Graphic displays of MLB pitching mechanics and its evolutions 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/x database. These evolving patterns across different pitchers or seasons are represented through three exclusively developed graphic displays. Understanding on such patterned evolutions will be beneficial for pitchers’ wellbeing in signaling potential injury, and will be critical for expert knowledge in comparing pitchers. Based on data-driven comput- ing, a universal composition of patterns is identified on all pitchers’ mutual conditional entropy matrices. The first graphic display reveals that this universality accommo- dates physical laws as well as systemic characteristics of pitching mechanics. Such visible characters point to large scale factors for differentiating between distinct clus- ters of pitchers, and simultaneously lead to detailed factors for comparing individual pitchers. 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-cloud geometries. The third graphic display exhibits exquisitely a pitcher’s idiosyncratic pattern-information of pitching across seasons by demonstrating all his pitch-subtype evolutions. These heatmap-based graphic displays are platforms for visualizing and understanding pitching mechanics.

1 Introduction

Both 2017 Cy Young Awards in Major League Baseball (MLB) have coincidentally gone to two previous awardees: Corey Kluber (winner in 2014) of the Cleveland Indians for American League, and Max Scherzer (winner in 2013 and 2016) of the Washington Nationals for National League. Winning multiple Cy Young awards is not rare in MLB history, but is not prevalent at all among MLB pitchers. This 2017 coincidence is far from implying that MLB 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 selected for being one of top-3 candidates for this award through 2012 to 2017 seasons, only Clayton Kershaw is a two-time winner (2013 and 2014). Indeed only three more pitchers: Justin Verlander, David Price and Adam Wainwright, have received top-3 candidacies more than once. 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 few pitchers 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 pitching is 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 biomechanical arXiv:1801.09126v1 [stat.AP] 27 Jan 2018 parameters: like pitcher’s physical power and arm strength; and by many non-mechanical parameters: 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 feet 6 inches(18.39m) journey that lasts only 0.5 second or less to go from pitcher’s mound to home plate. On top of such a convoluted complexity, a starting pitcher typically has three to 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 mechanics looks like is unrealistic, if not unimaginable.

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

1 On the other hand, the MLB is at its state of completely missing key information and understanding of pitching mechanics of its pitchers. This dire state is detrimental to this 150-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 also causes many critical issues of pitcher’s pitching well-being can’t be addressed and dealt with in time. Issues like: Are all pitch-types delivered by a pitcher’s in tune? Are there signs of injury? 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 offer pertinent information and relevant understanding is no availability of proper data. Baseball data 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 next to nothing. However, a revolutionary theme of pitching data collection was finally completed in MLB in 2008. Such a theme has never been seen before. Through 2006-2008, MLB and Sportvision, a TV broadcast effects company based in Mountain View, California, installed cameras 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 hand to home base, see Mike Fast (2010) [2]. These images are used to reconstruct 21 features on various speeds, accelerations, curving and spinning characteristics and coordinates of its release point. Such a database, named PITCHf/x, is now made available by Sportvision to Major League Baseball Advanced Media (MLBAM) to merge together with traditional manual 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 record batted baseballs and all players’ movements in the field. Starting from 2015, the Sportvision’s system has been further upgraded by MLBAM’s Statcast tracking system based on Trackman’s Doppler radar system and ChyronHego’s video system. By combining the radar and video technologies, data of all pitched and batted baseballs and all players’ movements is recorded in databases. Such a PITCHf/x database is not a data set per se: not only because of its largeness in size, but because of its entire information contents being just too diverse and too vast to be covered and reported in one single paper. In fact the goal of computing upon this database is set to build various platforms, on which all people can get access to their own information, and to foster their own understanding, and then to collectively produce fundamental and brand new insights into the sport. By now this database has been publicly available for almost 10 years. It is the right time that we build such platforms. Hopefully people can explore and look into all pitcher’s pitching mechanics, and see this sport from new perspectives. With the aforementioned goal in mind, we focus this paper on developing data-driven computational techniques to extract pertinent information of pitching mechanics based on PITCHf/x database. Then we build graphic displays as platforms for representing computed information in both systemic and idiosyncratic 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 across all seasons of all 24 starting pitchers. We discover a universal composition of patterned blocks upon all matrices, which is a structural manifestation of aerodynamics of baseball pitching. This universality also frames a graphic display that contains interacting relations capable of characterizing large and small scales distinctions among clusters of pitchers. For the idiosyncratic perspective, this universality also implies which features can exhibit evolving characteristics as a quick graphic display of a pitcher’s pitching mechanics through visible 3(+2)D geometries (size and color). For detailed evolutions within all pitch-types in a pitcher’s repertoire, a graphic display of categorical likelihood processes of all identified pitch-subtypes along their pitch-by-pitch temporal coordinates is constructed. Such a graphic display offers complete information of evolutions of a pitcher’s pitching mechanics across all seasons. Hence it is a platform for seeking what characters are contributing to his career peaks, and what characters are causing up-and-down along a pitcher’s career path. It can also 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 them as a baseline. In this big-data era, our data-driven computing and graphic displays can transform PITCHf/x database into “modern scientific platforms” that are sophisticated enough at

2 this computer age in order to attract the young minds, and at the same time are insightful enough 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 several physical laws and many natural and bio-mechanical parameters. It is not only too complex to describe in full, but also too expensive to record and store in fine detail. Further a starting pitcher typically throws about 100 pitches per game. With four or five starting pitchers rotating throughout the 162 games in a season, a pitcher can produce above 3000 pitches a year, not including play-off games. So, in order to store coherent information of a pitcher’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 the pitcher’s hand to the moment passing through home plate. Ideally these 21 features across approximately 3000 pitches should contain each pitcher’s pitching mechanics. It is well known that several key characteristics of pitching mechanics are governed by a physical law, called the Magnus effect. This effect primarily prescribes the complicate nonlinear 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 very close to the release point “(x0, y0, z0)” of a pitch. The spin direction (“spin dir”) and spin rate (“spin rate”) are attributed to complex factors, such as, how a baseball is held in a pitcher’s hand; how it is released; how pitcher’s fingers touch and slide against the surface and seams of a baseball. Particularly the ways of holding and sliding against baseball seams are responsible for various pitch-types contained in a pitcher’s repertoire. Indeed a MLB pitcher 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 air resistance. That is, spin makes the effects of baseball’s uneven surface insignificant at a high speed. That is why a 100 mph fastball, typically having high spin rate, looks straight. In sharp 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 the catch. When a baseball’s traveling speed is gradually reduced, the Magnus effect begins to show on its trajectory. The backspin fastball will go against the gravity and move upward when arriving at the home plate. In contrast, the topspin curveball will go downward and drop more than the effect caused by gravity. So topspin and backspin cause positive and negative vertical movements, which are measured by a feature at the home plate and denoted by “pfx z”. Thus this feature has a high association with “start speed” for pitchers, who has the high speed fastball as his chief pitch-type in his repertoire, than for pitchers, who doesn’t. The feature “pfx z” is also associated with features related to how a baseball trajectory curves. A baseball trajectory from release point to the home plate is coupled with two straight lines: the tangent line at the release point “(x0, y0, z0)” and the line links the release point and 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 is called 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 baseball trajectory indeed swerves to the right or left sides of home plate. Such horizontal swerving is 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 produces the 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 makes use of certain “spin dir” to produce a large “pfx x” value to deal with a batter. That is why a right-handed pitcher can be more effectively to against a right-handed batter, but less effective to left-handed one. A right-handed pitcher likely throws pitches with large

3 horizontal movements to the left. Such an effect explains why proportions of left-handed batters and pitchers in MLB are as high as 40%, which is twice the proportion of left-handed people in general population. In summary, the aerodynamics or pitching mechanics of a baseball pitch are basically governed 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 either highly 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/x data The object of data-driven computing here is information of pitching mechanics. The objective of such computing is the understanding of the information contained in PITCHf/x data. To stimulate such understanding we employ graphic displays as representations of computed systemic and idiosyncratic pattern information of MLB pitching mechanics. For systemic information, our graphic display is composed of a clustering composition of pitchers featured with computed patterns on: how some pitchers aggregate, some set apart, and why one pitcher appears on different branches over different years. For idiosyncratic information, our graphic display represents the evolving processes of all pitching-subtypes on: when it goes extinct; what are remaining stable; how only a few are created or re-borne. Also for idiosyncratic information, the graphic display via serial 3(+2)D geometries are designed to reveal 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 formidable visual and mental processing capabilities. In this computer era, our brain coupled with visual sensory systems might be still the most efficient device for recognizing and organizing patterns 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 explore potential non-linear associations among these features based on combinatorial information theory. For this application, we transform each real valued feature into a categorical one as a way of renormalization. This categorization also prepare us for calculations of mutual con- ditional entropy. The real-to-categorical transformation is performed by applying Analysis of Histogram (ANOHT) algorithm, which builds a possibly-gapped histogram upon 1D real valued data set. So one bin is one category. Such a possibly-gapped histogram is originally designed to reveal pattern information contained in 1D data via uniformity within all bins of various sizes. And a gap is an extreme form of uniformity. The validity of such a histogram is visible through its corresponding possibly-gapped piecewise-linear approximation to its empirical distribution, see details in Fushing 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 and start 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, for instance, with break length’s categories being arranged along the row axis and start speed’s categories on the column axis. Within a given category of break length, equivalently being fixed on one row of the contingency 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 Shannon entropy of start speed. This ratio is the relative local conditional entropy of start speed given

