Tag Archives: mathematical proof

Brute Force Proof of Pythagorean Expectation

Except for stats that occur in the future, the set of baseball statistics is finite so we should be able to run the Pythagorean Expectation formula through Proof by Exhaustion or Brute Force Proof.   First let’s run through an example of the original Bill James’  simple PE formula:

We’ll use the 2013 Chicago Cubs as an example.

2013 Chicago Cubs

The Cubs won 66 and lost 96 games in 2013.  This means

W-L = Actual WAA = 66 – 96 = -30

Actual WAA is the WAA not estimated, the WAA that really happened.  We will call the WAA as estimated by Pythagorean Expection PE WAA.

In 2013 the Cubs scored 602 runs and gave up 689 runs.  Thus:

Rs = 602

Ra = 689

Based upon the simple PE formula stated above

PE Win% = (Rs)**2/(Rs**2 +Ra**2) = (602)**2/(603**2 + 689**2) = 0.433

#Wins = PE Win% * (Number Games) = 0.433 * 162 = 70.15

#Loss = (Number Games) – #Wins = 91.85

PE WAA = #Wins – #Loss = 70.15 – 91.85 = -21.7

There is a difference between estimated WAA (PE WAA) and Actual WAA.   This difference in the estimation happens because other factors also contribute to generating wins and losses.  We can guess at some of those factors like efficient field managers, players that choke under pressure, or simple bad luck but none of those factors are part of the formula we want to prove.

The only thing we know for fact is its error.

Error = | Actual WAA – PE WAA | = | -30 – (-21.7) | = 8.3

The summation of players who played for the Cubs in 2013 add up to the PE WAA (-21.7) and not the Actual WAA.   There is a proof of the formula used by this data model to compute WAA that shows the above to be true.

Now that we know how to calculate error we can run these numbers for each team in 2013, add them together and get a total error for all 30 teams.  In the next post we will show error results for 3 different variations of Pythagorean Expectation including the original, the one we showed in the above example.

The predictive qualities of advanced pitching statistics have failed the Orioles

Ever since Billy Beane and the Moneyball A’s famously exploited a market inefficiency by noticing that ballclubs undervalued walks and extra-base hits, a statistical movement in baseball has emerged to look for the next glaring weakness in baseball analytics. When it comes to pitching, the common application of these techniques is to look past ERA (the results on the field) and instead focus on factors like batting average on balls in play (BABIP), left on base percentage (LOB%), groundball-to-flyball ratio (GB/FB), and the like — many of which are encapsulated in fielding-independent pitching (FIP) and expected fielding-independent pitching (xFIP), ratios designed to resemble ERA but to take luck out of the equation. The theory goes, then, that pitchers with an ERA below their FIP/xFIP are likely to regress, while pitchers with an ERA above their FIP/xFIP are likely to improve naturally. And guess what? This theory was wrong (at least partially) about every single member of the 2013 Orioles’ Opening Day rotation. Let’s take a look at them, one by one. All 2013 stats are as of Monday.

via The predictive qualities of advanced pitching statistics have failed the Orioles – Camden Chat.

The above article written July 31, 2013.  I have been seeing people toss around this stat called FIP so I did a quick search and found this formula.

FIP=\frac{13HR + 3BB - 2K}{IP}

FIP stands for Fielding Independent Pitching and this formula comes up with some sort of number that can be used to rank pitchers regardless whether they had superb or incompetent fielding behind them.  According to the Wikipedia definition BB does not include hit by pitch or intentional walks which seems odd.

The factors 13, 3, and -2 are the first thing that jump out at me.  Where did those factors come from?  How do base on balls and strikeouts factor into an equation that is supposed to be comparable to an ERA?  Is there a relation between ERA and FIP?

Let’s take a look at the assumptions about FIP outlined in this Fangraph article.

1) FIP is a perfect statistic that accurately measures a pitcher’s true talent level.
2) ERA equals FIP + ε, where ε can be seen as the luck or error term. => ε = (ERA-FIP)
3) ε is independent and symmetrically distributed around FIP.
4) There are 100 starting pitchers in the league (There are in fact about 150, but we’ll use 100 for simplicity)

According to the above there is a direct relation ERA = FIP + Constant.  How do base on balls and strikeouts relate to runs scored against?  Where is the proof that derived this FIP formula?

Update 12/25/2013

FIP does not pass the unit test

After writing the above analysis it occurred to me that FIP fails in the most fundamental way mathematically.   It does not pass the unit test.  In order to add values their units of measurements must be the same.  For example, you can’t add a measurement of 1 meter to another measurement of 1 foot because the result doesn’t make sense.  In order to add these two values either the value stated in meters must be converted to feet or vice versa.

The unit of measurement for FIP as stated in the above formula is in PA/IP (plate appearances per innings pitched).  HR, BB, and strikeouts are all types of plate appearances.   The unit of measurement for ERA is ER/game,  earned runs per game where earned runs is a type of run.  Since one game equals exactly 9 innings pitched by definition ERA takes on the units of Runs/IP.

The units Runs/IP do not match PA/IP thus FIP cannot be added or subtracted with ERA unless conversion is made upon their units.   The above theory on FIP is disproven based on misuse of units alone.  It is possible those 13, 3, and -2 factors have units attached like (runs/PA) converting the numerator of the FIP formula back to runs but that isn’t stated in the above formula.  And if these are some kind of coefficients there certainly isn’t any proofs as to their derivation.

The next set of posts will begin an exploration into the WAR stat.