Appendix F: Notation Guide

This appendix provides a comprehensive reference for all mathematical, statistical, and basketball-specific notation used throughout this textbook.

F.1 General Mathematical Notation

Basic Symbols

Symbol Meaning Example
$=$ Equals $a = b$
$\neq$ Not equal to $a \neq b$
$\approx$ Approximately equal $\pi \approx 3.14$
$\equiv$ Identically equal, defined as $TS\% \equiv \frac{PTS}{2(FGA + 0.44 \cdot FTA)}$
$<, >$ Less than, greater than $x < 5$
$\leq, \geq$ Less than or equal, greater than or equal $n \geq 30$
$\pm$ Plus or minus $\bar{x} \pm 1.96 \cdot SE$
$\propto$ Proportional to $y \propto x$
$\infty$ Infinity $\lim_{n \to \infty}$

Arithmetic Operations

Symbol Meaning Example
$+, -, \times, \div$ Basic arithmetic $a + b$, $a \times b$
$\cdot$ Multiplication (dot notation) $a \cdot b$
$a^n$ Exponentiation $x^2$
$\sqrt{x}$, $\sqrt[n]{x}$ Square root, nth root $\sqrt{4} = 2$
$|x|$ Absolute value $|-3| = 3$
$\lfloor x \rfloor$ Floor function (round down) $\lfloor 3.7 \rfloor = 3$
$\lceil x \rceil$ Ceiling function (round up) $\lceil 3.2 \rceil = 4$
$\log x$ Logarithm (base 10 or natural) $\log 100 = 2$
$\ln x$ Natural logarithm (base $e$) $\ln e = 1$
$e^x$ or $\exp(x)$ Exponential function $e^{0} = 1$

Set Notation

Symbol Meaning Example
$\{a, b, c\}$ Set containing elements $\{PG, SG, SF, PF, C\}$
$\in$ Element of $x \in \mathbb{R}$
$\notin$ Not an element of $x \notin \emptyset$
$\subseteq$ Subset of $A \subseteq B$
$\cup$ Union $A \cup B$
$\cap$ Intersection $A \cap B$
$\emptyset$ or $\{\}$ Empty set $A \cap B = \emptyset$
$\mathbb{R}$ Set of real numbers $x \in \mathbb{R}$
$\mathbb{R}^n$ n-dimensional real space $\mathbf{x} \in \mathbb{R}^5$
$\mathbb{Z}$ Set of integers $n \in \mathbb{Z}$
$\mathbb{N}$ Set of natural numbers $n \in \mathbb{N}$

Summation and Product

Symbol Meaning Example
$\sum_{i=1}^{n} x_i$ Sum from $i=1$ to $n$ $\sum_{i=1}^{n} x_i = x_1 + x_2 + \cdots + x_n$
$\prod_{i=1}^{n} x_i$ Product from $i=1$ to $n$ $\prod_{i=1}^{n} x_i = x_1 \times x_2 \times \cdots \times x_n$
$\displaystyle\sum$ Display-style summation Used in equations

F.2 Linear Algebra Notation

Vectors

Symbol Meaning Example
$\mathbf{x}$ or $\vec{x}$ Column vector (bold/arrow) $\mathbf{x} = (x_1, x_2, \ldots, x_n)^T$
$\mathbf{x}^T$ Transpose of vector Row vector
$x_i$ The $i$th element of vector $\mathbf{x}$ $x_3$ is the third element
$\|\mathbf{x}\|$ Euclidean norm (L2 norm) $\|\mathbf{x}\| = \sqrt{\sum x_i^2}$
$\|\mathbf{x}\|_1$ L1 norm (Manhattan distance) $\|\mathbf{x}\|_1 = \sum |x_i|$
$\|\mathbf{x}\|_\infty$ L-infinity norm (max norm) $\|\mathbf{x}\|_\infty = \max_i |x_i|$
$\mathbf{x} \cdot \mathbf{y}$ Dot product $\mathbf{x} \cdot \mathbf{y} = \sum x_i y_i$
$\mathbf{0}$ Zero vector $(0, 0, \ldots, 0)^T$
$\mathbf{1}$ Vector of ones $(1, 1, \ldots, 1)^T$