4 Break length pertaining to a specific category. It is taken as a directed local association from start speed to break length, that is, this directed local association is meant to evaluate the degree 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 sum of 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 relative mutual condition entropy is the average of global conditional entropy from break length to start speed and global conditional entropy from start speed to break length. This is how the non-linearity is accommodated in a measure of association between start speed and break 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 between start speed and break length. A pictorial illustration of this mutual conditional entropy can be found in an introductory paper Fushing et al. (2017) [5]. Likewise a 21X21 matrix of mutual conditional entropy is constructed, see examples in Figure 1. This matrix will be taken as a platform for collectively revealing the dependency structures 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 the 21 features. Upon this distance matrix, Hierarchical Clustering (HC) algorithm is applied to construct a HC tree on this ensemble of features. By superimposing the HC tree on the matrix’s row and column axes, multi-scale block-patterns are framed and revealed, see examples in Figure 1. Each block along the diagonal corresponds to a cluster of features by having uniformly low mutual conditional entropies (high associations). Such a feature-cluster is called a synergistic feature-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 these 21 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 all pitcher-year’s 210-dim vectors together, a rectangle matrix with 210 columns is constructed. Next, for computational simplicity and costs, we adapt hierarchical clustering algorithm on building a tree on row axis of pitcher/year and another tree on column axes. These two trees are supposed to be coupled because of interacting relations between pitcher-seasons on row-axis and pairwise mutual conditional entropies on column-axis. We then construct these two trees in an iterative fashion: 1) starting from building a HC tree on column axis with Euclidean distance; 2) define a new distance between row vectors as the sum of 210 dimensional Euclidean distance and another Euclidean distance of extra dimensions based on a clustering composition of cluster-averages with respect to a selected tree-level of the HC on column axis, and the build a HC tree on row axis; 3) the distance between column vectors is updated in the same manner with respect to the tree derived in Step 2. A distance being updated 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, the resultant 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), reveal the 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 Data Mechanics, see Fushing and Chen (2014) [6] on binary rectangular matrix and Fushing, et al. (2015) [7] on real-value matrix. In Data Mechanics, Ultrametric trees are derived based on 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 tree information contained in the data. It is contrasting with the man-made artificial binary splitting on each HC tree branch. A DCG tree will require a higher computing cost than

5 HC tree will do. However the key step of Data Mechanics is the iterative procedure. Hence the 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. This exploration into pitching mechanics is pitch-type specific. Consider a pitch-type specific rectangular matrix with 21 or a set of selected features being arranged on column axis and all 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 each pitch has a pitch-subtype-ID. At the same time, the clustering tree on column-axis reveal the visible characters that define each pitch-subtype in an explicit compositional fashion, see examples in Figure 6. Among all involving seasons, a subset is selected to serve as the baseline seasons, in which the pitcher of interest is considered being healthy. Then the 6 subtype-specific proportions of pitches belonging to the baseline seasons are calculated. Each proportion is termed a likelihood of a pitch pertaining to its specific pitch-subtype. Together they form a generically called 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 pitcher continues to pitch a high potential subtype of the baseline seasons. While a zero, or an extremely small likelihood value indicates a brand new subtype being created outside of the pitching repertoire employed in the baseline seasons. Therefore, when a pitch is displayed with its pitch-subtype-ID and likelihood value along with its coordinate on the temporal axis, 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 universally show a composition of two serial blocks on two distinct scales along the diagonal: the large scale one is a single 11x11 sub-matrix; the small scale ones are: one 4x4, two 3x3 and one 1x1 submatrices contained within the 11x11block, and two 3X3 and one 2x2 outside the large block, see Figure 1. (All 24 pitchers’ evolving mutual conditional entropy matrices across a series 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 via synergistic feature group:{pfx z, az, break length} depicts how and how much the baseball trajectory 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 the horizontal directions: left or right. These two mechanical components are closely associating with 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 pitching mechanics of the 24 pitchers, while the remaining 10 features outside the 11x11 block are relatively low in overall association because they are more related to pitchers’ idiosyncratic pitching gestures than pitching mechanics per se. This universality confirms the point-cloud geometry of 21 dimensional features of all pitches from each pitcher within one single season indeed contains all physical laws underlying pitching mechanics. Two important merits are implied by such a universality. First, essential features can be selected out of the fine scale blocks to facilitate 3(+2)D geometric display of a pitcher’s seasonal pitching mechanics and its evolution across a series of seasons, as seen in Figure 2. The three chosen coordinates are: {start speed, pfx z, break angle} plus two extra-dimensions of color-coded pitch-type on each focus season and {spin-rate} for sizes of balls. This choice of coordinates provides a visualization of separating pitch-types into cloud-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 within a cloud, then this pitcher’s corresponding pitch-type in this focal season is coherent with

6 iue1 uulcniinletoymtie:()CatnKrhwa 03sao;(B) season; 2016. Hendricks 2013 Kyle at (D) Kershaw 2016; Clayton Verlander (A) Justin (C) matrices: 2014; entropy Kershaw conditional Clayton Mutual 1: Figure (A) (C)

Count 0 100 Count

and Histogram 0 100 0 and Histogram 0 Color Key Color Key 0.4 V 0.4 V alue alue 0.8 0.8

spin_rate break_angle break_angle spin_dir pfx_x ax ax pfx_x end_speed end_speed vy0 start_speed start_speed vy0 spin_dir spin_rate break_length break_length az az pfx_z pfx_z x x0 px vx0 y x pz px ay y break_y pz x0 ay z0 break_y vz0 vz0 vx0 z0 spin_r break_angle pfx_x ax end_speed vy0 star spin_dir break_length az pfx_z x px y pz a break_y x0 z0 vz0 vx0 y break_angle spin_dir ax pfx_x end_speed star vy0 spin_r break_length az pfx_z x0 vx0 x px y pz a break_y vz0 z0 y t_speed t_speed ate ate 7 (B) (D)

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end_speed spin_dir vy0 break_angle start_speed pfx_x break_length ax az end_speed pfx_z vy0 spin_dir start_speed break_angle spin_rate ax break_length pfx_x pfx_z vx0 az x vx0 px x vz0 px pz pz y y z0 ay x0 break_y spin_rate x0 ay vz0 break_y z0 end_speed vy0 star break_length az pfx_z spin_dir break_angle ax pfx_x vx0 x px vz0 pz y z0 x0 spin_r a break_y y spin_dir break_angle pfx_x ax end_speed vy0 star spin_r break_length pfx_z az vx0 x px pz y a break_y x0 vz0 z0 y t_speed t_speed ate ate his in the previous seasons. Such phenomena are consistently seen in Kershaw’s pitching mechanics from 2012 through 2017. On the other hand, if balls of one color are evidently seen aggregating outside the gray-colored cloud, then we should suspect that this pitcher likely being out of tune often in the focal season, if not injured. (See DM website for all 24 pitchers’ 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 of the whole 21X21 mutual conditional entropy matrix. Two versions of systemic comparisons based on “pitcher-season” are reported in first two panels of Figure 3. The two heatmaps have 55 (=11x10/2) and 210 columns, respectively.

(A) (B)

(C) (D)

(E) (F)

Figure 2: Clayton Kershaw’s progressive data cloud geometries of pitch-types from season 2012-2017. (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 as baseline. 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 the marked-blocks, which are framed by the clustering tree of pitchers on the row axis, and another clustering tree of feature-pairs on the column axis, reveal information not only about who is close to and far away from whom, but also, more importantly, about whether a pitcher is close to himself across a series of season. Understanding of information of similarity and 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,

8 (A) (B)

A row a2 a row B2 B row

b row

c3 c row C3 C row

d row

e1 e row D row 1 3 4 5 1 2 3 4 5 (C) 2

Figure 3: Two versions of heatmap-based systemic patterns: (A) All feature-pairs among 21 features; (B) feature-pairs among 11 selected features; (C) The tree locations of Clayton Kershaw and R. A. Dickey.

