Checking the Numbers: Perceived Velocity

Few pitchers utilize their fastballs more frequently than J.A. Happ of the Phillies does, as he throws his four-seamed heater 71 percent of the time. Unlike Max Scherzer, who throws his fastball at a similar rate but routinely registers 95+ miles per hour on the gun, Happ averages a relatively modest 89.7 mph with rather pedestrian movement. Despite these facts pointing towards the idea that Happ's chief pitch is thus somewhat average or below, his plate discipline data has trended in the opposite direction: Happ ranks amongst the leaders in zone percentage yet has very low rates of both swings induced and contact made on pitches in the zone, performance characteristics that portend an ability to deceive hitters when coupled with his velocity and movement marks. Unless we accept that Happ's numbers are fluky, something about his delivery is preventing hitters from picking the ball up and reacting in appropriate fashion, whether that's a question of his hiding the ball well, or having a release that's closer to home plate than hitters are accustomed to seeing.

That latter component of possible deceptiveness comes up almost automatically in discussions of what has made Chris Young of the Padres so effective; Young has had no problem retiring hitters despite throwing a fastball with velocity that makes it seem like it's impersonating Jamie Moyer's luke-warm heat. The ideas for why Young can do this revolves around his height and his wingspan, and how he releases pitches much closer to home plate, allotting less time for the hitters to react. Warren Spahn used to say that "Hitting is timing; pitching is upsetting timing," and the shorter reaction time supposedly allowed by Young's long-armed release closer to home plate increases the perception of his fastball's velocity to the hitters. In other words, a radar gun and the PITCHf/x start speed may clock him throwing at 86 mph, yet the hitter actually "sees" a 91 mph pitch. To get a better sense of what's involved, I reached out to Mark Reynolds of the Diamondbacks for his thoughts on the matter:

Some guys hide the ball well. [Dan] Haren does his little pause, Doug [Davis] has his eight-second windup. Pitchers make stuff up because it's their job to deceive us, and we try to hit them. Randy Wolf, Chris Young, those are two that come to mind off the bat. I'm sure there's more. Some guys throw 95 and it looks like 85 because it's straight and you can time them. You've just got to stay back and get your foot down and see them. Wolf throws 88-90, but I don't have very good numbers against him because he is one of those guys who hides the ball and short-arms it.

The final portion of his quote deals with deception, which we will tackle a bit later on this month, but with his confirmation that the theory of perceived velocity is more than just analytical hullabaloo, I set out to find the extent to which pitchers exceed or fall short of their real velocity based on the length of their deliveries and the actual points of release relative to home plate.

Applying PITCHf/x to the Chris Young Conundrum

The PITCHf/x system continues to revolutionize baseball research, and since my foray into perceived velocity is reliant upon this particular dataset, a brief review of how everything operates is in order. Essentially, PITCHf/x uses a best-fit algorithm to compute constant acceleration along the flight path of a pitch, which is determined by two perpendicularly angled cameras recording images of the journey of a pitch to home plate. With constant acceleration computed based on the image data, the system is able to calculate velocity in feet per second as well as both of the movement components.

Both the PFX_X and PFX_Z-the horizontal and vertical components of movement, respectively-are measured 40 feet from home plate, while the velocity is measured 50 feet away, a critical piece of information for this article, as not every pitcher releases the ball at that exact spot. The images below, courtesy of Sportvision itself, offer a better sense of how the system functions. The first image shows a sample of where the two cameras would be placed and their lines of positioning from afar:

This next image shows the lines of positioning from the much closer view of behind the plate:

The system is shooting a laser from each camera to the perceived image of the ball. The two-dimensional points that result are the assumed screen positions of the ball. The software then takes these images from both of the tracking cameras and uses the camera registration and some tricky math to fit these two-dimensional points into a three-dimensional trajectory. The final image shows the system in action in an actual game setting:

From this angle we can see both home plate and the pitcher's windup, which comes in handy when trying to quantify something not currently extractable from the dataset. Now that we have the idea in mind and an understanding of how the system tracks pitches, the task at hand involves gauging where the pitcher actually released the ball and substituting that data for the automated 50 feet results. Thankfully, Greg Moore of Sportvision was kind enough to send me actual game videos for this study, offering assistance at every corner, in part out of his active interest in my findings.

Determining Actual Release Points

When initially developing this idea, determining the actual release points seemed like a daunting task. Some geometry and photogrammetry would be required to calculate field distance from a picture, and common ground would need to be abundant to make an accurate scale. While in Pittsburgh at our PNC ballpark event a couple of weekends ago, Will Carroll and I discussed a way of scaling the mound based on the knowledge that all rubbers are 60'6" from home plate and the suspicion that mound circumferences were uniform across the league. If so, the distance from rubber to the edge of the mound could be determined, affording the opportunity to scale our area of interest.

