Modeling the Relationship Between MLB Ballparks and Home Team Performance. Using Shape Analysis. Sara Biesiadny, B.S. A Thesis

Transcription

1 Modeling the Relationship Between MLB Ballparks and Home Team Performance Using Shape Analysis by Sara Biesiadny, B.S. A Thesis In Mathematics and Statistics Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCES Approved Dr. Leif Ellingson Chair of Committee Dr. Jingyong Su Dr. Alex Trindade Dr. Mark Sheridan Dean of the Graduate School May, 2016

2 c 2016, Sara Biesiadny

3 ACKNOWLEDGMENTS First and foremost, I would like to acknowledge my committee chair, Dr. Leif Ellingson, for the continual support and guidance throughout this process. This undertaking would have certainly not been possible without your ideas and direction and I would like to thank you for your encouragement and for helping me grow so much as an academic. I would also like to acknowledge Dr. Jingyong Su and Dr. Alex Trindade for taking time out of their busy schedules to serve as my committee members. Thank you both for making my defense such an enjoyable, proud, and unique experience. I would be remiss to not also acknowledge my multiple generous support systems, both academic and personal. On the academic side, I would like to thank everyone who has ever played a role in making this dream of pursuing statistics a reality; whether it be a high school calculus teacher or a graduate professor, your contributions can not be understated. I would like to thank everyone I have met throughout this process that have helped me in any capacity along the way: you know who you are. Finally, I would like to extend my sincerest gratitude and appreciation to my incredible family. To my brother, who instilled in me the competitive nature I so graciously exhibit today, having grown up next to you was the greatest gift any sibling could ask for. And to my mother and father, who kindly provided a daily ear though which I could relax my burdens and regain my composure, your continual support has been unbelievable and I thank you. I do not say it nearly enough, but I appreciate everything you have ever done for me. For better or for worse, you have shaped me into the person I am today, and I strive to reach half the level of compassion and patience you exhibit on a daily basis. ii

4 CONTENTS ACKNOWLEDGMENTS ii ABSTRACT v LIST OF FIGURES v 1 INTRODUCTION PRINCIPAL COMPONENT ANALYSIS Covariance Matrices Variance Covariance Correlation Covariance Matrices Eigen-Decomposition Eigenvalues Eigenvectors Eigen-Decomposition PROCRUSTES ANALYSIS Translation Uniform Scaling Rotation BOOKSTEIN REGISTRATION Translation Uniform Scaling Rotation APPLICATION TO BASEBALL Aligning the Ballparks Translation Scaling and Rotation Obtaining Ballpark Outlines iii

5 5.3 Calculating the Mean Ballpark Principal Component Analysis on the Ballparks Covariance Matrix Regression Over the Principal Components ANALYSIS OF RESULTS CONCLUSION AND DISCUSSION BIBLIOGRAPHY APPENDIX iv

6 ABSTRACT In the last few decades, there has been a sharp rise in the amount of work done in analyzing statistics related to the sport of baseball. These studies typically use previous performance figures to predict a team or player s production in future games. Here, we approach baseball statistics in a different manner. Interestingly, baseball is the only major American sport whose primary professional league does not standardize the dimensions of its playing arenas. This thesis plays on this idea by using various shape analysis techniques to model how home team performance and ballpark shape are related. Particularly, this piece implements the techniques of Bookstein Registration and Principal Component Analysis to investigate a possible relationship between home team performance and ballpark shape for the 2015 Major League Baseball regular season. Notably, two common performance aspects are considered: home runs and on-base percentage. The preliminary results of this study show that the ideal ballpark to maximize home team on-base percentage would have a deeper-than-average left field wall and a shallower-than-average center field wall, whereas the ideal ballpark to maximize home team home runs would have a shape very close to that of the mean Major League Baseball ballpark. v

7 LIST OF FIGURES 3.1 Original Shapes Translated Shapes Scaled Shapes Rotated Shapes Coors Field vs. Mean Ballpark General Baseball Field Layout Original 30 Ballpark Outlines Translated Ballpark Outlines Translated and Scaled Ballpark Outlines Fenway Park vs. U.S. Cellular Field MLB Ballparks with Mean Scree Plot Principal Component Principal Component 1 (Superimposed) Principal Component Principal Component 2 (Superimposed) Principal Component Principal Component 3 (Superimposed) Principal Component Principal Component 4 (Superimposed) Diagnostic Plots for On-Base Percentage Log-Likelihood Plot for On-Base Percentage Diagnostic Plots for Home Runs Log-Likelihood Plot for Home Runs Principal Component Principal Component Principal Component Principal Component vi

8 CHAPTER I INTRODUCTION In America, there is arguably no greater source of nostalgia and patriotism than through the game of baseball. Baseball has had such a great impact on American history that it is even said that baseball is the American pastime. The sport of baseball has had a vast effect on everyday American life, indicated in this quote by iconic American author Mark Twain: Baseball is the very symbol, the outward and visible expression of the drive and push, and rush, and struggle of the booming 19 th Century. Baseball evolved into the favorite American sport because it is fast-paced and physical[38]. It is even said that American visionary and sixteenth president of the United States, Abraham Lincoln, was participating in a baseball pickup game when his messenger arrived to notify him of his nomination for president by the Republican Party. Moreover, Lincoln told the messenger to hold off on giving him the news until he had taken his turn at bat[29]. Due to its many statistical aspects and relations, baseball is a very common instrument in investigative research and analysis. In almost all manners of life, one thrives to be the best. Oftentimes, this lends its way to disciplined leverage and the longing for an upper-hand on the competition by any means necessary, especially when there are millions of dollars at stake. These two motivations are acknowledged in the novel Moneyball: The Art of Winning an Unfair Game, by Michael Lewis [37], which is the main inspiration behind this paper. This best-selling novel chronicles the success of Oakland Athletics general manager Billy Beane in the late 1990s and the early 2000s. Beane pioneered a new sabermetric 1 approach to achieve success despite the California-based team s strapped 1 Sabermetrics is a term referring to the empirical, evidence-based approach to baseball statistics. 1

