Appendix F: Mathematical and Code Notation Guide

This appendix provides a comprehensive reference for all mathematical symbols, statistical notation, coordinate conventions, and code notation used throughout the textbook. For detailed derivations, see Appendix A (Mathematical Foundations).

F.1 Greek Letters

Symbol Name Meaning in This Text Primary Chapter
$\alpha$ Alpha Significance level; learning rate in gradient descent 3, 19
$\beta$ Beta Regression coefficient; Type II error probability 3, 19
$\gamma$ Gamma Discount factor; regularization parameter 9, 19
$\delta$ Delta Small change or difference; Kronecker delta 3
$\epsilon$ Epsilon Error term in regression; small positive constant 3, 19
$\zeta$ Zeta Noise term in state-space models 18
$\eta$ Eta Learning rate (alternative notation) 19
$\theta$ Theta Angle to goal; generic model parameter 7, 19
$\lambda$ Lambda Poisson rate parameter; hazard rate in survival analysis; regularization strength 3, 19, 26
$\mu$ Mu Population mean; expected value 3
$\nu$ Nu Degrees of freedom 3
$\pi$ Pi Mathematical constant; policy function in RL --
$\rho$ Rho Correlation coefficient 3
$\sigma$ Sigma Standard deviation; sigmoid function 3, 19
$\sigma^2$ Sigma squared Variance 3
$\tau$ Tau Time lag; Kendall's rank correlation 3, 18
$\phi$ Phi Basis function; probability density of standard normal 19
$\chi^2$ Chi-squared Chi-squared distribution or test statistic 3
$\psi$ Psi Generic function symbol --
$\omega$ Omega Angular frequency; event space 3

F.2 Statistical Notation

Probability and Random Variables

Notation Meaning
$P(A)$ Probability of event $A$
$P(A \mid B)$ Conditional probability of $A$ given $B$
$P(A \cap B)$ Probability of both $A$ and $B$
$P(A \cup B)$ Probability of $A$ or $B$ (or both)
$X \sim \text{Normal}(\mu, \sigma^2)$ Random variable $X$ follows a normal distribution
$X \sim \text{Poisson}(\lambda)$ Random variable $X$ follows a Poisson distribution
$X \sim \text{Bernoulli}(p)$ Random variable $X$ follows a Bernoulli distribution
$X \sim \text{Binomial}(n, p)$ Random variable $X$ follows a binomial distribution
$\mathbb{E}[X]$ Expected value (mean) of $X$
$\text{Var}(X)$ Variance of $X$
$\text{Cov}(X, Y)$ Covariance of $X$ and $Y$
$\text{Corr}(X, Y)$ Pearson correlation coefficient

Estimation and Inference

Notation Meaning
$\hat{\theta}$ Estimator or estimated value of parameter $\theta$
$\bar{x}$ Sample mean of $x$
$s$ or $s^2$ Sample standard deviation or variance
$n$ Sample size (number of observations)
$p$ Number of features (predictors)
$H_0$ Null hypothesis
$H_1$ or $H_a$ Alternative hypothesis
$p\text{-value}$ Probability of observing data at least as extreme as observed, given $H_0$
$\text{CI}_{95\%}$ 95% confidence interval
$t_{\alpha/2, n-1}$ Critical value of the $t$-distribution
$z_{\alpha/2}$ Critical value of the standard normal distribution

Regression and Machine Learning

Notation Meaning
$y = \beta_0 + \beta_1 x_1 + \cdots + \beta_p x_p + \epsilon$ Linear regression model
$\text{logit}(p) = \log\frac{p}{1-p}$ Log-odds (logit) function
$\sigma(z) = \frac{1}{1 + e^{-z}}$ Sigmoid (logistic) function
$\hat{y}$ Predicted value
$R^2$ Coefficient of determination
$\text{MSE} = \frac{1}{n}\sum(y_i - \hat{y}_i)^2$ Mean squared error
$\text{LogLoss} = -\frac{1}{n}\sum[y_i \log \hat{p}_i + (1-y_i)\log(1-\hat{p}_i)]$ Binary cross-entropy loss
$\text{AUC}$ Area under the ROC curve

F.3 Linear Algebra Notation

Notation Meaning
$\mathbf{x}$ Column vector (bold lowercase)
$\mathbf{A}$ Matrix (bold uppercase)
$\mathbf{x}^\top$ Transpose of vector $\mathbf{x}$
$\mathbf{A}^{-1}$ Inverse of matrix $\mathbf{A}$
$\|\mathbf{x}\| = \|\mathbf{x}\|_2$ Euclidean ($L^2$) norm
$\|\mathbf{x}\|_1$ Manhattan ($L^1$) norm
$\mathbf{x} \cdot \mathbf{y}$ or $\mathbf{x}^\top \mathbf{y}$ Dot product
$\mathbf{I}$ Identity matrix
$\text{diag}(\mathbf{v})$ Diagonal matrix with vector $\mathbf{v}$ on the diagonal
$\text{det}(\mathbf{A})$ Determinant of $\mathbf{A}$
$\text{tr}(\mathbf{A})$ Trace of $\mathbf{A}$ (sum of diagonal elements)

F.4 Pitch Coordinate Conventions

Standard Coordinate System (StatsBomb / This Textbook)

