Archive for Percentages and Such

This is the long-delayed fifth part in the xHR series. If you really want to read the first four parts, they can be located here, here, here, and here.

More than a month late, the highly anticipated follow-up to the first iteration of xHR has arrived. Once more, that increasingly trivial metric will grace the page of FanGraphs, wallowing in the mostly prestigious Community Research section (on the other hand, this section is most definitely the best section on the World Wide Web for experimental metrics and amateur analyses).

Unless the reader has an impeccable memory for breezily scanned, frivolous articles, he or she likely needs a reminder as to what xHR% is and aims to be. xHR% is a metric that describes at what rate a player should have hit runs over a given season. From this, expected home runs, a more understandable counting statistic, can be found by multiplying plate appearances by xHR%. It cannot be emphasized enough that the metric is not predictive; it only aims to describe. Without further ado, the formula is here:

I know that’s a lot to look at, and it isn’t exactly self-evident what all of the variables mean. As such, an explication of each part is necessary and provided below. (For logical rather than chronological purposes, the Kn variable will be analyzed last.)

AeHRD – One of the biggest differences between this formula and the last one is that this one does not use home run distance. This iteration uses expected distance, rendering it a combination of simple math, sabermetric theory, and physics. As such, expected home run distance strips out one of the biggest factors in luck — the weather.

Expected home run distance is found by utilizing a method taken from Newtonian Mechanics to calculate how far objects go. By using ESPN’s HitTracker website, I was able to obtain launch angles and velocities for nearly every home run hit in 2015. From this, I was able to resolve velocity into its respective parts, velocity in the x-direction (Vx) and velocity in the y-direction (Vy). After that, I calculated the amount of time the ball would be in the air with the formula vf=vi+gt, where vf is final velocity (0 m/s), vi is initial velocity (Vy), and g is simply the gravitational acceleration constant. Finally, I multiplied Vx by time in order to get the total expected distance.

I repeated that process for every home run hit by a given player in order to find his average expected home run distance. By doing this, I was able to strip out all weather-related components.

AeHRDH – Utilizing the same process as above, I found the average expected home run distance for every stadium. This is the player’s home stadium’s average home run distance, regardless of team.

AeHRDL – The same as above, but done for every home run hit in the majors last season.

When put together in the numerator and the denominator, the above variables serve as a “distance constant” of sorts that will at most adjust the resulting expected home runs by plus or minus two. Occasionally, the impact is negligible because the average expected distance is very close to that of the player’s home stadium and the league. Averaging the mean expected home run distance of the league and of the home stadium allows the metric to paint a more accurate picture of where the player hit his home runs and whether or not they should have left the park. Nevertheless, it’s important to note that this formula still fails to account for fly balls that fell just short of the wall due to the wind and other factors, meaning that there are still expected home runs unaccounted for.

FB% – If you remember correctly, or took the time to briefly review the previous posts, then you will recall that in the prior iteration of the formula there was a section very similar to this one. The only differences are that the weights on each year of data have changed (those are still somewhat arbitrary, however, but I am working on getting them to more precisely reflect holdover talent from past years) and the primary statistic used.

Previously, HR/PA was used, but it had to be abandoned because the results were too closely correlated with reality. This time, I looked at how similarly descriptive formulas were quantified. Oftentimes, those metrics did not use the target expected metric in their formulas. Rather, they utilized other metrics that correlated moderately well or strongly with their expected metric. In this case, I decided to use FB% because it’s a relatively stable metric (especially in comparison with HR/FB), and it has a strong correlation with HR% (about .6).

As a clarification, the subscript Y3, Y2, and Y1 indicate the years away from the season being examined, where Y1 is really Y0 because it’s zero years away. So just to be clear, Y1 is the in-season data from the year being examined. In the data to be examined, for example, Y1 is 2015, Y2 is 2014, and Y3 is 2013.

Kn – As you can well imagine, FB% numbers are always far greater than HR% numbers*, resulting in some truly ridiculous results if a constant isn’t applied that relates HR% to FB%. For instance, without a constant to modify the results, Jose Bautista would have been expected to hit 304 home runs last season. That’s a lot of home runs. Just two and a half seasons of playing at that level and he’d have the home run record in the bag. Luckily, I’m not stupid enough to think that that’s actually possible, and so I initially related FB% and xHR% with a constant, called KCon.

