Graphic displays of MLB pitching mechanics and itsevolutions in PITCHf/x data.

Fushing Hsieh∗, Kevin Fujii∗, Tania Roy∗, Cho-Jui Hsieh∗†, Brenda McCowan‡

Abstract

Systemic and idiosyncratic patterns in pitching mechanics of 24 top starting pitch-ers in Major League Baseball (MLB) are extracted and discovered from PITCHf/xdatabase. These evolving patterns across different pitchers or seasons are representedthrough three exclusively developed graphic displays. Understanding on such patternedevolutions will be beneficial for pitchers’ wellbeing in signaling potential injury, and willbe critical for expert knowledge in comparing pitchers. Based on data-driven comput-ing, a universal composition of patterns is identified on all pitchers’ mutual conditionalentropy matrices. The first graphic display reveals that this universality accommo-dates physical laws as well as systemic characteristics of pitching mechanics. Suchvisible characters point to large scale factors for differentiating between distinct clus-ters of pitchers, and simultaneously lead to detailed factors for comparing individualpitchers. The second graphic display shows choices of features that are able to ex-press a pitcher’s season-by-season pitching contents via a series of 3(+2)D point-cloudgeometries. The third graphic display exhibits exquisitely a pitcher’s idiosyncraticpattern-information of pitching across seasons by demonstrating all his pitch-subtypeevolutions. These heatmap-based graphic displays are platforms for visualizing andunderstanding pitching mechanics.

1 Introduction

Both 2017 Cy Young Awards in Major League Baseball (MLB) have coincidentally goneto two previous awardees: Corey Kluber (winner in 2014) of the Cleveland Indians forAmerican League, and Max Scherzer (winner in 2013 and 2016) of the Washington Nationalsfor National League. Winning multiple Cy Young awards is not rare in MLB history, but isnot prevalent at all among MLB pitchers. This 2017 coincidence is far from implying thatMLB pitchers all have steady pitching careers. In fact pitchers’ careers mostly go up-and-down from season to season. For instance, among the 24 starting pitchers, who were selectedfor being one of top-3 candidates for this award through 2012 to 2017 seasons, only ClaytonKershaw is a two-time winner (2013 and 2014). Indeed only three more pitchers: JustinVerlander, David Price and Adam Wainwright, have received top-3 candidacies more thanonce.

If being one of the top-3 candidates is taken as being at the peak of a pitcher’s career,then a natural question is: what characterizes pitching mechanics that only enables a fewpitchers to stay at their peaks, and what makes majority of elite pitchers’ peaks come and go?These characteristics are at best elusive in this sport. On one hand, aerodynamics of pitchingis no doubt primarily governed by Newton’s law of gravity and Magnus effect on spin-vs-speed interactions, see Briggs (1959) [1]. It is also critically influenced by biomechanicalparameters: like pitcher’s physical power and arm strength; and by many non-mechanicalparameters: like baseball’s skin surface, pitcher’s finger conditions and weather conditions.These laws, effects and parameters combine to crucially influence each every pitch’s 60 feet6 inches(18.39m) journey that lasts only 0.5 second or less to go from pitcher’s mound tohome plate. On top of such a convoluted complexity, a starting pitcher typically has threeto five pitch-types in his pitching repertoire, and each pitch-type contains several subtypes.So one explicit and detailed aerodynamic formula for what a pitcher’s pitching mechanicslooks like is unrealistic, if not unimaginable.

∗Department of Statistics, UC Davis†Department of Computer Science, UC Davis‡School of Veterinary Medicine, UC Davis

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On the other hand, the MLB is at its state of completely missing key information andunderstanding of pitching mechanics of its pitchers. This dire state is detrimental to this150-year-old sport. Being unable to explicitly reveal the complexity of pitching mechanics,which is the heart of the game, not only diminishes the basis to excite young people, but alsocauses many critical issues of pitcher’s pitching well-being can’t be addressed and dealt within time. Issues like: Are all pitch-types delivered by a pitcher’s in tune? Are there signs ofinjury? These natural and serious issues are ought to have proper answers after each game.But they are not.

One primary reason behind lacking rigorous scientific studies on MLB pitching to offerpertinent information and relevant understanding is no availability of proper data. Baseballdata has been summarized in the format of a box-score, which is at least as old as the sport.In a box-score out of a game, the amount of information related to pitching mechanics is nextto nothing. However, a revolutionary theme of pitching data collection was finally completedin MLB in 2008. Such a theme has never been seen before. Through 2006-2008, MLB andSportvision, a TV broadcast effects company based in Mountain View, California, installedcameras in all 30 MLB stadiums to track each pitch in every MLB game.

Approximately 20 images are acquired during a baseball’s flight from a pitcher’s handto home base, see Mike Fast (2010) [2]. These images are used to reconstruct 21 featureson various speeds, accelerations, curving and spinning characteristics and coordinates of itsrelease point. Such a database, named PITCHf/x, is now made available by Sportvisionto Major League Baseball Advanced Media (MLBAM) to merge together with traditionalmanual recording of Umpire’s calls: strike, ball; and batter’s results: hit, home-run, strike-out,etc. Sportvision more recently also created databases, HITf/x and FIELDf/x, to recordbatted baseballs and all players’ movements in the field.

Starting from 2015, the Sportvision’s system has been further upgraded by MLBAM’sStatcast tracking system based on Trackman’s Doppler radar system and ChyronHego’svideo system. By combining the radar and video technologies, data of all pitched and battedbaseballs and all players’ movements is recorded in databases. Such a PITCHf/x databaseis not a data set per se: not only because of its largeness in size, but because of its entireinformation contents being just too diverse and too vast to be covered and reported inone single paper. In fact the goal of computing upon this database is set to build variousplatforms, on which all people can get access to their own information, and to foster their ownunderstanding, and then to collectively produce fundamental and brand new insights into thesport. By now this database has been publicly available for almost 10 years. It is the righttime that we build such platforms. Hopefully people can explore and look into all pitcher’spitching mechanics, and see this sport from new perspectives. With the aforementioned goalin mind, we focus this paper on developing data-driven computational techniques to extractpertinent information of pitching mechanics based on PITCHf/x database. Then we buildgraphic displays as platforms for representing computed information in both systemic andidiosyncratic fashions.

