The motion of a car (or a car relative to a lead car) can be modeled using kinematic equations of motion as described below:
where,
for single car (for a pair of cars)
- u = inital speed (or initial speed difference)
- v = final speed (or final speed difference)
- a = acceleration (or acceleration difference)
- t = time
- s = displacement (or distance between the front bumper of the following car and rear bumper of the lead car)
To me, it looks like that equation 1 is a simple regression model with u as an intercept and a as a slope. Thus, the explanatory variable is time. As time goes on, the values of v and s change, depending on the value of a.
However, one stark difference between equation 1 and simple linear regression is that the intercept remains the same in the latter, but not in equation 1. This is because equation 1 is evaluated at every time frame (say 1 second). So, e.g. at t = 0, if u = 0 and a = 1, then v = 0. At t = 1, v = 1. At t = 2, the u is now equal to old v i.e. 1 and v gets updated to 2. In simple words, the speed is updated by 1 m/s every second because the acceleration rate is 1m/s per second.
Apart from this, the acceleration a does not necessarily remains constant. It can change sign (braking, pushing gas pedal) and/or change magnitude (driving freely or in heavy traffic) depending on the conditions and the driving style.
Therefore, in my opinion, time is a confounding factor in this case. My question is:
If one has to define a statistical model that captures car motion (with or without reference to a lead vehicle), what would be the best way to go about it? What framework should be used? My experience in statistical modeling has been limited to regression modeling only.
NOTE: I am aware of the body of literature on car-following models. But I'm interested in a statistical approach to look at this problem.
Best Answer
you're confounding, a bit, (1) cross-sectional regression and time series regression; and (2) deterministic and stochastic models. Equation (1) is your "core" equation; equation 2, for example, is simply its integral. All other equations are simply restatements from Newtonian mechanics using a different parameterization.
So, in eq. #1, you are stating that (1) a is constant (implicitly); (2) u(0) is known; (3) and v(0) is known. It is simple physics then to determine v(t) assuming u(t) is known. In fact, if a(t) is not a constant but known functionally, that, too, can be worked in.
It is a very different question to say that u(0) is known only as a random variable and u(t) is the same. (Let's assume a(t) is a constant.) Then you could use regression.
To understand what is going on, you must get beyond regression alone, and learn mathematical statistics: the calculus of (formally defined) random variables. Regression can be explained via mathematical statistics, but not the other way round.