delay . Goh and Wong take a more sophisticated approach . They define a transitional zone (TZ) based on the time headway between the driver and the traffic signal at the time that it changes to yellow. This TZ is “an empirically calibrated range of time headways suitable for identifying drivers with realistic stop-or-cross decisions” . Essentially, to estimate response times they limit the sample to those cars with a time headway of 4 seconds. Nearly all cars that chose not to stop at the light were within the 4-second threshold; thus, this threshold includes cars with a “real” choice between stopping and continuing on. This is illustrated in Fig. 3 above.
This analysis does suffer from some limitations. First, by restricting the sample to those cars within the TZ, they lose the information contained in those other data points. This is a particularly critical problem in our application, where we wish to learn about response times for a particular driver. We may not have the chance to observe response times very frequently for a single driver; it would therefore be helpful to be able to use all observed data points rather than just those with a time headway of 4 seconds or less. Second, although the relationship between time headway and BRT is reduced when the sample is restricted to cars with time headway of 4 seconds, a relationship can still be seen in the plot in Fig. 3. This suggests that some of the measured response times may still include a delay even within the TZ. In this paper, we pursue the idea that there is a significant relationship between BRT and these other variables, since multiple studies have supported this claim.
Iii Proposed Method for Brake Response Time Estimation
For the purposes of the accident warning system, we wish to learn about the distribution of response times for an individual when the car in front of them brakes. As discussed in section II-B, Zhang and Bham have proposed an effective method for estimating BRT when the cars are in steady state. However, this situation may be relatively rare in real-life driving situations, so that we may not make many observations of the BRT for an individual driver under this setting. Therefore, in practice it could be difficult to learn about the distribution of response times using only the method proposed by Zhang and Bham. Our proposed approach is to establish relationships between the distributions of reaction times under different circumstances. This will allow us to use measures of a driver’s response times under a variety of circumstances to estimate the distribution of an individual’s response times when the car in front of them brakes.
We will attempt to measure brake response times in three settings:
The cars are in steady state and the leader brakes.
The cars are not in steady state, the follower is driving faster than the leader, and the leader brakes.
The car approaches a traffic signal which changes from green to yellow.
In this section we discuss specific ideas for reaction time estimation in each of these settings For now we concentrate on methods to obtain a point estimate for a driver’s BRT to a particular event. Methods to combine these point estimates to estimate the distribution of PBRTs will be discussed in the next section.
Although it is not mentioned in any of the algorithms below, we suggest that response times should only be recorded if the driver is travelling faster than some cutoff speed such as 20 miles per hour.
Iii-a Steady State, Leader Brakes
In this case we use the algorithm developed by Zhang and Bham:
Identify when a pair of cars is in steady state for 4 seconds:
Separated by ft.
Speed and acceleration of leader and follower are equalized (speeds must be within ).
In the text, it is not clear whether specific limits are placed on acceleration, but it seems clear that if the distance and speed conditions are satisfied for 4 or more seconds, the cars’ accelerations must be approximately equal. However, there does seem to be a limit on the follower’s acceleration of , from the second step below.
Observe a time A when the the distance between the cars starts to decrease while the follower has an acceleration rate of . This change in distance between the cars is caused by acceleration or deceleration of the leader.
Observe the time B when the follower decelerates at a rate .
Zhang and Bham do not specify how they determined when the distance between the cars had started to decrease for step 2 in this algorithm. Several methods are possible. One simple idea is to determine at each time point whether the distance between the cars is less than it was at the previous measurement. If this is sustained for a sufficient length of time (such as a quarter-second), the starting point A is the time at which the distance first started decreasing. If limitations of the measurement instrumentation mean that we may observe an increase or no change in the distances between the cars for one time point when they are actually decreasing, this approach could be replaced by regressing distance on time over a quarter-second period to determine if they have a negative association on average over that time.