From systemic perspective, a mutual conditional entropy matrix, which reveals the depen-dency among all 21 features based on combinatorial information theory, is computed acrossall seasons of all 24 starting pitchers. We discover a universal composition of patternedblocks upon all matrices, which is a structural manifestation of aerodynamics of baseballpitching. This universality also frames a graphic display that contains interacting relationscapable of characterizing large and small scales distinctions among clusters of pitchers.

For the idiosyncratic perspective, this universality also implies which features can exhibitevolving characteristics as a quick graphic display of a pitcher’s pitching mechanics throughvisible 3(+2)D geometries (size and color). For detailed evolutions within all pitch-types ina pitcher’s repertoire, a graphic display of categorical likelihood processes of all identifiedpitch-subtypes along their pitch-by-pitch temporal coordinates is constructed. Such a graphicdisplay offers complete information of evolutions of a pitcher’s pitching mechanics across allseasons. Hence it is a platform for seeking what characters are contributing to his careerpeaks, and what characters are causing up-and-down along a pitcher’s career path. It canalso reveal emergent patterns for potential injures or being out-of-tune. In similar fashion,we are able to compare several pitchers’ pitching mechanics in detail by having one of themas a baseline.

In this big-data era, our data-driven computing and graphic displays can transformPITCHf/x database into “modern scientific platforms” that are sophisticated enough at

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this computer age in order to attract the young minds, and at the same time are insightfulenough to educate the young brains as well as to challenge the old wisdoms.

2 Materials and Methods

2.1 Aerodynamics of baseball pitching and its characteristic fea-tures

The aerodynamics of a baseball pitch in reality is rather complex, since it involves severalphysical laws and many natural and bio-mechanical parameters. It is not only too complexto describe in full, but also too expensive to record and store in fine detail. Further astarting pitcher typically throws about 100 pitches per game. With four or five startingpitchers rotating throughout the 162 games in a season, a pitcher can produce above 3000pitches a year, not including play-off games. So, in order to store coherent information of apitcher’s pitch-by-pitch aerodynamics, a private company, Sportvision contracted by MLB,has chosen 21 features out of each pitch’s flight from the moment the baseball leaving thepitcher’s hand to the moment passing through home plate. Ideally these 21 features acrossapproximately 3000 pitches should contain each pitcher’s pitching mechanics.

It is well known that several key characteristics of pitching mechanics are governed bya physical law, called the Magnus effect. This effect primarily prescribes the complicatenonlinear interacting relations between the baseball’s traveling speed and spin under influ-ences derived from forces of gravity and air resistance. The starting speed (“Start speed”)is measured when the ball is at the point 50 fts away from the home plate, which is veryclose to the release point “(x0, y0, z0)” of a pitch. The spin direction (“spin dir”) and spinrate (“spin rate”) are attributed to complex factors, such as, how a baseball is held in apitcher’s hand; how it is released; how pitcher’s fingers touch and slide against the surfaceand seams of a baseball. Particularly the ways of holding and sliding against baseball seamsare responsible for various pitch-types contained in a pitcher’s repertoire. Indeed a MLBpitcher can be seen as an “expert” manipulator of speed and spin of a baseball.

This effect of a baseball pitch is briefly described as follows. With a high enough“start speed”, spin tends to stabilize a baseball’s trajectory linearly even against the airresistance. That is, spin makes the effects of baseball’s uneven surface insignificant at a highspeed. That is why a 100 mph fastball, typically having high spin rate, looks straight. Insharp contrast, a knuckleball, typically having zero spin, has a rather unpredictable trajec-tory when it arrives at the home plate. It often causes an experienced catcher to miss thecatch. When a baseball’s traveling speed is gradually reduced, the Magnus effect begins toshow on its trajectory. The backspin fastball will go against the gravity and move upwardwhen arriving at the home plate. In contrast, the topspin curveball will go downward anddrop more than the effect caused by gravity. So topspin and backspin cause positive andnegative vertical movements, which are measured by a feature at the home plate and denotedby “pfx z”. Thus this feature has a high association with “start speed” for pitchers, whohas the high speed fastball as his chief pitch-type in his repertoire, than for pitchers, whodoesn’t.

The feature “pfx z” is also associated with features related to how a baseball trajectorycurves. A baseball trajectory from release point to the home plate is coupled with two straightlines: the tangent line at the release point “(x0, y0, z0)” and the line links the release pointand the trajectory’s end point. The angle between these two lines is termed “break-angle”,while the maximum distance between the baseball trajectory and the second straight line iscalled and denoted as “break length”. Therefore the three features: “pfx z”, “break-angle”and “break length”, are highly associated with each other.

It is also intuitive that this “break angle” could be critically affected when the baseballtrajectory indeed swerves to the right or left sides of home plate. Such horizontal swervingis caused by side-spin, which is often induced in pitch-type: like slider (SL) or change-up(CH). So side-spin makes the baseball to swerve to the corresponding side and producesthe horizontal movement, which is denoted as “pfx x”. Therefore the three features: “pfx x”,“break-angle” and “spin dir”, are mechanistically associated. A skilled pitcher usually makesuse of certain “spin dir” to produce a large “pfx x” value to deal with a batter. That iswhy a right-handed pitcher can be more effectively to against a right-handed batter, butless effective to left-handed one. A right-handed pitcher likely throws pitches with large

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horizontal movements to the left. Such an effect explains why proportions of left-handedbatters and pitchers in MLB are as high as 40%, which is twice the proportion of left-handedpeople in general population.

In summary, the aerodynamics or pitching mechanics of a baseball pitch are basicallygoverned by six features: “start speed”, “spin-rate”, “spin-dir”, “pfx z”, “break angle” and“break length” in an intricate and connected fashion. The remaining 15 features are eitherhighly associated with these six features individually and collectively, such as “end speed”and three directions of speeds and accelerations at the lease point, named “vx0, vy0,vz0”and “ax, ay, az”, respectively, or play only auxiliary roles, like “break y” and “(x0, z0)”.

