Wednesday, September 09, 2009

The Pomeroy Index

The Pomeroy Index is the primary means of measuring the relative speed of Formula 1 cars which not only raced in different years, but in different eras of the sport. Remarkably, it is capable of comparing the relative speeds of cars which never even raced on the same circuits. To achieve this, it uses a daisychaining technique, similar to the manner in which dendrochronology uses the overlap between tree rings from different eras to extend its dating technique all the way back from the present to prehistoric times (see The Greatest Show on Earth, Richard Dawkins, pp88-91). In both cases, it is the overlap principle which is vital. In the case of Formula 1, the daisychaining is achieved by identifying the circuits which are common to successive years of Grand Prix racing. Speed differences between successive years are averaged over these overlapping circuits, and the speed differences can then be daisychained all the way from the inception of Grand Prix racing in 1906 to the present day.

The index was invented by engineer and motoring journalist Laurence Pomeroy, and updated by Leonard (L.J.K.) Setright in 1966. (Another motoring journalist, Setright was hard to miss, "with his long, wispy beard, wide-brimmed hat, cape and black leather gloves, he looked like 'an Old Testament prophet suddenly arriving at a Hell's Angels meeting'." (On Roads, Joe Moran, p172)).

The index was resurrected and updated again more recently by Mark Hughes. The algorithm for calculating the index is as follows:

1) Identify the fastest car from each year by averaging the qualifying performance of all the cars over all the races.

2) For each pair of successive years, identify the overlapping circuits in the respective calendars. In other words, identify the circuits which were used in both years, in unaltered form.

3) Take the fastest car from the first year of Grand Prix racing, Ferenc Szisz's 1906 Renault, and assign it a Pomeroy Index of 100.

4) For year t+1, calculate the speed difference between the fastest car that year and the fastest car from year t, averaged over the overlapping circuits (and eliminating spurious cases where speed differentials were skewed by rain conditions). Express this speed difference as a percentage, and add it to the Pomeroy Index of the fastest car in year t to find the Pomeroy Index of the fastest car in year t+1. For example, if the fastest car in year t+1 is 2% faster than the fastest car from year t, and the fastest car in year t had a Pomeroy Index of 150, then the fastest car in year t+1 had a Pomeroy Index of 152.

5) Repeat step 4 until one reaches the current year.

An on-line version of the index from 1906 to 1966 exists for perusal, and Hughes's updated version in Autosport magazine obtained a value of 234.7 for Michael Schumacher's 2004 Ferrari. (Speeds have since fallen due to the imposition of smaller engines, rev-limits, a control-tyre formula, and a generally more restrictive set of technical regulations).

This doesn't mean, however, that the 2004 Ferrari was 2.347 times faster than the 1906 Renault. This would be to underestimate the speed difference between Herr Schumacher and Ur Szisz's respective steeds. Perhaps the crucial point to digest here is that average speeds in Formula 1 have historically increased, not in a linear fashion, and not even according to a power law; rather, average speeds in Formula 1 increase exponentially. Hence, the percentage speed increments tallied in the Pomeroy Index are akin to the yearly interest rates of a compound interest account. The 1935 Mercedes-Benz was 3% faster than the 1934 Auto Union, and the 1936 Auto Union was 5% faster than the 1935 Mercedes-Benz, but the 1936 Auto Union was more than 8% faster than the 1934 Auto Union because the 5% increase was 5% of a speed greater than the speed of the 1934 Auto Union.

Such an exponential increase in speed can be represented by the formula:

Q(t) = Q(0) (1 + r(t))t ,

where Q(t) is the speed in year t, Q(0) is the speed in year 0, t is the discrete year number, and r(t) is the interest rate in year t, expressed as a decimal. Thus, for example, if the year-on-year increase in speed were a constant 2%, then speeds would increase exponentially according to the formula:

Q(t) = Q(0) (1 + 0.02)t.


Luis Cayetano said...

I'm still not 100% clear on why we need to compare the times for successive years. Why not just take all the unaltered tracks that all cars have raced on, find their average qualifying times and call it a day? (also, does the index take into account that cars in qualifying trim are often massively overpowered relative to those in racing trim?)

According to the Wikipedia entry for the Ferrari F2002, it has the highest Pomeroy Index for any F1 car. Do you know if this still holds? I keep hearing contradictory statement about F1 cars getting slower (due to regulations) and faster (due to less energy being wasted in the form of sound energy, better aerodynamics, etc).

Gordon McCabe said...

Hi Luis.

You have to daisy-chain the comparisons, because there generally won't be any unaltered tracks in the Grand Prix calendar at the beginning and end of a long period of time.

On the subject of 2002 being the fastest year, remember that from 2003 the drivers were forced to qualify with the fuel-loads they started the race with.

Luis Cayetano said...

Thanks Gordon,

I understand now.