Parameter Value Unit
Origin Bottom-left corner of the pitch --
$x$-axis Along the length of the pitch (goal-to-goal) yards
$y$-axis Along the width of the pitch yards
$x$ range $[0, 120]$ yards
$y$ range $[0, 80]$ yards
Attacking direction Left to right (increasing $x$) --
Goal center $(120, 40)$ for the attacking goal yards
Penalty spot $(108, 40)$ yards
Center spot $(60, 40)$ yards
Goal width 8 yards (9.32 meters) yards
Goal post positions $(120, 36)$ and $(120, 44)$ yards
Penalty area $x \in [102, 120]$, $y \in [18, 62]$ yards
Six-yard box $x \in [114, 120]$, $y \in [30, 50]$ yards
Center circle radius 10 yards (11.15 meters) yards

Metric Coordinate System (Alternative)

Some data providers use meters with a $105 \times 68$ pitch:

Parameter Yards System Meters System
Pitch length 120 105
Pitch width 80 68
Goal width 8 9.32
Penalty spot distance 12 12.97

Conversion: $x_{\text{meters}} = x_{\text{yards}} \times \frac{105}{120}$, $y_{\text{meters}} = y_{\text{yards}} \times \frac{68}{80}$

Common Pitch Zones

Zone $x$ Range (yards) $y$ Range (yards) Description
Defensive third $[0, 40]$ $[0, 80]$ Own third
Middle third $[40, 80]$ $[0, 80]$ Midfield
Attacking third $[80, 120]$ $[0, 80]$ Final third
Left channel $[0, 120]$ $[0, 27]$ Left flank
Central channel $[0, 120]$ $[27, 53]$ Center
Right channel $[0, 120]$ $[53, 80]$ Right flank
Half-space (left) $[0, 120]$ $[18, 30]$ Left half-space
Half-space (right) $[0, 120]$ $[50, 62]$ Right half-space
Zone 14 $[80, 102]$ $[24, 56]$ Central area in front of the box

F.5 Key Formulas

Distance to Goal Center

$$d = \sqrt{(x - 120)^2 + (y - 40)^2}$$

Angle to Goal

$$\theta = \arctan\left(\frac{9.32 \cdot (120 - x)}{(120 - x)^2 + (y - 40)^2 - 3.66^2}\right)$$

Expected Points (Poisson Model)

$$\text{xPts} = 3 \cdot P(\text{win}) + 1 \cdot P(\text{draw})$$

where $P(\text{win}) = \sum_{g_h > g_a} \text{Poisson}(g_h; xG) \cdot \text{Poisson}(g_a; xGA)$

Acute:Chronic Workload Ratio

$$\text{ACWR} = \frac{\text{Rolling mean}_{7\text{d}}}{\text{EWMA}_{28\text{d}}}$$

Cosine Similarity

$$\text{sim}(\mathbf{a}, \mathbf{b}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \cdot \|\mathbf{b}\|}$$

PPDA

$$\text{PPDA} = \frac{\text{Opponent passes allowed (own half)}}{\text{Defensive actions (opponent half)}}$$

F.6 Code Notation Conventions

Python Naming

Convention Usage Example
snake_case Variables, functions, modules shot_distance, compute_xg()
PascalCase Classes PlayerPose, EthicsAssessment
UPPER_SNAKE_CASE Constants FEATURE_COLS, MAX_GOALS
_leading_underscore Private/internal functions _classify_buildup_speed()

Type Hint Conventions

Type Meaning Example
int, float, str, bool Scalar types n_splits: int = 5
List[str] List of strings feature_cols: List[str]
Dict[str, float] Dictionary with string keys weights: Dict[str, float]
Tuple[float, float] Fixed-size tuple position: Tuple[float, float]
Optional[str] Value or None output_path: Optional[str] = None
pd.DataFrame Pandas DataFrame df: pd.DataFrame
np.ndarray NumPy array X: np.ndarray

DataFrame Column Naming

Pattern Example Meaning
metric_p90 goals_p90, pressing_p90 Per-90 normalized metric
metric_pct pass_completion_pct Percentage metric
is_condition is_goal, is_header Binary indicator (0/1)
metric_rolling ppda_rolling Rolling average
metric_norm xg_norm Normalized to [0, 1]

F.7 Abbreviations

Abbreviation Full Term
ACWR Acute:Chronic Workload Ratio
AUC Area Under the Curve
CV Cross-Validation or Coefficient of Variation
ECE Expected Calibration Error
EWMA Exponentially Weighted Moving Average
FPR False Positive Rate
GDPR General Data Protection Regulation
GNN Graph Neural Network
GPS Global Positioning System
KDE Kernel Density Estimation
KS Kolmogorov-Smirnov (test)
LLM Large Language Model
MSE Mean Squared Error
NIRS Near-Infrared Spectroscopy
PCA Principal Component Analysis
PCI Player Contribution Index
PPDA Passes Per Defensive Action
ROC Receiver Operating Characteristic
TPR True Positive Rate
VAEP Valuing Actions by Estimating Probabilities
xA Expected Assists
xG Expected Goals
xGA Expected Goals Against
xPts Expected Points
xT Expected Threat