9 financial situation. Although figures such as batting average, runs batted in, and stolen bases had previously typically been used to dictate a player s value, Beane found that on-base percentage and slugging percentage were much better indications of a player s offensive potential. The Oakland Athletics made the Major League Baseball (MLB) playoffs in 2002 despite having a team salary of about $40 million, the third-lowest in the league. For comparison, the New York Yankees finished the 2002 regular season with the best record in the league, doing so with a team salary of approximately $120 million. Apart from Moneyball, these methods have been used many times in works such as [7] and [48]. While Beane and others have made a living performing analysis to predict team success using previous statistics for each player, there has been little done to investigate the impact that the team s home ballpark has on offensive capability. In an effort to examine this approach, this work evaluates baseball statistics in a different way. There are four major sports leagues in America: the MLB, the National Hockey League (NHL), the National Football League (NFL), and the National Basketball Association (NBA). Of these, the MLB is the only organization that does not completely standardize the size and dimensions of its official playing fields. While there are many measurements that are standardized in MLB ballparks, such as distance between bases, height of pitching mound, etc., the dimensions and height of the outfield walls are not. Even though there are minimum requirements (for instance, there must be at least 400 feet between home plate and the center field wall), ballparks deviate from each other quite a bit in terms of shape, and it is this idea of deviation that we attempt to exploit in this thesis. The main objective of this paper is to model if and how a ballpark s size and dimensions impact its home team s performance in different ways. First, Principal Component Analysis (PCA), a procedure typically used to locate variation in a data set, is detailed. To begin with, the notions of variance, covariance, correlation, and covariance matrices are explained. Variance is a scalar whose value describes variation in a random variable, while covariance and correlation 2

10 are scalars whose values describe how two or more random variables vary together. On the contrary, a covariance matrix is a matrix whose (i, j) th element describes the covariance between the i th and j th random variables. Following this, the notion of Eigen-Decomposition, an important aspect of Principal Component Analysis with be explained; in addition, the concepts of Eigenvalues, Eigenvectors, and Eigen- Decomposition are explored. These are yet additional concepts dealing with variation among variables. Next, we illustrate a couple of techniques from image and shape analysis; namely, Procrustes Analysis and Bookstein Registration. These are both techniques for superimposing images in such a way that they are comparable to one another, all the while maintaining the integrity of the features of each shape. While we will not use Procrustes Analysis in the study of baseball statistics, it is useful to consider as somewhat of a bridge to Bookstein Registration. Finally, these concepts are applied to Major League Baseball and the results are analyzed. In this paper, various statistics from all 30 teams for the 2015 MLB regular season are analyzed against home team ballpark shapes and dimensions. Firstly, the ballparks are superimposed using Bookstein Registration so that they are comparable. Next, an equally spaced 100 landmark points are taken from each ballpark. These 100 landmarks for each ballpark are then used to construct the shape of the mean ballpark as well as perform PCA on the covariance matrix for the 30 shapes. The first four principal components are used and the distance between different ballparks constructed from these principal components are used as independent variables in a linear regression model. Possible transformations are observed, backward model selection is used to find a reduced model, and regression is then performed to determine coefficients of the predictors in the reduced model. These regression coefficients are then interpreted geometrically and approximate shape features for optimal ballparks are given for different responses. 3

11 CHAPTER II PRINCIPAL COMPONENT ANALYSIS When analyzing a data set, in addition to measuring the magnitude of variation present, it is oftentimes equally important to evaluate how the different variables are working together to form said variation. Principal Component Analysis (PCA) is a technique used in statistics to analyze directions of variation in a data set. This technique was first invented in 1901 by Karl Pearson [39] and later independently developed and named by Harold Hotelling in the 1930s [32]. PCA is particularly useful when dealing with high-dimensional data because the human imagination can only visualize anything in up to three dimensions; any data in the fourth dimension or higher is impossible to visualize. Before Principal Component Analysis can be described in further detail, some notions related to matrix algebra should be established. In the following subsections, the notions of Covariance Matrices and Eigen-Decomposition will be illustrated. 2.1 Covariance Matrices Analysis of variation between two predictors 1 can reveal a lot about a data set. If two independent variables are highly correlated, they may be redundant in the data set. Covariance and correlation 2 are two measures that describe how much correspondence is present between two covariates. However, most data sets have large quantities of variation and well more than two independent variables. In this case, it is useful to express where, exactly, the variation takes place. A Covariance Matrix is one way to formulate just that. A covariance matrix is a square, symmetric, positive definite matrix whose (i, j) th entry represents the covariance between the i th and j th independent variables. The notions of variance, covariance, correlation, and covariance matrices are detailed below. 1 Also known as covariates or independent variables. 2 By correlation here, we mean the statistical notion correlation. 4

12 2.1.1 Variance Variance is a quantity that measures how much the observations of a variable vary from the mean observation of that variable. In other words, the variance of data set is a measure of how spread out the data is. For a sample with n observations, (X 1, X 2,..., X n ), and a sample mean X calculated by X = n i=1 X i, n the variance of that data set is denoted by S 2 and can be computed using the formula S 2 = n ( i=1 Xi X ) 2, n 1 where X is the sample mean and n is the sample size. In other words, the variance of a data set is equal to the sum of the squared deviations from the mean divided by n 1. It is worth noting here that variance is denoted as S 2 because S has its own label, namely the standard deviation 3 of the data set. The unit of measurement for standard deviation is equivalent to the unit of measurement of the variable itself, while the unit of measurement for variance is the square of the unit of measurement used to measure the data Covariance While variance is certainly a useful measure for analyzing the variability of a univariate 4 data set or the variation present in one of many variables in a data set, it may also be advantageous to consider how pairs of random variables differ from the mean with respect to each other. Perhaps somewhat trivially, the formula for sample variance, S 2, can be rewritten in the following form: S 2 = n i=1 ( Xi X ) ( X i X ) n 1 3 The variance is simply the square of the standard deviation. 4 A data set dealing with one variable.. 5

