# The Hardball Times

## Curveball command: once more, with feeling

by Max Marchi
March 10, 2011

Note: If you have not read Curveball command from a few weeks ago, you are invited to do so before proceeding with this one, as it is strongly based on concepts explained there and not repeated here.

In the Curveball command article of a few weeks ago, one strong assumption was made: That pitchers always aim at one spot when delivering curveballs.

To back up the assumption, observations on catchers' target (by Mike Fast and Nick Steiner) were related, and the resulting symmetrical single-peaked distributions of pitch locations were used as supporting evidence—the reasoning being that if pitchers aimed at different spots, either a multimodal or a skewed distribution would have emerged.

As one reader suggested, it's still hard to believe the premise to be true, thus more investigating is due.

### Reality vs. theory

Let's start with comparing observed pitch locations and locations based on theoretical distributions. In the scatter plot below, a bivariate normal distribution is compared to Roy Halladay's 2009-2010 curveballs against right-handed batters.

Note: The theoretical values come from a bivariate normal distribution, defined by standard deviations of 0.48 and 1.03 (lateral and upright command values calculated for Halladay in the previous article), and rotated by 41.6 degrees, the estimated arm angle for Doc.

So far so good. Doc's pitches don't seem to differ from a random sample from a bivariate normal distribution. However we have to dig deeper. Since the vertical location is more important for curves, let's compare the upright command (as defined in the previous article), again observed versus theoretical.

Principal Component histograms were used in the previous post (look there for reference). Looking at the pair above, there seems to be little difference between Halladay's actual and the theoretical distribution. However, when using density plots instead of histograms, something emerges.

While the theoretical distribution is a neat bell curve, a little bump can be noticed on the "reality line." Thus, a careful observation reveals a slight asymmetry in Halladay's locating along the arm-angle.

While working on future articles on change-up and cutter command, I began separating two-strikes counts from the other counts, as it was apparent (and expected) that pitchers expand their zone (and thus locate pitches differently) when they have opposing batters one strike away from elimination. This wasn't done for the Curveball command article, and the scatter plot below clearly shows it was a mistake.

Doc definitely locates his bender lower when there are two strikes in the count. This makes sense, since (unless the count is full) a wasted pitch is more affordable, and the batter can't take the chance of letting a borderline pitch go by.

Here is a comparison of the average locations.

Let's assume for now Halladay always aims at the lower location on two-strike counts and to the higher one on the other counts and perform separate Principal Component analyses. Here's the resulting upright command histogram for two-strikes counts.

The distribution is fairly symmetrical—statistical tests designed to look for asymmetry confirm the fact. On the contrary, the upright command histogram on other counts is undoubtedly skewed.

It seems the fairly symmetrical histogram we saw before separating by pitch counts is the byproduct of two distributions—one symmetrical and one skewed. Looking at the above two plots, we can speculate Halladay consistently aims at the low-outside corner when he has two strikes on the batter; on other counts he is more likely to vary his target, dropping some of his curves in the middle of the strike zone.

### Remaining untested assumptions.

There are at least a couple of assumptions still untested in this kind of analysis.

One is that pitchers can not choose which way to err. Stated differently, if you aim at a spot, you are equally likely to miss on one side or the other, at least voluntarily—maybe there is some mechanical stuff that makes you miss more on your throwing side (just guessing), just like the spread is bigger on the line parallel to the delivery angle than to the perpendicular.

Second, we are just looking at precision, while neglecting accuracy. It's really possible that a pitcher exists who can groove pitch after pitch in the very same spot—only the spot is half a foot from the intended target. The values appearing in the previous article on command measure the ability of pitchers to deliver their curves close together (precision); knowing whether the center of the cluster coincides with the intended location (accuracy) is impossible without knowing the desired destination of the ball.

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Recent news is that Greg Moore of Sportvision announced COMMANDf/x (i.e., catchers' glove tracking) at the MIT Sloan Sports Analytics Conference. Whether this info will be publicly available like PITCHf/x, or exclusive to clubs like HITf/x, has not been reported.

References and Resources
Thanks to Derek Neal for taking the time to thoughtfully comment on my previous article. His disagreements with my assumptions prompted me to further investigate the matter.

After creating a baseball rendition of The Beatles' Sgt. Pepper cover, Max began his baseball writing because he needed an excuse to show the picture. He wrote for an Italian audience for six years before making the jump to The Hardball Times. You can contact him by e-mail.