Anthony Rendon was ejected from a game earlier this month for … not contesting the strike zone. He flipped his bat down, faced away from the umpire, and did not visibly open his mouth. He was tossed by Marty Foster, for, what crew chief Joe West described incorrectly as ‘throwing equipment.’ (The pathologization of a non-white player’s actions after the fact to justify an ejection by a white ump is the subject of an entirely different set of analyses.)

After the game, Rendon actually went on record to say that umpires, like players, should be held to specific standards and demoted if they fail to meet those standards. This statement is remarkable for a couple of reasons. One, as most Nats fans know, getting Rendon to say anything, particularly anything of substance, to the media is pretty tough. He is, to forgive the pun, a pretty close-mouthed guy. For another, he points out that umpires, like players, are now doing their jobs in the Statcast era – we know, to a pretty refined degree, how well or not well they’re performing.

Players can be subject to replays that will tell them if their hand left the bag for the fraction of a fraction of a second, such as what happened to Jose Lobaton for the last out in the 8th in 2017 NLDS Game 5 (stay salty, my friends). But a home plate umpire’s word, particularly about the strike zone, is law. I understand ball vs. strike calls not being subject to replay. Even as someone who thinks most of the league’s pace-of-play ‘innovations’ are utter nonsense, I can’t see a good system in which every pitch could be subject to review. (Though, if the manager could make it one of their challenges, that’d be a start.) Umpires, therefore, should be held to the same standards, including performance reviews, as the players whose games they call.

The other thing that makes Rendon’s statement noticeable is that he’ll be facing the same umpiring crew in the final game against the Mets of this series and is likely to face them again this season. Saying that an umpire isn’t, in effect, doing their job commensurate with how Rendon is doing his is putting a pretty wide target on his own, and his team’s, back.

But beyond this instance, Rendon’s relatively mild approach to being struck out looking was disproportionately punished. He was ejected for not doing a whole heck of a lot, a punishment that seems incredibly disproportionate to a ‘crime’ that didn’t seem to go against MLB rules, written or unwritten.

I quickly tweeted out asking for an analysis of non-white versus white players in similar circumstances, because I had a hard time picturing a white player (like, say, Kris Bryant) being tossed for the same thing. Since no analysis existed, I did my own.

My analysis of available player ejection data from 2015-2017 led to the unmistakable conclusion: Non-white players, and Latino players in particular, are tossed at rates completely disproportionate to their representation in the league.

Methodology

Here’s a spreadsheet of data I compiled, mostly using Umpire Ejection Fantasy League data. I decided to limit it to 2015-7, in part because of use of Statcast and relatively consistent replay rules.

I also came at this analysis assuming any particular non-white and Latino player was as likely as any white, non-Latino player to be ejected, and so compared player ejections with league representation percentages for particular ethnicities. However, in doing analysis on position players only – that is, excluding pitchers – I didn’t have the league representation percentages adjusted for position players.

A major limitation in my data is having to hand-assign players as being white or non-white, and Latino or non-Latino. This was done using country of origin and knowledge of US-born players, and therefore is limited by my personal knowledge, particularly for US-born players. For instance, Marcus Stroman’s mother is from Puerto Rico and he was offered the chance to pitch for Team PR in the WBC. For the purposes of this analysis, he was classified as ‘nonwhite’ and ‘Latino.’

I also don’t know how players self-identify; I’m assuming Anthony Rendon, whose family is from Mexico and who was offered the opportunity to play for Team Mexico, self-identifies as Latino, but I don’t know if he’s stated that specifically. For non-US-born players, I also classified all players born in Latin American countries as Latino, but again, that’s not the same as asking for someone’s self-identification and that’s not the same as how any particular umpire perceives any particular player. For example, Francisco Cervelli, who is Italian and Venezuelan, was classified as non-white and Latino for this analysis.

I also classified Latino players as ‘non-white’ for the purposes of this analysis. While many Latinos self-identify as white, the Racial and Gender Report Card for Major League Baseball, where I got the league demographic data, identifies them as non-white and calculates them in the total of ‘players of color.’ So I maintained this classification for the purposes of this analysis. Any mistakes are unintentional; I welcome comments with suggestions for re-categorization.

Lastly, the umpiring corps has, as far as I know, not changed dramatically year to year. It’s a notoriously narrow pipeline and one almost entirely composed of white men. Analysis showed that some umpires toss players more than others, but this hasn’t been controlled for brawls. Additionally, the numbers of players tossed is a reflection of the number of games worked, which I haven’t controlled for.

This analysis isn’t meant to ascribe ejecting non-white and Latino players to any particular bad actor within the umpiring corps but to show a pattern of behavior.

The data:

Non-white players being ejected accounts for almost 50 percent of total ejections, despite players of color never being more than 42.5 percent of the league. Latino player ejections account for 38 percent of ejections, despite Latinos never being more than 31.9 percent of the league. Non-white, non-Latino players (of whom most are African-American), accounted for about 11 percent of ejections, fitting with representation in the league, except that no Asian players were ejected in this time period, and Asian players made up between 1.2 and 1.9 percent of the league. So, non-white, non-Latino, non-Asian players make up about 9-10 percent of the league and 11 percent of the ejections.

2017 is, therefore, a bit of a fluke. Of total players, nonwhite and Latino players were actually not tossed any more often (relative to their representation in the league) than their white peers.

I then controlled for two things:  pitcher ejections and ejections by non-home plate umpires, figuring that most pitcher ejections were as a result of beaning batters (which, yep, keep tossing them), and non-HP ejections might result from ejections during brawls, arguing slide calls, or in circumstances dissimilar to Rendon’s.

A few things became noticeable. One, overall ejections seem to be dropping, but non-pitcher, HP umpire ejections are holding pretty steady. Two, in 2015 and 2016, non-white position players comprised the majority of ejected players. Not only were non-white position players being ejected at a rate disproportionate to their representation in the league, they were being ejected more often than their white peers.