Iii-B Not In Steady State, Leader Brakes
In theory, it seems likely that a similar technique to the above can be used when the drivers are not yet in steady state and the lead car brakes. Note that we might only expect to observe a response in this situation if the follower is travelling at a higher speed than the leader. Also, the follower and leader should be near enough to each other that the follower will need to respond to the leader’s braking action. For example, we could measure response times only if the time headway between the leader and the follower is less than 10 seconds at the time that the leader brakes.
The key problems are selecting what measures to use in determining that the leader has braked and that the follower has responded. For deciding whether the leader has braked, it may be easiest to make use of the vector of accelerations of the lead car, and use a threshold value to decide when the leader has braked. We could simply use the value which was used above to detect when the following car reacted in the steady state setting.
To determine when the follower has responded, we would recommend first finding when a response has occurred in driving simulation trials by manually looking at the speed and acceleration profiles. This should allow you to select what variables to use to measure the response. One possibility that seems reasonable is a reduction in the acceleration of the follower.
Once this or some other similar quantity is determined to be the appropriate variable to use to detect the follower’s response, we will again need to choose what cutoff value for that variable indicates that the response has occurred. For example, we would need the cutoff value such that when the reduction in acceleration is less than , we say that the follower has responded. To find the value we could do a grid search, choosing candidate values and running the classification code for each value . For values of which are too close to 0, the threshold will be exceeded easily and the algorithm will say the response time was shorter than the manually determined value. For values of which are too far from 0, the threshold will be exceeded infrequently, and some of the manually determined responses will be missed. The objective is to select a value such that the results of the classification algorithm best match the manual classification results. This could be done informally, or formally by choosing to minimize a function such as the sum of squared differences between the manually determined response time and the algorithmically determined response time.
Iii-C Traffic Signal Changes from Green to Yellow
There are several factors to consider when estimating a driver’s brake response time to a traffic signal change. First, we should only expect the driver to respond to the signal change if they are within a reasonable distance of the signal. For that reason, we suggest a cutoff of 10 seconds in the time headway from the driver to the signal at the time it changes. Second, we should not record a response time if there is an intervening car between the driver and the traffic signal that also responds to the signal change. Finally, we should not record a response time if the driver turns at the intersection with the signal. This would not be an accurate measure of the driver’s response time since they would likely have been prepared to stop anyways.
We propose the following algorithm to estimate response times to traffic signal changes:
Log the time when the next traffic signal in front of the driver changes from green to yellow.
If the time headway between the driver and the traffic signal at the time of the signal change is large (e.g., over 10 seconds), stop looking for a reaction time.
If the leading car is also before the light, check to see if it decelerates. If it does, stop looking for a reaction time.
Check to see if the car decelerates. The difference between the time when the car decelerates and when the signal changed is the response time.
Follow up to see if the car turns at the intersection. If it does, ignore the measured reaction time.
In order to be successful in tuning ITS algorithms to individual drivers, we will need a model which provides us with an estimate of the average driver’s brake reaction time as well as the individual driver’s response time. The mix of drivers on the road is constantly changing, with new drivers joining and other, usually older, drivers leaving. Thus when there is no information on an individual, the average response times can be used. As more information about an individual driver’s response times becomes available, the system can switch from the general estimate of brake response time to the individual driver’s estimated brake response time.
Iv Estimating the Distribution of Potential Brake Response Times
Iv-a General Discussion
In this section we discuss the construction of a statistical model for the distribution of brake response times, and how this model can be used to estimate the distribution of potential brake response times for a particular individual.
As discussed earlier, previous researchers have consistently found that reaction times are skewed right and are approximated well by a lognormal distribution. The plot of brake reaction times reproduced in Fig. 3 above confirms that for a fixed value of time headway, the distribution of brake reaction times across individuals is skewed right. It is reasonable to assume that brake reaction times are skewed right within individuals as well. We therefore adopt a lognormal model for brake reaction times, modelling the logarithm of the observed BRT as normally distributed conditional on the time headway.