13 This provides a means of substituting one of the multiples of the variations of X with that of a different relevant random variable, say Y. If we replace one of the X deviation terms with a Y deviation term, we get exactly the formula for the covariance 5 of the variables X and Y : n ( i=1 Xi X ) ( Y i Y ) Ĉov(X, Y ) =, n 1 where now (Y 1, Y 2,..., Y n ) represent the n observations of the variable Y and Y denotes the sample mean of those observations. Unlike variance, which must always be non-negative (due to the squared terms in the numerator and the fact that sample size must be an integer greater than zero), covariance can take on any real value. A positive covariance between two variables signifies a direct relationship between them, meaning that as one variable increases, the other will generally increase, and vice versa. A negative covariance between two variables denotes that they have an inverse relationship, meaning that as one variable increases, the other will decrease, and vice versa. A covariance of zero between two variables indicates that the two variables are completely independent of one another; increasing or decreasing one variable will have no impact on the other variable. There are a couple of properties of covariance that will be useful in constructing our covariance matrix: 1. Ĉov(X, X) = V ar(x) 2. Ĉov(X, Y ) = Ĉov(Y, X) 5 The covariance between two variables X and Y is also sometimes denoted as σ XY. 6

14 If each variable is represented by a dimension, it can be said that covariance measures how much the dimensions vary from the mean with respect to each other. The unit of covariance between two random variables is the product of the units of the two variables involved. For instance, if the variable X is height, in feet, and the variable Y is weight, in pounds, the unit of the covariance between X and Y, Ĉov(X, Y ), is ft lb Correlation In the English language, the term correlation refers to the interrelationship between two parties. The notion of statistical correlation is very comparable, if not one and the same. Correlation in the statistics field is, more or less, the dimensionless analog of covariance. Because of this, correlation follows the same rules as covariance when it comes to sign: a positive correlation denotes a direct relationship between two variables, a negative correlation denotes an inverse relationship between two variables, and a correlation of zero denotes no relationship between two variables. The correlation coefficient between two predictors X and Y, represented by Corr(X, Y ) or ρ XY, is calculated by dividing the covariance of X and Y by the product of their two standard deviations. Expressly, if the covariance between X and Y is represented by σ XY and the standard deviations of X and Y and represented by σ X and σ Y, respectively, ρ XY = Corr(X, Y ) = Cov(X, Y ) σ X σ Y = σ XY σ X σ Y. There are a few properties about correlation coefficients that are worth noting. For any two independent variables X and Y, the following are known to be true: 1. ρ XY = ρ Y X 2. 1 ρ XY 1 3. ρ XX = ρ Y Y = 1 7

15 2.1.4 Covariance Matrices For a data set pertaining to more than two dimensions, it is especially essential to organize each possible covariance value between each pair of dimensions. For instance, a three-dimensional data set introduces six different covariance terms: the covariance between each of the three pairs of dimensions as well as the three variance terms. A useful way to organize covariance terms present in a data set is by using a matrix. This practice results in an aptly named covariance matrix, which is usually denoted by Σ, or simply Cov(X) for a data set X. For a p-dimensional data set, a covariance matrix will be of dimension p p. Say a data set has three dimensions: X, Y, and Z; then the covariance matrix will be Cov(X, X) Cov(X, Y ) Cov(X, Z) Σ = Cov(Y, X) Cov(Y, Y ) Cov(Y, Z). Cov(Z, X) Cov(Z, Y ) Cov(Z, Z) However, since Cov(X, X) = V ar(x), Cov(Y, Y ) = V ar(y ), and Cov(Z, Z) = V ar(z), the covariance matrix can be written as V ar(x) Cov(X, Y ) Cov(X, Z) Σ = Cov(Y, X) V ar(y ) Cov(Y, Z). Cov(Z, X) V ar(z, Y ) V ar(z) There is one more alteration that can be made to this matrix that will reduce the number of unique terms in the matrix. Since there is a property that says that Cov(A, B) = Cov(B, A), the covariance matrix can be rewritten as V ar(x) Cov(X, Y ) Cov(X, Z) Σ = Cov(X, Y ) V ar(y ) Cov(Y, Z), Cov(X, Z) V ar(y, Z) V ar(z) 8

16 and in alternate notation can be written as σ X 2 σ XY σ XZ Σ = σ XY σy 2 σ Y Z. σ XZ σ Y Z σz 2 Hence, any covariance matrix is symmetric. The covariance matrices up until this point have dealt with a three-dimensional data set with dimensions X, Y, and Z. However, it is also common that an entire data set will be denoted as X and will have dimensions X 1, X 2,...,X n. In this case, the covariance matrix will appear as σ 1 2 σ σ 1n σ Σ = 12 σ σ 1n σ 1n σ 2n... σ nn Note that correlation matrices, denoted by Corr(X), can also be constructed, where the (i, j) th cell represents ρ ij. Since the correlation between any random variable and itself is equal to 1, the principal diagonal 6 of any correlation matrix will be made up of ones. Correlation matrices, like covariance matrices, are always symmetric. 2.2 Eigen-Decomposition While the notion of covariance matrices is going to be very important when it comes to techniques later described in this document, equally paramount is the concept of of Eigen-Decomposition 7.Eigen-decomposition is a method for factoring a matrix into its canonical form. There are many different canonical forms of a matrix, but the canonical form of interest here is one in which the matrix is written in terms 6 Also sometimes referred to as the main diagonal. 7 Also called spectral decomposition. 9