Unfortunately, KCon didn’t work as well as I’d hoped because it skewed expected home run results way up for terrible home run hitters and way down for the best home run hitters. By skewed, I mean bad by more than six home runs. And so I, in my infinite (and infantile) amateur mathematical wisdom, made it into a seven part piecewise** function. By this, I mean that there’s a different constant for each piece of the formula, defined by HR% at somewhat arbitrary, though round points. For clarity, here they are:

K1 = HR%<1

K2 = 1≤HR%<2

K3 = 2≤HR%<3

K4 = 3≤HR%<4

K5 = 4≤HR%<5

K6 = 5≤HR%<6

K7 = 6<HR%

It works quite well. I am very excited about the current iteration of xHR%, its implications, and all it has to offer. Of course, it is not finished, but I think I’m getting closer. Please comment if you have any questions, an error to point out, or anything of that nature. There will be a results piece published soon on the 2015 season, so keep an eye out.

*It wouldn’t be surprising if Ben Revere became the first player to have a HR% equal to FB% (both at 0%, naturally).

**It is neither continuous nor differentiable.

Part 2 of a series of posts regarding a new statistic, xHR%, and its obvious resultant, xHR, this article will examine formula 1. The primer, Part 1, was published March 4.

As a reminder, I have conceptualized a new statistic, xHR%, from which xHR (expected home runs) can be derived. Furthermore, xHR% is a descriptive statistic, meaning that it calculates what should have happened in a given season rather than what will happen or what actually happened. In searching for the best formula possible, I came up with three different variations, all pictured below with explanations.

HRD – Average Home Run Distance. The given player’s HRD is calculated with ESPN’s home run tracker.

AHRDH – Average Home Run Distance Home. Using only Y1 data, this is the average distance of all home runs hit at the player’s home stadium.

AHRDL – Average Home Run Distance League. Using only Y1 data, this is the average distance of all home runs hit in both the National League and the American League.

Y3HR – The amount of home runs hit by the player in the oldest of the three years in the sample. Y2HR and Y1HR follow the same idea. In cases where there isn’t available major league data, then regressed minor league numbers will be used. If that data doesn’t exist either, then I will be very irritated and proceed to use translated scouting grades.

PA – Plate appearances

(Apologies for my rather long-winded reminder, but if you really forgot everything from Part 1, then you should really invest in some Vitamin E supplements and/or reread the first post.)

The focus formula of this post is the first one, which also happens to be the one I think will work the least well because it relies too heavily on prior seasons to provide an accurate and precise estimate of what should have happened in a given season.

In the second piece of the formula, with only fifty percent of the results from the season being studied taken into account, it likely fails to take into account the fact that breakouts occur with regularity. As a result, it probably predicts stagnation rather than progress.

Methodology

Luckily for myself and the readers, the process was an incredibly simple one. Pulling data from FanGraphs player pages, ESPN’s Home Run Tracker, and various Google searches, I compiled a data set from which to proceed. From FanGraphs, I collected all information for Part Two of the formula, including plate appearances and home runs. Unfortunately, because a few of the players from the sample were rookies or had fewer than three years of major league experience, I had to use regressed minor league numbers. In some cases, where that data wasn’t applicable, I dug through old scouting reports to find translatable game power numbers based off of scouting grades (and used a denominator of 600 plate appearances).

Then, from ESPN’s amazingly in-depth Home Run Tracker website, I obtained all relevant data for player home run distance, average home run distance for the player at home, and league average home run distance. Due to my limited time, I only used players that qualified for the batting title during the 2015 season, yielding an iffy sample of only 130 players. Additionally, before anyone complains, please realize that the purpose of my research at this point is only to obtain the most viable formula and refine it from there.

Results

Using Microsoft Excel, I calculated the resultant xHR% and xHR. Some key data points:

League Average HR% (actual):  3.03%

Average xHR%:  2.85%

Average Home Runs: 18.7

Expected Home Runs: 17.7

Please note that there is a significant amount of survivorship bias in this data. That is, because all of these players played enough to qualify for the batting title, they are likely significantly better than replacement level, which is why the percentages and home runs seem so high.

Clearly, the numbers match up fairly well, with this version of the formula expecting that the league should have hit home runs at a .18% lower clip, and one fewer per player, which amounts to a significant difference. Over the course of a 600 plate appearance season, the difference between them is still only a little more than one home run, an acceptable distance.

