# “Expected Hitting”: Motivation Plus A Refinement On HR Weighting

A topic that has really captured my interest, over the past six months or so, is the idea of luck-neutralized hitting stats. (“xHitting,” as I’ve been calling it with friends.) This actually is now my third post on the topic. In fairness to readers, I will back up a little bit with some motivation, before presenting the newest refinement. Those curious to read the full sequence can find parts one and two here.

Basically, the motivation is that, when a pitcher performs surprisingly well/poorly, people readily consult FIP (and its variants) to help check whether it stems from luck or something more fundamental. But this is not so readily available for hitters. The most popular hitter metrics, like wOBA and wRC+, continue to rely on realized hit outcomes, which are sensitive to luck. While many people know to look at BABIP to see if the hitter has gotten lucky/unlucky, this has a few shortcomings: (1) baseline BABIP differs across hitters; (2) even if you’re certain a hitter has gotten lucky/unlucky, this still doesn’t specify the precise luck-neutral level of performance.

Even existing xBABIP calculators seem only a partial solution to this problem (for reasons explained in piece one linked above.) Specifically, I find that xBABIP has trouble with xSLG, and anything that involves either slugging or a complete distribution of hit types — e.g. OPS and wOBA. So I’ve taken a different approach: to map a hitter’s peripheral performance to a *complete distribution of hit outcomes*, which can subsequently be used to compute expected versions of many statistics, such as AVG, OBP, SLG, ISO, OPS, and wOBA. Again, more details of this process can be found in pieces one and two.

Expected versions of these statistics seem to outperform their realized values in predicting future performance (aligning with known findings for pitchers), although in some cases you need to make a slight modification for home runs. That is, for home runs specifically, it seems best to use a *combination* of actual and predicted home runs. The following table makes this more clear.

I used 50/50 there just to keep things simple. But this begs a question: **What is the best weighting between actual and predicted home runs?** This latest refinement seeks to answer that question.

Intuitively, it seems the answer may be a function of plate appearances. That is, if a player outperforms his “expected” home run rate over 100 plate appearances, that’s really not much evidence to say that the model is wrong. Doing it over 300 plate appearances is a little less fluky, however, and over 600 plate appearances a bit less fluky still. In the extreme case, if a player outperforms his expected home run rate across thousands and thousands of plate appearances, then it’s clear that the model simply misses something about that player’s home run abilities.

How I actually search for the optimal weighting by PA is to:

1. Chop the sample into a bunch of 50-PA ranges (100-149 PA, 125-174, 150-199, etc.)

2. For each range find the weighting of actual versus predicted home runs that minimizes root mean squared error (RMSE) in a regression on the subsequent year’s home run rate

3. Make a scatterplot of these seemingly optimal weights, and fit a continuous function through them

(Note that these PA counts are for a single season. Throughout, the player-year (e.g. 2013 **Mike Trout**) is the micro-unit of observation.)

Example: 100-149 PA (N=111)

So for this PA range, it seems best to put 20% weight on actual HR rate, and 80% weighted on predicted HR rate. If you do this for a whole bunch of PA ranges, you get a scatterplot that looks like:

It’s not the cleanest pattern you’ve ever seen, but if you fit a line through it, it’s pretty clearly upward-sloping. Something else that intrigues me: it looks like the optimal weight on actual HR rate might get up to “full speed” by about 375 PA, and then “plateau” from there. To investigate this, I also fit a (cubic) polynomial through the points.

Hm, the polynomial version actually *decreases* at one point for higher PA. That’s a bit counter-intuitive, though not impossible. Of the two, empirically the linear version appears better for forecasting. And it (the linear version) improves upon the blanket 50/50 weighting — which does not take plate appearances into account — no matter which of the listed outcomes one cares about.

So, as suspected from intuition, the optimal weight to put on actual home run rate is increasing in plate appearances. Taking the (linear) trendline at face value, a 50/50 weighting is approximately correct for players with 375 PA. Those with more plate appearances should put more weight on actual home run rate, while those with fewer should put more weight on predicted home run rate. The exact weight is given by the equation *Optimal Weight = 0.0527102 + 0.0012157*PA.* (This is about 17% at 100 plate appearances, and 78% at 600 plate appearances.)

Granted, the method here probably is not the most elegant/sophisticated way to search for the optimal weight by PA range. Subsequent research is likely to be able to find improvements. The current result, though, seems to get the job done for the most part, and for now represents the most accurate specification of the “expected hitting” model.

As always, reader comments and suggestions are welcome!

*The following article was originally published on Batting Leadoff. For more information please visit us at www.battingleadoff.com or follow us on Twitter @Batting_Leadoff. *