17 of its eigenvalues and eigenvectors. These notions will be described below in further detail Eigenvalues Every square, say n n, matrix has n eigenvalues. An eigenvalue is also sometimes known as a characteristic value or a characteristic root of a matrix. Each eigenvalue of a matrix will have a corresponding eigenvector (which is elaborated on below), and these eigenvectors denote an axis along which variation is present. Say a matrix A has dimensions n n. Then the eigenvalues of A can be represented by λ 1, λ 2,...,λ n, where the λ i s are not necessarily real-valued. These eigenvalues can be calculated by solving for λ in the equation det(a λi) = 0, where det() refers to the notion of determinant from linear algebra and I denotes the n n identity matrix. The identity matrix is a square matrix made up of ones along the principal diagonal and zeroes elsewhere. In other words, if A is the n n matrix with the value a ij in the (i, j) th position, then the eigenvalues of A can be calculated by solving for λ in the following determinant: (a 11 λ) a a 1n (a 11 λ) a a 1n a det 21 (a 22 λ)... a 2n a..... = 21 (a 22 λ)... a 2n = 0.. a n1 a 2n... (a nn λ) a n1 a 2n... (a nn λ) There are n solutions for λ that will satisfy this equation. These solutions for λ are called the eigenvalues of the matrix A and are denoted as λ 1, λ 2,...,λ n. It is possible that not all values of λ are real and it is possible that λ i = λ j for i j. By convention, the eigenvalues are typically listed in ascending order, with λ 1 being the least and λ n being the greatest. Moreover, we have λ 1 λ 2... λ n 10

18 2.2.2 Eigenvectors Each eigenvalue of a matrix has a corresponding eigenvector, also sometimes called a characteristic vector. Mathematically, an eigenvector is a vector that, when multiplied on the left by its parent matrix, results in a vector that is a scale multiple of itself. Geometrically, an eigenvector represents a direction of elongation, where its corresponding eigenvalue represents the scale of said elongation. For example, if the parent matrix is a 2 2 matrix having an eigenvector a, then that vector is b pointing from the origin, (0, 0), to the point (a, b) in the two-dimensional cartesian plane. If the eigenvalues of an n n matrix A are represented as λ 1, λ 2,..., λ n, then the eigenvectors of A are represented as x 1, x 2,...,x n, where the eigenvector x i is an n 1 vector corresponding to the eigenvalue λ i. Mathematically, the eigenvectors x 1, x 2,...,x n are non-zero vectors that satisfy the equation Ax i = λ i x i. These eigenvectors x 1, x 2,...,x n are mutually orthogonal, meaning that x i x j for i j. It is common practice to normalize 8 these eigenvectors by dividing each vector by its length. Normalized eigenvectors are typically denoted as u 1, u 2,..., u n, where u i = x i x i. These u i now have the same direction representation as their corresponding x i, but with unit length. Therefore, since the directions of the unitized eigenvectors are the same as the directions of the original eigenvectors, the u i s are also mutually orthogonal. This means that u 1, u 2,...,u n form an orthonormal basis of R n. 8 Set length equal to one. 11

19 2.2.3 Eigen-Decomposition Once all n (not necessarily unique nor real-valued) eigenvalues have been equated for our n n matrix A and their corresponding n non-zero unit eigenvectors are constructed, the parent matrix A can be rewritten as a product of matrices represented by the eigenvalues and eigenvectors. Let us define the matrix D to be a diagonal 9 n n matrix made up of the eigenvalues λ 1, λ 2,...,λ n. Again, the eigenvalues are still in ascending order with λ 1 λ 2... λ n. Let us also define the matrix U to be the n n matrix made up of the unit eigenvectors u 1, u 2,...,u n where the i th column of U is the i th unit eigenvector u i. Then eigen-decomposition refers to the fact that the original parent matrix A can be rewritten as A = UDU 1 where U 1 denotes the inverse of the matrix U. 9 A diagonal matrix is a matrix with non-zero values on the principal diagonal and zeroes elsewhere. 12

20 CHAPTER III PROCRUSTES ANALYSIS A common practice in Shape Analysis is to compare a number of similar shapes to analyze the variability, or differences, among them. One issue that arises in this aspect of Shape Analysis is the issue of verifying that all shapes being examined are measured relative to each other. To measure the variability in a set of shapes, the shapes need to first be superimposed 1. However, to ensure that the shapes are all analyzed on the same scale, the shapes need to be aligned. One technique for aligning shapes is by using Procrustes 2 Superimposition. Procrustes superimposition has three main aspects to it: Translation, Uniform Scaling, and Rotation. Each of these facets are discussed in further detail below. Let us look at an example of two shapes we may wish to compare, depicted below in Figure 3.1. In each of the three steps below, we will alter these images to make the analysis more pleasant. Figure 3.1: Original Shapes 1 Superimposition is the process by which an image is layered on top of another image in order to obscure some or all of the initial image or analyze both images simultaneously. Here we are referring to the latter. 2 The name Procrustes comes from a fairly morbid story in Greek mythology. 13

21 3.1 Translation The first step in Procrustes analysis is to pick a number of landmark points 3 for each shape. These landmark points should correspond with the same feature among all shapes. The translation step consists of choosing one landmark corresponding to the same feature for each image and aligning those landmarks so that the images are now all overlaying. Procrustes uses the center of mass for each shape, as aligning all respective centers of mass would ensure that the shapes are all aligned at the center. Figure 3.2: Translated Shapes In Figure 3.2, above, we see that the center of each object was aligned at the origin. This is achieved by using the following algorithm: 1. For argument s sake, take k landmark points on an object (x 1, y 1 ), (x 2, y 2 ),..., (x k, y k ). 2. Then the mean of these k landmark points can be expressed as (x, y) where x and y are, respectively, the mean of the horizontal components of the shape and the mean of the vertical components of the shape and can be defined by and x = x 1 + x x k k y = y 1 + y y k. k 3 A landmark point is also sometimes simply referred to as a landmark. 14

