Progressive Pitch Projections: Cut Fastballs

In our previous installments, we examined the sliders and four-seam fastballs ofYu Darvish, and other MLB right-handers. The method of analysis was to look at the pitches incrementally, starting at 50 feet away from home plate, and project their location at the plate by removing the remaining PITCHf/x definition of movement. This was done to get an idea of how a batter might extrapolate the location of an approaching pitch. (For a full description of the method, strike zone definition, PITCHf/x simulations, or other aspects of the below research, see the previous two articles linked to above.)

From these studies, we found that for pitches outside the strike zone, those that were thrown in the direction of their movement were more likely to be swung at than those that were not. In this article, we will complete our analysis of Darvish by looking at his cut fastball, which was his third-most-thrown pitch of 2013 according to PITCHf/x. As before, since we ran these algorithms prior to the completion of the 2014 season, the data used is from 2013. In future articles, we will switch to the 2014 data set.

We will start by simulating a cut fastball from Darvish to see how such a pitch appears from the catcher’s perspective. The pitch chosen was thrown approximately down the middle of the plate, and the red circle indicates the location of the projection relative to the pitch’s distance from home plate. For the purposes of the simulation, the pitch starts at 55 feet and ends at the front of home plate. The strike zone shown is the standard width of the plate by 1.5 to 3.5 feet vertically. Initially, the cut fastball projects low and to the left of the strike zone, with the movement bringing the pitch toward the center of the plate.

Today, we will split the data set into left-handed and right-handed batters, starting with the former. Pitches are classified as a ball or a strike depending on whether the location where the pitch crossed the front of home plate has a 0.5 probability or higher of being called a strike, based upon pitches taken by similar-handed batters in 2013. The 0.5 probability contour is used to mark the “strike zone” in subsequent images, creating an empirical boundary between ball and strike.

With this in mind, each cut fastball thrown by Darvish will be classified as a ball or a strike, and by the batter’s response of either “swing” or “take.” This places each pitch into one of four possible groupings: strike/swing, strike/take, ball/swing, ball/take.

Versus Left-Handed Batters

For each of these four cases, we will, starting at 50 feet and ending at the front of the plate, remove the remaining movement and find the probability that the projection from that point on is a strike. Performing this for each case’s pitches, and averaging the results, produces four curves.

The swing/strike curve (green) and take/strike curve (red) are very close together and remain so to the plate. They start out projecting around probability 0.5 and level off around 20 feet. Therefore, the cutters in the zone that are not swung at are only slightly less attractive in terms of being called strikes. The nodes on each curve indicate the distance at which the projections have the highest average probability of being called a strike. For the swing/strike curve, this occurs at 1.417 feet, which is the front of home plate, with probability 0.928. The take/strike curve also peaks at the front of the plate, with probability 0.918. The swing/ball curve (blue) reaches its maximum further out, at 29.333 feet with accompanying probability 0.13. For balls that are taken (orange), the highest point on the curve is at 50 feet with probability 0.127.

If we forgo classifying the pitches as strikes or balls, we can plot the called-strike probability for the projections based solely on pitches that were swung at and those that were taken.

Both the swing and take curves are maximal at the front of home plate, with called-strike probabilities of 0.761 and 0.431, respectively. As opposed to sliders and four-seamers from Darvish, the swing curve does not show a peak prior to reaching the plate. This may indicate that batters are better able to account for the pitch movement, relative to the other two pitches. The idea is that if there is a peak prior to the plate, followed by a decline in the strike probability of the projection, batters may only be able to project/track the location of the pitch well up to that distance.

We can now break down the data to individual pitches and plot their projections as they approach the plate in order to gain a better understanding of how these curves are formed. The color scheme used is the same as for the four-curve plot.

From the above GIF, note that the projections of pitches taken in the “strike zone” (red), where pitches have a 0.5 probability or higher of being called strikes, are predominantly toward the left side of the plate, from the catcher’s perspective. The cut fastballs swung at outside the zone (blue) are relegated to the lower-right corner below the strike, projecting up and to the right. In addition, very few cut fastballs thrown up and outside to lefties are swung at.