Matrices

Symbol Meaning Example
$\mathbf{A}$ or $A$ Matrix (bold capital) $\mathbf{A} \in \mathbb{R}^{m \times n}$
$a_{ij}$ or $A_{ij}$ Element in row $i$, column $j$ $a_{23}$ is row 2, column 3
$\mathbf{A}^T$ Transpose of matrix $(A^T)_{ij} = a_{ji}$
$\mathbf{A}^{-1}$ Inverse of matrix $\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}$
$\mathbf{I}$ or $\mathbf{I}_n$ Identity matrix $n \times n$ identity
$\det(\mathbf{A})$ or $|\mathbf{A}|$ Determinant Scalar value
$\text{tr}(\mathbf{A})$ Trace (sum of diagonal) $\text{tr}(\mathbf{A}) = \sum a_{ii}$
$\text{rank}(\mathbf{A})$ Rank of matrix Number of linearly independent rows/columns
$\mathbf{AB}$ Matrix multiplication $(\mathbf{AB})_{ij} = \sum_k a_{ik}b_{kj}$
$\mathbf{A} \odot \mathbf{B}$ Element-wise (Hadamard) product $(A \odot B)_{ij} = a_{ij} \cdot b_{ij}$

Eigenvalues and Eigenvectors

Symbol Meaning
$\lambda$ Eigenvalue
$\mathbf{v}$ Eigenvector
$\mathbf{A}\mathbf{v} = \lambda\mathbf{v}$ Eigenvalue equation
$\mathbf{\Lambda}$ Diagonal matrix of eigenvalues
$\mathbf{V}$ Matrix of eigenvectors

F.3 Probability and Statistics Notation

Probability

Symbol Meaning Example
$P(A)$ Probability of event $A$ $P(\text{Made Shot}) = 0.45$
$P(A|B)$ Conditional probability $P(\text{Made}|\text{Open})$
$P(A \cap B)$ Probability of $A$ and $B$ Joint probability
$P(A \cup B)$ Probability of $A$ or $B$ $P(A) + P(B) - P(A \cap B)$
$P(A^c)$ or $P(\bar{A})$ Probability of not $A$ $1 - P(A)$
$\Omega$ Sample space Set of all outcomes

Random Variables

Symbol Meaning Example
$X, Y, Z$ Random variables (capitals) $X = $ Points scored
$x, y, z$ Specific values (lowercase) $x = 25$
$p(x)$ or $P(X = x)$ Probability mass function (PMF) Discrete
$f(x)$ Probability density function (PDF) Continuous
$F(x)$ or $P(X \leq x)$ Cumulative distribution function (CDF) $F(x) = \int_{-\infty}^{x} f(t) dt$

Expectation and Moments

Symbol Meaning Formula
$E[X]$ or $\mathbb{E}[X]$ Expected value (mean) $E[X] = \sum_x x \cdot p(x)$
$\mu$ Population mean $\mu = E[X]$
$\bar{x}$ or $\bar{X}$ Sample mean $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$
$\text{Var}(X)$ Variance $E[(X - \mu)^2]$
$\sigma^2$ Population variance $\sigma^2 = \text{Var}(X)$
$s^2$ Sample variance $s^2 = \frac{1}{n-1}\sum(x_i - \bar{x})^2$
$\sigma$ Population standard deviation $\sigma = \sqrt{\sigma^2}$
$s$ Sample standard deviation $s = \sqrt{s^2}$
$\text{SD}(X)$ or $\sigma_X$ Standard deviation $\sqrt{\text{Var}(X)}$
$\text{SE}$ Standard error $SE = \frac{s}{\sqrt{n}}$
$\text{Cov}(X, Y)$ Covariance $E[(X - \mu_X)(Y - \mu_Y)]$
$\rho_{XY}$ or $\text{Corr}(X, Y)$ Correlation coefficient $\frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$
$r$ Sample correlation Estimated $\rho$