Correlation between xHR% and HR%: 0.960506092

R² for above: 0.922571953

HR% Standard Deviation: 1.5769373

xHR% Standard Deviation: 1.3883746

Correlation between xHR and HR: 0.966224253

R² for above: 0.933589307

HR Standard Deviation:  10.43771886

xHR Standard Deviation: 9.201355342

While xHR% using this formula apparently explains about 92% of the variance, correlation may not be the best method of determining whether or not the formula works adequately. This holds at least for between xHR% and HR%, because there’s only a minuscule difference between their numbers (but one that matters), meaning it’s not a particularly explanatory method and that it may not have the descriptive power I’m looking for. Nevertheless, it is important to note that the correlation is not a product of random sampling, as p<.005. Unsurprisingly, the standard deviation for xHR% is smaller than that of HR% (nearly insignificantly so), indicating that the data is clumped together close to the mean as a result of using this formula, a potentially good thing (in terms of regression).

A better indicator of the success of the formula is the correlation between xHR and HR, a relatively high value of ≈.97. Here, presumably because the separation between home runs and expected home runs is greater, the formula ostensibly explains approximately 94% of the variance in outcomes and resultant data. However, in this case, the standard deviation for actual home runs is about 10.4, while for xHR it’s about 9.2, suggesting that, after being multiplied out by plate appearances, xHR is spaced nearly as evenly as HR. Ergo, it likely serves as a decent predictor of actual home runs.

Players of Interest

Mr. Bryce Harper – It’s likely there isn’t a better candidate for regression according to this formula than Bryce Harper, who the formula says have hit only 32 home runs as opposed to his actual total of 42. While he did lead his league in “Just Enough” home runs with 15, he’s also always been known for having prodigious power (or at least a potential for it). Furthermore, Mr. Harper dramatically changed his peripherals last season to ones more conducive to power. Suggesting this are the facts that he increased his pull percentage from 38.9% to 45.4%, his hard hit percentage from 32% to 40%, and his fly ball percentage from 34.6% to 39.3%. On their own, all of the previous statistics lend credence to the idea that Harper changed his profile to a more home-run-drive one, but when taken together they significantly suggest that. His season was no fluke, and the formula certainly failed him here because it weighted prior seasons far too heavily.

Mr. Brian Dozier – No surprises here. Mr. Dozier has certainly been trending upward for a long time, and in a model that heavily weights prior performance such as this one, upticks in performance are punished. Nevertheless, the data vaguely supports the idea that Dozier should have hit 24 home runs instead of 28. While he did significantly increase his pull percentage to an incredibly high 60% from 53%, he did play in a stadium where it’s of an average difficult to hit pull home runs as a right-handed hitter. Moreover, 10 of his 28 home runs were rated as “Just Enough” home runs, in addition to his average home-run distance being 12 feet below average (admittedly not a huge number, nor a perfect way of measuring power). If I were a betting man, I’d expect him to hit 4-6 fewer home runs this coming season.

Keep watch for Part 3 in the coming days, which will detail the results of the other formulas. Something to watch for in this series is the issue that the results of the formula correspond too closely to what actually happened, which would render it useless as a formula.

Note that because I have never formally taken a statistics course, I am prone to errors in my conclusions. Please point out any such errors and make suggestions as you see fit.

One of the most important developments in statistics — and its subordinate field, sabermetrics — is the usage of multiyear data to produce an expected outcome in a given year. It’s an old concept, one that’s been around for centuries, but it likely originated in sabermetrics circles with Bill James. In Win Shares (arguably the birth of WAR), the sabermetric response to Principia Mathematica, he details a procedure of finding park factors wherein the calculator uses a weighted average of several years of data in conjunction with league averages to find park factors for a certain ballpark.

Methods such as Mr. James’s allow the amateur sabermetrician (and even the mighty professional statistician) to determine what ought to have happened over a specific time period. Essentially, a descriptive statistic. The best example of a descriptive statistic for the unlearned reader is xFIP, which basically describes what a pitcher’s fielding-independent average runs allowed would have been if the pitcher had a league-average home runs per fly ball rate.

Several statistics fluctuate greatly from year to year and are thus considered unstable. Examples include BABIP, HR/FB% for pitchers, and line-drive percentage. HR/FB% in particular is very fluid because all sorts of variables go into whether a ball leaves the park or not. For instance, on a particularly windy day, an otherwise certain dinger might end up in the glove of an expectant center fielder on the warning track instead of in the beer glass of your paunchy friend in the cheap seats. Rendered down, xFIP takes the uncontrollable out of a pitcher’s runs-allowed average.

