# Attention Horizon as a Predictor for the Fuel Consumption Rate of Drivers
## 2 Materials and Methods
### Extraction of Red Light-Approaching Trajectories
If we want to calculate the FCR of a driver regardless of the vehicle, for a period of time (e.g., a day), we obtain the average FCR percentile index for different vehicles for that period of time and driver and then we compute their average, weighted by the distance traveled by each vehicle. After these normalization steps, we are finally able to investigate the effect of AH on FCR through time, for each driver. In particular, we expect that if a driver has a relatively high (i.e., good) AH during a period of time, it will correspond to a relatively lower FCR. For this reason, for every driver, we took into account normalized FCR and AH values during each day, as described before. Then, we collected these values for all the days and obtained a single correlation value across days between FCR and AH. Since we are calculating correlations, we require drivers with at least two days for which FCR and AH values exist. We perform the same procedure across weeks and also months to know how long the time period needs to be in order to obtain a good correlation between AH and FCR. We calculated the histograms of these per-driver correlation values and the results are shown in Figure 5. In this figure, we show different histograms when FCR and AH are obtained for calendar days, calendar weeks, or calendar months. The histogram generated for calendar days shows that most of the drivers' correlations are very close to the value 0. This means that it is not readily possible to connect the normalized AH value of a day to its normalized FCR value for a driver. Too many different factors affect both these measures, apparently. However, as we increase the length of the time period to weeks and months, the histograms show that the correlations progressively shift towards clearly negative values. In fact, the longer the time periods are, the more negative the correlation coefficients become. A negative correlation means that the larger the AH becomes for a person, i.e., the sooner they react to a red traffic light, the lower their overall FCR is. For extra visualization, we also fitted a normal distribution to the samples of the drivers' correlation values in each histogram. It can be seen that the standard deviation of the correlation values increases by increasing the length of the time periods. The reason for this could be that more data points become concentrated at the lower end of the possible values' interval (i.e., \(-1\)). Another explanation can be that other factors exist that we have not taken into account. Finally, mean and median of the correlation values are also overlaid on each histogram. The mean and median values for different time periods are additionally shown in Table 1.

It can be seen that both mean and median move closer to \(-1\) the larger the time period becomes. The median of the correlations appears to be decreasing even more than the mean when using months as time period. This, again, can be explained by the fact that the whole distribution of the correlation samples moves to the left for larger time periods; however, since the minimum possible value is \(-1\), the samples become concentrated at the lower end of the possible interval. In the next section, we discuss our results in more depth. ## 4 Discussion

In the Results section, we showed that the variability of AH caused by different traffic lights is higher than the one caused by different drivers. The difference was so significant that we had to replace the AH values by their corresponding percentile indices within each traffic light. We hypothesize that this high cross-light variability is caused by the different environmental aspects surrounding traffic lights. This could be due to differences in, for example, the visibility of the light on the road, the speed limit before the crossing, the dimensions of the road, its slope, traffic conditions in the area, or any other environmental aspect. Because of this, we believe that normalization based on traffic light is a very important step to make the AH values usable.