Common Distributions

Notation Distribution Parameters
$X \sim N(\mu, \sigma^2)$ Normal Mean $\mu$, variance $\sigma^2$
$Z \sim N(0, 1)$ Standard normal Mean 0, variance 1
$X \sim \text{Binomial}(n, p)$ Binomial Trials $n$, probability $p$
$X \sim \text{Poisson}(\lambda)$ Poisson Rate $\lambda$
$X \sim \text{Uniform}(a, b)$ Uniform Range $[a, b]$
$X \sim \text{Exponential}(\lambda)$ Exponential Rate $\lambda$
$X \sim t_{df}$ Student's t Degrees of freedom $df$
$X \sim \chi^2_{df}$ Chi-square Degrees of freedom $df$
$X \sim F_{df_1, df_2}$ F-distribution Numerator and denominator df
$X \sim \text{Beta}(\alpha, \beta)$ Beta Shape parameters $\alpha, \beta$

Hypothesis Testing

Symbol Meaning
$H_0$ Null hypothesis
$H_1$ or $H_a$ Alternative hypothesis
$\alpha$ Significance level (Type I error rate)
$\beta$ Type II error rate
$1 - \beta$ Statistical power
$p$-value Probability of observing result under $H_0$
$z$, $t$, $\chi^2$, $F$ Test statistics
$z_{\alpha/2}$ Critical z-value for significance level $\alpha$
$t_{\alpha/2, df}$ Critical t-value

Confidence Intervals

Symbol Meaning
$CI$ Confidence interval
$(1 - \alpha)$ Confidence level
$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$ CI for mean (known $\sigma$)
$\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$ CI for mean (unknown $\sigma$)

F.4 Regression Notation

Linear Regression

Symbol Meaning
$Y$ Dependent variable (response)
$X$ Independent variable (predictor)
$\beta_0$ Intercept (population)
$\beta_1, \ldots, \beta_p$ Slope coefficients (population)
$\hat{\beta}_0, \hat{\beta}_1$ Estimated coefficients
$\varepsilon$ or $\epsilon$ Error term
$\hat{Y}$ or $\hat{y}$ Predicted value
$e_i = y_i - \hat{y}_i$ Residual
$Y = \beta_0 + \beta_1 X + \varepsilon$ Simple linear regression model
$Y = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}$ Matrix form
$\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}$ OLS estimator

Model Evaluation

Symbol Meaning Formula
$R^2$ Coefficient of determination $1 - \frac{SS_{res}}{SS_{tot}}$
$R^2_{adj}$ Adjusted R-squared $1 - \frac{(1-R^2)(n-1)}{n-p-1}$
$SS_{tot}$ Total sum of squares $\sum(y_i - \bar{y})^2$
$SS_{reg}$ Regression sum of squares $\sum(\hat{y}_i - \bar{y})^2$
$SS_{res}$ Residual sum of squares $\sum(y_i - \hat{y}_i)^2$
$MSE$ Mean squared error $\frac{SS_{res}}{n - p - 1}$
$RMSE$ Root mean squared error $\sqrt{MSE}$
$MAE$ Mean absolute error $\frac{1}{n}\sum|y_i - \hat{y}_i|$

Regularization

Symbol Meaning
$\lambda$ Regularization parameter (penalty)
Ridge: $\sum(y_i - \hat{y}_i)^2 + \lambda\sum\beta_j^2$ L2 penalty
LASSO: $\sum(y_i - \hat{y}_i)^2 + \lambda\sum|\beta_j|$ L1 penalty
Elastic Net: L1 + L2 combined Mixed penalty
$\alpha$ Elastic net mixing parameter

Logistic Regression

Symbol Meaning
$p = P(Y = 1|X)$ Probability of success
$\text{logit}(p) = \log\frac{p}{1-p}$ Log-odds (logit function)
$\text{odds} = \frac{p}{1-p}$ Odds ratio
$\sigma(z) = \frac{1}{1 + e^{-z}}$ Sigmoid function

F.5 Machine Learning Notation

General

Symbol Meaning
$\mathbf{X}$ Feature matrix (design matrix)
$\mathbf{y}$ Target vector
$n$ Number of observations (samples)
$p$ or $d$ Number of features (dimensions)
$\mathcal{D}$ Dataset
$\mathcal{D}_{train}$, $\mathcal{D}_{test}$ Training and test sets
$\hat{f}$ or $h$ Learned function (hypothesis)
$\theta$ or $\mathbf{w}$ Model parameters (weights)
$\eta$ Learning rate
$J(\theta)$ or $L(\theta)$ Loss/cost function
$\nabla J$ Gradient of loss function