This lognormal model also has the advantage of automatically correcting for some differences in the variance of the BRT distribution at different time headways and across individuals. From the plot in Fig. 3, we can see that as the time headway increases, the mean BRT and the variance of the BRTs both increase. Similarly, it seems likely that some individuals have lower or higher mean reaction times than other drivers, and that the variance in the BRT distribution varies across individuals as well. Specifically, it is likely that individuals with a low mean reaction time also have a low variance in their reaction times, whereas individuals with a high mean reaction time also have a high variance in their reaction times. These differences in the variance of brake reaction times will be approximately corrected by modelling the logarithm of the BRT.
It also seems likely that the mean and variance of the brake response time distribution depend on several other variables. An important factor that will be accounted for in our model is the stimulus type (e.g. traffic signal vs. lead car decelerates). Reaction times also depend on a large number of other factors such as weather conditions and demographic characteristics of the driver. However, these variables will not generally be available to the accident warning system, so their effects will be absorbed into the error term of our model.
Iv-B The Model
Using just the time headway as an explanatory variable, the general ideas above can be formalized in the following model:
In this model,
indexes the driver
is a vector of the logarithms of observed reaction times for a particular driver.
is a matrix of covariates, detailed further below.
is a fixed vector of unknown coefficients.
is an unknown scalar.
is a random vector of unknown coefficients.
is an unknown matrix.
The basic idea of this model is that, conditional on the time headway, the distribution of BRTs for an individual driver has a mean which is given by an overall population mean, , plus an offset due to the particular characteristics of that driver, . This is illustrated in Fig. 5. It is assumed that the parameters determining the individual’s offset to the overall mean follow a multivariate Normal distribution in the population. This is a linear mixed effects model; three recommended references with further information about these models are McCulloch et al., Ravishanker and Dey, and Searle et al , , . A key assumption made in this model specification is that after the log transformation, the covariance matrix has the simple form . This assumption could fail to hold in a number of ways, but it makes the calculations much easier.
We now consider the form of the mean in more detail. From the plot in Fig. 3, we saw that the mean brake reaction time was an increasing function of time headway. Since the logarithm is a monotonically increasing function, it follows that the logarithm of the BRT is also an increasing function of time headway. For flexibility, we allow the possibility that the log BRTs are a quadratic function of time headway. We also allow for the possibility that the relationship between time headway and BRT is slightly different for each of the different stimulus types. For instance, it could be that drivers have a faster BRT at low time headways and the average BRT increases more rapidly as a function of time headway when the stimulus is a lead car braking than when it is a traffic signal changing to yellow. These considerations lead to the following possible form of the mean log-BRT as a function of time headway:
In equation (2), indexes the driver, indexes the stimulus type, and indexes the observation (so if we have 5 different BRT observations for a particular driver and stimulus type, will vary from 1 to 5). As before, is the log brake reaction time, and is the time headway at the time of the stimulus. The subscript on the and terms indicate that the values of those coefficients depend upon the stimulus type . To make this concrete, if this mean function is adopted and there are different stimulus types under consideration with observations for driver under stimulus type , and are vectors and the portion of the matrix corresponding to observations for driver will be of the following form:
Iv-C Training the Model: A Fit Using Data from Driving Simulations
For training the model, we assume data are gathered for subjects in a driving simulation. If possible, we prefer to gather data from real drivers on the road, but this is likely to be too difficult to be feasible. This being the case, we will take precautions to address concerns about using results from a driving simulation to learn about response times for drivers in real life driving situations. The subjects in the study will be a representative sample of the overall population of drivers who will be using the accident warning system. Brake responses for each subject will be elicited at a variety of levels of expectancy. To improve the statistical analysis, responses will also be collected at a range of time headways for each stimulus type. To separate the effects of expectancy and any other variables that may be included in the model, the combinations of these factors will be randomized (for example, we will have some observations where the braking stimulus was more and less surprising at different levels of the time headway variable).