Latino position players were also being ejected by home plate umpires at rates disproportionate to league representation in 2015 and 2016 – 40 percent of ejections in 2015, despite Latino players being 29 percent of the league, and 42 percent of ejections in 2016, despite being 28.5 percent of the league.

For 2017, non-white and Latino players were ejected slightly more frequently than representation would account for.

So, is 2017  a step in the right direction or a flukey year or something else? No idea, and with the 2018 season being nascent, it’s hard to say. If there have been interventions on the part of the MLB or the umpire’s union, fantastic, but those interventions have not, as far as I’m aware, been made public.

I also know we won’t know for a while about 2018 ejections because ejections aren’t all timed equally. One of the weird things about this data is that white players tend to be ejected in early months, and non-white and Latino players make up the majority (or at least a disproportionate percentage) of ejections after May.

What this means is that the early ‘eye test’ for white and non-white players being ejected at similar rates won’t bear out in later months.

What if they deserve it?

None of this has addressed a fundamental question in considering ejections: Some guys have it coming. I tried to control for this in considering repeat offenders – that is, if there are certain players who, by virtue of reputation and absent any racial dynamics, just get tossed a lot.

Of the guys who’ve been tossed more than three times, the results are … very unsurprising:

Of these repeat offenders, Escobar and Kemp are non-white, and the former is Latino. The rest are the kind of love-’em-or-love-to-hate-’em white guys you might expect to make up such a list. So again, the eye test of ‘Bryce gets tossed too,’ doesn’t bear out when you look at the number of different players tossed total.

For ‘three-peaters’ – guys tossed 3 times in the past three seasons – of the 13 tossed three times, only two, Joey Votto and Justin Turner, aren’t Latino. And for players tossed once or twice – so not for having a rep as a showboat or arguer or ‘disrespectful’, 57 Latino and 115 non-Latino players have been tossed in 3 years. So, 33 percent of ejections have been for Latino players, despite the fact that Latinos averaged at 30 percent of the league’s players during this time. For non-white players, 76 non-white and 96 white players were ejected once or twice, meaning 44 percent of players ejected once or twice weren’t white, with the league averaging 41.5 percent non-white players during this time.

In totality, 36 percent of the players being tossed are Latino, and 46.5 percent of the players being tossed are non-white, both higher than their representation in the league.

If ejections are the league’s way of dealing with argumentation at the plate, we should consider that Latino players and non-white players are already disproportionately disciplined by their fellow players – and brawls are more likely to break out between players of different ethnicities.

We should also consider why players are perceived to ‘have it coming’ to them for arguing, ‘showboating,’ or other displays of either enthusiasm or disrespect, depending on your perspective, and why Latino and non-white players are dinged for it so more than their white peers for what are likely similar behaviors.

Umpiring by largely white umpires on increasingly non-white players is a cross-cultural conversation, one that’s monitored by 40,000 fans, TV viewers, and the ever-watchful eye of Statcast. The league has a vested interest in solidifying its presence in Latin American countries and in trying to encourage African-American players – who are a decreasing percentage of players overall – to continue with the game.

I don’t pretend to know what’s in an umpire’s heart (I assume pine tar and certificates for failed eye exams). I didn’t do this analysis to say that any particular umpire is actively thinking that they should eject a non-white or Latino player because they are non-white or Latino. What I discovered in doing all of this is that there is a very clear pattern of behavior among umpires when it comes to player ejections when the Statcast era is taken in its totality. An action may cause harm – in this case, an ump being more likely to throw out a non-white player – without any specific racist intent.

Additionally, the idea that umpires are enforcing ‘respect’ (and Joe West said that was Foster’s intent in tossing Rendon – “You have to do something or he loses all respect from the players.”) on non-white and Latino players is particularly galling. If non-white and Latino players are disproportionately perceived as ‘disrespectful’ of the game for similar actions as their white peers, such as tossing a bat after a strikeout, then the issue is perceptions and not players.

This, of course, is a societal issue beyond baseball. Analyses of behavioral perception by white teachers show that they tend to ascribe disrespectful, aggressive behaviors to non-white students at higher rates than they do to white students or than black teachers do with black students. Analysis of school punishments shows that black and Latino students are suspended and expelled at much higher rate than their white peers without any evidence they’re misbehaving more. So this is not a problem unique to player-umpire dynamics, but instead is one indicative of broader structural societal dynamics.

To work on addressing this as a structural issue, the league can change how it handles ejections. A few proposals:

• All plays over which a player is ejected are automatically reviewable, including balls vs. strikes. If an umpire ejects a player on a strike call that, on review, is revealed to be a ball, the player isn’t ejected. If a player makes contact with an umpire, they should be ejected but if players see that there is a clear and objective appeals process to an ejection, my guess is that they’re more likely to calmly walk off than explode.
• All ejections should be reviewed as part of a rigorous rating process for umpires. Umpires who repeatedly eject players for calls that, on review, they should not have made (such as a bad ball vs. strike call) should experience some form of penalty – by being demoted, retrained, or fined.
• Umpires’ ability to call balls vs. strikes compared with what Statcast determines is in or out of the zone should be made publicly available. If an umpire is consistently below a certain percentage of accuracy, they should be demoted or retrained.
• Player strikeout rates should be adjusted for umpire accuracy the way player defense is adjusted for particular ballparks.
• Diversify the umpire corps. Currently, umpires are generally older white men who feel tasked with enforcing ‘respect’ from young, increasingly non-white players. I’m not saying that simply hiring more people of color (including women of color) is a cure-all for these kinds of issues, but diverse perspectives may mean a decrease in unintended slights between players and umpires, and a general change in player-umpire dynamics.
• Radically, I would also like strike calls to be reviewable. They would cost a manager a challenge if incorrect like any other play. If a manager challenges – and is correct in challenging – strike calls repeatedly, then the umpire, and not the player or manager, should be held at fault.

If this sounds like we can replace a home plate umpire with Statcast for calling balls vs. strikes, then I’m for it. As the cliche goes, I didn’t watch the game for the umpiring, and if a computer can do what the umpires are doing in a fashion that doesn’t disproportionately penalize players of color, then I don’t see a downside.