After consulting Matt Thomas, a photogrammetry expert, much of my excitement dissipated. The potential for angle errors and mound-height discrepancies would prevent accurate scaling of the field, which in turn would preclude us from measuring the exact distance from release to home. However, before even attempting to circumvent these issues I reached an epiphany-velocity equals distance traveled divided by the time it took to travel that distance. My focus had been spent on measuring distance this entire time, when the more feasible methodology centered on accounting for time. With the videos provided, I could literally count frames, use the appropriate frame rate and determine the amount of time a pitch took from release to home relative to the projected time set from 50 feet away.

Drawing on my experiences as an undergraduate film student, I loaded a sample video into a film-editing program, placed the clip onto the timeline, and noted the frames at which the pitch was released as well as when it crossed home plate. According to Moore, the PITCHf/x system records at a 29.99 frame rate, so dividing the total number of frames between Point A and Point B by 29.99 produces the actual flight time of the pitch. Now, one thing to keep in mind is that the editing program does not measure partial frames, so I used my best judgment in certain cases of determining when the actual pitch was released. In some cases, the pitch appeared to be released between frames five and six, venturing closer to the latter; in situations like that I may have jotted down the start frame as 5.7 frames.

After formulating the idea I consulted with Dr. Alan Nathan, and soon became able to take attributes actually recorded by PITCHf/x for a certain pitch as well as my new flight time, and calculate new initial and average outputs based on the change in flight time from projected to actual. We'll run through an example to explain how things worked a bit more efficiently.

Chris Young Under the Microscope

The data below belongs to a Chris Young fastball thrown against the Angels on June 14, 2009:

  x0   y0   z0   vx0    vy0    vz0   ax     ay     az
-0.77  50  6.63 1.05 -123.23  -7.1 -4.13  25.12  -12.43

x0, y0, z0: the release point in three dimensions; PITCHf/x always sets y0=50 feet.
vx0, vy0, vz0: the feet per second velocity in three dimensions as measured 50 feet away.
ax, ay, az: acceleration in feet per second squared, a constant.

The first step involves converting the feet-per-second velocity figures into miles per hour, which can be accomplished by summing the squares of vx0, vy0 and vz0, taking the root and dividing by 1.467, the conversion rate of feet per second to miles per hour. In this case, Young threw the pitch with an initial start speed of 84.1 miles per hour and an initial average velocity of 80.7 miles per hour. Average velocity looms quite large in this study because, as counterintuitive as it may sound, hitters are not seeing the start speed. Instead, if the idea of perceived and effective velocities-the latter of which we will discuss throughout the remainder of the year-were fictitious, the hitter would see the average velocity, which is the total distance divided by the total time. Back to the scorecard, we converted the three velocity components into Young's original start speed as well as his original average velocity.

Were we interested in modifying the actual starting distance, the next step involves calculating the new start speed in the y-dimension; however, since our focus deals with time, the y0 of 50 feet remains valid, meaning that initial start speed equals new start speed in the y-dimension, or vy0=vy1. Next, calculate the vyF, or the end speed in feet per second of the y-dimension. The vyF can be found through the following formula:

vyF = -(SQRT(vy0^2+2*ay*(1.417-y0)))

Reverting to Young's sample data, the vyF equals -112.70. Calculating end velocity and relating it to the starting speed from the y-dimension allows for the calculation of original flight time, which is found by subtracting vyF from vy0 and dividing by the y-acceleration component; the formula produces 0.4119 seconds for the sample pitch, meaning that the pitch traveled for 0.4119 seconds based on the standard PITCHf/x output.

The subsequent step involves actually counting the frames. Carefully using the direction arrows I found that Young released the ball slightly before the fifth frame, crossing the plate at exactly the 16^th frame. Subtracting 4.8 from 16 and dividing by the 29.99 frame rate results in an actual flight time of 0.3734 seconds. Essentially, by measuring Young's pitch from 50 feet away, the PITCHf/x system overstated his flight time by 0.0384 seconds, because Young releases the ball closer than 50 feet from home plate.

By subtracting 1.417 feet-the front of home plate-from 50 feet and dividing the difference by the actual flight time of 0.3734 seconds, further dividing that quotient by the 1.467 conversion rate we can produce the perceived average velocity of Young's pitch. The perceived average velocity comes out to 88.7 miles per hour, but this does not mean that hitters saw an 88.7 mph pitch compared to the recorded 84.1 mph, as the comparison is apples to oranges between the start speed and average velocity. Since the idea of average velocity likely feels weird, a short-hand for turning the perceived average velocity into the perceived initial velocity involves finding the delta between the originally reported values-the 84.1 mph and 80.7 mph average velocity, a delta of 3.4 mph-and adding that to the new perceived average velocity. For this specific pitch from Young, the perceived average velocity of 88.7 mph is added to the 3.4 mph delta to produce a perceived initial velocity of 92.1 mph.