9 Kershaw-12 through Kershaw-17 exclusively occupy the “d-row” branch. This fact means that Kershaw’s pitching mechanics is rather steady throughout the 2012-17 seasons from the mechanical 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 are found in “a-row” for 2012-14 seasons, 2016 in “b-row” and 2015 and 2017 in “c-row”, likewise for Kluber’s 2012 through 2017 ,and Verlander’s 2011 through 2017 seasons. However, by adding biomechanical aspects, that is, pertaining to the 21 features, the heatmap in panel (B) and its clustering tree in panel (C) of Figure 3 can provide different version of comparison among pitcher-seasons. The 6 seasons of Kershaw are divided into two three-season branches: 2012-14 vs 2015-17 (marked by two upward-pointing arrows). The detailed 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 out as an outlier branch away from the rest of pitcher-season, including his own Dickey-11. The distinct separation can be attributed to the fact that R. A. Dickey had been developing his knuckleball through 2011 (marked by a small star). His knuckleball became very effective in 2012 season (together with 2013 marked by a star), in which he received the Cy Young award. Since the 2012 season, he has been recognized as one of the best knuckleball pitchers in American baseball history. There must be many patterns and related understanding waiting 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 for comparing a chosen set of pitchers. For instance, three pitchers: Hendricks, Kershaw and Verlander, are to be compared upon four features, {break length, Spin rate, pfx z, az}, which constitute 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-16 at “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 the three pitchers in 2016 season on row-axis. The 5 resultant heatmaps from Data Mechanics computations for four Kershaw’s pitch-types and one pooled pitch-type beyond his are shown in Figure 4.

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161216081594 1237 1086 376 1713 16311617 1323 1099 382 1714 208818291828 1326 388 17221721 1330 1104 389 1723 1334 1105 392 1724 1336 1118 1726 1338 1121 980 1732 1340 989 17381737 1368 1122 990 1739 1372 1131 1136 1740 1377 1134 1137 1744 1380 17521748 1383 1753 1391 1754 1392 1756 1394 17641758 1395 az az az az ate az ate ate ate ate pfx_z pfx_z pfx_z pfx_z pfx_z spin_ r spin_ r spin_ r spin_ r spin_ r break_length break_length break_length break_length break_length

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 four features: 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 (Yellow colored) 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 the four 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}, are highly associative as forming a 3x3 diagonal block within the universality, we understand that Hendricks’ fastball pitching mechanics has much smaller vertical acceleration (az) and vertical movement (pfx z) and spin-rate. From outlook of a pitch, his fastball is not as sharp as 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 that Verlander’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 red colored, that is, Kershaw and Verlander have large overlaps on all 6 subtypes of slider. This fact means that their pitching mechanics for slider are rather comparable. Slider is not in Hendricks’ 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 dominated by Hendricks. The smallest cluster: CH3, is shared by Verlander and Kershaw. This rather distinct 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 among these three pitchers. The four subtype-clusters: CU1-CU4, are color-coded with orange and yellow, while two subtype-clusters: CU5 and CU6, are entirely color-coded with red. These 6 curveball subtypes are relatively similar with respect to the features and their corresponding mechanism. The only difference is that Kershaw’s curveballs curve more than Verlander’s and Hendricks’. The panel (E) of Figure 4 shows that Hendricks has certain pitch-types that are not in Kershaw’s or Verlander’s repertoires.

kershaw−verlander−hendricks 2014−2016 Subtypes never thrown by Kershaw: CH2,CH5,CH6,CU3,FF3 1.0

CH3 ●●●●●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● 0.8

FF2 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ●● ● ● ●● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ●

CU6 ● ●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●● ●●● ● ● ●● 0.6 Likelihood 0.4

CU5 ●●●●●●● ●●●●●● ●●● ●● ●● ● ●●●●●●●●●●● ● ● ● ●●●● ●●●● ●●●●● ● ●●●●●●●●● ●● ●●●●●●●●●●●●● ● ● ●●●●●●●●●●●●● ●●●●●●

SL6 ●●●●●●●●●● ●●●●●●●●●●●●●●●● ●● ●●●● ●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●● ●● ●●●●●●●●●● ●●●● ●●●● ●● ●●●● ●●●●●●●●● ●●●●●● ●●●●●● ●●●●●●●●●● ●●●● ● ● ● ●● ● ● ● ●●●● ●● ●●● ●●●●●● ● ●● ● ●● ●●●●●● ●●●●●●● ●● ●●●●● SL2 FF6 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●● ● ● ●●●●●●● ●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●● ●●●● ● ● ●●●● SL4 ●●●●●●● ●●●●●●● ●● ●● ●● ●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●● ●●● ●●●●●● ● ● ●●●●●● ● ●● ●●●● ●● ●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ● ●●● ● ● ● ● ●●● ● ●● ●●●●● ● ● ●● 0.2

● ●● ●●●●●●● ● ● ● ● ●● ● ● ●● ●● ●● ●● ●● ● ● ●● ● ● ● ●●●●● ●● ●● ● ● ●●● ● ● ●●●●● ● SL1 CH1 ● ● ●● ●●●●●●●●●●● ● ● ●●● ●●● ●●●● ● ●● ● ●●●●● ●●● ●●●●●●●● ●●● ● ●●●●●●●●●●●●●● ●●●●●●●●●●● ● ● ● ●●●● ●●●●●●●●● ●●●●● ●● ●●● ●●●●● ●● ● ●●●●● ●● ● ●●●● ●●●● ●●● ●●● ●● ● ●●●● ● ● ●● SL5 ● ● ● ●● ●● ● ● ● ●●●●●● ● ●● ● ●●●● ●●● ●●●●● ●● ●●●●●●●●●●●●●● ●●●● ● ● ● ●●●●●● ●●●● ●● ● ●●● ● ●● ●● ● ● ●●●● ● ● ●●●● ●●● ●●●●●●● ●●●●●● ●●●●●●●●●● ●●●●●●●●●●● ● ●●●● ●●●● ●●● ●●●●● ●●●● ●●●●●●●● ●●● ●●●● ●●●●●●● FF4 ● ● ● ● ●●●●●●●●●● ● ●●●● ●●●●●●●●●●●●●●●●●●● ● ● ●●●●●●●●● ●●●● ●●●●●●● ● ● ● ● ●● ●● ● ●● ●●●●●●●●●● ●●●●●● ● ● ●● ● ● ● ●●● ●● ● ● ●●●● ● ●● ● ● ●●●● ●●●●●●●●●● ● ●●● ● ●●●● ● ●● ●● ●●●●● ●

SL3 CU1 CH4 ●●●●●●●● ● ● ●● ●● ●●●● ● ●●●●● ●● ●● ●●● ● ● ●●● ●● ● ● ●● ●●● ●●●● ●● ●●●●● ● ● ● ● ● ●●●●● ● ●●●●●●●●●● ●●● ●●●● ● ● ● ●●●●●●●●●●●●●● ● ● ●●●● ●●●●●●● ●●●●● ● ●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

● ● ● ● ●● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● FF5 ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●●● ●●● ● ● ● ● ● ● ●●● ● ● ● ● ●●●● ●●●● ●●● ●●● ●●●●●●●● ● ●●●●● ●●● FF1 CU4 CU2 ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0.0 kershaw verlander hendricks