I co-host Resting Pitch Face, a bi-weekly baseball podcast with a Nationals bias. I can be reached on Twitter at @sydrpfp.

As one of the remaining top free agents, Cain agreed to a 5yr/$80M deal with the Milwaukee Brewers. Reports indicate that Cain will earn $75M from 2018 to 2022 ($13M/$14M/$15M/16M/$17M) plus incentives, in addition to a deferred yearly payment of $1M for years 2023-2027 ($5M total). Based on Cain’s projected Market Value, he should be worth close to 9.6 sWAR for the next 3 years ($84M approx. Assuming 5% inflation per year). Understandably, Cain’s history regarding injuries and durability is a considerable factor moving forward.

A new home should benefit Cain’s offensive output. For the past 3 years, Miller Park has averaged a 106 HR Factor for RHB; which is significantly (P = 0.03) higher than Kauffman Stadium’s 88 (RHB) in the same period. Moreover, Cain has not “slowed” down, his sprint speed for last year (29.1 ft./sec) ranked him in the top 20 overall, in addition to being ranked in the top 10 among Center-Fielders as well.

Based on the aforementioned factors, Cain’s 2018 updated projections represent an increase in both OPS and ISO from last season. He should be able to get on-base at an above-average rate (0.358 OBP) in addition to an increase in SLG from last year. Although his wOBA is projected to regress marginally, Cain’s updated projections still aim toward a 3.7 sWAR (as a CF) for this upcoming season. Please find Cain’s updated 2018 projections in the chart below.

———————————————————————————————————————————————————Disclaimer: SEG projections are computer-based projections of performance based on the “SEG Projections System” framework. Regarding Wins Above Replacement (WAR), sWAR is the “SEG Projection System” calculation of WAR. sWAR will always stand for WINS ABOVE REPLACEMENT (“WAR”), unless noted otherwise.

J.D. Martinez had another great year in 2017. With 3.9 sWAR[1] and a .430 wOBA, J.D. contributed well above average once again. Offensively (wOBA) he has been able to consistently contribute year after year since 2014. J.D. does carry some defensive shortcomings, yet he is an excellent asset in any lineup.

For the past three years he has been able to get on base at an above-average rate (.364 OBP), alongside an excellent .289 ISO and a .587 SLG. He does carry a lifetime 25% K-rate (approx.), but as long as he is able to produce and contribute the way he has, he should be able to make an impact in any organization.

In 2018[2], J.D. should see a slight decrease in wOBA (.395). Based on the 2018 projections, both OPS and ISO should decline marginally; nevertheless, J.D. should be able to perform as a top-caliber player.

Please find J.D.’s 2018 projections in the table below.

J.D. Martinez’s estimated AAV is around $27M, based on a five-year/$135M contract. J.D. is projected for 14.6 sWAR for the next five years.

[1] sWAR = “SEG Projection System” calculation of WAR

[2] 2018 Projections: JD Martinez (SEG Projection System)

Hosmer certainly had his best season so far, with a 4.0 sWAR[1] and a .376 wOBA. Overall, consistency has not been there; over the past three years his offensive output has fluctuated, and that is something that can be said for his entire career. When looking at his offensive contribution, it seems that he has a “quality” season every other year. Nonetheless, Hosmer has been able to get on-base at an above-average rate of .359 OBP for the past three seasons. Also, he has managed to strike out (K%) at an average rate of 17.2% for the same period of time.

Moving forward, Hosmer’s offensive output for 2018 is projected[2] to see a slight decline. As previously mentioned, consistency is not his strength, and this should be reflected on his overall contribution for next year. A decline in wOBA (.351) from last year, alongside an increased K% (17.1%) will negatively impact his sWAR (2.6) in 2018.

Below you can find a detailed 2018 projection.

Eric Hosmer’s estimated AAV is $21M, based on a five-year/$105M contract. He should be worth about 11.5 sWAR over the next five seasons. There has been a lot of noise regarding dollar amount and duration of contract. Going up to a seven-year agreement, he should be worth no more than $124M.

[1] sWAR = “SEG Projection System” calculation of WAR

[2] 2018 Projections: Eric Hosmer (SEG Projection System)

After a strong 2017 (.347 wOBA, 4.1 sWAR[1]), Lorenzo Cain is one of the top remaining free agents. As a plus center fielder, defense is one of Cain’s greatest assets. On the other hand, Cain’s durability is a big question, having played just once over 140 games in a single season (2017). Injuries and age are both substantial concerns moving forward.

If able to stay healthy for at least 130 games in 2018, Cain is projected[2] to get on-base at an above-average rate (.356 OBP). Based on the projections, Cain should see a slight increase in both SLG and ISO from last year. Nonetheless, his wOBA should see a decrease in conjunction with an increase in K%. An overall decrease in offensive output will impact Cain’s sWAR (3.7) for 2018.

Lorenzo Cain’s estimated AAV is around $21M per year, based on a four-year/$84M contract. He should be worth about 10 sWAR over the next three years. Staying healthy is crucial; as long as his speed does not drop dramatically, he should be able to significantly contribute for the next 2-3 seasons.

[1] sWAR = “SEG Projection System” calculation of WAR

[2] 2018 Projections: Lorenzo Cain (SEG Projection System)

Spring-training games in the Cactus League are a unique joy, especially for baseball fans (like me) who hail from colder climes. Unlike the Grapefruit League, which features stadiums separated by hundreds of miles of humid Florida air, the Cactus League consists of a compact cluster of stadiums bathed in sunshine and desert-dry air. Spectators and players alike can enjoy the spring conditions (and for some, including myself and Carson Cistulli, Barrio Queen guacamole and sangria) in the Valley of the Sun for weeks before teams return to their home stadiums across the country in late March.

