Model for predicting world records

[ribod]

hello,
i hope I'm finding the right forum for my question.

The question is quite simple: how to set a mathematical model for estimating the lowest possible time a world record in a computer game track, and with this model also being able to predict world records in the future.

I don't need a complicated model, but just a means for how I could plot this graph with a computer program. I have found a site which does basically what I would like to do. It has the different world records on 100 metres sprinting, and the times they were made. By creating a logistic equation they could predict the time limit and when it would be achieved. (link: they have some explanation on this page, but I couldn't really understand it: http://www.condellpark.com/kd/sprintlogistic.htm)

My question is how to create such a formula like this, with a best-possible-time limit, from a dataset of earlier world records and the time they were made.

 

[p1ayaone1]

I think Matlab can fit exponentials to datasets. Evaluating the asymptote could perhaps predict the fastest possible time.

 

[Mute]

hello,
i hope I'm finding the right forum for my question.

The question is quite simple: how to set a mathematical model for estimating the lowest possible time a world record in a computer game track, and with this model also being able to predict world records in the future.

I don't need a complicated model, but just a means for how I could plot this graph with a computer program. I have found a site which does basically what I would like to do. It has the different world records on 100 metres sprinting, and the times they were made. By creating a logistic equation they could predict the time limit and when it would be achieved. (link: they have some explanation on this page, but I couldn't really understand it: http://www.condellpark.com/kd/sprintlogistic.htm)

My question is how to create such a formula like this, with a best-possible-time limit, from a dataset of earlier world records and the time they were made.

As p1ayaone1 said, you can use a program like MATLAB to do a curve fit once you have your fit equation. To get the equation you want to fit to is a little more complicated. The authors of the link you gave basically assumed two things:

1) that the rate of change of performance in the sprinting event at a given time was a function of the performance at that time

2) the performance eventually reaches some limit that cannot be exceeded, at which point the rate of change of performance is zero.

The measure of performance is the time of the world record. Mathematically, this gives us a differential equation,

$$\frac{dP}{dt} = f(P)$$
where dP/dt is the rate of change of the performance, P, and f(P) is some function of the performance. Basically, we want to know what functional form of P(t) will have a derivative equal to f(P). We have no idea what f(P) is, though, so the authors assume they can expand the function as f(P) = aP + bP^2. For values of P such that |aP| > |bP^2| (and all higher order P6n terms), this would be valid. You could have a constant term, and P^3 and higher terms, but they're not necessary. The P^2 term is necessary because dP/dt = aP does not have a solution that tends to a non-zero value of P as t -> infinity. Solving the equation

$$\frac{dP}{dt} = aP + bP^2$$

gives the logistic equation they use to fit the data on the page you linked to. The solution is

$$P(t) = \frac{a}{b + (a/P_0 - b)e^{-at}},$$
where a > 0 and b < 0 (to get a limiting value that is positive), and P_0 isthe value of P at t = 0.

Their derivation of this equation didn't really have anything to do with sprinting, so you could use it for pretty much any kind of performance-record event that satisfies the assumptions (1) and (2) above. However, note that there are some caveats. First, that page doesn't give the actual values the authors obtain for a and b in their fit. That means it's hard to judge if their expansion of f(P) was valid, because for it to be valid we expect |aP| to be much larger than |bP^2|.

The other thing to note is that this "model" doesn't really tell you anything about why this is the fundamental performance limit. You have no way of knowing what changes could invalidate your model. For example, in the context of a computer game, a set of players using keyboards to play the game may reach a fundamental performance limit than a set of players using joysticks. The difference between using keyboards and joysticks will be expressed quantitatively in the values of the fit parameters a and b, but such a model can't tell you anything beyond the numbers being different.