To quantify these observations, let’s divide the area at the front of the plate into nine regions in a three-by-three configuration with the strike zone in the middle. For each region, we will place an arrow at the average location of the projections, pointing in the direction the projections are moving, and with a number indicating their quantity.

We will also display the percentage of pitches swung at in each region.

Darvish – Cutters vs. LHB
2.3	8.3	20
0	63.4	75
42.5	36.7	33

Among the well-populated regions, the highest percentage of swings is in the strike zone. Next is directly beneath the strike zone at 36.7 percent followed by down and to the right at 33 percent. All other regions with high percentages have too small samples sizes to draw any meaningful conclusions. Based on these results, the most effective areas outside the strike zone for Darvish to get swings on cutters were down and down-and-in to lefties, letting the movement carry the pitches toward/parallel to the strike zone.

We will switch to a different visualization technique for all 2013 MLB right-handed pitchers to get an idea of how the projections influence swings. We will use a heat map in which each pitch projection occupies a baseball-sized circle at the front of the strike zone. We then, over a sufficiently refined grid, find the percentage of overlapping projections that are swung at at that location. This forms a heat map running from 0.0 (zero swings at the location) to 1.0 (100 percent swings). We are primarily interested in when the high-swing, say between 0.8 and 1.0, region of the heat map occupies a majority of the strike zone and where it goes afterward.

Note that early on, the high-swing region condenses and then drifts upward. Also, similar to Darvish, it appears that throwing cut fastballs on the outer edge of the strike zone to lefties is a good strategy for getting called strikes, as there are relatively few swings in this area. From around 20 feet from the back of home plate onward, the projections of the cutters swung at seem to occupy a large portion of the strike zone, with most of the overflow being inside to lefties.

Since it is difficult to tell the reason why a batter may have swung at or taken a pitch in the strike zone, outside of a two-strike count, we will focus our subsequent analysis on pitches that end up outside the 0.5 probability contour. From this, we hope to gain some insight into how pitch movement influences decisions to swing or not on cutters.

We will use, as a metric, the angle between the movement vector — in this case, we take it to be the vector (pfx_x, pfx_z) from the PITCHf/x data — and a vector stretching perpendicularly from the edge of the 0.5 contour to the final position of the pitch at the front of home plate. This metric runs from 0 to 180 degrees. An angle near 0 indicates the movement is carrying the pitch perpendicularly away from the strike zone, and 90 means the pitch projection moves parallel to the strike zone. A 180-degree angle indicates the pitch movement is carrying the pitch perpendicularly toward the strike zone. We will examine this angle versus distance from the zone (in feet) both for Darvish and all MLB righties in 2013, starting with the results against left-handed batters.

Darvish – Cutters Outside the Zone v. LHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		16.6		0.657 ft
Less Than 90 Degrees		40.7		0.562 ft
All		100		0.614 ft

For all pitches outside the strike zone, there does not appear to be a distinct pattern present for Darvish’s cut fastballs, expect slightly more pitches thrown above the 90-degree mark. The table bears out similar results, with 16.6 percent of the pitches thrown outside with an angle of less than 45 degrees between the movement and strike zone vector, and 40.7 percent less than 90 degrees. So there are more pitches thrown outside moving toward the strike zone than moving away. The average distances from the strike zone are all more than 0.5 feet outside, with no trend as the range of angle is increased.

MLB RHP 2013 – Cutters Outside the Zone v. LHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		26.1		0.508 ft
Less Than 90 Degrees		57.8		0.479 ft
All		100		0.481 ft

The MLB right-hander data generates one dense cluster just below 90 degrees (movement parallel to the strike zone) for pitches less than half a foot away from the zone. There is also a much smaller cluster around 15 degrees and possibly around 170 degrees. Compared to Darvish, the percentages, relative to angle, are much higher. The pitches outside also are closer to the strike zone by an average of one to two inches.

In order to see whether or not throwing cut fastballs with these characteristics outside the strike zone is effective at inducing swings from batters, we can take the hexplots for both Darvish and RHP and isolate swings and pitches taken.

