Estimation of the deceleration profiles

One of the objectives of this research is to estimate the deceleration profiles in different scenarios (e.g. in different road types or in different elements). The literature review yields a variety of deceleration models (from really simple, constant deceleration to more complex linear and polynomial models (Akçelik and Besley, 2002). Deceleration

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value, distance and duration, together with the initial and final speeds of the event are necessary for modelling the deceleration of vehicles.

Within this PhD, three different functions are tested to represent the deceleration profile for each event, which can be assumed as typical braking patterns. For the better fit and interpretation of the functions, the deceleration event is split into two parts (Regime I and Regime II). The first part starts with the beginning of the deceleration event as was defined in the previous section and ends when the maximum deceleration occurs and in real life; this is the part where the driver presses the brake or releases the throttle. The second part begins from the maximum deceleration of the event and ends with the end of the event and depicts the release of the brake from the driver or the repress of the throttle.

The split is performed by the algorithm that was developed and implemented in Matlab. So, after the algorithm has detected the deceleration event as described in the previous section and has saved every event separately in a different file, it calculates the maximum deceleration of the event. Then, starting from the first observation of the event, it runs through the file till it meets the deceleration value that equals to the maximum one and marks that as the end of the first part of the event and the beginning of the second one. Finally, it saves each part in a different file in order to later estimate the profile for each of them. An example is shown in Figure 4.3 below.

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Figure 4.3: Example of the separation of the deceleration event

The next step is the estimation of the deceleration profiles. Hence, a curve fitting algorithm was developed. Three functions are tested for both parts of the deceleration events since it is of interest to understand the whole picture of the braking, i.e. how the driver press and release the brake or the throttle. The first function is the simplest and has a linear relationship between deceleration value (a) and elapsed deceleration time (t). The function is 𝑎 = 𝑝1× 𝑡 + 𝑝2 (linear equation), where p1 and p2 are the

coefficients of the equation. In real traffic, this reflects the driver braking gradually and releasing the brake gradually too. The second function is 𝑎 = 𝑝1× 𝑡2 + 𝑝2× 𝑡 + 𝑝3

(Parabola 1-red colour in Figure 4.4) where 𝑝₁ is negative for the first part and positive for the second one. In real traffic, Parabola 1 represents for the first part of the deceleration event the situation where the driver brakes smoothly at the beginning considering enough space to stop the vehicle, though this is followed with a harder brake due to lack of space and time; for the second part it depicts that the driver

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presses the brake hard for some more time and after he releases it slowly. Finally, the last function is 𝑎 = 𝑝₁× 𝑠𝑞𝑟𝑡(𝑡) + 𝑝₂ (Parabola 2-green colour in Figure 4.4) and represents a firm brake at the beginning of the event due to a sudden obstacle appearing, followed by gradually smoother braking since there is plenty of space to stop. As far as the releasing of the brake concerns, it illustrates that the driver releases the brake firmly. The hard press or release of the brake may also indicate that the driver obtains an aggressive driving style (Figure 4.4).

Figure 4.4: Tested functions for the first part (above diagram) and the second part (below diagram) of the deceleration event

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The next step is to obtain the reference function from the tested functions described above. To judge which of the three abovementioned functions fits best to each deceleration event, the algorithm calculates the appropriate coefficients and the adjusted R square by fitting each function to the deceleration data of each event. The adjusted R2 _{is a goodness of fit measure that takes into consideration the number of}

predictors, making it more reliable than R2_{. However, to calculate the adjusted R}2_{, the}

R2 _{should be found first. The R}2 _{is a ratio between the regression variance and the}

total variance of the data and is estimated by:

𝑅2 = 1 −∑ (𝑦𝑖− 𝑦̂𝑖) 2 𝑁 𝑛=1 ∑𝑁 (𝑦_𝑖− 𝑦̅)2 𝑛=1 (4.1)

where 𝑦_𝑖 is the actual observation; 𝑦̅ is the mean value of the observations and 𝑦̂_𝑖 is the predicted value.

Then, the adjusted R2 _(𝑅̅2_{) is calculated by:}

𝑅̅2 _{= 1 − (1 − 𝑅}2_{) [} 𝑛 − 1

𝑛 − (𝑘 + 1)]

(4.2)

where n is the sample size and k the number of independent variables in the regression equation.

Therefore, by comparing the adjusted R squared, the function with the maximum value is the most appropriate to represent the deceleration profile of that event and is saved as a new variable.

Up to this point, the best-fit deceleration functions for all the deceleration events have been obtained, but there is one different function for every deceleration event. Since the aim is to achieve deceleration reference functions that could describe the deceleration events in general for different scenarios, it is essential to calculate an average one for each function (i.e. linear, parabola 1 and parabola 2). Using the average of the coefficients of each event for the best-fitted function will lead to an average reference function. Aiming to explore in-depth those profiles and since the

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duration plays an important role, a cluster analysis was performed in SPSS for each function concluding in equations for long, medium and short deceleration events.