Since rectangles are used to construct the inconsistency index *I*_{Rn}, it measures inconsistency under the assumption (stipulated in the rules of baseball) that the true strike zone is a rectangle. However, as we will see in Section 3, strike zones in practice tend to be rounded at the corners. In this section we will introduce inconsistency measures that relax the assumption of a rectangular zone. Instead, we assume that a consistent zone will have the property that any pitch landing between two called strikes will also be a called strike. In other words, we assume that the established strike zone is convex.

Given a discrete set $P\subseteq {\mathbb{R}}^{2}$ representing the locations of called strikes during a game, there is a natural geometric definition for the established strike zone, namely, the *convex hull* of *P*. We can define the convex hull *S* as the intersection of all closed half planes that contain *P*:

$$S=\bigcap _{\{{H}_{l}\mid {H}_{l}\cap P=\mathrm{\varnothing}\}}{H}_{l}^{c},$$

where ${H}_{l}^{c}$ denotes the complement of the open half-plane bounded by the line *l*.

Using the convex hull as our established strike zone, we can define the *convex hull inconsistency index I*_{CH} analogously to the one-rectangle inconsistency index. Now an inconsistent ball is one that lies within the convex hull of strikes, and *I*_{CH} is given by

$${I}_{CH}=\frac{\text{number of inconsistent balls}}{\text{total number of called balls}}.$$

For example, see Figure 4. There were five inconsistent balls versus left-handed batters, and one versus right handed batters, out of a total of 118 called balls. Therefore *I*_{CH} = (5 + 1)/118 ≈ 0.051.

Like the one-rectangle index, the convex hull inconsistency index can fail to account for multiple bad strikes in the same location. It can also be unaffected by outlying strikes, depending on their location. For example, in Figure 4 versus right-handed batters, the strike at (−0.01, 1.31) has no effect on *I*_{CH}; removing this point would shrink the convex hull without changing the number of called balls enclosed. However, this call seems inconsistent, given its proximity to several called balls. The problem is that a vertex of the convex hull can lie in a region populated by called balls, yet fail to enclose any. Creating smaller convex hulls inside the first (as we did to define *I*_{Rn}) will not address this issue.

To account for this phenomenon, we can use the locations of called balls to define a *called-ball region*. Instead of counting called balls within the established strike zone, we can measure the area of the overlap between the called-ball region and the convex hull of strikes.

Figure 4: The established strike zone is the convex hull of called strikes.

Unlike the established strike zone, the called-ball region will typically not be convex, or even simply connected. Given a set $Q\subseteq {\mathbb{R}}^{2}$ representing the locations of called balls during a game, and given some radius *α* > 0, define

$$X=\bigcap _{\{{B}_{x,\alpha}\mid {B}_{x,\alpha}\cap P=\mathrm{\varnothing}\}}{B}_{x,\alpha}^{c},$$

where ${B}_{x,\alpha}^{c}$ denotes the complement in the plane of the open disk of radius *α* centered at the point *x*. The region *X*, which will serve as our called-ball region, is called the *α**-convex hull* of *Q* (Pateiro-López and Rodrıguez-Casal 2010). Note that the *α*-convex hull is not convex, in general.

Let *a*_{L} and *a*_{R} be the areas of the intersection of the convex hull of called strikes and the *α*-convex hull of called balls, for left-handed and right-handed batters, respectively. We define the *α**-convex hull inconsistency index I*_{ACH} to be a weighted average of these two areas. Let *n*_{L} be the number of called pitches thrown to left-handed batters, and let *n*_{R} be the number of called pitches to right-handed batters. Then

$${I}_{ACH}=\frac{{n}_{L}{a}_{L}+{n}_{R}{a}_{R}}{{n}_{L}+{n}_{R}}$$

Figure 5 shows the ball and strike calls for the same game as Figure 4, along with the *α*-convex hull of called balls, using *α* = 0.7. In this case, *I*_{ACH} = 0.127. Notice that the called strike at (−0.01, 1.31) now has a significant effect on *I*_{ACH}, since it causes a large region of overlap between the convex hull and the *α*-convex hull, which contains the nearby called balls.

Figure 5: The established strike zone is the convex hull of called strikes (in red), and the called-ball region is the *α*-convex hull of balls (in blue), where *α* = 0.7.

One limitation to this choice of inconsistency metric is that there is no canonical choice for the constant *α*. Figure 6 illustrates the issues involved. If the radius *α* is too small, the *α*-convex hull will contain isolated points and small disconnected regions. Large values of *α* (such as *α* = 0.9 in this example) will produce a single, simply-connected *α*-convex hull, making the called-ball region completely cover the established strike zone. Generally speaking, the larger the value of *α*, the tougher the metric *I*_{ACH} is on the umpires.

Figure 6: Six different called-ball regions (*α*-convex hulls) for different choices of *α*.

In the analysis that follows, we have chosen to use *α* = 0.7, based largely on qualitative inspections of various game examples, as in Figure 6. A correlation analysis can lend some empirical support to this choice. gives the pairwise correlations for *I*_{ACH} computed over all 2017 regular season games using six different values of *α* between 0.4 and 0.9, in increments of 0.1. For these six values, the greatest correlation is between *α* = 0.6 and *α* = 0.7, and we observe that once *α* exceeds 0.7, the correlations between adjacent values begin to decrease. These results confirm the observation that choosing *α* in the range 0.6 ≤ *α* ≤ 0.7 tends to give similar measures of *I*_{ACH}.

Table 1: Correlation matrix for *I*_{ACH} calculated with different values of *α* over all games in the 2017 season.

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