Pitch Movement, Spin Efficiency, and All That

In the past few years, the concept of “spin efficiency” seems to have taken hold. I first wrote about this subject in the article “All Spin Is Not Alike,” which appeared at Baseball Prospectus back in 2015. The essence of the article is that there are two types of spin. First is the so-called “transverse spin,” which directly leads to movement; second is “gyrospin,” for which there is no movement.

The Trackman radar system, which is an integral part of the Statcast system installed in every major league park, measures the full trajectory of the pitched ball, from which the movement can be determined and the transverse spin inferred. Moreover, it also measures the total spin rate for the pitch (i.e., the Pythagorean sum of transverse spin and gyrospin). Although the term “spin efficiency” was not used in the article, it is simply the ratio of transverse to total spin.

The concept is simple: For a given total spin rate, a larger spin efficiency means more movement. The primary goal of this article is a critical examination of two different techniques for extracting the movement and transverse spin from the trajectory. One of these techniques is the one actually used by Trackman; the other is an alternate. As we shall see, the latter produces more accurate results than the former.

The first step in the analysis was to simulate a set of 2110 “typical” pitches using data downloaded from Baseball Savant and taken from games in June 2018 at Tropicana Field, where the atmospheric effects are both known and constant. The trajectory of each pitch is fully determined by the position at release (x_R,y_R,z_R), the velocity at release (v_xR,v_xR,v_xR), the spin rate, the spin axis, and a drag coefficient. The mixture of the pitches among fastballs, curveballs, etc. was those typically found in major league games, as shown in Figure 1. Each pitch was assumed to have only transverse spin (i.e, a spin efficiency of 100 percent), so that the movement is determined by the spin rate and axis.

Using these parameters, the equations of motion were solved numerically using standard techniques to find the trajectory of the pitch from release to home plate in 0.01-sec intervals. This represents the trajectory Trackman measures. To find the movement for each pitch, the spin rate is set to zero, and the equations are solved again. The difference between the actual x & z positions at home plate and those positions with zero spin is what is meant by the movement. For ease of discussion, I will call these “exact” quantities M_xEX and M_zEX, where “exact” means they are the expected movements in the context of the trajectory model I have used.

The standard Statcast/Trackman procedure is to fit the “raw” trajectory to a smooth function, the so-called nine-parameter (9P) fit using a constant acceleration model, the same model used in the older PITCHf/x system. With the exception of the total spin, which is separately determined, all of the important parameters of the pitch, such as the home plate location and the movement, are determined directly from the 9P fit rather than from the raw trajectory.

The nine fitted parameters are the position at release (x₀,y₀,z₀), the velocity at release (v_x0,v_y0,v_z0), and the average acceleration (a_x,a_y,a_z). The first six of these are similar to the parameters that determine the exact trajectory. However, given that the constant acceleration model is only an approximation to the actual trajectory, for which the acceleration is not constant, there is no guarantee the inferred position and velocity at release will coincide with the actual values. This is not just an academic statement and will play a critical role in the analysis below.

As already mentioned, the x and z movements are derived from the 9P fit. I will now describe two different techniques for determining the movement from the 9P fit. Since the 9P fit is an approximation to the actual trajectory, the movements derived from that fit are also necessarily approximations to the actual movement. Accordingly, I will evaluate these two techniques by comparing them to the exact movements, M_xEX and M_zEX, discussed above. I omit most of the technical details, which will be described in a separate article I will post on my website.

Technique 1

The first technique is the one actually used by Trackman. The resulting movements are identical to the fields “pfx_x” and “pfx_z” at Baseball Savant and are the movement measured from the release point to the front edge of home plate. The technique uses the argument that, in the absence of forces, the ball would follow a straight-line trajectory; movement represents the deviation from a straight-line trajectory.

There are only three forces acting on the baseball in flight: the drag force, the Magnus force due to spin, and gravity. The drag slows the ball down but does not change its direction and, therefore, does not cause any movement. The other two forces change the direction of the ball, resulting in movement. Since we are only interested in movement due to the spin, the effect of gravity is removed.

Therefore, the TM movement is calculated as the deviation of the trajectory from a straight line between the release position and home plate position, with the effect of gravity removed. It is a perfectly reasonable definition of movement. I will refer to these movements as M_x1 and M_z1.

Technique 2

The second technique is one I suggested many years ago in the early days of the PITCHf/x era and more recently described in an unpublished companion piece to my Baseball Prospectus article (see my work on determining 3D spin axis from Statcast data, , including the references to earlier work).

The essence of the technique is to separate the average accelerations into contributions due to drag, Magnus (due to spin), and gravity. Once the Magnus contribution is isolated, it is straightforward to determine the movements due to spin. I will call these M_x2 and M_z2. Note that since neither the drag nor the Magnus forces are constant throughout the trajectory (they depend on the square of the velocity), the technique used to find M_x2 and M_z2 is also approximate. The nature of the approximation is described in the companion piece, just after Eq. 5.

So now we have two approximate methods of determining the movements from the 9P fit, both of which can be compared to the exact movements. Moreover, the movements from both techniques can be used to find the transverse spin, S₁ and S₂, respectively, which can be compared to the exact spin S_EX. The results of these comparisons are shown in Figure 2.

The results show a remarkable difference in the overall accuracy of the two approximation methods, with the clear winner being Technique 2. At the risk of boring some of my readers, it is worth explaining at least briefly why Technique 1 works less well than Technique 2. There are two reasons, one of which can be relatively easily explained and the other of which is very subtle even for physicists. Let’s talk about the easy one first.