0 2000 4000 6000 8000

Index

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 four features: 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 pattern distributions 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 every pitch from a pitcher has a temporal coordinate. Therefore, when all pitches of a pitcher can be laid out with respect to its ID, likelihood value and temporal coordinate, as such a graphic display is constructed, as shown in Figure 5. This graphic display provides the 4-feature-view of pitching mechanics comparison from the perspective of Kershaw. Similar viewing 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 mechanics evolves across a series of consecutive seasons, particularly including his career peaks. Here we construct and show Kershaw’s pitching mechanics from 2012 through 2017 seasons. There are 5 pitch-types: FF, FT, SL, CH, CU, in his pitching repertoire. His fastball is subdivided into: 4-seam fastball (FF) and 2-seam fastball (FT). Data Mechanics computations are applied onto the five ensembles of pitches collected from 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: 11 features contained in the universality and {x0, z0} the horizontal and vertical coordinates of its releasing point. The reason for including these two features is that they are closely related to pitching gesture at large. Kershaw is known for his rather distinct gesture particularly for 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 6 subtypes are displayed on the heatmap in individual and associative manners. Across these 6 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 higher end. 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 large value of horizontal movement {pfx x} and a distinctively small value of {break angle}, as seen in subtypes FF5 and FF6. In a reverse fashion, small values of {x0} and {ax} will result into small {pfx x}, but large {break angle}, as seen in subtypes FF3 and FF4. And median values of {x0} and {ax} lead to median values of {pfx x} and {break angle}, as seen in subtypes FF1 and FF2. Further the uneven proportions of color-coding across the 6 subtypes also bring out the 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 effective pitch-type in his pitching repertoire. The 2-seam fastball (FT) was only added to Kershaw’s repertoire in season 2015. The number of pitches in FT is much smaller than that in FF. However, its evolving patterns are evident and interesting to see, as shown in panel (B). The interacting relational patterns in panel (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}, as seen 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, are dominate in 2015-16 seasons. This modification on {z0} has gone further down in 2017 season, 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 with a 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 retained the interacting patterns seen in pitching mechanics of fastball in panels (A) and (B), but in a slight scale: SL1 vs SL4 on seasons 2012-2015; SL2 vs SL5 on season 2016; and SL5 vs SL6 on season 2017. Beyond the interacting pattern, like the shadow of fastball, we also see a gradual, but persistent pattern: values of {Start speed, pfx z, az, end speed} become slightly larger in values, while values of {break angle, break length, spin-dir, vy0, z0} become slightly smaller

12 esn(aeie ih21-07saos A F B T()S;()C;()CU. (E) CH; (D) SL; (C) FT (B) 2012-2014 FF; Kershaw (A) Clayton between seasons. comparisons 2015-2017 specific with pitch-type seasons(baseline) of Heatmaps 6: Figure (C) (A)

SL6 SL5 SL4 SL3 SL2 SL1 FF6 FF5 FF4 FF3 FF2 FF1 pfx_x break_length

ax 1 vy0 1 (E) spin_rate spin_dir end_speed z0

start_speed break_angle 2 2 pfx_z x0 hi 3 az hi pfx_x

x0 3 ax CU6 CU5 CU4 CU3 CU2 CU1 end_speed z0 end_speed

start_speed break_angle start_speed 1 4

4 pfx_z pfx_z spin_dir spin_rate az break_length vy0 az 87 86 85 84 81 74 68 65 62 56 54 50 48 45 41 40 39 36 33 32 29 25 24 23 22 21 19 18 12 7 6 13 pfx_x 4484 4479 4477 4467 4459 4457 4456 4455 4447 4445 4443 4437 4435 4433 4424 4420 4419 4416 4404 4401 4399 4398 4397 4396 4395 4394 4391 4389 4385 4379 4377 4366 4355 4349 4347 4346 4342 4333 4330 4328 4327 4326 4325 4324 4323 4322 4321 4320 4319 4317 4316 4314 4313 4312 4311 4310 4309 4308 4307 4306 4305 4304 4303 4302 4301 4300 4298 4297 4296 4295 4294 4293 4292 4291 4290 4289 4288 4287 4286 4285 4284 4283 4282 4281 4280 4279 4278 4275 4273 4268 4267 4266 4265 4264 4263 4262 4261 4253 4252 4251 4250 4249 4248 4247 4246 4245 4244 4243 4242 4239 4237 4236 4235 4234 4233 4231 4229 4228 4225 4223 4221 4220 4219 4218 4217 4216 4215 4214 4213 4212 4211 4210 4209 4208 4207 4206 4205 4203 4202 4200 4195 4194 4193 4191 4190 4188 4187 4186 4185 4184 4183 4179 4178 4177 4176 4175 4174 4173 4172 4171 4170 4169 4168 4167 4166 4165 4164 4161 4156 4155 4153 4151 4149 4148 4145 4143 4140 4138 4137 4136 4135 4134 4133 4132 4131 4130 4129 4127 4126 4125 4121 4118 4117 4116 4114 4113 4112 4108 4107 4105 4104 4101 4100 4097 4096 4095 4091 4089 4087 4086 4083 4082 4081 4079 4078 4077 4076 4075 4074 4073 4072 4071 4070 4069 4068 4067 4066 4065 4064 4063 4062 4061 4060 4059 4058 4057 4056 4055 4054 4053 4051 4050 4049 4048 4047 4046 4045 4044 4043 4040 4038 4037 4036 4035 4032 4031 4030 4029 4027 4026 4024 4022 4017 4015 4012 4011 4010 4004 4000 3994 3991 3990 3988 3985 3984 3981 3980 3979 3978 3977 3976 3973 3969 3965 3961 3960 3959 3956 3954 3953 3951 3950 3949 3947 3945 3942 3940 3935 3927 3926 3923 3922 3919 3913 3912 3911 3910 3903 3901 3896 3894 3892 3890 3868 3864 3859 3857 3855 3854 3853 3852 3851 3850 3849 3848 3847 3846 3844 3841 3839 3838 3836 3833 3832 3831 3830 3828 3827 3823 3822 3821 3818 3817 3814 3813 3812 3811 3810 3809 3807 3806 3805 3804 3803 3799 3798 3797 3796 3788 3786 3780 3779 3778 3776 3772 3771 3768 3767 3762 3758 3755 3754 3753 3752 3743 3741 3740 3737 3736 3735 3734 3731 3724 3730 3723 3721 3719 3718 3714 3713 3710 3706 3704 3701 3700 3697 3694 3693 3691 3692 3689 3688 3687 3681 3680 3679 3678 3668 3664 3661 3660 3657 3656 3647 3645 3643 3639 3638 3637 3630 3627 3624 3623 3617 3609 3593 3591 3590 3580 3577 3462 3344 3343 3333 3325 3318 3275 3250 3248 3243 3222 3204 3203 3202 3190 3189 3187 3186 3077 3068 3010 3007 2935 2744 2221 2184 2183 1955 1315 1314 1313 903 519 355 350 344 338 337 336 151 4487 4486 4485 4483 4478 4476 4475 4474 4473 4471 4466 4465 4463 4462 4460 4458 4454 4453 4451 4450 4448 4446 4444 4440 4438 4436 4434 4432 4431 4430 4426 4423 4417 4415 4414 4413 4411 4410 4409 4408 4407 4405 4402 4400 4386 4384 4383 4382 4381 4378 4376 4368 4367 4365 4361 4359 4358 4357 4356 4353 4352 4350 4348 4345 4344 4343 4340 4339 4338 4337 4336 4335 4334 4332 4331 4329 4315 4299 4277 4276 4274 4272 4271 4270 4269 4260 4259 4258 4257 4256 4255 4254 4241 4240 4238 4232 4230 4227 4226 4224 4222 4204 4201 4199 4198 4197 4196 4192 4189 4182 4181 4180 4163 4162 4160 4159 4158 4157 4154 4152 4147 4146 4144 4142 4141 4139 4128 4123 4122 4119 4115 4111 4110 4109 4106 4103 4102 4099 4094 4093 4092 4090 4088 4085 4084 4080 4052 4042 4041 4039 4034 4033 4028 4025 4023 4021 4020 4019 4018 4016 4014 4013 4009 4008 4007 4006 4005 4003 4002 4001 3999 3998 3997 3996 3995 3993 3992 3989 3987 3986 3983 3982 3975 3974 3972 3971 3970 3968 3967 3964 3963 3962 3958 3957 3955 3952 3948 3946 3943 3941 3937 3936 3934 3933 3931 3929 3928 3920 3915 3914 3909 3907 3905 3904 3900 3899 3895 3891 3889 3888 3886 3883 3882 3880 3875 3872 3871 3867 3865 3863 3862 3860 3856 3842 3837 3802 3801 3800 3795 3792 3791 3790 3789 3787 3785 3784 3783 3782 3781 3775 3774 3773 3770 3769 3766 3765 3764 3763 3761 3760 3757 3756 3751 3750 3747 3746 3745 3744 3742 3739 3738 3733 3732 3729 3728 3727 3726 3725 3720 3712 3711 3709 3708 3707 3705 3703 3702 3696 3695 3690 3685 3684 3683 3682 3677 3676 3675 3672 3671 3670 3669 3662 3658 3654 3652 3651 3650 3644 3642 3641 3636 3635 3633 3634 3632 3631 3629 3628 3626 3625 3622 3621 3620 3619 3618 3616 3614 3612 3586 3594 3568 3564 3558 3541 3540 3539 3538 3537 3529 3527 3524 3523 3522 3517 3515 3513 3507 3502 3501 3500 3499 3498 3497 3496 3494 3492 3491 3489 3488 3487 3486 3485 3484 3481 3480 3479 3478 3477 3476 3475 3474 3473 3472 3471 3470 3468 3467 3466 3465 3464 3460 3458 3381 3377 3374 3372 3368 3366 3359 3358 3352 3339 3317 3310 3305 3304 3302 3301 3300 3299 3295 3287 3286 3283 3280 3273 3264 3249 3247 3225 3218 3217 3201 3199 3196 3184 3176 3169 3168 3167 3166 3157 3156 3154 3153 3152 3151 3146 3139 3137 3136 3133 3132 3131 3130 3127 3126 3125 3115 3110 3100 3097 3094 3072 3051 3050 3049 3047 3045 3043 3040 3037 3031 3030 3029 3027 3