Figure 0: Your author enjoying the 82-degree sunshine (and probably a juicy IPA, not pictured) at Hohokam Stadium, March 2017

Some teams will return to relatively warm and dry climates (Arizona Diamondbacks, who have to trudge the 20 freeway miles to Chase Park), but others will return to retractable domes (Seattle Mariners) or cold conditions where snowed-out games are certainly not out of the question (Cleveland). Given that the point of spring training is to get players ready for 81 games at their home ballpark, are two months of baseball in dry, sunny paradise the best way to prepare players for opening day at home? Short of building exact climate-controlled replicas of Kauffman Stadium and Wrigley Field in the Phoenix Metro, how could teams better prepare their players for the start of the season at their own home ballpark? Enter an unlikely hero, the great “Rocky Mountain equalizer”: the humidor.

Figure 1: Climatology of Phoenix, AZ (Feb-Mar) and the home locations (ICAO Airport codes) of the 15 Cactus League teams (Apr-May)

Just by eyeballing the graphs in Figure 1, without wading into the different lines and the specific airports (some lines switch to larger airports with RH), no stadium’s meteorological conditions are close to those in the Phoenix area. With the exception of the Rangers, no team plays in a stadium with an average May high temperature greater than the average March high temperature in Arizona. And only the “high desert” of Colorado comes close in RH to the dry air in Arizona March. Clearly, the opening day meteorological conditions will be significantly different from those Cactus League players see during spring training (Figure 2).

Figure 2: Changes in climate between April (major airport nearest home stadium) and March (PHX), with larger markers indicating larger temperature differences (dotted markers indicate increased T) and blue markers indicating more humid conditions (orange being drier)

This drastic change in temperature and humidity (Figure 2) is likely to have a major impact on how the ball plays once teams leave Arizona. Like many baseball physics researchers before me, I will once again heavily rely on the work previously done by Dr. Alan Nathan to inform my physical exploration herein. As shown in Nathan, et al. (2011), the two crucial meteorological factors of temperature (T) and relative humidity (RH) have a strong impact on both aerodynamic factors (such as drag) AND contact factors (such as coefficient of restitution, COR) that determine how far a batted ball travels. Rather than run afoul of the copyright of the American Journal of Physics by reproducing the figures here, I highly encourage you to check out Figures 2-4 in Nathan, et al. (2011) to see these relationships.

Equation Block 1: Calculating the effect of COR changes on “effective” exit velocity of a batted ball

The eternally relevant Baseball Trajectory Calculator developed by Alan Nathan has the ability to adjust aerodynamic factors associated with stadium altitude, barometric pressure, temperature, and relative humidity. Combined with the equations from Block 1 above, the changes in COR as a result of meteorological changes can be simply approximated in the Nathan Calculator as a manual change in the rebound (exit) velocity of the ball off the bat.

Great, simply smash aerodynamic and COR changes together and we’re in business, right? Well, almost…it seems every baseball physics article could have all the baseball-specific details stripped out and what would remain is a meditation on linearity and covariance. This example is no different. While we might expect meteorologically-induced aerodynamic and contact factors to vary independently, in real on-the-field situations, balls will be affected by not only their current conditions but also their recent history of past conditions. Absent experimental data on the time scale of such internal ball changes, we can still get a general sense of what could happen when multiple changes overlap. Let’s dive into some colorful 3-D contour plots of results using the default batted ball parameters of the Trajectory Calculator (100 mph pitch, 100 mph exit velocity, 30 degree launch angle) and see what happens!

Figure 3: Effects of meteorological T and RH on fly ball distance, including COR effects equal to ambient conditions (as if balls were kept in the same conditions)

We aren’t too far afield from the basic variables one can change in the Nathan Calculator, so the results from Figure 3 aren’t terribly surprising. Baseballs travel further through warm and dry air. In addition, dry/warm baseballs are bouncier than cold/wet baseballs. It’s unlikely that equipment managers are keeping baseballs outside, so they probably aren’t going to actually experience changes in COR associated with extreme conditions due to the time necessary for water vapor to diffuse into the guts of the baseballs and soften them. But absent a sense of how equipment managers store baseballs, let’s explore the possible impact that a spring training humidor could have.

Figure 4: Effects of humidor-like T and RH on fly-ball distance, with aerodynamic effects equal to PHX March average but COR changing with humidor conditions

Figure 4 shows what would happen if we changed the internal ball T and RH but continued to play in the average Phoenix-area meteorological conditions in March. The weakness of the temperature effect compared to the strength of the humidity effect can be predicted with the slope of each experiment in Nathan, et al. (2011). It’s unlikely, though, that T and RH both have, when combined, a linear effect on COR. For example, it’s unclear whether this linear model captures the hot/wet and cold/dry combinations correctly. This indicates the need to inspect the covarying relationship between T and RH on COR (and therefore, fly-ball distance) more deeply than the simple linear combination I used in this model.

Table 1: Monthly climate, elevation, default fly ball distance using the Nathan Calculator and monthly climate, and scale factors for conversion of March fly ball distance (at PHX) to April fly ball distance (at home).

With the data from Figures 3-4, we can figure out an appropriate scaling factor (Table 1) to translate the dimensions of each team’s spring training stadium and compare them to the dimensions of their home stadium (Figure 5).

Figure 5: Surprise Stadium (KC) and Scottsdale Stadium (SF) scaled to April climatology in KC and SF (no humidor)

After comparing the “effective dimensions” of the Cactus League stadiums to the home stadiums of each team, one can’t help but wonder if the teams had a hand in the way the stadiums in Arizona were constructed. Some teams, such as the Royals, share a stadium with another team (Texas Rangers); therefore, this clearly can’t explain all of the similarities between stadium shapes.

Figure 5 shows that in Arizona during the month of March, the spring training stadiums play much “smaller” compared to other stadiums than their physical dimensions might indicate. By slightly lowering the COR of the ball by using a humidor, teams could cause their spring training stadiums to play with effective dimensions approximately equal to those of their home stadiums. If the Royals were to store their spring training baseballs in a humidor at approximately 70% RH, the differences between the distance up the lines (longer at Surprise than Kauffman) and the distance to straightaway center (shorter at Surprise than Kauffman) would yield around the same “effective surface area” of the scaled outfield.

