Archive for Research

Every franchise is looking for that player who seems to come out of nowhere to be a major contributor in their lineup. Players like José Bautista, who went from 1.8 WAR in 2009 to 6.5 WAR in 2010, or Justin Turner, who jumped from 0.5 WAR in 2013 to 3.4 WAR in 2014. The cost for acquiring these players was affordable because they were no longer prospects and most of the league had written them off as potential everyday players.

If a team had the ability to identify which players are most likely to exceed industry expectations, they would have a significant advantage over their competition. That is why I decided to create a model that tries to identify potential breakout performers.

Methodology

The first thing I needed to do was to define what constitutes a breakout performance. I thought of several different definitions, but I decided to define a breakout performance as any player that exceeded their career high WAR in a single season by at least 2.0 WAR. So if a player had recorded a season of 0.0 WAR, they would need to have at least a 2.0 WAR season. If a player had recorded a season of 1.0 WAR, they would need to have at least a 3.0 WAR season and so on and so forth. Read the rest of this entry »

There are few things I enjoy in baseball more than the pitcher vs. hitter dynamic. Everyone likes to see highlight plays like a great catch or a mammoth home run, but those plays are few and far between. I believe that the tension created in a drawn-out plate appearance is where baseball is most enjoyable. Every pitch is meaningful, and the strategy of the game is on full display. The pitcher is trying to decide the best way to get the hitter to produce an out and the hitter is doing everything he can to thwart the pitcher.

This dynamic of baseball has always fascinated me. I was curious how pitchers and catchers decided which pitch was correct to throw in a situation. There are plenty of tools available to them that were not readily available when I was a child, like heat maps made from pitch-tracking data, but they show results without the context of what previous pitches were thrown in the plate appearance. Heat maps provide useful data, but the real art of pitching is being able to set up a hitter to take advantage of their weaknesses. If a pitcher throws the same pitch in the same location every time, eventually the hitter is going to catch on and change his strategy accordingly. So which sequence of pitches is the most effective at retiring hitters? This is the question I attempted to answer with this article. Read the rest of this entry »

Rebuilding has become the popular way for MLB franchises to construct a World Series contender. Considering the league’s structure of compensating the worst teams with the best draft picks, it seems like a viable strategy to maximize your losses in order to obtain the services of the best amateur talent available. The Astros and Cubs are two of the more recent franchises to successfully cap their extensive rebuilding process with a World Series victory, and both franchises acquired top-10 draft picks for several years before they turned the corner and became champions, but how often does this strategy work and how long does a rebuild take?

If an organization’s strategy is to not win games right away, when do the fans and ownership realize that the rebuilding process has failed and that their team is in the middle of a downward spiral of ineptitude? I am sure there are fans of the Pittsburgh Pirates and Kansas City Royals from the 1990s and 2000s that know how difficult it is to build a contender and cringe whenever they hear the term rebuild. Hopefully this article can provide a reasonable timeline for contention and an objective overview on how a franchise’s rebuilding effort should be progressing.

For my dataset, I gathered the GM or President of Baseball Operations for each organization since 1998. I chose 1998 because it was the first year the league consisted of 30 teams and it also happened to be the first full season for the current longest-tenured executives, Billy Beane and Brian Cashman. If an executive’s tenure with the team started before the 1998 season, their entire tenure was included in the dataset. This means Braves GM John Schuerholz’s regime is measured in its entirety from 1991-2007 and not just from 1998-2007. Read the rest of this entry »

Serious prospect hounds know that only about 10% of minor leaguers ever participate in a major league game. However, even the most discerning fans can be deluded into believing that their team’s farm system can overcome the odds and build a perennial contender based on their prospects alone.

I decided to investigate how much average WAR a prospect generates based on their ranking in Baseball America’s Prospect Handbook. I used a similar process in a previous article in which I calculated the amount of WAR based on the next six seasons of a player’s career since being listed (instead of when a player makes their major league debut). This means that players closer to the majors get a boost to their value, since they will have more opportunities to accumulate WAR than players in the lower minors.

Next, I grouped the players by their ordinal ranking in their organization from the 2001-2015 seasons and calculated each group’s average WAR to create the visualization below. Read the rest of this entry »

Ever since the MLB draft was created in 1965, teams have been searching for any competitive edge to separate themselves from the rest of the league. After all, it is one of the best ways to acquire young affordable talent for your organization. Not picking the best players available is a huge missed opportunity for any club and can set the organization back for years. It can also exasperate even the most devoted fans. It is imperative to have successful drafts every year, but what constitutes a successful draft? How many major leaguers are available in a draft and where can you find these players? These are some of the questions I hope to answer.

