In the last installment, I explored using the Cigol dataset to estimate the Pythagorean exponent. Alternatively, we could sidestep the attempt to estimate the exponent and try to directly estimate the z parameter in the Pythagenpat equation x = RPG^z.

The positives of this approach include being able to avoid the scalar multipliers that move the estimator away from a result of 1 at 1 RPG, and also maintains a form that has been found useful by sabermetricians in the last decade or so. The latter is also the biggest drawback to this approach--it assumes that the form x = RPG^z is correct, and foregoes the opportunity of finding a form that provides a better fit, particularly with extreme datapoints. It’s also fair to question my objectivity in this matter, given that a plausible case could be made that I have a vested interest in “re-proving” the usefulness of Pythagenpat. That’s not my intent, but I would be remiss in not raising the possibility of my own (unintentional) bias influencing this discussion.

Given that we know the Pythagorean exponent x as calculated in the last post, it is quite simple to compute the corresponding z value:

z = log(x)/log(RPG)

For the full dataset I’ve used throughout these posts, a plot of z against RPG looks like this:

A quick glance suggests that it may be difficult to fit a clean function to this plot, as there is no clear relationship between RPG and z. It appears that in the 15-20 RPG range, there are a number of R/RA pairs for which a higher z is necessary than for the pairs at 20-30 RPG. While I have no particular reason to believe that the z value should necessarily increase as RPG increases, I have strong reason to doubt that the dataset I’ve put together allows us to conclude otherwise. Based on the way the pairs were chosen, extreme quality differences are overrepresented in this range. For example, there are pairs in which a team scores 14 runs per game and allows only 3. The more extreme high RPG levels are only reached when both teams are extremely high scoring; the most extreme difference captured in my dataset at 25 RPG is 15 R/10 RA.

The best fit to this graph comes from a quadratic regression equation, but the negative coefficient for RPG^2 (the equation is z = -.0002*RPG^2 + .0062*RPG + .2392) makes it unpalatable from a theoretical perspective. The apparent quadratic shape may well be an accident of the data points used as described in the preceding paragraph. Power and logarithmic functions fail to produce the upward slope from 5-10 RPG, as does a linear equation. The latter has a very low r^2 (just .022) but results in an aesthetically pleasing gently increasing exponent as RPG increases (equation of .2803 + .00025*RPG). The slope is so gentle as to result in no meaningful difference when applying the equation to actual major league teams, leaving it as useless as the r^2 suggests it would be (RMSE of 4.008 for 1961-2014, with same result if using the z value based on plugging in the average of RPG of 8.805 for that period).

It’s tempting to assume that z is higher in cases in which there is a large difference in runs scored and runs allowed. This could potentially be represented in an equation by run differential or run ratio, and such a construct would not be without sabermetric precedent, as other win estimators have been proposed that explicitly consider the discrepancy between the two teams (explicitly as in beyond the obvious truth that as you score more runs than you allow, you will win more games). (See the discussion of Tango’s old win estimator in part 7).

First, let’s take a quick peak at the z versus RPG plot we’d get for the limited dataset I’ve used throughout the series (W%s between .3 and .7 with R/G and RA/G between 3 and 7):

The relationship here is more in line with what we might have expected--z levels out as RPG increases, but there is no indication that z decreases with RPG (which assuming my reasoning above is correct, reflects the fact that the teams in this dataset are much more realistic and matched in quality than are the oddballs in the full dataset). Again, the best fit comes from a quadratic regression, but the negative coefficient for RPG^2 is disqualifying. A logarithmic equation fits fairly well (r^2 = .884), but again fails to capture the behavior at lower levels of RPG, not as damaging to the fit here because of the more limited data set. The logarithmic equation is z = .2484 + .0132*ln(RPG), but this produces a worse RMSE with the 1961-2014 teams (4.012) than simply using a fixed z.

Returning to the full dataset, what happens if we run a regression that includes abs(R - RA) as a variable alongside RPG? We get this equation for z:

z = .26846 + .00025*RPG + .00246*abs(R - RA)

This is interesting as it is the same slope for RPG as seen in the equation that did not include abs(RD), but the intercept is much lower, which means that for average (R = RA) teams, the estimated z will be lower. This equation implies that differences between a team and its opponents really drive the behavior of z in the data.

Applying this equation to the 1961-2014 data fails to improve RMSE, raising it to 4.018. So while this may be a nice idea and seem to fit the theoretical data better, it is not particularly useful in reality. I also tried a form with an RPG^2 coefficient as well (and for some reason liked it when initially sketching out this series), but the negative RPG^2 coefficient dooms the equation to theoretical failure (and with a 4.017 RMSE it does little better with empirical data):

z = .24689 - .00011*RPG^2 + .00378*RPG + .00183*abs(R - RA)

One last idea I tried was using (R - RA)^2 as a coefficient rather than abs(R - RA). Squaring run differential eliminates any issue with negative numbers, and perhaps it is extreme quality imbalances that really drive the behavior of z. Alas, a RMSE of 4.014 is only slightly better than the others:

z = .27348 + .00025*RPG + .00020*(R - RA)^2

If you are curious, using the 1961-2014 team data, the minimum RMSE for Pythagenpat is achieved when z = .2867 (4.0067). The z value that minimized RMSE for the full dataset is .2852. This may be noteworthy in its own right -- a dataset based on major league team seasons and one based on theoretical teams of wildly divergent quality and run environment coming to the same result may be an indication that extreme efforts to refine z may be a fool's errand.

You may be wondering why, after an entire series built upon my belief in the importance of equations that work well for theoretical data, I’ve switched in this installment to largely measuring accuracy based on empirical data. My reasoning is as follows: in order for a more complex Pythagenpat equation to be worthwhile, it has to have a material and non-harmful effect in situations in which Pythagenpat is typically used. If no such equation is available (which is admittedly a much higher hurdle to clear than me simply not being able to find a suitable equation in a week or so of messing around with regressions), then it is best to stick with the simple Pythagenpat form. If one a) is really concerned with accuracy in extreme circumstances and b) thinks that Cigol is a decent “gold standard” against which to attempt to develop a shortcut that works in those circumstances, then one should probably just use Cigol and be done with it. Without a meaningful “real world” difference, and as the functions needed become more and more complex, it makes less sense to use any sort of shortcut method rather than just using Cigol.

Thus I will for the moment leave the Pythagenpat z function as a humble constant, and hold Cigol in reserve if I’m ever really curious to make my best guess at what the winning percentage would be for a team that scores 1.07 runs and allows 12.54 runs per game (probably something around .0051).

The “full” dataset I’ve used in the last few posts is available here.

## Monday, February 04, 2019

### Enby Distribution, pt. 9: Cigol at the Extremes--Pythagenpat Exponent

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