ax (D) (B) hi spin_rate 2

break_angle CH6 CH5 CH4 CH3 CH2 CH1 FT6 FT5 FT4 FT3 FT2 FT1

break_angle 1 break_angle

x0 1 end_speed z0 spin_dir 3 start_speed spin_dir z0 2 break_length vy0 2 break_length 4 spin_dir break_length vy0

2307 2049 2037 2035 2028 1952 1931 1929 1926 1916 1915 1898 1894 1890 1886 1884 1850 1835 1834 1833 1831 1829 1825 1815 1802 1792 1791 1774 1771 1765 1764 1752 1747 1746 1737 1736 1730 1717 1711 1705 1703 1702 1700 1699 1690 1688 1686 1685 1682 1679 1677 1674 1647 1645 1644 1640 1639 1635 1632 1631 1616 1611 1610 1609 1606 1605 1604 1603 1601 1600 1599 1598 1596 1595 1593 1592 1590 1588 1587 1586 1585 1583 1582 1581 1579 1578 1577 1575 1574 1573 1572 1571 1570 1568 1567 1562 1560 1559 1555 1554 1552 1551 1550 1549 1545 1544 1543 1537 1536 1535 1533 1532 1531 1527 1522 1518 1516 1514 1513 1512 1510 1509 1505 1503 1501 1499 1497 1494 1493 1492 1491 1490 1486 1485 1484 1483 1482 1481 1478 1477 1476 1475 1474 1473 1472 1470 1455 1449 1443 1439 1438 1436 1434 1433 1432 1430 1429 1427 1426 1423 1421 1418 1417 1415 1412 1406 1405 1403 1401 1392 1391 1389 1387 1385 1383 1382 1381 1380 1377 1376 1371 1370 1369 1366 1365 1364 1363 1361 1360 1358 1357 1356 1350 1349 1347 1346 1345 1343 1341 1339 1338 1337 1336 1335 1334 1332 1326 1325 1323 1321 1320 1319 1318 1316 1315 1314 1311 1309 1308 1307 1304 1303 1302 1301 1300 1299 1298 1297 1296 1295 1291 1290 1289 1285 1283 1282 1281 1279 1278 1276 1272 1268 1260 1258 1257 1255 1251 1247 1243 1242 1241 1240 1239 1238 1236 1234 1233 1229 1225 1224 1223 1209 1206 1199 1197 1196 1194 1193 1192 1191 1188 1186 1183 1182 1181 1179 1178 1177 1175 1174 1173 1172 1171 1166 1164 1163 1162 1160 1158 1157 1152 1150 1149 1144 1138 1137 1136 1135 1134 1133 1131 1130 1129 1127 1125 1119 1117 1114 1111 1110 1109 1103 1101 1092 1090 1085 1084 1082 1081 1080 1079 1078 1077 1075 1074 1071 1070 1069 1068 1067 1065 1063 1061 1058 1057 1055 1052 1049 1047 1043 1039 1037 1036 1034 1032 1030 1029 1028 1027 1025 1024 1021 1017 1015 1013 1012 1010 862 1008 861 1007 860 1006 859 1005 858 1003 857 998 850 997 849 995 848 993 847 986 845 985 844 984 842 983 841 978 834 972 833 968 825 959 824 958 822 956 820 954 818 953 814 952 810 951 809 950 807 948 804 944 802 920 801 917 798 916 791 914 777 912 771 905 769 904 759 899 758 896 724 894 717 883 698 882 669 877 658 875 657 869 653 868 651 865 650 603 601 599 598 597 595 594 591 590 581 578 577 570 562 561 560 550 549 548 543 539 538 537 536 535 529 526 524 522 520 502 493 487 486 483 458 446 444 442 432 429 428 422 421 420 418 417 414 413 412 410 409 405 387 382 381 380 379 376 375 374 373 367 366 357 356 355 350 347 346 343 341 331 327 324 321 320 319 317 316 315 314 313 312 310 309 308 306 296 290 263 262 229 228 224 192 190 184 182 179 174 170 166 159 158 157 152 150 148 147 146 145 144 143 142 141 139 138 137 136 135 134 133 130 129 127 125 123 121 118 78 54 53 2344 2341 2337 2336 2335 2334 2332 2326 2325 2324 2323 2322 2300 2299 2298 2296 2295 2293 2292 2290 2287 2286 2284 2283 2282 2280 2279 2278 2277 2276 2275 2274 2273 2272 2271 2270 2269 2268 2267 2266 2265 2264 2263 2262 2261 2259 2258 2257 2256 2255 2254 2253 2252 2251 2250 2248 2247 2242 2241 2240 2233 2232 2231 2230 2229 2228 2227 2226 2225 2224 2223 2222 2218 2216 2215 2209 2208 2206 2204 2203 2202 2201 2200 2199 2195 2194 2193 2190 2186 2183 2182 2179 2177 2173 2172 2171 2169 2168 2166 2165 2164 2163 2162 2160 2159 2156 2155 2153 2152 2151 2150 2149 2148 2147 2146 2145 2143 2142 2141 2140 2139 2138 2137 2133 2129 2127 2126 2123 2122 2121 2120 2117 2115 2108 2105 2101 2099 2097 2094 2093 2092 2090 2088 2087 2086 2082 2080 2079 2078 2076 2075 2074 2072 2071 2070 2067 2063 2062 2059 2051 2050 2048 2046 2045 2044 2039 2038 2036 2034 2033 2032 2031 2030 2029 2027 2026 2025 2024 2023 2022 1927 2021 1917 2020 1914 2017 1913 2016 1912 2014 1910 2011 1909 2010 1908 2009 1907 2008 1906 2007 1905 2004 1904 2002 1903 2000 1902 1999 1901 1994 1900 1992 1899 1991 1896 1990 1895 1977 1893 1975 1892 1974 1889 1970 1888 1968 1887 1965 1885 1964 1883 1962 1882 1961 1880 1959 1879 1958 1877 1957 1876 1956 1873 1955 1872 1953 1870 1951 1869 1950 1855 1949 1854 1948 1852 1942 1851 1939 1849 1934 1848 1933 1847 1932 1846 1928 1844 1843 1842 1840 1839 1837 1822 1821 1820 1819 1817 1816 1814 1813 1812 1811 1809 1808 1807 1806 1804 1803 1801 1800 1799 1798 1796 1795 1794 1793 1787 1783 1782 1780 1778 1777 1775 1773 1772 1770 1769 1768 1767 1766 1763 1762 1761 1760 1759 1758 1757 1756 1755 1753 1751 1750 1749 1748 1745 1744 1743 1742 1741 1740 1739 1738 1735 1734 1733 1732 1731 1729 1728 1727 1726 1725 1723 1722 1721 1718 1716 1714 1710 1709 1708 1706 1704 1701 1698 1697 1694 1693 1692 1691 1689 1687 1684 1683 1681 1680 1678 1665 1659 1658 1657 1656 1652 1651 1648 1646 1643 1642 1641 1638 1629 1623 1620 1613 1612 1608 1607 1602 1591 1589 1566 1564 1561 1557 1556 1547 1541 1539 1538 1534 1530 1529 1526 1525 1519 1515 1506 1502 1496 1495 1489 1488 1479 1471 1454 1446 1425 1424 1422 1420 1419 1416 1413 1 hi x0 pfx_z hi

spin_rate az 3

vy0 3 end_speed pfx_z start_speed az x0

pfx_x pfx_x 4 4 ax ax 201 181 179 178 137 129 127 126 125 124 123 122 118 117 112 110 104 103 101 100 97 87 78 72 71 67 66 65 62 46 19 176 175 92 41 39 38 37 35 34 206 205 204 203 202 200 199 198 197 196 195 194 193 192 191 190 189 188 187 186 185 184 183 182 180 102 52 51 50 48 47 32 31 30 27 26 24 21 17 16 14 13 9 7 5 2 1 149 131 115 114 107 94 90 89 77 76 75 69 49 28 25 23 22 15 12 11 10 8 6 4 3 207 173 162 148 135 133 96 91 86 85 82 81 79 74 73 70 68 64 61 59 58 57 177 174 172 171 170 169 168 167 166 165 164 163 161 160 159 158 157 156 155 154 153 152 151 150 147 146 145 144 143 142 141 140 139 138 136 134 132 130 128 121 120 119 116 113 111 109 108 106 105 99 98 95 93 88 84 83 80 63 60 56 55 54 53 45 44 43 42 40 36 33 29 20 18 22 21 26 25 1 67 48 36 32 31 28 27 24 23 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 66 64 62 54 53 40 33 58 52 51 38 37 35 34 30 29 65 63 61 60 59 57 56 55 50 49 47 46 45 44 43 42 41 39 kershaw 2012−2017 Subtypes never thrown in baseline (2012−2014): FT1,FT2,FT3,FT4,FT5,FT6