Cross-Validation

Symbol Meaning
$k$ Number of folds
$k$-fold CV k-fold cross-validation
LOOCV Leave-one-out cross-validation

Clustering

Symbol Meaning
$K$ Number of clusters
$\mu_k$ Centroid of cluster $k$
$c_i$ Cluster assignment for observation $i$

F.6 Basketball Statistics Notation

Traditional Box Score Statistics

Symbol Meaning Notes
$PTS$ Points Total points scored
$REB$ Rebounds ORB + DRB
$ORB$ Offensive rebounds
$DRB$ Defensive rebounds
$AST$ Assists
$STL$ Steals
$BLK$ Blocks
$TOV$ Turnovers
$PF$ Personal fouls
$FGM$ Field goals made
$FGA$ Field goals attempted
$FG\%$ Field goal percentage $\frac{FGM}{FGA}$
$3PM$ or $FG3M$ Three-pointers made
$3PA$ or $FG3A$ Three-pointers attempted
$3P\%$ or $FG3\%$ Three-point percentage $\frac{3PM}{3PA}$
$FTM$ Free throws made
$FTA$ Free throws attempted
$FT\%$ Free throw percentage $\frac{FTM}{FTA}$
$MIN$ or $MP$ Minutes played
$GP$ Games played
$GS$ Games started

Rate Statistics

Symbol Meaning Formula
$PPG$ Points per game $\frac{PTS}{GP}$
$RPG$ Rebounds per game $\frac{REB}{GP}$
$APG$ Assists per game $\frac{AST}{GP}$
$SPG$ Steals per game $\frac{STL}{GP}$
$BPG$ Blocks per game $\frac{BLK}{GP}$
$MPG$ Minutes per game $\frac{MIN}{GP}$
Per 36 Statistics per 36 minutes $\frac{Stat}{MIN} \times 36$
Per 100 Statistics per 100 possessions Pace-adjusted

Efficiency Metrics

Symbol Meaning Formula
$TS\%$ True Shooting Percentage $\frac{PTS}{2(FGA + 0.44 \cdot FTA)}$
$eFG\%$ Effective Field Goal Percentage $\frac{FGM + 0.5 \cdot 3PM}{FGA}$
$ORtg$ Offensive Rating Points per 100 possessions
$DRtg$ Defensive Rating Points allowed per 100 possessions
$NetRtg$ Net Rating $ORtg - DRtg$
$PER$ Player Efficiency Rating See Chapter 9
$USG\%$ Usage Rate $\frac{(FGA + 0.44 \cdot FTA + TOV) \cdot TmMP/5}{MP \cdot (TmFGA + 0.44 \cdot TmFTA + TmTOV)}$

Rebounding Percentages

Symbol Meaning Formula
$ORB\%$ Offensive Rebound Percentage $\frac{ORB \cdot TmMP/5}{MP \cdot (TmORB + OppDRB)}$
$DRB\%$ Defensive Rebound Percentage $\frac{DRB \cdot TmMP/5}{MP \cdot (TmDRB + OppORB)}$
$TRB\%$ Total Rebound Percentage

Playmaking and Defensive Percentages

Symbol Meaning
$AST\%$ Assist Percentage
$TOV\%$ Turnover Percentage
$STL\%$ Steal Percentage
$BLK\%$ Block Percentage

Plus-Minus Metrics

Symbol Meaning Notes
$+/-$ Raw plus-minus Point differential on court
$RAPM$ Regularized Adjusted Plus-Minus Ridge regression adjusted
$BPM$ Box Plus-Minus Box score estimate
$OBPM$ Offensive Box Plus-Minus Offensive component
$DBPM$ Defensive Box Plus-Minus Defensive component
$VORP$ Value Over Replacement Player $(BPM + 2.0) \cdot \frac{MP}{3936}$