Darvish – Cutters Swung At Outside the Zone v. LHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		9.8		0.249 ft
Less Than 90 Degrees		27.5		0.261 ft
All		100		0.361 ft

For Darvish’s 2013 cutters to lefties, the percentage of swings is very low, at 9.8 percent, for pitches thrown in the direction of movement (45 degrees or less) and only 27.5 percent for pitches thrown with an angle of 90 degrees or less. The average distances for each of the three cases range from three to four inches outside. Of note is that the distance outside increases as the angle increases, which is atypical based on previous results. Compared to all outside pitches, the percentages are lower across the board. This may be due to the large distance that the average cutter for Darvish is from the strike zone, relative to the MLB right-hander data.

Going back to the four-color GIF of individual projections, Darvish did not throw many pitches in the direction of movement, up and to the right of the strike zone, which explains why both the swing and all-pitch hexplots are not heavily populated for small angles.

MLB RHP 2013 – Cutters Swung At Outside the Zone v. LHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		29.4		0.299 ft
Less Than 90 Degrees		65		0.29 ft
All		100		0.289 ft

The distribution for MLB righties very much mirrors the distribution for all pitches, with a large number of pitches swung at just below 90 degrees and within six inches of the strike zone and a small grouping of pitches swung at around 15 degrees. As opposed to Darvish, the percentages for less than 45 degrees and less than 90 degrees are both slightly up from all pitches outside. The average distances in all three cases are in the range of three to four inches. So, as in the case of sliders and four-seamers, there is a cluster of pitches with a small angle between the movement and strike zone vectors, but the angle that brings about the most swings is around 90 degrees, which is congruent with where a large number of cut fastballs are thrown.

Let’s move on to pitches taken.

Darvish – Cutters Taken Outside the Zone v. LHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		18.9		0.73 ft
Less Than 90 Degrees		45.3		0.625 ft
All		100		0.7 ft

Darvish’s cut fastballs to lefties do not appear to show any specific clusters for the pitches taken outside. The percentages indicate that these pitches were predominantly thrown with the movement carrying them toward the strike zone. The average distances are much higher than for pitches swung at, coming in around 7.5 to 9 inches.

MLB RHP 2013 – Cutters Taken Outside the Zone v. LHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		24.7		0.62 ft
Less Than 90 Degrees		54.6		0.579 ft
All		100		0.567 ft

The data for MLB right-handers loosely resemble that of all pitches, both in terms of the hexplot and the table, with a large cluster around 90 degrees and within six inches of the zone. Note that the small cluster around 170 degrees for all cut fastballs outside is only present for pitches taken and not for swings, indicating that unlike for Darvish, pitches thrown where the movement carries them toward the strike zone are not as good at creating swings from batters. Another noticeable difference is that the average distances are over six inches from home plate, which is about twice as far outside compared to swings.

Cut Fastball Examples

Before moving on to the results versus righties, we will stop to observe a few characteristic pitches outside of the strike zone from Darvish where the angle between the movement vector and the vector perpendicular to the strike zone was near 0, 90, and 180 degrees. Hopefully this will give the reader a better intuition of how to interpret the meaning of the angle metric.

The first cut fastball profiled has an angle between the strike zone vector and movement vector of 3.976 degrees, meaning the movement is strongly in the direction perpendicularly away from the strike zone (as can be seen from the red circle representing the projection). This is a pitch that was fouled off, and since the projection swept through the strike zone from bottom to top, it seems reasonable that a batter might swing at it.

The next cut fastball is parallel to the corner of the zone, with an angle of 97.944 degrees.

This pitch never really appears to be a strike but does run near the lower-right corner of the strike zone and only finishes up two inches outside the 0.5 probability contour (again, the rectangle featured is the textbook strike zone).

For the 180-degree case, the pitch projection moves perpendicularly toward the strike zone.

The angle, in this case, is 178.302 degrees. The pitch projection starts out looking to be in the dirt and moves up toward the bottom of the strike zone.

Versus Right-Handed Batters

Switching to right-handed batters, the called-strike probability of Darvish’s projected cut fastballs is broken down into four cases.