This analysis, much like my earlier piece on fly-ball precession, neglects many physical variables that would impact the actual games being played. In this example, I have neglected the effects of wind and day-to-day changes in barometric pressure. Prevailing winds due to stadium orientation and location would make this experiment much more realistic. For variations in pressure due to synoptic weather systems (cold fronts, warm fronts, etc.), however, “averages” over an entire month inform us less in terms of the baseline environments of each stadium than monthly averages of temperature and relative humidity. The model also assumes that the balls are essentially stored in temperatures and humidities equal to the ambient conditions in the home stadiums; equipment managers likely store them in some indoor location, but it’s unclear whether they are treated to the exquisite RH control seen with the humidor at Coors Field. Such confounding factors will be explored in future follow-ups to this piece.

In addition to physical assumptions made here, it’s quite possible that baseball operations departments in teams have goals in spring training other than closely approximating the hitting conditions in their home stadiums. But if they want to see who will have power that plays well in their home stadium, the humble humidor could play a key role in moderating the enhanced fly-ball distance that comes naturally with the warm, dry spring air of paradise (Cactus League baseball, that is).

In the chase to break the story of the “smoking gun” behind the recent surge in MLB home runs, many a gallon of digital ink hath been spilt exploring possible modifications to the MLB balls, home-run-optimized swing paths, and even climate change. In my field of Earth Science (atmospheric chemistry, to be more exact), it’s rare that a trend in observations can be easily attributed to a single causal factor. Air quality in a city is driven by emissions of pollutants, wind conditions, humidity, solar radiation, and more; this typically leads to a jumble of coupled differential equations, each with a different capacity to impact overall air quality. To my untrained eye, agnostic to the contents of the confidential research commissioned by MLB and others, this problem is no different: a complex mixture of factors, some compounding each other and some canceling others, is likely fueling the recent home-run spike.

This article will examine the potential for a change in the MLB ball minimally explored thus far: reduction of precession due to decreased internal mass anisotropy. What a mouth full! “Precession” and “anisotropy” don’t have the same ring as “juiced ball” or “seam height” (though they may be on par with “coefficient of restitution”). But these words can be replaced with a more familiar (though funny-sounding) word: wobble. This wobble can occur for many reasons, but the most probable explanation in baseball is that the internal baseball guts are slightly shifted from the center of the ball. This could be due to manufacturing imperfection, or in the course of a game, contact-induced deformation of the ball.

Precession, in general, occurs when the rotational axis of an object changes its own orientation, whether due to an external torque (such as gravity) or due to changes in the moment of inertia of the rotating object (torque-free). Consider a spinning top: the top spins about its own axis (symmetrically spinning about the “stem” of the top) while the rotational axis itself (as visualized by the movement of the stem) can trace out a coherent pattern. If imparted with the same initial “amount” of spin in different ways, the total angular momentum (from both rotation and precession) of the top will be the same whether it’s spinning straight-up or precessing (wobbling) in an elliptical path.

Figure 0: Perhaps the most hotly debated spinning top in the world

As with other potential explanations relating to a physical change in the ball, a change in mass distribution could have occurred unintentionally due to routine improvements in manufacturing processes. By getting the center of mass (approximately, the cork core of the baseball) closer to the exact geometric center of the ball, backspin originally “lost” to precession (in the form of wobble-inducing sidespin) could remain as backspin while conserving total angular momentum; increased backspin has been shown to increase the “carry” of a fly ball, therefore increasing the distance (potentially extending warning-track shots over the fence). A deeper discussion of angular momentum can be found in any mechanics textbook or online resource (such as MIT OCW handouts), but the key takeaway when considering a particular batted fly ball is that productive backspin gets converted to non-productive precession (roughly approximated as sidespin in one axis) when mass is not isotropically (uniformly from the center in every direction) distributed. This imparts a torque-free precession on the spinning ball, causing the rotational axis to trace out a coherent shape.

Precession in baseball has not been deeply studied; in fact, when explicitly mentioned in seminal baseball physics resources, it is noted as a potential factor that will be ignored to simplify the set of physical equations. Together, dear reader, we shall peek behind the anisotropic veil and explore how precession might impact fly-ball distance, and by extension, home-run rates.

***

For those of us with some experience throwing a football, even just in the park, we can picture the ideal “backyard Super Bowl” pass: a tight spiral that neatly falls into the outstretched hands of the intended receiver. The difficulty of executing such a perfect throw is evident in the number of nicknames for imperfect throws that wobble (precess) on their way up the field short of their intended target (see “throwing ducks” re: Peyton Manning). In football, the wobbly precession of a ball in flight is typically blamed on the passer or credited to a defender for deflecting it (or in some cases, allegedly, a camera fly wire). It’s not as easy to imagine such behavior in baseball: even in slow-motion video shots of fly balls, the net spin of the ball is dominated by backspin. In addition, the nearly-spherical shape of a spinning baseball has significantly different aerodynamics than the tapered ellipsoid used in football. However, even a small amount of precession has the potential to shave yards off the distance of a football pass; therefore, impacts of precession are certainly worth exploring in the game of baseball.

As a sometimes-teacher (I have taught two laboratory classes at MIT), I strongly believe in the power of simple physical models to qualitatively inform trends in the not-so-simple real world. Therefore, for the first step of exploring the effect of ball precession in the game of baseball, I have turned to the wonderful Trajectory Calculator developed by Dr. Alan Nathan. The Calculator numerically solves the trajectory of a batted ball by computing key physical properties in discrete time steps. While many physical attributes of the ball are calculated in the various colored fields, any of them can be overwritten with custom values.

Figure 1: Fly Ball Distance with Nathan Trajectory Calculator defaults, conversion of backspin to sidespin

In Figure 1, I use the Trajectory Calculator to explore the effect of sidespin conversion on a single fly ball with the same initial contact conditions as the default (100mph exit velocity, 30-degree launch angle, default meteorological conditions), with the total spin set to 240 radians per second. Backspin is not converted to sidespin in a one-to-one fashion: because of the Pythagorean relationship between these factors, total spin is equal to the square root of the sum of the squares of sidespin and backspin. Therefore, to conserve angular momentum, a 10% reduction in backspin (216 rad/s) yields 104.6 rad/s of sidespin, which together lead to a ~1% decrease in fly ball distance from 385.3 ft to 381.3 ft.