22 These means can be used to center each shape at the origin by simply subtracting the horizontal component of the mean from the horizontal component of each landmark point and subtracting the vertical component of the mean from the vertical component of each landmark point. If (x, y) represents the geometric center 4 of a shape, then the k landmark points for that shape can now be translated and represented as (x 1 x, y 1 y), (x 2 x, y 2 y),...,(x k x, y k y). This procedure will keep the structure, configuration, and orientation of each shape intact, but the shape will now be centered at the origin. 3.2 Uniform Scaling Now that the shapes have each been translated so that they are individually centered at the origin, the shapes must now be analyzed at the same scale. Comparing shapes of different sizes can prove to be very difficult, while standardizing all shapes to have the same scale may be much less arduous. A common method to obtain a universal scale among all shapes is to standardize the shapes by removing each shape s scale component. This can be done by setting the root mean square 5 distance (RMSD) from the landmarks to the translated origin equal to 1. 4 The geometric center of a shape is also sometimes referred to as its centroid. 5 The root mean square is defined to be the square root of the mean of the squares of a set of numbers. 15

23 Figure 3.3: Scaled Shapes In Figure 3.3, above, we see that, while the shapes have not changed in structure, they have both been condensed down to a smaller scale. This is established by calculating the afore mentioned root mean square distance for each shape and dividing the coordinates of each landmark point by the root mean square distance for that shape. In other words, the procedure would resemble the following: 1. The root mean square distance for any two-dimensional shape with k landmark points can be calculated by (x1 x) s = 2 + (y 1 y) (x k x) 2 + (y k y) 2 k where (x 1, y 1 ),...,(x k, y k ) are the coordinates for the k landmark points for that shape. 2. The root mean square distance for a shape is now used to scale the shape by dividing each landmark point by that shape s RMSD. The scale for that shape then becomes standardized (equal to 1) when the coordinates for the landmark points are divided by the shape s initial scale. This is achieved by now having the k landmark points take coordinates ( x 1 x s, y 1 y s ), ( x 2 x s ), ( y 2 y s ),...,( x k x s, y k y s ). 16

24 3.3 Rotation The shapes have now been superimposed onto the same set of axes, translated so that they are all centered at the origin, and calibrated to such an extent that the scale of each image is equal to 1. All shapes are now centered at the origin and bear a common scale; however, there is still the issue of orientation: their rotational aspects need to be congruent. Particularly, the shapes need to all be oriented 6 in the same manner. In a euclidean space, a rotation matrix is used to rotate a shape an angle θ counter-clockwise. A rotation matrix is usually denoted by R(θ), where θ is the desired counter-clockwise rotation, and for a p-dimensional euclidean space is a p p matrix. When considering a two-dimensional euclidean space, the rotation matrix is rather straightforward: R(θ) = cos(θ) sin(θ) sin(θ) cos(θ) where, again, θ is the angle of desired counter-clockwise rotation. In the twodimensional case, the rotation matrix is implemented using the following algorithm: 1. Represent each landmark point (x i, y i ) as a column vector x i. y i 2. Then the vector x i y i = R(θ) x i = cos(θ) sin(θ) y i sin(θ) x i cos(θ) y i represents the rotated point (x i, y i ). 6 Here we are referring to rotational orientation. 17

25 Figure 3.4: Rotated Shapes The issue that then presents itself is the issue of rotational magnitude; it is not always easy to estimate how much rotation is needed from the shapes themselves. In Figure 3.4, above, both shapes have been rotated so that their major axes 7 align with the plot s horizontal axis. Rotating both shapes in this manner is convenient in practice, but the process can be much of a chore. One way to simplify this is to hold orientation constant in one shape and to rotate the ancillary shape to match its orientation to that of the first shape. If the k landmark points of the first shape are represented by (u 1, v 1 ), (u 2, v 2 ),..., (u k, v k ) and the k landmark points of the shape to be rotated are represented by (x 1, y 1 ), (x 2, y 2 ),...(x k, y k ), then, as before, rotation of the second shape by an angle θ gives the coordinates (x i, y 1) = (cosθ x i sinθ y i, sinθ x i + cosθ y i ) as its new, rotated landmark points. The squared Procrustes Distance between the two shapes can be denoted as d 2, where d 2 = (x 1 u 1 ) 2 + (y 1 v 1 ) 2 + (x 2 u 2 ) 2 + (y 2 v 2 ) (x k u k ) 2 + (y k v k ) 2. 7 A major axis of a elliptic shape refers to the axis along which the shape is the most elongated. The major axis of an ellipse passes through its foci. 18

26 Taking the derivative of d 2 and setting equal to zero to solve for θ gives ( k ) θ = tan 1 i=1 (x iv i y i u i ) k i=1 (x. iu i + y i v i ) This newly derived angle θ represents the optimum angle of rotation such that the sum of squared distances (SSD) between the two objects corresponding landmarks points is minimized. 19

27 CHAPTER IV BOOKSTEIN REGISTRATION Another way to superimpose a set of shapes for the purpose of analysis involves lining up exactly two landmarks that correspond to the same feature in each shape. Bookstein Registration (Bookstein, 1986: [21]) is the process by which the landmarks of an object are all first translated so that Landmark 1 is placed at the origin, which in a two-dimensional space is the point (0, 0), and second scaled and rotated so that Landmark 2 is placed at a point one unit away. In a two-dimensional space, the latter step usually involves aligning Landmark 2 at either the point (1, 0) or the point (0, 1). Chronically, the algorithm for Bookstein Registration is detailed below. 4.1 Translation The first phase of Bookstein Registration involves the translation of all shapes such that each respective Landmark 1 is put at the origin. The below examples deal with shapes in the two-dimensional space, so the origin will be defined as the point (0, 0). To establish the translation phase of Bookstein Registration, one would proceed as follows: 1. Select a landmark point common to all shapes to be centered at the origin. 2. Subtract that point s coordinates from the coordinates of all other landmarks for that shape. For instance, if a shape in a two-dimensional space has the k landmark points (x 1, y 1 ), (x 2, y 2 ),..., (x k, y k ) and the point (x 1, y 1 ) is the one to be placed at the origin, the new landmark points for this shape become (x 1 x 1, y 1 y 1 ), (x 2 x 1, y 2 y 1 ),..., (x k x 1, y k y 1 ). 20