As with lefty hitters, both curves corresponding to strikes are close together and peak at the front of the plate with probabilities 0.927 and 0.901 for swings (green) and pitches taken (red), respectively. The swing/strike probability at the plate is virtually the same as for lefties and the take/strike is lower for righties. For balls, both probabilities peak at 50 feet and drop off after that, with the maximum being 0.196 for swings (blue) and 0.136 for pitches taken (orange). The shape of the curve for right-handed batters is markedly different for swings on balls compared to lefties, but this may be due to the small sample size of 10 pitches.

Next we consider the cases of “swing” or “take” for righties versus Darvish.

Both curves reach their maximum at the front of the plate, with a peak called-strike probability of 0.729 for swings and 0.341 for pitches taken. So, as before, the average pitch in both cases appears more and more to be a strike as it nears the plate. Both of these peaks are lower than for lefty hitters, especially for cutters taken.

Because we only have 103 cut fastballs thrown to righties by Darvish in 2013, the data appear rather sparse, but we still can contrast this with the lefties case to see if the same trends hold. Here, despite their small number, the pitches taken in the strike zone (red) appear less skewed toward the left. In addition, the swings outside the zone are below or down and to the right of the strike zone, as was the case to lefties.

Darvish – Cutters vs. RHB
0	28.6	0
100	60.4	X
X	33.3	16

The average location of the projections for cut fastballs taken in the strike zone is to the right of those swung at for right-handers, the opposite of the result for left-handers. So in either case, cutters taken from Darvish in the zone tend to be on the outer half of the strike zone. Due to the dearth of data in many regions, we can only really examine the results in the zone and down and to the right. In the strike zone, the swing percentage is slightly down from 63.4 percent to lefties. Down and to the right, the percentage has dropped from 33 percent to lefty hitters to 16 percent for righties.

For the MLB righty pitcher/righty batter data, compared to the heat map against lefty batters, the cluster of pitches in the 0.8-to-1.0 range are far more centered in the strike zone, but it still has a slight skew toward the right side, which would be outside to righties. So the possible advantage of pitching to one side of the plate seen against lefties does not seem present for righty-righty match-ups.

Darvish – Cutters Outside the Zone v. RHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		25		0.867 ft
Less Than 90 Degrees		40.4		0.711 ft
All		100		0.646 ft

For Darvish’s cutters to right-handers outside the zone, 25 percent of the pitches are congregated below 45 degrees and 40.4 percent are below 90 degrees. The average distances from the plate are all in the range of 7-10 inches outside. Compared to left-handed batters, the percentage below 45 degrees is higher and the average distances are larger for the 45- and 90-degree cases.

For MLB right-handed pitchers to right-handed batters, the distribution of the hexplot looks similar to left-handed batters.

MLB RHP 2013 – Cutters Outside the Zone v. RHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		22.9		0.527 ft
Less Than 90 Degrees		53.3		0.526 ft
All		100		0.549 ft

As before, there is a large cluster of pitches near 90 degrees, with signs of a small cluster around 15 degrees. The percentages are slightly down from lefty batters, and the average distances are a bit larger.

Darvish – Cutters Swung At Outside the Zone v. RHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		20		0.558 ft
Less Than 90 Degrees		30		0.406 ft
All		100		0.403 ft

Because of the size of the data set for cut fastballs, the hexplot for Darvish’s swings is sparsely populated at 10 pitches and therefore hard to draw any conclusions from. It is of note that, even in this sample, the average distances are much greater than for the left-handed batter case.

Expanding the data set back to all MLB righty pitchers hopefully will provide more insight.

MLB RHP 2013 – Cutters Swung At Outside the Zone v. RHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		28.6		0.299 ft
Less Than 90 Degrees		59.6		0.3 ft
All		100		0.3 ft

The hexplot shows a large concentration of swings just south of the 90-degree mark and within six inches of the strike zone, in addition to a small concentration around 15 degrees and close to the strike zone. The average distances are approximately 0.3 feet for the cases considered. Both the percentages and average distances are very similar to those against lefty batters.