With this, and an excellent article about xLOB% from The Hardball Times, in mind, I started developing my own statistic a few days ago. xHR%, as I dubbed it, attempts to find an expected home-run percentage, and from there one can easily find expected home runs (xHR) by multiplying xHR% by plate appearances, a more understandable idea to the casual baseball fan. In order to calculate this, I wrote several different (albeit very similar) formulas:

More likely than not, your eyes glazed over in that section, so I will explain.

HRD – Average Home Run Distance. The given player’s HRD is calculated with ESPN’s Home Run Tracker.

AHRDH – Average Home Run Distance Home. Using only Y1 data, this is the average distance of all home runs hit at the player’s home stadium.

AHRDL – Average Home Run Distance League. Using only Y1 data, this is the average distance of all home runs hit in both the National League and the American League.

Y3HR – The amount of home runs hit by the player in the oldest of the three years in the sample. Y2HR and Y1HR follow the same idea. In cases where there isn’t available major-league data, then regressed minor-league numbers will be used. If that data doesn’t exist either, then I will be very irritated and proceed to use translated scouting grades.

PA – Plate appearances

(For the uninitiated, HR% is HR/PA)

Essentially, what I have created is a formula that describes home-run percentage. First off, I used (.5)(AHRDH) + (.5)(AHRDL) in the denominator of the first part because a player spends half his time at home and half on the road. If I were so inclined, I could factor in every single stadium that gets visited, weight the average of them, and make that the denominator, but that’s just doing way too much work for a negligible (but likely more accurate) effect. Besides, writing that out in a formula would be a disaster because then there essentially couldn’t be a formula. Furthermore, having half of the denominator come from the player’s home stadium factors in whether or not the stadium is a home-run suppressor or inducer, which helps paint a more accurate picture of the player.

Dividing the player’s average HRD by(.5)(AHRDH) + (.5)(AHRDL) allows the calculator to get a good idea of whether or not the player was “lucky” in his home runs. If his average home-run distance is less than the average of the league and his home stadium, then it follows that he is a below-average home-run hitter and his home-run totals ought to be lesser.

Since the values in the numerator and the denominator will invariably end up close in value to each other, I decided that this part of the formula could be used as the coefficient (as opposed to just throwing it out) because it will change the end number only slightly. Moreover, the xCo (as I call it) acts as a rough substitute for batted-ball distance and park dimensions in order to factor those into the formula.

The second part, the meat of the formula, uses a weighted average of multiple years of home-run-percentage data to help determine what should have been the home-run percentage in year one (the year being studied). Basically, it helps to throw out any extreme outlier seasons and regress them back a little bit to prior performance without stripping out everything that happened in that season (notice that in every formula the biggest weight is given to the season studied).

At this juncture, I cannot say for certain how much weight ought to be given to prior seasons. Obviously, a player can have a meaningful and lasting breakout season, with continued success for the rest of his career, making it inaccurate to heavily weight irrelevant data from a season two years ago. On the other hand, a player can have a false breakout, making it better to include more data from previous seasons. Undoubtedly that will be the subject of future posts. At present, the formula is a developmental one that will no doubt experience heavy changes in the future.

For the interested reader, some prior iterations of the formula are below:

As a reminder, with some small addenda, here is the explanation for each variable:

HRDY3 – Average Home Run Distance Year Three (year three being the oldest of the three years in the sample). HRD is calculated with ESPN’s home run tracker. HRDY2 and HRDY1 follow the same idea.

AHRDH – Average Home Run Distance Home. Using only Y1 data, this is the average distance of all home runs hit at the player’s home stadium by any player.

AHRDL – Average Home Run Distance League. Using only Y1 data, this is the average distance of all home runs hit in both the National League and the American League.

Y3HR – The amount of home runs hit by the player in the oldest of the three years in the sample. Y2HR and Y1HR follow the same idea. n cases where there isn’t available major league data, then regressed minor league numbers will be used. If that data doesn’t exist either, then I will be very irritated and proceed to use translated scouting grades.

PA – Plate appearances

(You should be initiated at this point, so figure out HR% for yourself.)

The reason these formulas were thrown out was that the xCo relied too heavily on seasons past to provide an accurate estimate. When I briefly tested this one on a few players, it delivered incredibly scattered results. Furthermore, there wouldn’t be any data available for rookies to use these iterations on because there’s no such thing as a minor-league or high-school home-run tracker (and if there were I probably wouldn’t trust it). The first formulas described are overall more elegant and more accurate.

Stay tuned for Part 2, when results will be delivered instead of postulations.