28 Though, of course, (x 1 x 1, y 1 y 1 ) is equal to (0, 0). This will maintain the integrity of the structure of the shape, but it simply translates the shape so that the first landmark point lies at the point (0, 0). 4.2 Uniform Scaling The next phase of Bookstein Registration is concerning the scaling of the shapes such that Landmark 2 for each shape is placed at either the point (1, 0) or the point (0, 1) (in the two-dimensional case). In the application of this process discussed in a later section of this paper, the point (0, 1) is used for this phase. To scale the shapes in such a way that all of the second landmark points are aligned at the point (1, 0) but all of the first landmark point are still at the origin of (0, 0), we first need to scale the shapes such that the Landmark 2 points are one unit away from the origin, regardless of direction. The directional aspect will be controlled in the rotational phase of Bookstein Registration. The procedure involved in the scaling step is as follows: 1. Choose a second landmark point to serve as the scaling and rotation medium. This point will either be moved to the point (0, 1) or the point (1, 0) in the rotation phase, depending on personal preference and the problem at hand. As a reminder, the translated landmark points, before scaling, are: (x 1 x 1, y 1 y 1 ), (x 2 x 1, y 2 y 1 ),..., (x k x 1, y k y 1 ), or, in other words, (0, 0), (x 2 x 1, y 2 y 1 ),..., (x k x 1, y k y 1 ), where the first landmark point is located at the origin. 2. The coordinates of the now chosen Landmark 2 will be used to scale the landmark point in such a way that Landmark 2 is one unit away from Landmark 1 21

29 (the origin). This is done by dividing the coordinates of every landmark point by the distance between the first landmark point (at the origin) and the second landmark point. This results in the following landmark points: ( ) 0 (x, 0 2 x 1 ) 2 +(y 2 y 1 ) 2 (x 2 x 1 ) 2 +(y 2 y 1 ) 2, ( ) x 2 x 1 (x2 x 1 ) 2 + (y 2 y 1 ), y 2 y 1,... 2 (x2 x 1 ) 2 + (y 2 y 1 ) 2, ( ) x k x 1..., (x2 x 1 ) 2 + (y 2 y 1 ), y k y 1. (4.1) 2 (x2 x 1 ) 2 + (y 2 y 1 ) 2 However, again, we see that Landmark 1 is at the origin. Hence, the updated landmark points become ( (0, 0), ) x 2 x 1 (x, y 2 y 1 2 x 1 ) 2 +(y 2 y 1 ) 2 (x 2 x 1 ) 2 +(y 2 y 1 ) 2..., (,... ) x k x 1 (x2 x 1 ) 2 + (y 2 y 1 ), y k y 1. (4.2) 2 (x2 x 1 ) 2 + (y 2 y 1 ) 2 It can be confirmed here that the distance between the origin (Landmark 1 now) and the scaled Landmark 2 is equal to one unit. This proof can be quickly sketched as follows. Proof. ( ) 2 ( ) 2 d = x 2 x 1 (x2 x 1 ) 2 + (y 2 y 1 ) 0 y 2 y (x2 x 1 ) 2 + (y 2 y 1 ) 0 2 (x 2 x 1 ) = 2 (x 2 x 1 ) 2 + (y 2 y 1 ) + (y 2 y 1 ) 2 2 (x 2 x 1 ) 2 + (y 2 y 1 ) 2 (x 2 x 1 ) = 2 + (y 2 y 1 ) 2 (x 2 x 1 ) 2 + (y 2 y 1 ) = 1 = ±1 = 1 2 Note: The last equality follows because distance is a scalar and must always be 22

30 non-negative. 4.3 Rotation As was said earlier, the point (1, 0) is used for this phase of Bookstein Registration in the application portion of this paper (as opposed to the point (0, 1)). Therefore, this section will involve rotating the second landmark point such that it lies at the point (1, 0). Recall that Landmark 2 is now set exactly one unit away from Landmark 1, which lies at the origin, by virtue of the translation and scaling aspects of Bookstein Registration. It is true that for any two vectors u and v, the angle between them can be calculated by ( ) u v θ = sin 1, u v where the numerator is the dot product 1 of u and v and the denominator is the product of the norms, or lengths, of the two vectors. Using this information, we can easily calculate the angle of rotation needed such that Landmark 2 will lie at the point (1, 0) by letting u be the vector of coordinates of Landmark 2 and v be the vector of of coordinates of the point (1, 0). The same Rotation Matrix in two dimensions can again be utilized, once the angle of desired rotation is known, to place the second landmark point at the point (1, 0). matrix in two dimensions is the matrix R(θ) = cos(θ) sin(θ) sin(θ). cos(θ) As a reminder, the rotation Spelled out, the process by which to rotate the landmark points to achieve placement of Landmark 2 at the point (1, 0) is as follows: 1. Denote the coordinates of the newly translated and scaled second landmark point by (a, b). Then these coordinates can be represented as a column vector 1 Also sometimes called inner product. 23

31 as a. b 2. The angle of desired rotation can be calculated by ( ) θ = sin 1 a 1 + b 0 a 2 + b 2 = sin 1 (a), where the last equality holds because we already know that the length of Landmark 2 and the length of the point (1, 0) are both equal to Then the translated, scaled, and now rotated Landmark 2 can be represented by the coordinates (a, b ) where a and b are calculated by pre-multiplying the twodimensional rotation matrix upon the vector of coordinates (a, b). Thus, we have a = R(θ) a = cos(θ) b b sin(θ) sin(θ) a cos(θ) b where θ is the angle of rotation calculated in step 6. 24