Based on Young's size, stride, and release of the ball closer to home plate-which should be a clue that there are other factors potentially at work here capable of affecting the perceived velocity, all of which we will explore later on this year-the opposing hitter, Torii Hunter in this instance, saw a 92.1 mph pitch no matter what the radar gun speculated. Even if we adjust the frame count to counteract any sort of measurement error on my part, the fact remains that Chris Young threw this pitch with at least six more miles per hour of perceived velocity.

Plugging In Other Pitchers

Greg Moore sent me four games of data and videos, allowing for a diverse sample. I specifically requested games with both Young and Happ, but figured it would be interesting to run someone like Justin Verlander through the gauntlet as well. Taking five fastballs from nine different pitchers in the various games, the table below shows the averages in initial start and average velocity, perceived start and average velocity, and the delta between the perceived and initial start speeds.

Pitcher                V_IN   V_AVG   PV_IN   PV_AVG  DELTA
Chris Young            84.4    80.9    91.1    87.6    6.7
Heath Bell             92.6    88.7    97.7    93.9    5.2
Francisco Rodriguez    93.6    89.9    95.4    91.6    1.8
David Aardsma          94.6    91.0    96.0    92.4    1.4
Justin Verlander       96.1    92.5    95.6    92.0   -0.5
Johan Santana          90.9    87.2    89.9    86.2   -1.0
JA Happ                89.9    86.3    88.4    84.8   -1.5
Johnny Cueto           92.9    89.3    90.8    87.2   -2.1
Ian Snell              91.7    88.1    87.6    84.0   -4.1

According to these results, the theory on Young seems to ring true-on average he threw these five fastballs with a perceived velocity of 6.7 mph greater than what the PITCHf/x measured at 50 feet, a figure that correlates strongly to radar gun reports. Meanwhile, we struck out on Happ, as his five pitches averaged a real velocity of 89.9 mph that bested the perception by 1.5 mph. This does not necessarily indicate that Happ lacks deception or that his numbers are fraudulent, because there are a few aspects of perceived velocity left alone for the time being that will be discussed in a few weeks.

Two of the nine pitchers, both of whom are ironically members of the same team, really exceeded their recorded velocity, with Young and Heath Bell each averaging perceived velocities 5 mph or more better than recorded. On the flipside, Ian Snell averaged a recorded velocity of 91.7 mph, but based on the flight time of the pitch and the distance it actually traveled, the hitters saw 87.6 mph deliveries, which goes hand-in-hand with Mark Reynolds' suggestion that some pitchers throw slower than their gun readings. Truly interesting is the comparison between Justin Verlander and Francisco Rodriguez: Verlander averaged 96.1 mph to the 93.6 of Rodriguez, yet their perceived velocities were 95.6 mph and 95.4 mph respectively, a rather minuscule difference that goes a long way towards showing that two pitchers with a somewhat substantial difference in real velocity can create the same perception to the opposition based on their deliveries.

Moving Forward

Knowing the time from release to plate, or the actual spot at which the pitch left the pitcher's hand is incredibly valuable information. This study should serve as a stepping stone for the future in terms of what type of data could be available, because the radar gun velocity or start speed measured at 50 feet by PITCHf/x does not tell the entire story. There are several very real and scientifically proven factors that affect the perception of velocity to the hitter, making someone like Randy Wolf and his 91 mph fastball incredibly tough for Mark Reynolds, while Ian Snell becomes easier to time for one reason or another. Additionally, pitchers with divergent real velocities can converge on perceived velocity.

Today the exploration involved the first of the factors which affect the perception of velocity-the actual flight time as determined by the actual point of release. Next week we'll look at how the location of the pitch affects the perception of velocity in conjunction to the data presented in this piece. Following that, I'll eventually shift to a discussion of what is known as the hitter's 'attention zone, 'and how the perceived velocity measured here, the added or subtracted velocity based on the location of the pitch, and the preceding sequence of pitches all play into the batter/pitcher matchup, as well as which pitchers have truly maximized their efficiency based on these ideas of perception and setting up the hitters. For now, though, we can at least take away that Chris Young definitely throws harder than it looks from the stands, which at least partially explains how a non-Moyer can post very solid numbers with a very Moyer-like substandard four-seam fastball.