CU4 ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●● ●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●● ●●● ●● ●●● ● ●● ●●● ● ●●● ●●● ●● ● ●● ● ● ● ●● ● ●● 0.5

SL1 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●● ● ●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●● ●●● ● ●●● ●● ●● ● 0.4

CH1 ● ● ●●● ●● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●●●●● ●●●●●● ●● ● ● ● ● ●●●●● ●● ● ● ● ●● ● ● ● ● ● ●

FF1 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●● ●● ●●●● ● ●●● ●●●●●●●●●●●● ● ● ●●●●●●●●●●●●● ●●●● ● ●● ● ●

SL3 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●● ●●● ●●●●●●●●●●●●● ●●●●●●●● ●●● ●●●●●● ● ●●● ●●● ●● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ● ●●●●●●●● ● ● ● ● ● ●●● ● ● ●●● ●●●●●●●● ●●● ●●●●●● ● ● ● ●● ●● 0.3 Likelihood

FF3 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●● ●●●●●●●●●●●●● ●●●●●●● ● ● ●●●●●●●●●●●● ●● ●●●●● ●●●● ● ●●●●●●●●●● ● ●● ●●●●

CU6 ● ● ●●●●●●●●●●●●●●●● ●●●●●●● ●● ● ●●●●●●●●●●● ●●●●●●● ● ●●●●● ● ●●●●●●● ●●●● ● ●●● ● ●●●●●●●●●● ●● ●●●●●●●● ●● ●● ● ● ●● ● ●●● ●●●●●●●●●● ●●●●●●●●●●●●● ●●●● ●●● ●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●● ●●●● ●●● ● ●● ●●●● ● ●●●● ● ●●● ● ●● ● ●

● ● ●●●●●● ●● ● ●●●●●● ●●●●●●●●●●●●●●●● ●●●●● ●●●●●●● ● ●●●● ●●●● ● ● ●●●●●●● ● ●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●● ● ●●●● ● ● ●●●●● ●●●●●●●● ●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●● ●●● ● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●● ●●●●●●●●●●● ●● ●●●● ● ● ●●●●● ●●●●●●●●●●●●● ●●●●●● ●●● ● ●● ● ●●●● ●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●● 0.2 SL4 ● ●● ●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●● ● ●●●●●●●●●● ●●●●●●●● ●●●● ● ● ● ● ● ●●●● FF6 CU3 ●● ● ●●●●●●●●● ● ●●●● ●● ●●●●●●●●●●●●● ●●●●●●● ●● ●●● ●● ●● ●●● ● ● ●● ●●●● ●●● ● ●● ● ●●● ●●● ●●● ● ●● ●● ●●● ●●●●●● ● ● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ● ●●●● ●●●●● ●●● ●● ● ●● ● ● ●●●●● ● ● ●● ● ● ●●●●

CH6 ● ● ●●● ●● ● ● ●● ●● ● ● ●● ●●●●● ● ● ● ● CH3 ●● ●●●● ● ● ●●● ● ● ●● ●● ● ● ●● ● ●

CH4 ● ● ●● ●●●● ● ●● ●●● ● ●● ● ● ● ●● ● ● ●●● ● ● ●● ● ●●● ● ● ● ●● ●● CH2 ●●● ● ●● ● ●● ●●●● ● ● ●● ● ● ● ● FF5 ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●● ●●●●●●●●● ●●●● ●●● ●●● ●●●●●●●● ●● ● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ● ●●● ●●●●●●● ●●●●● ●●●●●●●●●●●●●●● ●●●●● ● ● ●●● ● ● ●●●●●●● ●●●●●●●● ●●●●●●● ● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●● ● ●● ●●●●●●●● ●● ●● ●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●● ●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● FF2 ●●●● ●● ●● ● ●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●● ●● ●●● ● ● ●● ●● ●●●● ●●●●●● ●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●● ●●● ●●●● ●●●●● ● ● ●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0.1

CH5 ●●●● ● ● ● CU1 ● ● ● ●●● ● ● ● ● ● ● ●● ●●●● ● ● ● ●●● ● ● ●● ● ●● ●● ●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●● ●●●●●●●● ● ●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●● ●●●● ●●●● ●●●● ●●●●●● ●●● ● ● ●●●●●● ●● ●● ●●● ● ● ●●●●● FF4 ● ● ●● ● ●●●● ●●●●●●●●●●●●●●●●●●● ●●●● ● ● ●● ●● ●●●●●● ● ●●●●●● ●●● ●●●●●● ●● ● ● ●●● ●● ●●●●●● ● ● ● ●●●● ● ●● ●● ● ● ●● ● ● ●●●● ● ● ●● ●●●●● ●●●● ● ● ●●●●● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● SL2 ● ●●●●●●●●● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ● ●●●●●●● ●●●●●● ●●●●● ●●●●●● ●●● ●● ●●● ●● ●●●● ●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●● ● ●●●●●●●●●● ● ● ● ●●●● ●● CU5 ●● ●●●● ● ● ● ● ● ●● ● ● ● ● ● ●●●●●●●●●●●●● ●● ●●● ●●●●●●●●●●● ●●● ●●●●●●● ●●●●●●● ●● ●●●●●●● ●●●●●●●● ●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● SL6 SL5 ● ●●●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●●●●● ● ●● ●● ● ●● ●●● ●●● ● ● ●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● CU2 ● ● ●●●● ●● ●● ● ●● ●● ●●●●●●● ●● ●●●●●●●● ●●●●●●●● ●●● ● ●●●● ● ●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ● ● ●●●●●●●●●●●●●●●● 0.0

2012 2013 2014 2015 2016 2017

0 5000 10000 15000

Index

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 the lower end, except the in the subtype SL6, which dominated by 2017 season. The evident idiosyncratic 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 a much reduced speed coupled with very diverse spin-direction. This diversity is carried out via many ways of holding a baseball when pitches a changeup. Different ways of holding induce different spin directions, and render different vertical and horizontal movements. So a changeup is disguised as a fastball in gesture, but has a drastically distinct and diverse movements. This is why it is an effective pitch-type for many MLB pitchers, but it is not a major pitch-type in Kershaw’s repertoire.

verlander 2011−2017 Subtypes never thrown in baseline (2011−2014): CU1,FC1,FC2,FC3,FC4,FC5,FC6,FT6,SL1,SL4

SL3 ●●● ●●● ●● ● ●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●● ●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● ● ● ● ● ● ● ● ● FT2 ●●●●●●●● 0.5 0.4

CU6●●●●●●●●●●●● ●●●●●●●● ●● ●● ●●● ●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●● ●●● ●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●● ● ● ● ●●●●●●●●●●● FF1 ● ●●● ● ●●●● ● ●●● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●● ●● ● ●●●●●●●●●●●●●●●●● ● ●● ●●●●● ●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ● ●●●● ● ●● ● ●●●●●●●

0.3 CU4 ● ● ●●●● ●● ●●●●● ● ● ●●●●●●● ● ●●●●●●●● ● ●●●●● ●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ●● ●●●●●●● ●● ● ●● ●●●●●●●●●●●●●●● ● ● ●●●●● ● ●● ● ● ●● ●● ● ● ●●● ● ● ●● ●●● CH4 ●●●● ● ●●●● ●●● ●●● ●●●●●● ●●● ●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ● ●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●● ● ● ●● ●●● ●●●●●● ●● ●●●● ●● ●● ● ● ●●● ●● ●●●●●●●●● ● ●●● ●●● ● ●●● ● ● ●●●●●●●● ●● ● ● ● ●