2.2 Graphic displays as Representations of knowledge in PITCHf/xdata

The object of data-driven computing here is information of pitching mechanics. The objectiveof such computing is the understanding of the information contained in PITCHf/x data. Tostimulate such understanding we employ graphic displays as representations of computedsystemic and idiosyncratic pattern information of MLB pitching mechanics. For systemicinformation, our graphic display is composed of a clustering composition of pitchers featuredwith computed patterns on: how some pitchers aggregate, some set apart, and why onepitcher appears on different branches over different years. For idiosyncratic information,our graphic display represents the evolving processes of all pitching-subtypes on: when itgoes extinct; what are remaining stable; how only a few are created or re-borne. Also foridiosyncratic information, the graphic display via serial 3(+2)D geometries are designed toreveal visible characters of pitching mechanics through key features across a series of seasons.

The guiding principle of employing graphic displays is to appeal to human’s formidablevisual and mental processing capabilities. In this computer era, our brain coupled withvisual sensory systems might be still the most efficient device for recognizing and organizingpatterns into understanding and knowledge, see also Grenander and Miller (1994) [3].

3 Computing Methods

3.1 Possibly-gapped Histogram for categorical renormalization

Among 21 features, there are drastic different measurement units involved. We explorepotential non-linear associations among these features based on combinatorial informationtheory. For this application, we transform each real valued feature into a categorical one asa way of renormalization. This categorization also prepare us for calculations of mutual con-ditional entropy. The real-to-categorical transformation is performed by applying Analysisof Histogram (ANOHT) algorithm, which builds a possibly-gapped histogram upon 1D realvalued data set. So one bin is one category.

Such a possibly-gapped histogram is originally designed to reveal pattern informationcontained in 1D data via uniformity within all bins of various sizes. And a gap is an extremeform of uniformity. The validity of such a histogram is visible through its correspondingpossibly-gapped piecewise-linear approximation to its empirical distribution, see details inFushing and Roy (2017) [4].

3.2 Mutual conditional Entropy based on combinatorial informa-tion theory

A possibly-gapped histogram transforms the 1D feature into an (ordinal) categorical vari-able upon the ensemble of pitch-IDs. Two features, for instance, say, break length andstart speed, will give rise to a bivariate categorical variable upon the same pitch-ID ensem-ble. This bivariate categorical variable can be also expressed via a contingency table, forinstance, with break length’s categories being arranged along the row axis and start speed’scategories on the column axis.

Within a given category of break length, equivalently being fixed on one row of thecontingency table, we evaluate Shannon entropy with respect to categories of the start speed.Further we calculate the ratio of this Shannon entropy with respect to the overall Shannonentropy of start speed. This ratio is the relative local conditional entropy of start speed given

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Break length pertaining to a specific category. It is taken as a directed local association fromstart speed to break length, that is, this directed local association is meant to evaluate thedegree of exclusiveness of categories of start speed within a specific category of break length.

The global start speed to break length association will be calculated by a weighted sumof all local ones with weighting being the proportions of break length’s category sizes. Fi-nally the bi-directed global association of break length to start speed based on their relativemutual condition entropy is the average of global conditional entropy from break length tostart speed and global conditional entropy from start speed to break length. This is howthe non-linearity is accommodated in a measure of association between start speed andbreak length. The most important merit of this association is that, as an effective sum-marizing statistics of the contingency table, it conveys the authentic dependency betweenstart speed and break length. A pictorial illustration of this mutual conditional entropy canbe found in an introductory paper Fushing et al. (2017) [5].

Likewise a 21X21 matrix of mutual conditional entropy is constructed, see examples inFigure 1. This matrix will be taken as a platform for collectively revealing the dependencystructures among these 21 features.

3.3 Synergistic feature-groups and backbone of dependency

The 21X21 mutual conditional entropy matrix is taken as a distance matrix among the21 features. Upon this distance matrix, Hierarchical Clustering (HC) algorithm is appliedto construct a HC tree on this ensemble of features. By superimposing the HC tree onthe matrix’s row and column axes, multi-scale block-patterns are framed and revealed, seeexamples in Figure 1.

Each block along the diagonal corresponds to a cluster of features by having uniformly lowmutual conditional entropies (high associations). Such a feature-cluster is called a synergisticfeature-group. Thus a series of diagonal blocks will signal a series of synergistic feature-groups that potentially serve as one chain of mechanistic dependency structure among these21 features.

3.4 Data Mechanics for systemic characteristics of pitching me-chanics

To explore systemic characters of all involving pitcher-seasons, we convert the upper trian-gular part of 21X21 mutual entropy matrix into a 210-dimensional vector. By stacking allpitcher-year’s 210-dim vectors together, a rectangle matrix with 210 columns is constructed.

Next, for computational simplicity and costs, we adapt hierarchical clustering algorithmon building a tree on row axis of pitcher/year and another tree on column axes. These twotrees are supposed to be coupled because of interacting relations between pitcher-seasonson row-axis and pairwise mutual conditional entropies on column-axis. We then constructthese two trees in an iterative fashion: 1) starting from building a HC tree on column axiswith Euclidean distance; 2) define a new distance between row vectors as the sum of 210dimensional Euclidean distance and another Euclidean distance of extra dimensions basedon a clustering composition of cluster-averages with respect to a selected tree-level of the HCon column axis, and the build a HC tree on row axis; 3) the distance between column vectorsis updated in the same manner with respect to the tree derived in Step 2. A distance beingupdated with respect to a tree structure is designed as a way of enhancing the universal block-patterns found in the mutual conditional entropy matrix. After iterating once or twice, theresultant HC tree on the row axis gives rise to multiple scales of similarity among pitcher-seasons, while the block patterns on the matrix’s rectangle lattice, (or heatmap), revealthe systemic characteristics for each cluster of pitcher-season, see examples in in Figure 3.Likewise computations can be performed on a sub-matrix of the 21X21 matrix.