With all of the assumptions made here, notably that introduction of precession can be simulated as pure conversion to sidespin to conserve angular momentum, the effect of precession on the flight path is clear but rather modest in this simple approach. However, the Calculator results show that by reducing the “wobble” in a ball’s trajectory, it will carry further. A league-wide reduction in precession would mean that balls would, on average, travel further, leading to an uptick in home runs. If decreased precession would also decrease the effective drag the ball experiences in flight, the effect of increased fly-ball distance could be even further enhanced.

A more realistic exploration of precession will require further modification to the modeling tools at hand. Following Brancazio (1987), which studied the effects of precession on the trajectory of a football, and additional follow-on work, a precession-only physical model can be developed to explore more complex aspects of the problem posed here. Elements of this precession-only model can be fed back into the Nathan Trajectory Calculator, but without a full understanding of some unconstrained physical constants and mechanical aspects of the pitch-contact-trajectory sequence, a tidy figure in the style of Figure 1 will be difficult to produce.

Again, as I mentioned above, I find simple models to be effective tools for teaching concepts. Therefore, let’s consider a “perfect” baseball to be a completely uniform, isotropic sphere, as in Figure 2. This perfect ball is axially symmetric and should not have any precession in its trajectory due to changes in its moment of inertia (I). Now, let’s add a small “spot mass” (that doesn’t add roughness to the surface) on the surface of the ball along the axis of rotation corresponding to pure backspin (the x-axis here). This ball with a spot mass should approximately represent an otherwise-perfect sphere whose center of mass is slightly shifted in the x-direction.

Figure 2: (A) real baseball, (B) perfect sphere, (C) sphere with a point mass at the surface, and (D) sphere with slightly offset center of mass approximately equivalent to (C)

If the model ball has a mass m1 that is isotropically distributed through the entire sphere, and a point mass with mass m2 that is located on the surface along the x-axis, the moment of inertia can be calculated in each direction, summing the contributions from the bulk mass m1 and the point mass m2 (Figure 3).

Figure 3: Moments of inertia for isotropic ball (mass m1) with a point mass (m2) at the surface

Of course, the mass of a real baseball isn’t isotropically distributed, and there is no such thing as a “point mass” in reality; however, by exploring different combinations of m1 and m2 that sum to to mass of an actual MLB baseball (5.125 oz, as used in the Nathan Trajectory Calculator), the ball can be distorted in a controlled manner to explore the effects on precession and fly-ball distance.  Using a set of equations derived from Brancazio (1987) Equation #7, the initial backspin of a ball (omega_x0) can be calculated given an initial total spin (omega), the variable B (the “spin-to-wobble” ratio indicating the number of revolutions about the x-axis per precession-induced “wobble”, a function of the moments of inertia I_x and I_yz), and the angle of precession (built into the variable C, with theta being the angle between the x-axis and the vector of angular momentum when precessing, similar to the angle between a table and the “stem” of a spinning top).

Equation Block 1: Derivations from Brancazio (1987) used in a simple model of baseball precession

The limitation of this approach is that in order to explore the theta-m2 phase space, we must prescribe a priori an angle theta at which the precession occurs. By instead solving for theta from equation 5 above (Figure 4), we can get a sense of the possible values for theta by prescribing the fraction of omega that is converted to precession (the variable A, a mixture of omega_y and omega_z, also called “effective sidespin”).

Figure 4: Contour plot of theta (degrees) with respect to ranges of m2 and variable A (effective total sidespin)

Figure 4 shows that angles between 0 and 6 degrees are reasonable for the conditions explored using the approach from Brancazio (1987) as translated to baseball. So let’s turn to equation 6, using a range of angles from 0 to 6 degrees, to explore the effects of precession on backspin omega_x (Figure 5).

Figure 5: Contour plots of backspin (omega_x) and effective sidespin (variable A) with respect to m2 (as % of m) and theta (degrees)

Great, the effect of a point mass along the x-axis of the ball can be quantified in this model! The effect is modest, but has the potential to slightly decrease the distance of an identically struck isotropic ball. But there is one major limitation to the model as currently shown: when the angle theta is chosen a priori, there is no capacity of the model to correct to a more physically stable angle. In fact, along the entire x-axis of the plots in Figure 5, where m2 = 0, the ball should be completely isotropic and therefore no precession would occur; a small initial theta would likely be damped out over a small number of time steps. In addition, the contours of constant omega_x in Figure 5a curve in the opposite sense than might be expected: increasing m2 should lead to more pronounced procession. On the other hand, this very simple model does not take into account the possible effects of torque-induced precession caused by gravity (extending the effect of mass anisotropy alone), nor does it account for additional drag impacting a precessing ball. More study is needed to further elucidate the possibility of precession having a considerable impact on fly-ball distance; however, unlike the sometimes-empty calls for “further exploration” of minimally promising leads in academic journal articles, I intend to execute such investigation.

All of these limitations are inherent in the fact that, without outside data to constrain the physics of precession as it applies to baseball, the problem we are trying to solve with this simple model is an ill-posed problem in which there is not a unique solution for a given set of initial conditions. Luckily for us, we live in the Statcast age where position, velocity, and spin of the baseball are all continuously measured (if not fully publicly available). In addition to benefits gained from Statcast data, this problem can also be further constrained by experimental data on MLB balls. Finally, an opportunity to put my skills as an experiment-first, computational-modeling-second scientist, to use! Stay tuned to these pages for follow-up experiments and data analysis in this vein.

The conspiratorial allure of an intentional ball modification directly induced by Commissioner Rob Manfred is visible on online comment sections far and wide; however, many of the most credible explanations for ball changes are benign in Commissioner intent and perhaps attendant with improvements in ball-manufacturing processes. In any case, there are likely multiple facets to the current home-run surge. Ball trajectory effects due to precession have traditionally been ignored to simplify the problem at hand; this initial exploration shows that due to the difficulty of the problem, that was likely a good trade-off given the data available in the past. In the future, however, past work in diverse areas from planetary dynamics to mechanics of other sports can be used alongside new and emerging data streams to help determine the impact of precession on fly-ball distance.