For each driver, we have multiple observations of reaction times for each stimulus type. These data can be used to estimate the unknown quantities , , and in this model using standard statistical techniques implemented in the lmer function of the lme4 library in R. We will use a subscript of to indicate quantities obtained from this training data set; in particular, let be the covariate matrix obtained using data from this data set and denote the estimates by , , and . can be written as , where and the superscript denotes a generalized inverse. The estimates and can be found through numerical maximum likelihood techniques. A study conducted by McGehee et al. has found that the population average brake response time was about 0.3 seconds faster in driving simulations than it was in real life driving studies . This difference was found at time headways of approximately 2 seconds. It is difficult to account for this effect in a rigorous way, especially since this observed difference may be due in part to methodological differences between the simulator trials and the real car driving trials. One ad hoc solution would be to increase the estimated value of by an amount such that the estimated population mean reaction time at a time headway of 2 seconds increases by 0.3 seconds.
Iv-D Real Time Estimation of the PBRT Distribution for One Driver
We estimate the distribution of PBRTs for a particular driver in two steps. First, we establish the relationship between the covariates and BRT for that driver. Then we use this relationship to estimate the distribution of PBRTs by using values of the covariates at which the BRT does not include an intentional delay to braking.
Iv-D1 Estimating the Relationship Between Time Headway and BRT for One Driver
As data are gathered in real time for an individual driver , our goal is to estimate the driver’s offset to the population-average regression coefficients . This is estimated by the Best Linear Unbiased Predictor (BLUP).
Intuitively, we might expect that if a particular driver has a higher than average brake response time in one stimulus type, they are likely to have a higher than average brake response time in other stimulus types as well. Similarly, if they are particularly sensitive to the time headway in one situation, they are more likely to be sensitive to the time headway with other stimulus types. This intuition suggests that the covariance matrix will have non-zero off-diagonal entries; that is, there is some degree of correlation among the coefficients. Because of this correlation, observations from one stimulus type can give us information about the coefficients in the other stimulus types. For example, if we make some observations of driver brake response times in the traffic light setting which give positive estimates of the coefficients for that stimulus, a positive correlation between the coefficients might lead to positive estimates of the coefficients for other stimuli as well.
To reduce the computational complexity of computing the BLUP, we assume that the information about the unknowns , , and that is provided by the training data set from the driving simulator is much greater than the information provided by the data from this individual driver. That is, the estimates , , and obtained from the training data set above are very similar to what we would obtain if we estimated them using the combined training data set with the observations for this driver. If this assumption holds, we can approximate the BLUP using the estimates of these quantities found with the training data set, which saves the computational effort of re-fitting the model every time we observe a new reaction time.
Let be the covariate matrix as in the full model, but formed using only the data from driver . The BLUP of is
where . Ordinarily would be estimated from all of the data, but by our assumption above we will instead use the estimate . The formula for the BLUP still involves the unknowns and . We estimate the BLUP by plugging in the estimates of these quantities obtained from the training data above. Denoting this estimated BLUP by , we have:
where The covariance matrix of the BLUP is given by
To estimate the covariance matrix of , we plug our approximation to , , and our estimates of , , and into this formula. Denote this estimated covariance matrix by .
When no data have been gathered yet, the best predictor is just the vector 0, with covariance matrix . In this case, the estimated mean for the individual is equal to the estimated mean for the population of all drivers.