Look at the left-middle plot in Figure 2, which remarkably shows that the difference M_z1– M_zEX is consistently positive. Recall that the z movement is the deviation from a straight line with the effect of gravity removed. The method of removing the effect of gravity is to add to the total z movement the positive number 0.5gt², where g is the acceleration due to gravity and t is the total flight time from release to home plate.

To demonstrate this is wrong, consider a ball released horizontally with neither backspin nor topspin, for which one expects zero z movement due to spin. Since the ball is released horizontally, the total z movement (including the effect of gravity) is simply the negative of the amount by which the ball drops between release and home plate. In the absence of drag, the ball falls by exactly 0.5gt² so that adding that amount to the total movement produces exactly zero movement due to spin, as expected.

But the drag is not zero. It acts mainly in the +y direction, opposite to the primary direction of motion. However, there is a component of drag in the +z direction, opposite to the direction of the falling ball and opposing the downward pull of gravity. As a result, the drop is actually less than 0.5gt². Therefore, adding 0.5gt² to the total z movement results in a net positive movement due to spin rather than the zero.

This is exactly what the middle-left plot of Figure 1 shows: M_z1 is consistently more positive than M_zEX. Therefore, for pitches thrown with backspin, the magnitude of the z movement is consistently overestimated, while for pitches thrown with topspin (e.g., curveballs), the magnitude of the z movement is consistently underestimated. The middle-right plot of Figure 2 shows no such problem exists for Technique 2.

The improper accounting for drag when determining the movement on a pitch is not unique to the Trackman system. The PITCHf/x system suffered from the same problem. Indeed, I first wrote about this problem over 10 years ago in an article I posted on my web site — Effect of the Magnus Force in the PITCHf/x Tracking System — discussed it at the very first PITCHf/x summit in 2008.

I won’t go into much detail about the more subtle problem, which affects both the x and z movements. Suffice it to say it has to do with the fact that the Magnus acceleration is not constant but proportional to the square of velocity. As a result, the average Magnus acceleration (which the 9P fit determines) is consistently less in magnitude than the Magnus acceleration at release.

One consequence is that v_x0 is consistently greater (i.e., more positive) than v_xR for M_xEX>0 and consistently less (i.e., more negative) than vxR for M_xEX<0. Therefore, regardless of the sign of M_xEX, using the initial direction of the pitch from the 9P fit to determine a straight-line path (as Technique 1 does) will consistently underestimate the magnitude of the horizontal movement. Similarly for the vertical movement, although that effect is mostly masked by the effect of gravity and is therefore less apparent in the plots. Once again, Technique 2 does not suffer the same problem.

Another way to look at the results is to do a linear regression of the x and z movements for the two techniques to the exact values, obtaining the results:

M_x1=0.00 inches+0.963*M_xEX

M_x2=0.00 inches+1.007*M_xEX

M_z1=1.78 inches+0.938*M_zEX

M_z2=-0.03 inches+1.000*M_zEX

In all cases, R² exceeds 0.99. The 1.78-inch intercept for M_z1 is a direct consequence of the gravity correction. The less-than-one slope for both M_x1 and M_z1 is mainly due to the more subtle effect described above and confirms the magnitude of the x movement is consistently underestimated. For Technique 2, the intercepts and slopes are essentially 0 and 1, respectively.

At this stage, it is worth emphasizing that the errors in Technique 1 have nothing to do with calibration issues or random noise. It has to do only with the methodology itself. Or, said differently, it is a physics/analysis issue, not a measurement issue.

Now let’s return to the discussion of spin efficiency, referring to Figure 3, which is a box plot of the ratio of inferred spin to exact spin (essentially, the inferred spin efficiency) for each pitch type for each of the two techniques. Recall that the actual spin efficiency is exactly 1.0 (i.e., 100 percent) for every pitch in the simulation, so any deviation from 1.0 on the plot represents an error.

While Technique 2 gets within about one percent of one for nearly every pitch, Technique 1 does much worse, especially for curveballs (CU), for which the transverse spin is consistently underestimated by about 25 percent due to the incorrect gravity correction discussed earlier. In other words, if a pitcher actually threw his curveball with 100 percent spin efficiency, the inferred spin efficiency using Technique 1 would only by about 75 percent. Likewise, the transverse spin is consistently overestimated by about 10 percent for four-seam fastballs (FF).

One may argue that small errors in the precise values of the movement are not really all that important from an analytics point of view. Perhaps that is correct. However, as I pointed out in “All Spin Is Not Alike,” small errors in the movement can translate to big errors in the inferred transverse spin and therefore to the spin efficiency. To the extent that anyone considers spin efficiency to be an important feature of a pitch, errors as large as those obtained using Technique 1 should be deemed unacceptable.

As one who spent a career doing experimental physics, I learned early on that before using a measurement instrument, it is important to understand the overall accuracy and precision of that instrument. The same can be said of using approximation methods, such as the 9P fit, to characterize data.

My recommendation to Trackman, MLBAM, MLB analysts, and anyone else doing analytics based on spin and movement is to use Technique 2. I made the same recommendation to Sportvision 10 years ago, although my advice was not taken. Nevertheless, some analysts have followed my advice, including Pitch Info, several major league clubs, and Glenn Healey in the recent analysis he presented at the Saberseminar. Those who would like to use Technique 2 for obtaining movement and transverse spin might find my spreadsheet template helpful.

I would like to thank Profs. David Kagan and Glenn Healey for their critical reading of this article and some helpful suggestions.