Methodology

Much of my analysis in this article will include references to team-controlled WAR. I calculated each draftee’s WAR total by summing their pitching and hitting WAR totals for the first seven years of their career to estimate the amount of value they provided their clubs before the players were eligible for free agency. This method is not perfect, because it does not consider demotions to the minor leagues, and it incorrectly assumes that every team would keep their prospects down in the minors to gain an extra year of control. However, I believe that the first seven years of WAR in a player’s career is a valid estimation of the value a player provides his organization before he exhausts his team-controlled seasons.

The drafts being examined are the drafts that took place from 1965 to 2004. I chose to stop at 2004 because that was the last year that had every player in its draft class exhaust his team-controlled seasons. If I were to include more recent drafts that still have active players, I could draw erroneous conclusions, since these players still have time to make their major league debuts and accumulate more WAR in their team-controlled seasons. Read the rest of this entry »

It’s a cool and breezy April afternoon down by Baltimore’s Inner Harbor, and the mid-rebuild Orioles are taking on the division-winning and record-breaking Minnesota Twins. Trying to salvage the final contest of a three-game series, the O’s — to no one’s surprise — find themselves trailing in the bottom of the ninth. But not all hope is lost. The Twins’ lead is small — two runs — and the Orioles have some of their best players due up. Out of the gate, Twins pitcher Taylor Rogers hits the first Orioles batter, Joey Rickard, in the foot. Then, after a Chris Davis lineout, Jesús Sucre resurrects the inning with a single to left that advances Rickard to third. The comeback is on.

Hanser Alberto then plunges the Orioles hopes back down to earth with a swinging strikeout that gives his team just one more out with which to work. But then comes Jonathan Villar, who rips a double to deep left, scoring Rickard and advancing Sucre to third. The Twins lead is cut in half. After an intentional walk to Trey Mancini that loads the bases, the game now rests in Pedro Severino’s hands. With two outs and the bases loaded, still down by one, Severino manages to work the count to 3-0. His team is one pitch away. The crowd is on its feet. Rogers winds and delivers his pitch. It’s outside! “Ball 4!” the commentator exclaims. The fans cheer, Severino begins to walk towards first, and the tying run starts his trot towards home. But suddenly, the umpire punches his arm through the air. He called it a strike. Severino walks back towards home plate, distraught. He pops up the very next pitch, and just like that, the game is over.

***

Using data from Baseball Savant’s pitch-by-pitch library, we can begin to understand the role that these incorrect calls play in baseball. By matching up the database’s pitch locations to the calls associated with those pitches, we can see which calls were supposedly correct, and more importantly, which were not. The results are pretty astounding. Last year, by this data, MLB umpires made a total of 33,277 incorrect calls. That’s good for 13.8 per game, or just over 1.5 per inning. While not every bad call is a comeback-killer, these mistakes have the ability to greatly alter an at-bat, a game, and maybe even a season. Read the rest of this entry »

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.

In 2015, the Boston Red Sox had one of the worst bullpens in Major League Baseball. Red Sox relievers were worth -1.3 WAR with  a FIP of 4.64, finishing 30^th in the league in both measures. They allowed opposing hitters to hit .261 with a BABIP of .300. Unsurprisingly last offseason, newly-installed president of baseball operations Dave Dombrowski set out to remake Boston’s bullpen. Throughout his long and storied career as a general manager, Dombrowski has consistently turned lagging franchises into contenders. His one weakness, as Dave Cameron pointed out last year, has been constructing bullpens. After examining Dombrowski’s tenure with the Detroit Tigers, Cameron wrote, “There was not a single aspect to pitching that the Tigers bullpen excelled at during Dombrowski’s tenure.” In the 2015 offseason, Dombrowski made two significant trades to bolster the back end of the Red Sox pitching staff. He shipped four prospects to the San Diego Padres for closer Craig Kimbrel and sent left-handed starter Wade Miley to the Seattle Mariners in exchange for reliever Carson Smith. Both of these moves reveal that despite his years of experience, Dombrowski still has difficulty properly valuing relievers.

THE KIMBREL TRADE

From 2011-2015, Craig Kimbrel led all relievers with 12.6 WAR. He struck out 40.9% of opposing hitters, allowing a .159 batting average with a 1.73 FIP. Only Aroldis Chapman struck out more hitters over the same time period. Kimbrel’s league-leading 224 saves were 58 more than the closest reliever, Huston Street. The difference between Kimbrel and Street is roughly equivalent to the difference between Street and Addison Reed, who had the 15^th-most saves from 2011-2015.

A closer examination of Kimbrel’s peripheral stats, however, reveals that he’s been slipping from his career peak in 2011 and 2012. In 2015, Kimbrel’s FIP rose to 2.68. Opposing hitters hit more home runs against him and their batting average against his four-seam fastball rose from .180 from 2011-2014 to .212 in 2015. In 2016, this decline has continued. Kimbrel’s walk rate has ballooned to 12.2%. His ground-ball and fly-ball rates have reversed themselves and he’s allowing much more hard contact. Just take a look at the chart below.