Iv-D2 Obtaining the Estimated PRBT Distribution
The final step is to estimate the distribution of potential brake response times for an individual driver, not including any delays. For the suggested model form above using a quadratic function of time headway, the intuitive idea is to pick a specific time headway value at which the driver does not have enough time to delay braking, and use that time headway value to evaluate the mean function. Based on the plots in Fig. 3, it appears that might be an appropriate value. We can then estimate the mean of the driver’s log-RTs by plugging into the estimated mean function: . This provides an estimated mean for the log-reaction time. There are several options for estimating the variance of the log-PBRT distribution. One simple idea would be to use the estimate of the quantity in the model statement 1. However, this does not take into account the uncertainty in our estimate . This uncertainty is captured by the prediction error, . It can be shown that , where
This covariance can be estimated by plugging in estimates of the unknown quantities , , and . An estimate of the variance of the distribution of log-PBRTs which takes into account our uncertainty about the value of the mean is then
When we do not yet have any data, the adjusted variance estimate is
The plot in Fig. 6 shows the resulting distribution estimates obtained in a simulation when these variance estimates are used as the parameters of the distribution of PBRTs. From this plot we can see that the estimates taking into account uncertainty in the coefficient estimates are more conservative. On the scale of these simulation results, the difference in the percentiles obtained from these estimates is just a fraction of a second, but the difference could be more significant with real data. We will use the more conservative value for the estimated variance since it more accurately reflects what we know about the distribution of response times based on the available data.
Fig. 7 shows how the estimated reaction time distribution changes with the sample size and the allocation of the sample among the different stimulus types. These results are dependent upon the parameter values used in the simulation, but they illustrate that observed reaction times for the stimulus type that is used in estimating the PBRT distribution contribute more information than observations in other stimulus types. This will generally be the case, but our simulation likely shows an extreme example since the correlation among the gamma coefficients for different stimulus types is very low in the simulation. It could be helpful to run a simulation like this once the training data has been gathered to determine what sample sizes are necessary to get good estimates of the “true” PBRT distribution.
We note that computation of the estimated PBRT distribution requires only the operations of matrix inversion and matrix multiplication. The matrix which must be inverted is , which has dimension , the number of observations for driver . The inversion operation has computational complexity . All of the matrix multiplication operations are between matrices of dimension , , , , or . Because multiplying an matrix by an matrix has complexity , this implies that the complexity of the “worst” matrix multiplication operation is (for the product ). Therefore the whole computation has complexity when .
Iv-E Estimated PRBT Distribution vs Population Distribution
Our goal in this section is to make a comparison between two types of systems.
Conventional systems which use the population distribution.
Customized systems which use an individual driver’s distribution.
We described our method  for estimating the PBRT distribution which can then be entered into collision warning algorithms. Next, we want to relate this distribution to the distribution of PRTs for the population to show how collision warning algorithms benefit from taking the estimated distribution for an individual driver into account. As discussed earlier, previous researchers have consistently found that reaction times are skewed right and are approximated well by a lognormal distribution. It is reasonable to assume that brake reaction times are skewed right within individuals as well.
In equation 3, we let be a random variable representing the PRT for an individual driver. Also, to analyze the situation, we let the mean of the distribution for an individual driver, denoted by , be a random variable. This assumption will represent the fact that different drivers in the population have different PRT means, and that more drivers have means within certain ranges than others. For our model of driver PRTs to be reasonable, the marginal distribution should closely match the log-normal distribution for the overall population found in the literature. Once a distribution for a driver’s reaction time has been established, we would like to use this information to improve safety and minimize the rate of false alarms. One simple method for doing this would be to give the driver a warning when they are approaching an obstacle, and there will not be enough time for them to react otherwise. As we mentioned,  established that the distribution of PRTs of drivers reacting to surprise events follows a log-normal curve with parameters and . Since failing to give a warning when one is needed could be very dangerous, we will assume that the percentage of possible collisions that the system fails to provide warning for is fixed at a small number (e.g. at 1%), and then try to minimize the frequency of false alarms that the system gives subject to this constraint. If the system detects that the driver has less than his or her PRT to react to an obstacle, it should give the driver a warning. We can only state the probability that any PRT is above or below a certain value. Thus, the constraint states that we must calculate some threshold above which there is only chance that a PRT will be , and send a warning whenever a driver has less than this amount of time to react. Therefore, we can calculate the threshold to send the warnings using the distribution for the entire population:
Also, we can calculate warning threshold using the distribution for an individual driver:
Now that we have established the thresholds for sending collision warnings, we can calculate the false alarm rates that will result from using the different systems. A false alarm occurs whenever a warning is sent, but it is not needed. To best explain this problem, let us consider the scenario that a vehicle is following another vehicle on a one-lane roadway when the lead vehicle suddenly begins to decelerate to avoid an unexpected obstacle. Suppose that the system has calculated that the following driver has seconds to react, and that is less than , therefore a warning has been sent. Then, the false alarm rate is the probability that the driver’s reaction time, X, will be less than . Let denote the cumulative distribution function for this distribution then is the total false alarm rate.