Since there were very few pitches swung at outside by lefties for Darvish, the hexplot for pitches taken outside closely resembles that for all pitches.

Darvish – Cutters Taken Outside the Zone v. RHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		26.2		0.923 ft
Less Than 90 Degrees		42.9		0.762 ft
All		100		0.704 ft

For pitches taken outside by righties, the 45-degree case has a higher percentage of pitches compared to lefties while the 90-degree case has a slightly lower percentage. The average distances in all three cases are larger than their lefty counterparts.

MLB RHP 2013 – Cutters Taken Outside the Zone v. RHB
Angle		Percentage		Average Distance
Less Than 45 Degrees		20.9		0.637 ft
Less Than 90 Degrees		51.1		0.618 ft
All		100		0.637 ft

For righty pitchers to righty batters in 2013, a single large cluster is present in the 90-degree range. Both pitches with an angle of 45 and 90 degrees or less show a decrease in percentage of pitches in contrast to the lefty data. The average distances for the three cases considered are all in the area of 7.5 inches from the strike zone contour. Compared to lefty batters, the distances are a bit higher.

As before, the small cluster of pitches around 15 degrees that shows up in the all-pitch data is a component of the swing data, meaning that there is some preference for swings at pitches thrown outside in the direction of movement, while the overall preference is still toward cut fastballs with movement paralleling the strike zone.

Discussion

From this analysis of cut fastballs, we can make a few observations. First, unlike for sliders and four-seam fastballs, Darvish’s cut fastballs achieve their maximum attractiveness, in terms of the probability that their projections are called a strike, at the front of the plate for all swing cases. Therefore, it would seem — provided our projection technique is reasonable — batters are better able to track/project the final location of his cutters. It also might mean the PITCHf/x definition of movement for the cut fastball, at least for Darvish, does not have a large influence on batter judgment, in terms of balls and strikes. Also, Darvish’s average cut fastball finishes around 0.75 probability of a called strike for swings for both types of batters and roughly half that for pitches taken, with right-handed batters receiving less attractive pitches overall.

If we focus on the results for pitches outside the strike zone, Darvish’s cut fastball actually showed a decrease in the percentage of swings for pitches thrown in the general direction of movement (90 degrees or less). This runs counter to the observations for sliders and four-seamers that had an increase in swings for pitches with an acute angle between the movement and strike zone vectors. In fact, other than inside the 0.5 probability contour that we are labeling as the strike zone, the most advantageous places for Darvish to throw his cut fastball is down and down-and-right of the strike zone, where the movement carries the pitch projections either parallel or toward the zone.

For all MLB right-handed pitchers in 2013, against left-handed batters, the cut fastball was predominantly swung at when its projection was near the inner half of the plate, leaving pitches on the outer edge of the strike zone vulnerable to called strikes. This is in line with the behavior seen for Darvish. For right-handed batters, the projections of pitches swung at were more centered but still skewed slightly more toward the right side of the plate, from the catcher’s viewpoint.

For both left-handed and right-handed batters, cut fastballs parallel to the strike zone elicited a large number of swings, followed by pitches in the direction of the pitch movement. The latter observation is in line with results from sliders and four-seamers, but the former is new. It may be that pitch movement does not pose a large obstacle to batter perception for the cut fastball and so the angle that has the most pitches thrown there also has the most swings.

Future Work

At this point, we will conclude our analysis of Darvish’s 2013 PITCHf/x data, as any other pitches will live in the realm of small sample size when split between the handedness of the batters (as we have started to encounter in this study). In future articles, we will venture out in two different directions: one toward focusing on a single batter and one comparing drag in the projection versus no drag (which is the case for the current algorithm).

For a batter, since the variability in speed for a single type of pitch is much greater among a group of pitchers than for a single pitcher, we will switch our method of discretization from distance to time (monitoring pitches as being t seconds from home plate rather than y feet). For drag versus no drag, we will see if including drag affects the results greatly. We can preview the difference this adjustment makes in the GIF below. For the cut fastball shown, the red circle still represents the projection with drag removed as part of the PITCHf/x movement, and the green circle is with drag left in.