This iterative HC algorithm mimics the newly developed computing paradigm called DataMechanics, see Fushing and Chen (2014) [6] on binary rectangular matrix and Fushing, etal. (2015) [7] on real-value matrix. In Data Mechanics, Ultrametric trees are derived basedon a data-driven algorithm called Data Cloud Geometry (DCG), see Fushing and McAssey(2010) [8] and Fushing, et al. (2013) [9]. DCG was designed to extract the authentic treeinformation contained in the data. It is contrasting with the man-made artificial binarysplitting on each HC tree branch. A DCG tree will require a higher computing cost than

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HC tree will do. However the key step of Data Mechanics is the iterative procedure. Hencethe iterative procedure based on HC algorithm is still termed Data Mechanics (DM).

[Data Mechanics for idiosyncratic characteristics: Pitch-subtype evolution.]Again Data Mechanics is applied to explore a pitcher’s idiosyncratic characteristics. Thisexploration into pitching mechanics is pitch-type specific. Consider a pitch-type specificrectangular matrix with 21 or a set of selected features being arranged on column axis andall pitches across a series of consecutive seasons on row axis. By applying DM, a 6 cluster-level of the clustering tree on the row axis is selected to define 6 pitch-subtypes, so eachpitch has a pitch-subtype-ID. At the same time, the clustering tree on column-axis revealthe visible characters that define each pitch-subtype in an explicit compositional fashion, seeexamples in Figure 6.

Among all involving seasons, a subset is selected to serve as the baseline seasons, in whichthe pitcher of interest is considered being healthy. Then the 6 subtype-specific proportionsof pitches belonging to the baseline seasons are calculated. Each proportion is termed alikelihood of a pitch pertaining to its specific pitch-subtype. Together they form a genericallycalled categorical pattern distribution.

Accordingly such a categorical pattern distribution depicts how a pitch relates to pitch-ing mechanics of the baseline seasons. A large likelihood value indicates that this pitchercontinues to pitch a high potential subtype of the baseline seasons. While a zero, or anextremely small likelihood value indicates a brand new subtype being created outside of thepitching repertoire employed in the baseline seasons. Therefore, when a pitch is displayedwith its pitch-subtype-ID and likelihood value along with its coordinate on the temporalaxis, a graphic display of evolution of pitch-subtype is constructed for a specific pitch-type.

4 Results

4.1 Universality

For all 24 pitchers, their 108 pitcher-seasonal mutual conditional entropy matrices universallyshow a composition of two serial blocks on two distinct scales along the diagonal: the largescale one is a single 11x11 sub-matrix; the small scale ones are: one 4x4, two 3x3 and one 1x1submatrices contained within the 11x11block, and two 3X3 and one 2x2 outside the largeblock, see Figure 1. (All 24 pitchers’ evolving mutual conditional entropy matrices across aseries of seasons are reported in the DM website https://www.dm-mlbpitching.com.)

Each of small scale block reveals how one small set of synergistic features works as one me-chanical component within the pitching mechanics. For instance, the four mechanical com-ponents: {start speed, end speed, vy0}, {pfx z, az, break length}, {break angle, pfx x, ax,spin dir} and {spin rate}, are evident parts of pitching aerodynamics. The mechanism viasynergistic feature group:{pfx z, az, break length} depicts how and how much the baseballtrajectory would curve in the vertical direction. The synergistic feature group:{break angle,pfx x, ax, spin dir} depicts how and how much a baseball trajectory would move in thehorizontal directions: left or right. These two mechanical components are closely associatingwith Magnus effect. In contrast, the 11x11 block depicts how the four mechanical compo-nents collectively work together to form the aerodynamics commonly shared by all pitchingmechanics of the 24 pitchers, while the remaining 10 features outside the 11x11 block arerelatively low in overall association because they are more related to pitchers’ idiosyncraticpitching gestures than pitching mechanics per se.

This universality confirms the point-cloud geometry of 21 dimensional features of allpitches from each pitcher within one single season indeed contains all physical laws underlyingpitching mechanics. Two important merits are implied by such a universality. First, essentialfeatures can be selected out of the fine scale blocks to facilitate 3(+2)D geometric display ofa pitcher’s seasonal pitching mechanics and its evolution across a series of seasons, as seenin Figure 2. The three chosen coordinates are: {start speed, pfx z, break angle} plus twoextra-dimensions of color-coded pitch-type on each focus season and {spin-rate} for sizesof balls. This choice of coordinates provides a visualization of separating pitch-types intocloud-like clusters.

Further, as visualizing color-coded pitches (in ball shape) of current season against pre-vious seasons along the serial geometries, if balls of one color are only seen sparsely withina cloud, then this pitcher’s corresponding pitch-type in this focal season is coherent with

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Figure 1: Mutual conditional entropy matrices: (A) Clayton Kershaw at 2013 season; (B)Clayton Kershaw 2014; (C) Justin Verlander 2016; (D) Kyle Hendricks 2016.

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his in the previous seasons. Such phenomena are consistently seen in Kershaw’s pitchingmechanics from 2012 through 2017. On the other hand, if balls of one color are evidentlyseen aggregating outside the gray-colored cloud, then we should suspect that this pitcherlikely being out of tune often in the focal season, if not injured. (See DM website for all 24pitchers’ pitching mechanics evolution through a series of seasonal 3(+2)D animation.)

Secondly, the off-diagonal entries beyond the four small blocks within the 11X11 subma-trix collectively characterize how synergistic-group-based mechanical components interact.Such quantified interactions importantly provide information about how individual pitchers’pitching mechanics would differentiate from others. In fact more detailed individual differ-ences in pitching mechanics and biomechanics are summarized in the off-diagonal entries ofthe whole 21X21 mutual conditional entropy matrix. Two versions of systemic comparisonsbased on “pitcher-season” are reported in first two panels of Figure 3. The two heatmapshave 55 (=11x10/2) and 210 columns, respectively.