32 CHAPTER V APPLICATION TO BASEBALL There have been numerous studies performed on baseball statistics for the purpose of finding the optimal balance between maximizing success and minimizing cost. Most notably, the 2003 novel Moneyball: The Art of Winning an Unfair Game, written by Michael Lewis, chronicles the unparallelled sabermetric approach taken by Billy Beane, general manager of Major League Baseball s Oakland Athletics, in the late 1990s and early 2000s. Professional baseball remains one of the only sports in the world to not have a standardized size in place for its playing domains. While there are standards that must be met in terms of distance between bases, etc., there are few rules in place regarding dimensions of outfield walls. The goal of this paper is, as the title may suggest, to model the affect ballpark dimensions and shape have on different performance statistics. To achieve this, the above techniques were applied: namely, Principal Component Analysis and Bookstein Registration. The scope of this experiment extends across all 30 ballparks in North America 1 in which one of the 30 Major League Baseball (MLB) teams plays their home games. We used data from the MLB regular season. Location of these ballparks varies considerably: stretching from San Diego, California to Toronto, Ontario, Canada and from Seattle, Washington to Miami, Florida. Because of the varying locations, climate is a sizable factor in performance. For instance, Coors Field, located in Denver and home to the Colorado Rockies, lies at an altitude of 5200 feet, almost a mile above sea-level. This means that the air around the stadium is relatively quite thin compared to other ballparks, which contributes to much less air resistance of in-flight baseballs and consequently more home runs. As evidenced in Figure 5.1, below, the Colorado Rockies have a larger-than-average 1 Out of these 30 ballparks, 29 are in the United States and 1 is in Canada. At one point, Canada had a second MLB team, located in Montreal, Quebec, but they were relocated to Washington, D.C. in

33 ballpark. Despite this, because of Coors Field s altitude, the Rockies hit the fourthmost home runs at home in the league from April of 2011 to October of 2015 (the span of the seasons, excluding playoffs). Figure 5.1: Coors Field vs. Mean Ballpark On the opposite side of the spectrum, ballparks in cities on the west coast of the United States (such as San Diego, San Francisco, and Seattle) are at much lower altitudes, which means that the air is much thicker and that baseballs experience more resistance in-flight. Factors such as jet-streams and wall height are also possible contributors for or against home-team success in terms of different performance aspects, but these will not be talked about here. Below, we are only considering the two-dimensional dimensions of each ballpark s outfield walls, without the height component. 5.1 Aligning the Ballparks After gathering images of all 30 ballparks from various sources, the first step is to outline the walls that define the playing field for each ballpark; we will choose to do this using MATLAB. However, since not all ballpark source images are necessarily on 26

34 the same scale, either Procrustes Analysis or Bookstein Registration is going to have to be used to align the ballparks so they are comparably analyzable. Figure 5.2: General Baseball Field Layout [52] Recall that the placement of the bases and the pitching mound relative to one another is standardized among all ballparks in Major League Baseball. This means that it would be easy to choose common landmarks among all 30 ballparks that would correspond to the same feature for each of the 30 shapes. A general layout of the bases on a baseball field is depicted above in Figure 5.2. Also recall that for Bookstein Registration to be performed, some standardized landmarks must be chosen. Therefore, since the bases are standardized among all ballparks, they would serve as an excellent means by which to run the Bookstein Registration; we decided to use third base as the translation medium and first base as the scaling medium. Since all ballpark images used to generate outlines were oriented in the same fashion, with home plate at the lowermost point of the diamond and second base directly above it, no rotational aspect was needed in the Bookstein Analysis. For each ballpark, before outlining the baseball field itself (which is what we are interested in), we must construct landmark points that we can use for our Bookstein 27

35 Analysis. As mentioned before, third base is going to be used for translation and first base for scaling; we are going to let Landmark 1 correspond to third base for each ballpark and Landmark 2 correspond to first base for each ballpark. After those two landmarks, the number of landmarks needed to sufficiently outline each ballpark is completely dependent upon that ballpark s shape and features. Since only the twodimensional coordinates of these ballparks are being used, each of the 30 ballparks will have landmark points (x i,1, y i,1 ), (x i,2, y i,2 ),..., (x i,ni, y i,ni ) where (x i,1, y i,1 ) are the coordinates third base for ballpark i and (x i,2, y i,2 ) are the coordinates first base for ballpark i. Also note that i = 1,..., 30 and n i denotes the number of landmarks used to trace ballpark i. Once we have these outlines for all 30 ballparks with the first and second landmark points corresponding to third base and first base, respectively, we can apply Bookstein Registration. Since no rotating of the ballparks is necessary, Bookstein Registration here involves only two steps: translation and scaling. For reference, the original 30 ballpark outlines (excluding the first two landmark points) are depicted in Figure 5.3, below. 28

36 Figure 5.3: Original 30 Ballpark Outlines Translation We will use third base (our first landmark) as our means for translating the ballparks by way of moving third to the origin, (0, 0). This is performed by subtracting the the coordinates of Landmark 1 for each ballpark from the coordinates of every other landmark for that ballpark. Recall that the unmeddled landmark points had coordinates of (x i,1, y i,1 ), (x i,2, y i,2 ),..., (x i,ni, y i,ni ). Then, subtracting the third base coordinates of ballpark i from each landmark point corresponding to ballpark i in both the x-direction and the y-direction, the translated landmarks become (x i,1 x i,1, y i,1 y i,1 ), (x i,2 x i,1, y i,2 y i,1 ),..., (x i,ni x i,1, y i,ni y i,1 ), 29

37 or (0, 0), (x i,2 x i,1, y i,2 y i,1 ),..., (x i,ni x i,1, y i,ni y i,1 ). Figure 5.4, below, shows the ballparks after being translated in such a way. Note that third base is now at the origin for all 30 ballparks, represented by a black diamond on the image. Figure 5.4: Translated Ballpark Outlines Scaling and Rotation The next step in Bookstein Registration is scaling the shapes so that they are all analyzed at the same scope. Recall that we will be using first base (Landmark 2 ) as our scaling mechanism. By the fact that all images used to trace the ballparks had the same orientation (with home plate at the bottom of the diamond) and that, by design, first base and third base are on the same level vertically, it is sufficient to rescale each ballpark so that first base is at the point (1, 0). We currently have third base (Landmark 1 ) set at the origin for all ballparks and we now want first base (Landmark 2 to be at the point (1, 0) for all ballparks. Thus, we must simply divide 30