SL2 ● ●● ●● ● ● ● ●● ●●●●● ●●●●●●● ● ●●● ●● ●● ●●●●●●●●●● ●●●●●● ● ●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●● ●●● ●●● ● ●●● ●●●●●●● ●● ● ● ● ●● ●●●● ● ●

Likelihood ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●● ● ● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●● ●●●●●●●●● ●● ● ●● ● ● ●●●●●●●●●●●●●● ●●●●● ● ●●●●● ●●●●●● FT3FF4CH2●●●●●●●●●● ● ● ● ●● ●● ●●●● ●●●●●●●●●●● ●●● ●●● ●● ● ● ●●● ●●● ●● ●●●●●● ●●●●●●●●●●●●● ●●● ●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●● ●●● ●● ●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●● ● ● ●●●● ●●●●●●● ●●● ● ●●●●●● ●●●●●●●●●●●●●●●● ●● ●● ●●● ● ● ● ● ●● ●

FF2 ●●●●●●●●●●●●●●●●●●●●● ●● ● ●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ● ●● ●●●● ●●●● ●● ● ●●●●● ● ●●●●●●●●●●●●●● ●●● ●● ●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● CU3●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ● ●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●● ● ● ● ●● ● ●●●●● ●● ●●● ●● ● ●●●●●●● ● ●●●●●●●●● ●● ● ●●●●●●●● ● ●●●●●●● ● ●●●●● ● ●● ● ●●● ●● ● ● ● ● ● ● ●●●● ● ●● ● ● ●●●●● ● ● ●●●●● ●●●● ● ● ● ●●● ● 0.2

CH6 ● ● ●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●● ● ●●●●●●●●●●●●●●● ● ●●●● ●● ●● ●●●●●● ● ● ●● ●●● ●● ●●●●●● ● ●●●●●●●●●● ● ●●●●●●●●● ●●●●●●●● ● ● ●●●● ● ●●●● ● ● ● ● ● ●●●● ●● ●● ●● ● ● ●● ●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●● ●●●●● ●● ●● ● ●● ● ●●●● ●●●●●● ● ●● ●●●● ●●● ●●●● ●● ● ●

CU5CH5 ●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ●●●●● ●●● ●●●●●●●●● ●●● ●●●●●●●●●●● ●● ●●●● ●● ●● ●●●●●●●●● ●● ●●●●●●● ●●●●● ● ● ●● ● ●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●● ● ●● ●● ● ●● ●●● SL6 ●●●●●●●●●●●● ●●●●●●● ●● ●●● ● ● ● ● ● ● ● ● ● ●● ●●● ● ●●●● ●●● ●●● ●● ● ●● ●●●●●●●● ● ●●●● ● ●● ●●●●● ●● ●●●●●●● ●●●●● ● ● ●●●●●●●●●●●● ●●●●● ●●●●●● ● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

FT1 ●●●● ●● FF5●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●● ●●●●●●● ●●●●●●●●●●●●●● ● ● ●●●●●● ●●●● ●●● ● ●●●● ● ● ● ●● ●●● ●● ●●●● ●●●●●●●●●●●●●●●●●●● ●●● ● ● ●●●● ● ● ●●●● ●●● ●● ● ● ●●●●●●● ●● ●● ● ●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0.1 CH1 ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ●●●●●●● ●●●●● ● ●● ● ●●●● ●●●●●●● ● ●● ●● ●●●● ●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●● ●● ● ● ● ●●●● ●●●●●● ●● ●●●●●●●●●●●● ● ● ● ●●●●●●●● ●●● ●● ●● ●●●● ●

FF6●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●●●●●● ●● ●●●● ● ● ●●●● ●● ● ● ● ● ●● ● ●●●●● ●●●● ●●●● ●●●●● ●●● ●●● ● ● ● ●●● ●●●●●●●●●●●●●● ● ●●●● ●● ●●●●●● ●●●●●● ●●●●●●●●●●●● ●●● ●●●●●● ●●● ● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●● ●●● SL5 ●●● ● ●●●● ●●●●●●●●●●●●●●● ● ●●●●● ● ●●●● ●●●●● ●● ●●● ●●●● ● ● ● ●●● ● ● ● ● ●●●●●●●●●● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● FT5 ● ● ● ● ●●●● ● ●● ● ●●●

FT4FF3 ●●●●●●●●● ●●●●●●●● ●●●●●●●●● ●●●●●●●●● ● ● ● ● ● CH3 ●●●●●●● ●●●●●● ●●●●●●● ●●●●●● CU2 ● ●●●●● ●●●● ● ●● ● ● ● ●●● ●● ●● ● ● ●●● ● ●● ● ● ●●● ● ●● ●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●● ●●● 0.0

2011 2012 2013 2014 2015 2016 2017

0 5000 10000 15000

Index

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 seen only 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 Magnus effect of going downward is further aided by gravity, its trajectory usually has a drastic, if not sudden, drop when a baseball is entering the home plate. That is why it is an effective pitch-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 of curveball 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 has a hard time to guess which sides: right or left, his curveball will swerve to. This uncertainty makes 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 curveball seemingly is narrowing into two subtypes: CU2 and CU5, in recent reasons. This might be an alarm sign of losing his effectiveness on this pitch-type. Information contained in these five panels of Figure 6 is synthesized into a so-called likelihood plot, as shown in figure Figure 7, with 2012-2014 being used as baseline. It is recalled that each pitch has a pitch-subtype ID and a likelihood value, which are derived from a pitch-type specific heatmap and its categorical pattern distribution in one of the five panels in Figure 6. The likelihood plot displays each pitch’s subtype-ID and likelihood vale with respect to its temporal coordinate upon the entire axis of 6 seasons. Such a graphic display collectively reveals nearly all pattern information of Kershaw’s pitching mechanics, which evolves from 2012 through 2017. Many pitching subtypes go extinct, while many subtypes are created in later seasons. Also some subtypes even go into extinct, and then come 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, shows that Verlander has 5 pitch-types: {FF, FT, SL, CH, CU}. In fact he has 6 pitch-types in the PITCHf/x database, including the pitch-type: Cuter (FC), which was created only recently way after 2014 season. He also creates one new curveball subtype, one 2-seam fastball 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 separation can be somehow seen in this likelihood plot as well. The likelihood plot of Hendricks’ pitching mechanics on three seasons: 2015-2017 is shown in 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 2016 to 2017 season. Many subtypes are nearly extinct, while many sparsely used subtypes in the baseline seasons are heavily used in the 2017 season. This evident change indeed coincided with 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. Information of several aspects of pitching mechanics is genuinely extracted and then explicitly represented for systemic comparison among all 24 pitchers, and for individual pitcher’s idiosyncratic evolution across multiple seasons. Understanding of computed information via a graphic display as a platform seems to be able to flow smoothly. Nonetheless that is just the beginning of a new era into this 150 year old sport. The systemic comparison based on a heatmap, derived from mutual conditional entropy matrices of all pitcher-seasons, not only enables us to see who are close to whom, and far away from whom among MLB pitchers, but also pinpoints which features can be accounted for their differences in pitching mechanics. Equally importantly, the idiosyncratic evolution of 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 or declines. In this big-data era, it is critical to develop computing platforms in order to properly address the fundamental question: what and where is information contained in data? In this paper, the discovery of the universal block patterns, which embed with all involving physical

15 hendricks 2015−2017 Subtypes never thrown in baseline (2015−2016): FF5,FF6

FF4 ●●● ● ●● ● ●● ● ●● ●● ● ● ●●● ●● ● ● ●● ● ●●●●●● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ●●●● ● ● ● ● ●●● ●●●●●●●●●●● ● ●●● ● ●●● ● ● ●● ●● ●●●● ● ● ● ● ● ● ● 0.7 0.6 0.5

CU1 ● ●●●●●●● ● ● ●● ●● ●● ● ● ● ●● ●●●●●●● ●●● ●●●●●●● ● ●● ●● ●● ●●● ● ●● ●●●● ● ●● ●●● ● ● ●●●● ● ●●●● ●● ● ● ● ●● ● ●●●●● ●●● ● ●●● ● ● ● ● ● ● ●●●●●●●●● ●● ● ●● ● ●●●●●●●● ●●●●● ●●●●●●●●●● ● ● ● ● 0.4