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Figure 2: Clayton Kershaw’s progressive data cloud geometries of pitch-types from season2012-2017. (A) 2012; (B) 2013 with 2012 as baseline; (C) 2014 with 2012-13 as baseline; (D)2015 with 2012-14 as baseline; (E) 2016 with 2012-15 as baseline; (F) 2017 with 2012-16 asbaseline. Color-coding: FF in Red; FT in Orange: SL in Green; CU in Blue.

Each heatmap provides a visible platform for systemic comparisons among “pitcher-seasons” and for understanding how and why they are similar or distinct. Specifically themarked-blocks, which are framed by the clustering tree of pitchers on the row axis, andanother clustering tree of feature-pairs on the column axis, reveal information not onlyabout who is close to and far away from whom, but also, more importantly, about whether apitcher is close to himself across a series of season. Understanding of information of similarityand distinction can be easy seen through the coloring of the block.

For instance, upon the heatmap in panel (A) of Figure 3 pertaining to the 11 features,

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Figure 3: Two versions of heatmap-based systemic patterns: (A) All feature-pairs among21 features; (B) feature-pairs among 11 selected features; (C) The tree locations of ClaytonKershaw and R. A. Dickey.

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Kershaw-12 through Kershaw-17 exclusively occupy the “d-row” branch. This fact meansthat Kershaw’s pitching mechanics is rather steady throughout the 2012-17 seasons from themechanical aspect pertaining to the 11 features. Also his pitching mechanics is not extremely,but rather different from many pitchers’. In contrast, Scherzer-12 through Scherzer-17 arefound in “a-row” for 2012-14 seasons, 2016 in “b-row” and 2015 and 2017 in “c-row”, likewisefor Kluber’s 2012 through 2017 ,and Verlander’s 2011 through 2017 seasons.

However, by adding biomechanical aspects, that is, pertaining to the 21 features, theheatmap in panel (B) and its clustering tree in panel (C) of Figure 3 can provide differentversion of comparison among pitcher-seasons. The 6 seasons of Kershaw are divided into twothree-season branches: 2012-14 vs 2015-17 (marked by two upward-pointing arrows). Thedetailed information attributed to this separation would be seen in the later in this section.

It is also informative to note that a branch consisting of Dickey-12 and -13 stands outas an outlier branch away from the rest of pitcher-season, including his own Dickey-11. Thedistinct separation can be attributed to the fact that R. A. Dickey had been developing hisknuckleball through 2011 (marked by a small star). His knuckleball became very effectivein 2012 season (together with 2013 marked by a star), in which he received the Cy Youngaward. Since the 2012 season, he has been recognized as one of the best knuckleball pitchersin American baseball history. There must be many patterns and related understandingwaiting to be discovered from Figure 3 alone by devoted experts.

4.2 Comparing pitchers’ with specific feature-pairs

The heatmaps in Figure 3 also guide us to see: how to choose which set of features forcomparing a chosen set of pitchers. For instance, three pitchers: Hendricks, Kershaw andVerlander, are to be compared upon four features, {break length, Spin rate, pfx z, az}, whichconstitute three feature-pairs located in the cluster No.1 of feature-pairs in panel (A).

Specifically three significant different block values are visible: “very red” for Hendrick-16at “a-row”, “very blue” for Kershaw-16 at “d-row” and “intermedium” for Verlander-16 at“b-row”. The detailed comparison is performed by applying the Data Mechanics on pitch-type specific data matrices having 4 columns for the four features and all pitches of thethree pitchers in 2016 season on row-axis. The 5 resultant heatmaps from Data Mechanicscomputations for four Kershaw’s pitch-types and one pooled pitch-type beyond his are shownin Figure 4.

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OT6

OT5

OT4

OT3

OT2

OT1

1 2(A) (B) (C) (D) (E)

Figure 4: Heatmaps of pitch-type specific comparisons among Clayton Kershaw (baseline),Justin Verlander and Kyle Hendricks Hendricks on 2014-2016 seasons with respect to fourfeatures: Break-length, pfx z, az and spin-rate. (A) FF; (B) SL; (C) CH; (D) CU; (E) OT:all pitch-types out of Kershaw’s repertoire.

The panel (A) of Figure 4 shows the heatmap of fastball (FF), including 2-seam fastballs(FT), of these three pitchers, which are color-coded. Hendricks’ fastball pitches (Yellowcolored) only show up in subtype cluster FF4 together with Kershaw’s (Red colored) and

10

a rather small number of Verlander’s (Orange colored). Thus, it is evident that Hendricks’fastball pitches are drastically different from Kershaw’s and Verlander’s with respect to thefour features.

From the heatmap, we also see that this FF4 cluster has rather smaller values in features{pfx z, az, spin-rate}. Further, since the three features: {pfx z, az, break length}, arehighly associative as forming a 3x3 diagonal block within the universality, we understandthat Hendricks’ fastball pitching mechanics has much smaller vertical acceleration (az) andvertical movement (pfx z) and spin-rate. From outlook of a pitch, his fastball is not as sharpas Kershaw’s and Verlander’s.

On the same heatmap, we see that Verlander’s and Kershaw’s fastball pitching mechan-ics are distinct only on the {break length} feature. The three clusters: FF1, FF3 and FF5,are nearly completely dominated by Verlander’s, while clusters: FF2 and FF6 nearly com-pletely dominated by Kershaw’s. This slight difference in value of break length reveals thatVerlander’s fastball curves more than Kershaw’s fastball.

As for slider (SL) in panel (B) of Figure 4, we see that all clusters are orange and redcolored, that is, Kershaw and Verlander have large overlaps on all 6 subtypes of slider. Thisfact means that their pitching mechanics for slider are rather comparable. Slider is not inHendricks’ pitching repertoire.

In panel (C) for changeup (CH), two clusters: CH1 and CH2 are nearly completely dom-inated by Verlander, while 3 clusters: CH4, CH5 and CH6, are nearly completely dominatedby Hendricks. The smallest cluster: CH3, is shared by Verlander and Kershaw. This ratherdistinct subtype is in opposite characters of the rest of 5 subtypes.