Opposing hitters are now hitting more of Kimbrel’s pitches as fly balls, they’re grounding out less often, and they’re making more hard and less soft contact than ever before. These factors have turned Kimbrel from an otherworldly reliever to merely an effective one. Looking at his yearly WAR figures, we can see that this transformation has been underway for a while now.

In 2015, Kimbrel ranked 19^th in reliever WAR, right between Justin Wilson of the Yankees and Keone Kela of the Rangers. That’s hardly inspiring, especially since Kimbrel earned $9 million in 2015 while Wilson and Kela made the league minimum.

Considering the price in prospects the Red Sox paid to acquire Kimbrel, they need him to perform at an elite level. In November 2015, Boston sent 3B Carlos Asauje, SS Javier Guerra, OF Manuel Margot, and LHP Logan Allen to the Padres for Kimbrel. Asuaje profiles as a utility infielder. According to Ben Badler of Baseball America, Logan Allen, whom the Red Sox drafted in the 8^th round, had the talent of a 2^nd or 3^rd round pick. Margot and Guerra were both among the top 100 or even top 50 prospects in the minors depending on which prospect list you prefer. Using the prospect valuation system developed by Kevin Creagh and Steve DiMiceli (you can read about their methodology here), I’ve estimated the cost to the Red Sox in terms of the surplus value of Margot and Guerra. Due to the varying nature of prospect valuations I’ve included the players’ rankings in Keith Law’s Top 100 prospects and Baseball America’s Top 100 as of February 2016.

Even if Kimbrel were the pitcher of 2011-2012 that would still be an astronomically high price to pay for a reliever who throws 60-70 innings per year. Now that Kimbrel is a 2-WAR reliever, it’s even worse.

THE SMITH TRADE

After acquiring Kimbrel, Dombrowski wasn’t finished remaking the Red Sox bullpen. On December 7, 2015 he traded left-handed starter Wade Miley and right-handed reliever Jonathan Aro to the Seattle Mariners for right-handed reliever Carson Smith and left-handed pitcher Roenis Elias. Aro is currently pitching at Triple-A Tacoma and Elias has a grand total of three appearances for the Red Sox this season, so the crux of the trade is Smith for Miley.

Based on their salaries and performances in 2015, Smith and Miley were both valuable pitchers and trade assets. Relying heavily on his slider, Smith held opposing hitters to a .194/.278/.262 batting line. He struck out 32.4% of opposing hitters with a 2.12 FIP and finished fifth among relievers with a 2.1 WAR. Additionally, Smith comes with five more years of team control. He isn’t arbitration-eligible until 2018 and won’t become a free agent until 2021. In 2015, Miley was a 2.6-WAR pitcher, best among any qualified starter on the Red Sox. From 2012-2015, Miley threw an average of 198 innings per season. Prior to the 2015 season, he signed a team-friendly three-year, $19.5-million contract from 2015-2017 with a $12-million club option in 2018.

After signing David Price to a seven-year contract in December 2015, the Red Sox believed they had an excess of starting pitching. With Price, Rick Porcello, Miley, Clay Buchholz, Joe Kelly, and Eduardo Rodriguez, they had six starters for five rotation spots. Additionally they had prospects Henry Owens, Brian Johnson, and knuckleballer Steven Wright waiting in the wings. In order to bolster the bullpen, Dombrowski decided to trade Miley, recognizing that he was the most valuable trade chip among the remaining starters. Porcello had just underperformed in 2015 and was entering the first year of a four-year, $82.5-million extension. Joe Kelly, while having an electrifying arm, had not really shown himself to be an effective starter. While Buchholz had pitched well in 2015, he managed only 18 starts. And Eduardo Rodriguez, the 23-year-old left-hander and potential top-of-the-rotation starter, was untouchable. This left Miley as the most logical trade chip.

By trading Miley, a serviceable innings eater, the Red Sox left themselves open to injuries and ineffectiveness. While Steven Wright effectively stepped into the rotation after Rodriguez dislocated his kneecap in spring training, Buchholz and Kelly were disasters. In 22.1 innings as a starter, Kelly allowed opposing hitters to hit .316/.437/.564 for a wOBA of .419 or the equivalent of Mike Trout this season. He sported a walk rate of 16% and a 5.88 FIP. In his 88 IP as a starter, Buchholz allowed opposing hitters to hit .268/.347/.470, good for a .349 wOBA and a 5.68 FIP. Since 2010, Buchholz has never been healthy and effective at the same time. For all of the talk about Kelly improving last season, a look at his peripheral numbers revealed a pitcher that was merely getting lucky with stranding runners as opposed to improving his underlying performance. By trading away Miley, the Red Sox cost themselves a cushion for the failures of Buchholz and Kelly. In order to fill the rotation void, Dombrowski traded highly-regarded pitching prospect Anderson Espinoza (the 19^th-best prospect in baseball according to Baseball America) to San Diego for Drew Pomeranz. Carson Smith, meanwhile, underwent Tommy John surgery in May after straining a flexor muscle in spring training.