SI4 ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●● ●●●●●●●●●●●●●●●●● ● ●●●●●●●● ●● ●●●●●●●● ●●●●●●●●●● ● ●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ●●● SI6 ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ● ●●● ● ●●●●●●●●●● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ● ●● ● ● ● ●● ● ●●● ● ●●●●● ●●●●● ● ●●●●●● ●●●●● ●●●●●●●●● Likelihood

0.3 CU2 ●●●●●●●● ●●● ●●●●●● ●● ●●● ●● ● ●●●●●●● ●●●●●●●●● ● ● ● ●● ●●●● ●● ● ● ●●● ● ● ●● ● ●●●●●●●●●●●●●●●● ●●●● ●●● ●●● ● ● ● ●●● ● ● ● ● ● ● ● CH6 ●●●●●●● ● ● ●●●●●● ●● ●●●● ● ● ● ● ●●●●●●●● ●●●●● ● ●● ●●●●●●● ●●● ●●● ●● ● ●●●● ●● ●●● ●● ●●●● ●●● ●●●● ●●● ●●●●●●●●●●●● ● ●●●●●● ● ● ● ●● ●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●● ●● ●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●● ● ● ●●●●●●●● ● ● ● ● ● ● ●● ● ●●●●●●● ● ● ●● ● FF3 ●●●● ●● ● ● ● ● ● ● ● ●●●● ●● ● ●● ●●●●●● ● ● ● ● ● ● ● ●●● ● ● ● ● CH1 ● ●●●●●●●●●●● ●●●●●● ●● ●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ● ● ●●●●●●●● ● ●● ●● ● ●● ●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●● ●●● ●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●● ● ●● ● ●●●● ●●●●●●●●●●●●●●●●●●● ●●● ● ●●●●● ●●●●●●● ●● ●● ●●●●●●●● ●●● ●●●●● ● ● ● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●● ● ● ●●●●●● ●●●●●●●●●●●●● ● ●● ● ●●●●●●● ● ●●●●●●● ●● ● ●● ●●●● ●

FC6 ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●● ● ● ● ●● ●●●●● ●● ●●●●● ●●●●● ●●●●●●●● ●●●●●●●●●●● ● ●● ●●● ● ● ● ● ●●●●● ● ●●● ● ● ●● ● ● ● ● ● ●●●●●● ●● ● ●● ●●●●●●●●●● ●●

0.2 CH5 ● ●● ● ●●● ● ● ●● ●● ●● ● ● ●●● ●● ●● ●● ●●● ●●●● ●●●● ● ●●● ●●●● ● ●●● ●●● ●●● ●● ● ●● ●●●●● ● ● ● ●● ●● ●● ●● ●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●● ●●●●● ●● ● ●●●● ●●● ●● ●● ●● ● ● ● ● ● ● ● ● ● FC2 CU6 ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●●● ●●●●● ●●●●●● ●●●● ●●● ●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●● ● ● ● ●● ● ●●●● ● ●● ● ●●●●● ●●●●● ● ● ●●●●● ●● FC5 FC1 ● ● ● ● ●●● ● ●●●●● ● ● ● ● ● ●● ● ● ● ●●●●●● ● ● ● ●● ● ●● ●●●● ●●●●● ● ●● ● ● ●●● ●●●● ● ●●●●●●●●●●●●●●●●●● ● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ● ●●●●●● ●●●●●●●●●●●●●● ●● ●●●●● ● ●● ●● ● ●●●●●●● ● ●●● ●●●●●● ● ● ●●●●●●● ● CH2 ● ●●●●● ● ● ● ●●●● ●●●●●● ●●●●●● ● ●●●●● ●● ●●●●● ●● ●● ●●●●● ● ● ●● ●●●●●●● ●●●●● ●● ●● ●●●●● ●●●●●●● ●●●● ● ●● ● ● ●●●● ●● ● ●●● ●●● ●● ● ● ● ●●●●●●●●●●●●● ● ●● ●●●●●● ●● ●● ●●●●●●●●●●●● ● ● ●●●●●●●● ●

FC4 ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ●●●●●●● ●●●●● ● ●●●●●●● ●● ● ●● ● ● ● ● ● ●●●●● ●●●●● ● ●●● ●● ●● FC3 ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ●●● ●● ● ●● ●● ● ●●●●●●●●●●● ● ●●●●● ● ●●●●●● ● ● ● ● ●● ● ● ●●● ●●●●●●●●● ● ● ● ● ●● ●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●● ● ● ●●● ● ●●●●●●●●●● ● ●● ●● ● ●●●●●●●●●● ● ●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●● ●●●● ●●●●●●● ● ● ● ●●●●●●

0.1 SI5 SI2 ●●●●●●● ● ●●●●●● ● ●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●● ●●●●● ●●● ● ● ● ●●●● ●● ● ● ●●●●●●●● ● ● ●●●●● ●●●● ●●● ● ● ●●● ● ●●● ●● ●●●● ● ● ● ●●● ● ●● ●●●● ●●●●●●●● ●●●● ●●● ●●●●●●●●●●● ● ●●●●●●●●● ●●● ● ●● ● ●●●●●●●●● ● ● ● ● ●●● ● ● ●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ●● ● ●●● ●●● ● ● ● ●●● ● ● SI1 CU5 ●●●● ●●● ● ●● ●●●●● ●●● ●●●●●●●●●●● ●● ●●●●●●● ● ● ● ●● ●●● ● ●●●●● ● ●● ● ●●●●●●●●●●●●● ●●●●● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●● ●●●●● ● CH3 ● ● ● ● ● ● ● ●●● ●● ● ●●● ● ● ● ● ● ●●●●●● ●●● ●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●● ●●● ● ●●● ● ● CH4 ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●●●● ●●● ● ●● ● ● ● ●●●● ● ● ●●●● ●● ● ● ●● ● ● ● ●● ● ● ●● ●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● SI3 ● ● ● ● ● ● ●●● ● ●● ● ●● ●●●●●●●●● ●● ●●●●●● ● ●●●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ●● ● ●●● ● ●●● ●●●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●●● ●● ● ● ● ●●●●● ● ●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●● ●●● ●● ●● ● ●● ●●●●●●● ● ●● ●● CU4 CU3 ● ● ●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ● FF1 ● ●●●●●●● ●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ● ●●●●● ●●●●● ●●● ● ●● ● FF2 ●● ● ● ● ●●● ●●● ●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●● 0.0 2015 2016 2017

0 2000 4000 6000 8000

Index

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 a validity check of our data-driven computations. However we have to admit that results and understanding presented here fulfill only a small part of our primary objective that our computational endeavors intend to achieve upon PITCHf/x database. Our primary objective is to use our data-driven computing and graphic displays as platforms for all people to explore and discover information by themselves, and building 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 because of its rather limited space. Therefore it becomes necessary to create a website to facilitate all potential explorations and discoveries possibly accomplished by all curious minds. This is one way to accommodate all possible relevant information about pitching mechanics contained in this Big-Data era. People, who has a career centering around or in baseball pitching, are able to find the landscape of pitchers and their differences in pitching mechanical factors for management purposes 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 adversarial pitchers’ characteristics in pitching mechanics. People, who are interested in data-driven computing or data science in general, are to find inspirations from resolutions of various issues on high dimensional point-cloud geometries, and, at the same time, to discover a new direction of research into biomechanics of pitching. Particularly young people, who are guided by their own curiosity in baseball, are able to wonder around the new data-driven physical and aerodynamic laws embedded within pitching mechanics, and to appreciate their tangled complexity. The immediate sense of achievement 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 perceive this sport from totally different perspectives that their parents’, grand-parents’ generations never have imagined.

Ethics:

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

16 Data Accessibility:

The pitching data is available in PITCHf/x database belonging to Major League Baseball via 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 analyzed the data. H.F, K. F., T.R. C-J. H., and B. C. interpreted the results. H.F wrote the manuscript. H.F, K. F., T.R. C-J. H., and B. C. edited the manuscript. All authors gave final approval for publication.

Funding:

No financial funding received for this study.

Acknowledgement:

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

References

[1] Briggs L.J. Effect of spin and speed on the lateral deflection (curve) of a baseball; and the magnus effect for smooth spheres. American Journal of Physics, 27(8):589–596, 1959.

[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 of Histogram (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 bipartite networks. PLOS ONE, 9(8):1–11, 08 2014.

[7] Hsieh Fushing, Hsueh C-H., Heitkamp C., Matthews M. A., and Koehl P. Unravelling the geometry of data matrices: effects of water stress regimes on winemaking. Journal of 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. PLOS ONE, 8(2):1–10, 02 2013.

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