The curveball(CU) in panel (D) of Figure 4 shows a very interesting comparison amongthese three pitchers. The four subtype-clusters: CU1-CU4, are color-coded with orange andyellow, while two subtype-clusters: CU5 and CU6, are entirely color-coded with red. These 6curveball subtypes are relatively similar with respect to the features and their correspondingmechanism. The only difference is that Kershaw’s curveballs curve more than Verlander’sand Hendricks’. The panel (E) of Figure 4 shows that Hendricks has certain pitch-types thatare not in Kershaw’s or Verlander’s repertoires.

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0 2000 4000 6000 8000

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1.0

kershaw−verlander−hendricks 2014−2016

Index

Like

lihoo

d

Subtypes never thrown by Kershaw: CH2,CH5,CH6,CU3,FF3

kershaw verlander hendricks

CH1

CH3

CH4CU1

CU2CU4

CU5

CU6

FF1

FF2

FF4

FF5

FF6

SL1

SL2

SL3

SL4

SL5

SL6

Subtypes never thrown by Kershaw: CH2,CH5,CH6,CU3,FF3

Figure 5: Pitch-subtypes based evolution for comparing three pitchers: Clayton Kershaw(baseline), Justin Verlander and Kyle Hendricks on 2014-2016 seasons with respect to fourfeatures: Break-length, pfx z, az and spin-rate.

The aforementioned similarity and distinctions among these three pitchers’ pitching me-chanics from the aspect of the four features: {break length, Spin rate, pfx z, az}, are sum-marized in Figure 5. By taking Kershaw as the baseline for deriving his 5 categorical patterndistributions on the five pitch-types, each pitch then has acquired a pitch-subtype ID, such

11

as FF1, and a likelihood value according to a categorical pattern distribution. Also everypitch from a pitcher has a temporal coordinate. Therefore, when all pitches of a pitchercan be laid out with respect to its ID, likelihood value and temporal coordinate, as sucha graphic display is constructed, as shown in Figure 5. This graphic display provides the4-feature-view of pitching mechanics comparison from the perspective of Kershaw. Similarviewing perspectives from Verlander and Hendricks can be likewise constructed.

4.3 Individual pitcher’s pitching evolution

For one individual MLB pitcher, it is critical to be able to see how his pitching mechanicsevolves across a series of consecutive seasons, particularly including his career peaks. Herewe construct and show Kershaw’s pitching mechanics from 2012 through 2017 seasons. Thereare 5 pitch-types: FF, FT, SL, CH, CU, in his pitching repertoire. His fastball is subdividedinto: 4-seam fastball (FF) and 2-seam fastball (FT).

Data Mechanics computations are applied onto the five ensembles of pitches collectedfrom the 6 seasons. Each pitch’s season-ID is color-coded: red, orange, yellow, green, blue,purple, from 2012 to 2017 in increasing order. Each pitch is characterized by 13 features: 11features contained in the universality and {x0, z0} the horizontal and vertical coordinates ofits releasing point. The reason for including these two features is that they are closely relatedto pitching gesture at large. Kershaw is known for his rather distinct gesture particularlyfor his curveball pitching.

The five pitch-type specific heatmaps are reported in Figure 6. On the panel (A), all 4-seam fastballs (FF) are classified into 6 subtypes. The idiosyncratic characteristics of these 6subtypes are displayed on the heatmap in individual and associative manners. Across these6 subtypes, features: {break length, spin-dir, vy0, z0 } are kept constant at the lower end,features: {Start speed, spin rate, pfx z, az, end speed}, are also kept constant at the higherend. The idiosyncratic characteristics of the 6 pitch-subtypes are captured by the opposite-contrasting interactions between features: {pfx x, ax, x0} and feature: {break angle}.

When the release point of a fastball pitch has a large horizontal coordinate value {x0}companied with a large value of acceleration (ax), the fastball trajectory will have a largevalue of horizontal movement {pfx x} and a distinctively small value of {break angle}, asseen in subtypes FF5 and FF6. In a reverse fashion, small values of {x0} and {ax} willresult into small {pfx x}, but large {break angle}, as seen in subtypes FF3 and FF4. Andmedian values of {x0} and {ax} lead to median values of {pfx x} and {break angle}, as seenin subtypes FF1 and FF2.

Further the uneven proportions of color-coding across the 6 subtypes also bring outthe information of how Kershaw distributes 4-seam fastball subtypes across the 6 seasons.One important observation is that the aforementioned 3 interacting characteristics: FF1&2,FF3&4 and FF5&6 have been constantly and alternatively realized by Kershaw’s fastballs.Such an observation might explain why his 4-seam fastball is still one of his most effectivepitch-type in his pitching repertoire.

The 2-seam fastball (FT) was only added to Kershaw’s repertoire in season 2015. Thenumber of pitches in FT is much smaller than that in FF. However, its evolving patterns areevident and interesting to see, as shown in panel (B). The interacting relational patterns inpanel (B) somehow evolves and extends between features: {pfx x, ax, x0} and {break angle,z0}. The larger values of {pfx x, ax, x0} couple with smaller values of {break angle, z0}, asseen in three subtypes FT1-3 and FT6, and the reverse pattern is seen in subtypes FT4&5.

In summary, Kershaw created his 2-seam fastball by first reducing the vertical coordinate{z0} on the releasing point of his 4-seam fastball gesture. Such pitch subtypes: FT4&5, aredominate in 2015-16 seasons. This modification on {z0} has gone further down in 2017season, as seen in three subtypes FT1&2&3.

A slider in general is threw with its pitching gesture just like that of a fastball, but witha slightly reduced speed. Slider is one of three major pitch-types in Kershaw’s repertoire.Upon the panel (C), Kershaw’s pitching mechanics for slider over the 6 seasons has retainedthe interacting patterns seen in pitching mechanics of fastball in panels (A) and (B), but ina slight scale: SL1 vs SL4 on seasons 2012-2015; SL2 vs SL5 on season 2016; and SL5 vsSL6 on season 2017.