In trading for Craig Kimbrel and Carson Smith, Dave Dombrowski has revealed that his biggest weakness remains properly valuing bullpen talent. For a baseball executive with a generally sterling record, this may seem like a minor flaw, but it’s one that caused him to overpay for a declining closer, to trade Miley while relying on a pair of risky starters, and then to swap a prospect who garners comparisons to Pedro Martinez to fill the resulting void in the rotation. With Smith’s injury and the failings of Buchholz and Kelly, Dombrowski has little to show for all his bullpen efforts other than generously restocking the Padres’ farm system.

Take a look at Chris Davis’s stat line in August: 11 games, 45 PA, 14 Ks, 7 BBs, 6 HRs. Nothing really jumps out; it’s pretty typical for Chris Davis. Looking deeper though, this selection of plate appearances is actually quite remarkable. 27 out of the 45, or 60% of them, ended with a strikeout, walk, or home run, known as the “three true outcomes” where the ball does not end up in play.

As Baseball Prospectus explains in its definition of TTO, the statistic actually gained relevance with the introduction of DIPS, FIP, and other pitching estimators that ignored the outcomes of balls in play. While still not commonly used, it’s certainly interesting to take a look at once in a while to see what players are taking luck into their own hands.

Chris Davis is actually not the most extreme three true outcome player. Despite his 60 TTO% August, his season-long percentage through August 13 stands at 48.9%, good for 5th in baseball of those who have at least 300 plate appearances. The rest of the top-10 leaderboard features both good names and bad. On the good side, we have Giancarlo Stanton, the only player to feature a HR% over 8% (his is 8.5% , and he actually leads second-place Nelson Cruz by 1.4%). Other names you might associate with quality players are Bryce Harper, Joc Pederson, and George Springer, all of whom have a K% under 30% and a HR% of over 4%. The players who might not be as happy to be on this list include the aforementioned Chris Davis, Chris Carter, Steven Souza, Kris Bryant, and Colby Rasmus, who all feature a K% of 31% or higher. Mike Zunino, who comes in at 10th, sports a walk rate and home run rate of just 5.6% and 2.8%, respectively, but more than makes up for it with a 34.2% strikeout rate, second only to Souza.

Now that we’re done with the fun facts, let’s get into what it really means. TTO players are swing-for-the-fence players, those who aim to hit the ball over the wall every time they make contact. This is the cause behind their multitude of strikeouts. It also accounts for their walks, with the reasoning that pitchers are simply afraid to throw them hittable pitches.

The real question becomes “Are these TTO players valuable?” Looking at a graph comparing TTO% to wRC+ over the past 15 years, there is little correlation. It seems as though it is slightly more productive to be a TTO player, mainly because of the home runs and walks. This is far from a correlation though, as many bad players have a high TTO% and vice versa.

If we split it up into its parts, we might get a better view. League average TTO% has risen over the last decade, from 27.3% in 2005 to 30.3% this year (with a high of 30.5% in 2012).

We know the overall percentage has risen, but what’s driving it? If you’ve been following baseball, you know that the quality of pitchers has improved in recent years. Predictably, this has led to a decrease in walk rate and home run rate.

If 2/3 of the TTO% has decreased, but TTO% has still increased, that must mean the change in the third category must be drastic. This happens to be exactly the case. While BB% and HR% have fallen approximately a combined 1% over the past 10 years, league wide K% has risen by 4%.

What this means is that nowadays, if you are a TTO player, it’s likely much of that is coming from your strikeouts. In fact, out of the top-25 TTO% players with at least 200 PAs, only Paul Goldschmidt has a K% under 20%. Does this make high TTO% players bad? As I said before, there really isn’t a correlation, You’ll see players like Bryce Harper and Mike Trout with a high TTO%, while Buster Posey has one of the lowest because of his low K%.

The reality is, there are many different kinds of players. Some have adopted this TTO mentality, but others have stayed with a more conservative contact-focused approach. Without further information, it’s difficult to say which strategy is better. As a fan of statistics, I prefer the TTO players because it’s much easier to predict their performance. I don’t think they care much about that though.

Also, if you were curious, here’s a list of the top TTO% players with 200 PAs, created using FanGraphs data through August 13.