# Driving Behavior Modeling Based on Consistent Variable Selection in a PWARX Model
## 1 Introduction  
The automotive industry has seen significant advances in autonomous driving and advanced driving assistance systems (ADAS) in recent times [1], not only from a technological point of view but also from a regulatory point of view [2]. These advances have been spurred on by corresponding advances in sensor technology [3]. These sensors such as LiDAR (Light Detection and Ranging), radar, camera, ultrasonic, GPS (Global Positioning System), and IMU (Inertial Navigation System) in combination with sensor fusion have made automated driving possible. Generally speaking, there are six levels of automation in autonomous driving according to the US Society of Automotive Engineers [4]. To the best of our knowledge, as of 2019, level 3 automation has been the maximum level of automation of commercially available vehicles. However, in places like the US, Netherlands, Germany, and the UK, testing of level 6 (full) automation is allowed under certain conditions. The emergence of open-source driving simulators that support flexible specification of sensor suites and environmental conditions such as the CARLA Simulator [5], and LGSVL Simulator [6] have also contributed to the rapid development of autonomous driving. These simulators give full control of the traffic actors, thus making it possible for researchers to easily simulate different traffic scenarios for testing automated solutions. However, these simulators are still not able to fully express the dynamical environment and behavior of real-world traffic. Furthermore, recent advances geared towards decision-making architectures for automated driving [7; 8; 9] on a real road means that higher levels (level 4) of automation should be expected in the near future. Judging from these recent advances in the autonomous driving technology and design of advanced driving assistance systems (ADAS), the natural evolution will be geared towards mixed autonomous driving (i.e., a mixture of human-driven vehicles and automated vehicles), as well as how to reflect personalized driver characteristics and preferences in automated vehicles. Automated vehicles and ADAS need to understand and predict the cognitive and behavioral aspects of the human driver. This means that driving behavior based on real-world driving must be carefully analyzed and modeled. In other words, developing driver models that are more realistic, mathematically rigorous enough, and are capable of expressing the dynamical characteristics of the driving behavior will be the key to meeting this advancement. From the viewpoint of control technology, numerous ideas have been leveraged to model the human driving behavior [10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. In those works, to analyze the driving behavior, the concept of regarding the driver as a sort of controller in combination with linear control theory was adopted. However, a linear controller could be limited in the cases that the driver incorporates not only simple reflexive motion but also decision-making in operating the car. To this end, nonlinear and stochastic modeling techniques like neural networks and hidden Markov models (HMM) were developed for human behavior modeling [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. However, a careful consideration of some applications, such as trying to estimate both the mental and physical states of the driver or interpreting obtained information for ADAS design, reveal the shortcomings of these techniques. The works [33; 34; 35; 36; 37], have shown that, in analyzing the human behavior, the driver timely switches the simple control laws rather than using the complex nonlinear control law. The Hybrid System (HS) is the result of mathematical formulation of this idea, where the transition between modes regarded as the driver's decision-making, are expressed by a discrete-event-driven system and the dynamical characteristics basal to each mode is represented by the continuous-time-driven system.