38 the coordinates of each landmark for ballpark i by the translated coordinates of first base for ballpark i. Note that, hypothetically, since third base has a y-coordinate of zero for each ballpark, first base should have a y-coordinate of zero as well. Thus, before scaling, the landmark points for ballpark i are, theoretically, (0, 0), (x i,2 x i,1, 0), (x i,3 x i,1, y i,3 y i,1 ),..., (x i,ni x i,1, y i,ni y i,1 ). We see that incorporating the scaling component by dividing all landmark points by the current coordinates of first base is simply equivalent to diving the x-coordinates of all landmark points by the new x-coordinate of first base. This results in the following landmark points for ballpark i: ( ) ( ) ( ) xi,2 x i,1 xi,3 x i,1 xi,ni x i,1 (0, 0),, 0,, y i,3 y i,1,...,, y i,ni y i,1, x i,2 x i,1 x i,2 x i,1 x i,2 x i,1 or, equivalently, ( ) ( ) xi,3 x i,1 xi,ni x i,1 (0, 0), (1, 0),, y i,3 y i,1,...,, y i,ni y i,1, x i,2 x i,1 x i,2 x i,1 where (x i,j, y i,j ) represents the location of the j th landmark point of the i th ballpark. Here, i = 1,..., 30, j = 1,..., n i, and n i denotes the number of landmarks used for ballpark i. As mentioned, third base and first base are on the same plane vertically. Thus, rotation is not needed. 31

39 Figure 5.5: Translated and Scaled Ballpark Outlines This process completes the Bookstein Registration alignment of the ballparks. The final superimposed ballparks are depicted above in Figure 5.5, where the two diamonds denote third base (on the left) and first base (on the right) for each ballpark. 5.2 Obtaining Ballpark Outlines Since the general shape and dimensions of the field varies from ballpark to ballpark, some ballparks require more outlining landmarks than others to be ensure that unique features be properly conveyed. As shown below in Figure 5.6, some ballparks are very simple when it comes to dimensional features, while some are more sophisticated. In Figure 5.6, the ballpark outlined in black is U.S. Cellular Field, which is the home ballpark for the Chicago White Sox, and the ballpark outlined in red is Fenway Park, which is the home ballpark for the Boston Red Sox. While U.S. Cellular Field retains a very traditional shape 2, Fenway Park is more eccentric. With each ballpark requiring a seemingly different number of landmarks, it is not the case 2 Its shape is quite similar to the mean ballpark s. 32

40 that each landmark point corresponds to the same feature for each ballpark 3. Figure 5.6: Fenway Park vs. U.S. Cellular Field However, to compute the outline of the mean ballpark, we need common landmarks among the 30 ballparks so that we can take their average x-coordinate and their average y-coordinate. Some outlines required the use of as little as 36 landmark points, whereas some outlines required as many as 111, so taking their average coordinates would give us information that doesn t mean anything. However, a MATLAB program written by Chalani Prematilake of Texas Tech University nicely solves this problem. This program takes a column vector of points and interpolates k equallyspaced landmarks among those points, where k is specified by the user. This program was used to interpolate 100 equally-spaced landmarks for each ballpark, where each of the landmarks corresponded to the same feature for each. 5.3 Calculating the Mean Ballpark Ms. Prematilake s interpolation program, entitled INTP.m, requires input of a single column of one-dimensional data; however, since the landmark coordinates of 3 Other than the first two landmarks, which were used to superimpose the ballparks. 33

41 our ballparks lie in a two-dimensional coordinate system, they not one-dimensional. Thus, the two-dimensional coordinates of the landmark points must be somehow converted such that they are compatible with the interpolation. One way this can be done is by converting each coordinate pair to a single complex number, where the x-coordinate corresponds to the real component and the y-coordinate corresponds to the imaginary component. This maintains that the x-coordinates are read apart from the y-coordinates while formatting the data so that it can be inputted into the interpolation program. For instance, if a ballpark k had landmark points of (x k,1, y k,1 ), (x k,2, y k,2 ),..., (x k,nk, y k,nk ) after Bookstein Registration, the reformatted coordinates would become x k,1 + i y k,1, x k,2 + i y k,2,..., x k,nk + i y k,nk where i here represents the positive complex square root of 1. Once the landmark points for each ballpark are formatted in this way, we can use the INTP.m program to create n equally spaced landmarks along the ballpark outlines. Here we chose n = 100. The output from the program will again be complex numbers, but the coordinates can be reformatted reversely to obtain the traditional (x, y) coordinates. After all 30 ballpark outlines have been fitted with 100 equally spaced landmarks, we can now take the mean coordinates of the landmarks to calculate the mean ballpark shape. If the landmark shapes for ballpark k are (x 1,k, y 1,k ), (x 2,k, y 2,k ),..., (x 100,k, y 100,k ), then the mean ballpark would have landmarks of (x 1, y 1 ), (x 2, y 2 ),..., (x 100, y 100 ) where x k = i=1 x 1,i, k = 1, 2,..., 100. The mean ballpark can then be superimposed on top of the 30 ballparks, as shown in Figure 5.7, below. 34

42 Figure 5.7: 30 MLB Ballparks with Mean 5.4 Principal Component Analysis on the Ballparks Covariance Matrix To perform Principal Component Analysis on this ballpark data, we first need to construct a covariance matrix. Recall that a covariance matrix is a p p square matrix whose (i, j) th entry represents the covariance between the i th and j th random variables under consideration where i and j go from 1,..., p and p is the number of random variables. Here, the random variables we are concerned with are the 100 landmark points for each ballpark. But since these 100 landmark points each have an x-component and a y-component, we should analyze the covariance between each. This leaves us with 200 random variables (100 x-coordinates and 100 y-coordinates). Thus, the covariance matrix here will have dimensions Due to the vast size of this covariance matrix, it will not be included here. This is the matrix upon which we will perform Principal Component Analysis, which will result in 200 eigenvalues, 200 eigenvectors of dimension 200 1, and 200 principal components. A scree plot of these principal components is shown below in Figure 5.8. We see 35