Beyond the interacting pattern, like the shadow of fastball, we also see a gradual, butpersistent pattern: values of {Start speed, pfx z, az, end speed} become slightly larger invalues, while values of {break angle, break length, spin-dir, vy0, z0} become slightly smaller

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CU6

CU5

CU4

CU3

CU2

CU1

1 2 3 4

hi

(A) (B)

(C)

(E)

(D)

Figure 6: Heatmaps of pitch-type specific comparisons between Clayton Kershaw 2012-2014seasons(baseline) with 2015-2017 seasons. (A) FF; (B) FT (C) SL; (D) CH; (E) CU.

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0 5000 10000 15000

0.0

0.1

0.2

0.3

0.4

0.5

kershaw 2012−2017

Index

Like

lihoo

d

2012 2013 2014 2015 2016 2017

CH1

CH2

CH3

CH4

CH5

CH6

CU1

CU2

CU3

CU4

CU5

CU6

FF1

FF2

FF3

FF4

FF5

FF6

SL1

SL2

SL3

SL4

SL5SL6

Subtypes never thrown in baseline (2012−2014): FT1,FT2,FT3,FT4,FT5,FT6

Figure 7: Pitch-subtypes based evolution for Clayton Kershaw 2012-2014 seasons (baseline)through 2015-2017 seasons.

in values over the 6 seasons. The features: {pfx x, ax, spin rate} are kept constant in thelower end, except the in the subtype SL6, which dominated by 2017 season. The evidentidiosyncratic characteristics of these 6 subtypes are characterized by values of feature: {x0}.This character also contains a persistent decreasing pattern over the 6 seasons.

Like slider, ideally a changeup is also thrown in a pitching gesture of fastball, but has amuch reduced speed coupled with very diverse spin-direction. This diversity is carried outvia many ways of holding a baseball when pitches a changeup. Different ways of holdinginduce different spin directions, and render different vertical and horizontal movements. Soa changeup is disguised as a fastball in gesture, but has a drastically distinct and diversemovements. This is why it is an effective pitch-type for many MLB pitchers, but it is not amajor pitch-type in Kershaw’s repertoire.

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0 5000 10000 15000

0.0

0.1

0.2

0.3

0.4

0.5

verlander 2011−2017

Index

Like

lihoo

d

2011 2012 2013 2014 2015 2016 2017

CH1

CH2

CH3

CH4

CH5

CH6

CU2

CU3

CU4

CU5

CU6FF1

FF2

FF3

FF4

FF5

FF6

FT1

FT2

FT3

FT4

FT5

SL2

SL3

SL5

SL6

Subtypes never thrown in baseline (2011−2014): CU1,FC1,FC2,FC3,FC4,FC5,FC6,FT6,SL1,SL4

Figure 8: Pitch-subtypes based evolution for Justin Verlander 2011-2014 seasons (baseline)through 2015-2017 seasons.

In panel (D) of Figure 6, the signature interacting pattern of Kershaw’s fastball is seenonly slightly in the heatmap for changeup (CH). It seems that Kershaw is on his way of

14

phasing out this pitch-type altogether in the future season.In terms of pitching gesture, Kershaw’s curveball(CU) is thrown in a rather artificial

way to create top-spinning. It is not natural, so the speed is very low. Since its Magnuseffect of going downward is further aided by gravity, its trajectory usually has a drastic, ifnot sudden, drop when a baseball is entering the home plate. That is why it is an effectivepitch-type if a pitcher can introduce uncertainty into such a pitch, like Kershaw.

Curveball(CU) is one of Kershaw’s most effective pitch-types in dealing with batters.The panel (E) for curveball (CU) clearly indicates that he throws two distinct subtypes ofcurveball with two opposite spin-direction: CU1&2 with extremely low values of {spin-dir},and CU3&4&5&6 with extremely high values of {spin-dir}. That is, in general a batter hasa hard time to guess which sides: right or left, his curveball will swerve to. This uncertaintymakes his curveball well-known, even the height of release point of a pitch, i.e. feature {z0},is kept detectably higher than that of all other pitch-types of his. Further his curveballseemingly is narrowing into two subtypes: CU2 and CU5, in recent reasons. This might bean alarm sign of losing his effectiveness on this pitch-type.

Information contained in these five panels of Figure 6 is synthesized into a so-calledlikelihood plot, as shown in figure Figure 7, with 2012-2014 being used as baseline. It isrecalled that each pitch has a pitch-subtype ID and a likelihood value, which are derivedfrom a pitch-type specific heatmap and its categorical pattern distribution in one of the fivepanels in Figure 6. The likelihood plot displays each pitch’s subtype-ID and likelihood valewith respect to its temporal coordinate upon the entire axis of 6 seasons. Such a graphicdisplay collectively reveals nearly all pattern information of Kershaw’s pitching mechanics,which evolves from 2012 through 2017. Many pitching subtypes go extinct, while manysubtypes are created in later seasons. Also some subtypes even go into extinct, and thencome back on again.

Two likelihood plots of Verlander and Hendricks are given in Figure 8 and Figure 9,respectively. The likelihood plot with 2011-2014 being taken as the baseline seasons, showsthat Verlander has 5 pitch-types: {FF, FT, SL, CH, CU}. In fact he has 6 pitch-typesin the PITCHf/x database, including the pitch-type: Cuter (FC), which was created onlyrecently way after 2014 season. He also creates one new curveball subtype, one 2-seamfastball subtype and two slider subtypes. In the Figure 3, Verlander-2013 is located in the“a-row” branch, while the rest of 7 pitcher-seasons are in “b-row” branch. This separationcan be somehow seen in this likelihood plot as well.

