Sunday, February 21, 2010

Formula One and random processes

"There is still a desire to introduce mechanisms that create even greater abnormality to the grid, even more random processes." Ron Dennis, March 2003.

In the eyes of a statistician, a Formula One lapchart graphically represents a sample from twenty inter-dependent stochastic processes, (colloquially referred to as random processes).

Mathematicians define a stochastic process as a time-ordered family of random variables Xt upon a probability space Ω. This is a rather recondite definition, but the basic idea is that Ω is the space of histories for the system under consideration. In other words, each point in this probability space, ω ∈ Ω, represents a possible history of the system.

This space could be the space of possible share-price histories of a company, or it could be the space of possible lap-by-lap positions of a Formula One car starting from a particular grid position in a Grand Prix. Each actual share-price history, and each actual lap-by-lap history, is a sample from such a probability space. Run the process again, from the same starting point, and you'd get a different history. In the case of a Grand Prix, each different starting grid position generates a different random process. Run a Grand Prix again, with all the cars in the same starting grid positions, and you'd get a different sample of 20 histories from the joint probability space of these inter-dependent random processes.

By definition, a random variable X is a function on a probability space Ω, which possesses a probability distribution over its range of possible values by virtue of the probability measure on the subsets of the probability space Ω. In the case of a stochastic process, Xt is a function on the path-space of the system, which represents the position of the system at time t. The probability measure on Ω, the space of histories, determines the probability distribution over the range of each random variable Xt, and thereby determines a probability distribution over position at each time t. Different positions at time t have different probabilities because different histories have different probabilities.

In the case of a Grand Prix, there are twenty such inter-dependent random variables, corresponding to the number of different starting grid positions:

Xnt: n = 1,...,20.

Xnt is the random variable which represents the position on lap t of the car which started in the nth grid position. The probability distribution over the range of possible values of Xnt, specifies the probability of a car being in position x on lap t after starting nth on the grid.

Each different track on the Grand Prix calendar will define its own unique probability measure on the joint space of possible lap-by-lap histories. Thus, the probability distribution over the range of possible values of Xnt, will vary from one track to another. For example, the probability of being in 6th position on lap 40 at Monaco, after starting 16th on the grid, is different from the probability of being in 6th position on lap 40 at Monza, after starting 16th on the grid.

Randomicity, and the unpredictability which follows from it, is one of the ingredients which makes a sport exciting. Back in 2003, Ron Dennis was complaining about new qualifying regulations which introduced a degree of randomicity into the starting grid positions. The cars were forced to start a race with the fuel load they qualified with, and different cars would choose to start a race with different fuel loads. Hence, for the past seven years, qualifying grid positions have not necessarily been a reflection of true pace. For 2010, however, Formula One returns to low-fuel qualifying, hence that particular random element of the sport has been eliminated.

Anticipation is high for the 2010 season, with Michael Schumacher's return, and Jenson Button paired with Lewis Hamilton at McLaren. However, with overtaking as difficult as it has been for the past decade, and with the ban on re-fuelling removing another strategic dimension, one wonders if the resurgence of deterministic processes in Formula One might provoke a mid-season revision to the sporting regulations...

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