It is clear from Fig. 8 that when we use the population brake reaction the false alarm rate is higher by almost a factor of two than when we use the individual driver’s brake reaction time. Therefore, safety applications could potentially take full advantage of being customized to an individual’s characteristics. Regardless, there is an observable tradeoff between the false alarm rate and the probability of an accident, one that cannot be remediated by obtaining estimates of an individual’s brake reaction time. It’s worth mentioning that false alarms are not evenly distributed across the population. Drivers with fast reaction times will have very high false alarm rates, but drivers with slow reaction times will have lower rates. If the warnings turn out to be false alarms too frequently, drivers may begin to ignore them, and since novice drivers receive the most false alarms, the danger to ignore the safety system is higher for them. While it is known that collision warnings can offer a great help to older drivers, our method benefits novice drivers as well. If it were simply the case that novice drivers were careless, warnings might be of little use. But, the existing research suggests that many novice drivers are clueless, not careless, e.g. in . Thus, it is of vital importance to minimize false alarms so that the system sends warnings only when it is needed.
Also, we need to take into account the estimation errors since in real life our estimations are not accurate. Fig. 8 shows that by increasing the error (equation IV-E) of the estimated distribution, the individual curve approaches the population curve. Our error model can be described as:
High level of errors in estimating the distribution of individual drivers is almost equivalent to the case when no individual sample is available; thus, the system uses the population distribution similar to the conventional systems.
V Conclusion And Future Work
In this paper we discussed the need to adapt collision warning systems to drivers’ individual characteristics and proposed a method for doing this customization by estimating the distribution of potential brake response times for an individual driver in real time. Collision warning systems generally rely solely on the distribution of the entire population of drivers, thereby ignoring the distinct characteristics of individual drivers. They may frustrate the drivers with the overly high numbers of false alarms, causing them to ignore warnings or even disable the system. If drivers are distracted by overly frequent warnings, the safety benefits of the system are compromised or even lost.
Our proposed method uses a statistical model that was developed based on previously published results about the population-level brake response times. This model has not yet been validated with data that includes multiple reaction times for each driver. However, we demonstrated why employing this method will result in reducing the rate of traffic collisions, thereby dramatically improving the safety benefits for all drivers. In our future work, we will collect this data, fine-tune the model, and apply it to a collision warning system.
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Ali Rakhshan is currently pursuing a Ph.D. degree in electrical and computer engineering at the University of Massachusetts, Amherst. With a broader scope of communication systems, his main field of interest is Vehicular Ad Hoc Networks (VANETs).
Evan Ray is a Ph.D. student in the Statistics program at the University of Massachusetts, Amherst. He previously earned a M.S. in Statistics from the University of Massachusetts, Amherst and a B.S. in Mathematics from the University of Massachusetts, Boston. His research interests include state-space models and measurement error.
Hossein Pishro-Nik is an Associate Professor of electrical and computer engineering at the University of Massachusetts, Amherst. He received a B.S. degree from Sharif University of Technology, and M.Sc. and Ph.D. degrees from the Georgia Institute of Technology, all in electrical and computer engineering. His research interests include the mathematical analysis of communication systems, in particular, error control coding, wireless networks, and vehicular ad hoc networks. His awards include an NSF Faculty Early Career Development (CAREER) Award, an Outstanding Junior Faculty Award from UMass, and an Outstanding Graduate Research Award from the Georgia Institute of Technology.