The likelihood plot of Hendricks’ pitching mechanics on three seasons: 2015-2017 is shownin Figure 9. The 2015-2016 seasons constitute the baseline and there are five pitch-types:FF, FC, SI(sinker), CH, CU. This likelihood plot shows a visible evident change from 2016to 2017 season. Many subtypes are nearly extinct, while many sparsely used subtypes in thebaseline seasons are heavily used in the 2017 season. This evident change indeed coincidedwith his injury at early part of 2017 season.

5 Conclusion

Though MLB pitcher’s pitching mechanics are complex and somehow mysterious, our data-driven computing and graphic displays are able to lift the veil to certain extents. Informationof several aspects of pitching mechanics is genuinely extracted and then explicitly representedfor systemic comparison among all 24 pitchers, and for individual pitcher’s idiosyncraticevolution across multiple seasons. Understanding of computed information via a graphicdisplay as a platform seems to be able to flow smoothly. Nonetheless that is just the beginningof a new era into this 150 year old sport.

The systemic comparison based on a heatmap, derived from mutual conditional entropymatrices of all pitcher-seasons, not only enables us to see who are close to whom, and faraway from whom among MLB pitchers, but also pinpoints which features can be accountedfor their differences in pitching mechanics. Equally importantly, the idiosyncratic evolutionof pitching subtypes in a pitcher’s repertoire shows the history of how this pitcher maintains,creates or gives up pitching subtypes across seasons, along which his career peaks, slides ordeclines.

In this big-data era, it is critical to develop computing platforms in order to properlyaddress the fundamental question: what and where is information contained in data? In thispaper, the discovery of the universal block patterns, which embed with all involving physical

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0 2000 4000 6000 8000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hendricks 2015−2017

Index

Like

lihoo

d

2015 2016 2017

CH1

CH2

CH3CH4

CH5

CH6

CU1

CU2

CU3CU4

CU5

CU6FC1

FC2

FC3FC4

FC5

FC6

FF1FF2

FF3

FF4

SI1SI2

SI3

SI4

SI5

SI6

Subtypes never thrown in baseline (2015−2016): FF5,FF6

Figure 9: Pitch-subtypes based evolution for Kyle Hendricks 2015-2016 seasons (baseline)through 2017 season.

laws on all mutual conditional entropy matrices, is an answer to this question. It also is avalidity check of our data-driven computations.

However we have to admit that results and understanding presented here fulfill only asmall part of our primary objective that our computational endeavors intend to achieve uponPITCHf/x database. Our primary objective is to use our data-driven computing and graphicdisplays as platforms for all people to explore and discover information by themselves, andbuilding and constructing understanding and knowledge from the database for themselves.This whole scale objective can’t be achieved in a classic format of a published paper becauseof its rather limited space. Therefore it becomes necessary to create a website to facilitate allpotential explorations and discoveries possibly accomplished by all curious minds. This is oneway to accommodate all possible relevant information about pitching mechanics containedin this Big-Data era.

People, who has a career centering around or in baseball pitching, are able to find thelandscape of pitchers and their differences in pitching mechanical factors for managementpurposes as well as for self-evaluations linked to pitching wellbeing. People, who are inter-ested in baseball as sport fans, are able to find deeper insights into favorite or adversarialpitchers’ characteristics in pitching mechanics. People, who are interested in data-drivencomputing or data science in general, are to find inspirations from resolutions of variousissues on high dimensional point-cloud geometries, and, at the same time, to discover a newdirection of research into biomechanics of pitching.

Particularly young people, who are guided by their own curiosity in baseball, are ableto wonder around the new data-driven physical and aerodynamic laws embedded withinpitching mechanics, and to appreciate their tangled complexity. The immediate sense ofachievement to them is being able to go far beyond the more than 150-year-old Boxscores.Such an educational merit could be way bigger than the total sum of aforementioned ones.Young people likely rediscover baseball games with brand new insights, and then perceivethis sport from totally different perspectives that their parents’, grand-parents’ generationsnever have imagined.

Ethics:

No ethical approvals are needed in this study. All measurements pertaining to study subjectsare available from two public websites.

16

Data Accessibility:

The pitching data is available in PITCHf/x database belonging to Major League Baseballvia http://gd2.mlb.com/components/game/mlb/.

Competing Interests:

We have no competing interests.

Author’s contributions:

H.F. designed the study. K. F. collected all data for analysis. H.F., K. F.and T.R analyzedthe data. H.F, K. F., T.R. C-J. H., and B. C. interpreted the results. H.F wrote themanuscript. H.F, K. F., T.R. C-J. H., and B. C. edited the manuscript. All authors gavefinal approval for publication.

Funding:

No financial funding received for this study.

Acknowledgement:

We thank MLBAM for making the PITCHf/x data available.

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[2] Fast M. What the heck is pitchf/x? The Hardball Times Baseball Annual., 2010.

[3] Ulf Grenander and Michael I. Miller. Representations of knowledge in complex systems.Journal of the Royal Statistical Society. Series B (Methodological), 56(4):549–603, 1994.

[4] Hsieh Fushing. and Roy T. Complexity of Possibly-gapped Histogram and Analysis ofHistogram (ANOHT). ArXiv e-prints, February 2017.

[5] Hsieh Fushing, Liu S.-Y., Hsieh Y.-C., and McCowan B. From patterned response depen-dency to structured covariate dependency: categorical-pattern-matching. ArXiv e-prints,May 2017.

[6] Hsieh Fushing and Chen C. Data mechanics and coupling geometry on binary bipartitenetworks. PLOS ONE, 9(8):1–11, 08 2014.

[7] Hsieh Fushing, Hsueh C-H., Heitkamp C., Matthews M. A., and Koehl P. Unravellingthe geometry of data matrices: effects of water stress regimes on winemaking. Journalof The Royal Society Interface, 12(111), 2015.

[8] Hsieh Fushing and M. P. McAssey. Time, temperature, and data cloud geometry. Phys.Rev. E, 82:061110, Dec 2010.

[9] Hsieh Fushing, Wang H., VanderWaal K., McCowan B., and Koehl P. Multi-scale clus-tering by building a robust and self correcting ultrametric topology on data points. PLOSONE, 8(2):1–10, 02 2013.

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