Python code used to generate Figures 4-5 can be found at https://github.com/mcclellm/baseball-fg

Special thanks to Prof. Peko Hosoi (MIT) and Dr. Alan Nathan for providing feedback on early versions of this idea, which was born on a scrap of paper at Saberseminar 2017.

I went over the starters version of dScore here, so I’m not going to re-visit that here. I’ll just jump right in with the list!

Risers/Fallers

Danny Salazar (+16) – dScore never gave up on him, despite him being absolute trash early on this year. He came back and dominated, launching him up the ranks even farther in the process. Current status: injured. Again.

Sonny Gray (newly ranked) – If there were any doubts about the Gray the Yankees dealt for, he’s actually surpassed his dScore from his fantastic 2015 season. He’s legit (again).

Alex Wood (-8) – Looks like the shoulder issues took a bit of a toll on his stuff, but dScore certainly isn’t out on him.

Dallas Keuchel (-8) – Keuchel’s stuff isn’t the issue. He’s still a buy for me.

Lance McCullers (-11) – Poor Astros. Maybe not too poor though; their aces have gotten hammered but haven’t fallen far at all. McCullers is going to bounce back.

The Studs

Some light flip-flopping at the top, with Kluber taking over at #1 from Scherzer. The Klubot’s been SO unconscious. Everyone else is pretty much the usual suspects.

The Young Breakouts (re-visited)

Zack Godley (11) – He’s keeping on keeping on. He barely moved since last month’s update, and I’m all-in on him being a stud going forward.

Luis Castillo (9) – He’s certainly done nothing to minimize the hype. In fact, he’s added a purely disgusting sinker to his arsenal and it’s raising the value of everything he throws. Also, from a quick glance at the Pitchf/x leaderboards, two things stand out to me. He seems to have two pitches that line up pretty closely to two top-end pitches: his four-seamer has a near clone in Luis Severino’s, and his changeup is incredibly similar to Danny Salazar’s. That’s a nasty combo.

James Paxton (15) 

The Test Case

Buck Farmer (20) – Okay, so to be honest when he showed up on this list, I absolutely thought it was a total whiff. By ERA he’s been a waste, but he’s really living on truly elite in-zone contact management, swinging strikes, K/BB, and hard-hit minimization. His pitch profile is middling (not bad, but not great either), so I really don’t think he’s going to stay this high much longer. He’s certainly doing enough to earn this spot right now, and I’d expect him to not run a 6+ ERA for much longer.

The Loaded Teams

Yankees – Luis Severino (12), Sonny Gray (17), Masahiro Tanaka (25) / Some teams have guys higher up, but the Yankees are loaded up and down.

Astros – Dallas Keuchel (14), Lance McCullers (19), Brad Peacock (18), Charlie Morton (23) / Similar to the Yankees. Morton and Peacock are having simply phenomenal years.

The Dropouts

Rich Hill (39)

Trevor Cahill (35)

Marcus Stroman (28)

Poor Rich Hill. Lost his perfect game, then lost the game, then lost his spot in the top 25. Cahill’s regressed to #DumpsterFireTrevor since his trade to the Royals. Stroman really didn’t fall that far…and his slider is still a work of art.

The Just Missed

Jordan Montgomery (26) – Too bad the Yankees couldn’t send down Sabathia instead. This kid is good.

Aaron Nola (27) – #Ace

Carlos Martinez (29) – Martinez simply teases ace upside, but frankly I think you can pretty much lump him and Chris Archer (30) in the same group — high strikeouts, too many baserunners and sub-ace starts to move into the top tier.

Dinelson Lamet (32) – He’s absolutely got the stuff. He could stand to work on his batted-ball control though.

Jimmy Nelson (34) – dScore buys his changes. He finished at #148 last year. I’ll call him a #2/3 going forward.

Notes from Farther Down

Jose Berrios is all the way down to 47. His last month cost him 19 spots, but frankly it could be much worse: Sean Manaea lost 39 spots, down to 87. Manaea really looks lost out there. I don’t want to point at the shoulder injury he had earlier this year since his performance really didn’t drop off after that…but I’m wondering if he’s suffering from some fatigue that’s not helped by that. He’s pretty much stopped throwing his toxic backfoot slider to righties, and that’s cost him his strikeouts. Michael Wacha is another Gray-like Phoenix: he’s up to 52 on the list, once again outperforming his 2015 year. I’m cautiously buying him as a #3 with upside. And finally, buzz round: Mike Clevinger (33), Alex Meyer (36), Robbie Ray (38), Rafael Montero (41), and Jacob Faria (43) are already ranked quite highly, and outside of Montero and maybe Meyer I could see all of them bumping up even higher. Clevinger’s really only consistency away from being a legitimate stud.

My next update will be the end-of-season update, so I think I’m going to do a larger ranking than just the top 25; maybe all the way down to 100. Enjoy the last month-plus!

Baseball Prospectus, in their 2015 scouting report of Maikel Franco, had this to say:

“Extremely aggressive approach; will guess, leading to misses or weak contact against soft stuff; gets out in front of ball often—creates hole with breaking stuff away; despite excellent hand-eye and bat speed, hit tool may end up playing down due to approach…”

We saw early this year, and even last year, that exact prediction come to life. Franco seemed to be flailing about vs the soft stuff, beating too many pitches into the ground, and even popping too many up. He never really stopped hitting the ball hard, but we saw too many of those hit in non-ideal ways. For most of the first part of this year the slider gave him absolute fits, and Alex Stumpf wrote about that here. He’s striking out at a career-low rate (13% on the year), but he still isn’t really walking that much although it’s bounced up a percentage point from last year (7.3% in 2017).

Here’s a rundown of his career batted-ball profiles:

I was watching the Phillies game vs. the Marlins on the 18th, and Franco went 3-4 with the go-ahead HR off Dustin McGowan. His HR came on a slider middle-away — literally the exact pitch that’s done nothing but given him fits all year. I also noticed that his batting stance seemed to be different. More upright, quieter. I pulled up a highlight video of an at-bat from early May. Here’s a screencap of his stance just before the pitcher starts his delivery:

That AB ended in an RBI line drive to right. Here’s a screencap of the HR in question from Tuesday, at a similar point in the pitcher’s delivery:

Now if that’s not a mechanical change, I don’t know what is. He’s closed off his stance, eliminated a lot of the knee bend, and seems to have raised his hands juuuuuust a touch. It could be the difference in the camera angle though. Phillies hitting coach Matt Stairs mentioned they’d been trying to get Franco to cut down on his leg kick, so let’s look at that too:
Old leg kick:

New:

Shortly after contact, old:

and the recent HR, similar point:

The “leg kick” seems to be more of a toe tap, and hasn’t changed. What did change, though, is the quality of his follow-through. His head is on ball, he’s better transferred his weight to his front foot, and the results follow. The old AB was a line-drive single opposite field, which looks less of an intentional opposite-field hit and more of a product of bad mechanics. Being so open, he really could only go to right field with authority. If he tried to pull it he’d roll over the pitch. That also would cause him to struggle with the breaking pitch away, which he’d bounce to second. Closing off has allowed him to better get the bat head into a more ideal position to cover the whole plate with authority. He’s always had the bat control to make contact everywhere, but it looks now like he’s improved his chances of making quality contact all over the zone. Here’s the same look at his batted-ball profile since the start of July:

Here’s some assorted metrics, same time period:

vs. his career metrics:

He’s cut his grounders by over 10%, raised his liners by 3%, and turned the rest into fly balls (8%). He’s likely always going to have a pop-up issue, but his pull/center/oppo profile is back to where he was at in 15/16, and he’s hitting the ball hard at a higher rate than ever. Also, his strikeout rate is 6%(!!!!!!)!!!!! He’s making more contact than ever, and that contact is better than ever.

We’ve seen Franco get us hyped before, but never before has there been this type of major mechanical change to point to. Miguel Sano did something similar preseason by raising his hands and quieting his pre-swing load, and it’s paid dividends. Since I started this article, Franco went 2-4 with a single, double, and sac fly; and three of those batted-ball events were hit at 100+mph (the single and double; he was robbed by the 3B on a sharp liner as well).

Going back to his 2015 scouting report: Franco’s still aggressive, if not slowly becoming less aggressive the more he’s in the majors. By changing up his stance, however, he’s closed up the two major holes in his report: getting out in front of the breakers away, and bad contact on soft stuff. Keep an eye on this. One of the more frustrating hyped prospects seems to have made the transformation we all hoped he would, right in front of our eyes.

If you take a look at the leaderboards on FanGraphs for all triple-A pitchers this year, you’ll find a surprising pitcher in the lead in FIP who is above two Rays pitchers, MVP of the Futures Game Brett Honeywell and Yonny Chirinos, along with surprising pitcher Buck Farmer. It’s Andrew Albers, with a 2.58 FIP in triple-A in 77.1 IP, 20 appearances, and 11 starts, with a less impressive 3.61 ERA, along with a sterling 2.77 xFIP.

What’s driving this 2.58 FIP? A strikeout rate of 9.54 per 9, with a measly 1.40 walks per 9 and .58 homers per 9, which is shockingly low, even for him. The home runs will likely increase as he isn’t getting too many ground balls; 46.2% is all right, but not elite. He is also getting a ton of infield pop-ups, with a shockingly high 21.9%. He has had very high infield pop-up numbers in the minors before, which make it easier to do as well as he had, although some negative regression should be expected.

Why his ERA is too high: He generally runs a high BABIP as it has usually been above .330 in the minors since 2015. This year his BABIP is a ridiculous .372 which is inflating his numbers above where can can truly perform at. It should regress to normal levels, maybe even a .320 BABIP perhaps, since minor-league defenses are worse than big-league defenses are (even the A’s pitiful defense).

His strikeout and walk rates are exceeding previous levels; last year in triple-A his walk rate was a good but not great 2.17 per 9, while his strikeout rate was a disappointing 6.08 per 9. I think he’ll likely negatively regress in his K/9 to around 7.5 per 9, walks to around 1.9/9.

But, there’s a chance that Albers could just be a second coming of Jamie Moyer, which could be useful for a big-league team looking for a cheap player to be their fifth starter, since he wouldn’t cost much on a minimum MLB contract or in prospects, and for all intents and purposes is a poor man’s Jason Vargas, who has been surprisingly good this year and is a Comeback Player of the Year candidate. It seems like Albers has made a serious adjustment in performance. Quite an interesting buy-low opportunity for a playoff hopeful that is tight on prospects (Angels, Royals), or tight on cash (Brewers, Rays, Twins, Royals). The Braves should have an extra selling chip that they didn’t know about before. Granted, they might get a lottery-ticket prospect for him, but the Braves are rebuilding, so they need prospects to try out at the big-league level eventually since a lot will flame out. Another pitcher who is similar to Albers is Wade LeBlanc, who I feel should be a starting pitcher for the Pirates, especially considering their rotation issues. But it seems like the thought of him starting is scarier to them than being in a saw trap.

It’s an idea that teams like the ones above should use to get underrated players cheap, while teams that have players like that should sell them for more value than they invested in the player. His best comp is of a right-handed pitcher who is with the Blue Jays: Marco Estrada. They have similar velocities, similar lack of performance till they got older, and get lots of pop-ups. Essentially, he is a left-handed version of Marco Estrada, and Marco Estrada received $26 million over two years after the 2015 season — quite an interesting thought. Especially considering his unimpressive stats in the majors so far. Let’s see if anyone will be willing to give him a chance as a swingman, as he could be an amazing fit on the Nationals; way better than Jacob Turner, and he could start in place of Joe Ross if he performs the way he has so far.

All stats from FanGraphs as of 7-13-2017. I do not own any stats or pages used to help me write this article.