Win-Based Metrics

Symbol Meaning
$WS$ Win Shares
$OWS$ Offensive Win Shares
$DWS$ Defensive Win Shares
$WS/48$ Win Shares per 48 minutes
$WAR$ Wins Above Replacement

Team Statistics

Symbol Meaning
$Tm$ prefix Team total (e.g., $TmFGA$)
$Opp$ prefix Opponent total (e.g., $OppFGA$)
$Pace$ Possessions per 48 minutes
$Poss$ Possessions

F.7 Subscript and Superscript Conventions

Common Subscripts

Subscript Meaning Example
$i$ Observation/player index $x_i$ is the $i$th player
$j$ Feature/variable index $x_{ij}$ is player $i$, stat $j$
$t$ Time period $x_t$ at time $t$
$k$ Cluster/group index Cluster $k$
$\text{home}$, $\text{away}$ Game location $PTS_{\text{home}}$

Common Superscripts

Superscript Meaning Example
$T$ Transpose $\mathbf{A}^T$
$-1$ Inverse $\mathbf{A}^{-1}$
$(i)$ The $i$th iteration $\theta^{(i)}$
$*$ Optimal value $\theta^*$

F.8 Greek Letter Reference

Lowercase Greek Letters

Letter Name Common Uses
$\alpha$ alpha Significance level, learning rate, regularization mixing
$\beta$ beta Regression coefficients, Type II error rate
$\gamma$ gamma Discount factor, various parameters
$\delta$ delta Small change, Kronecker delta
$\epsilon$ epsilon Error term, small positive number
$\zeta$ zeta Various parameters
$\eta$ eta Learning rate
$\theta$ theta General parameter
$\iota$ iota Rarely used
$\kappa$ kappa Condition number
$\lambda$ lambda Eigenvalue, regularization parameter, Poisson rate
$\mu$ mu Population mean
$\nu$ nu Degrees of freedom
$\xi$ xi Random variable
$\pi$ pi Probability, proportion (also 3.14159...)
$\rho$ rho Correlation coefficient, autocorrelation
$\sigma$ sigma Standard deviation
$\tau$ tau Various parameters, Kendall's tau
$\upsilon$ upsilon Rarely used
$\phi$ phi Normal PDF, various functions
$\chi$ chi Chi-square related
$\psi$ psi Various functions
$\omega$ omega Angular frequency

Uppercase Greek Letters

Letter Name Common Uses
$\Gamma$ Gamma Gamma function
$\Delta$ Delta Change, difference
$\Theta$ Theta Parameter space
$\Lambda$ Lambda Diagonal matrix of eigenvalues
$\Xi$ Xi Rarely used
$\Pi$ Pi Product operator
$\Sigma$ Sigma Summation, covariance matrix
$\Phi$ Phi Normal CDF
$\Psi$ Psi Various functions
$\Omega$ Omega Sample space, big-O notation

F.9 Abbreviations Reference

Statistical Abbreviations

Abbreviation Full Term
CI Confidence Interval
CLT Central Limit Theorem
CDF Cumulative Distribution Function
CV Cross-Validation, Coefficient of Variation
df Degrees of Freedom
IID Independent and Identically Distributed
LLN Law of Large Numbers
MLE Maximum Likelihood Estimation
MSE Mean Squared Error
OLS Ordinary Least Squares
PDF Probability Density Function
PMF Probability Mass Function
RMSE Root Mean Squared Error
RSS Residual Sum of Squares
SE Standard Error
SD Standard Deviation

Machine Learning Abbreviations

Abbreviation Full Term
AUC Area Under the Curve
CNN Convolutional Neural Network
DT Decision Tree
GB Gradient Boosting
KNN K-Nearest Neighbors
LASSO Least Absolute Shrinkage and Selection Operator
NN Neural Network
PCA Principal Component Analysis
RF Random Forest
ROC Receiver Operating Characteristic
RNN Recurrent Neural Network
SVM Support Vector Machine

Basketball Abbreviations

See Section F.6 for comprehensive basketball statistics abbreviations.

This notation guide serves as a quick reference for symbols and conventions used throughout the textbook. When in doubt, context and chapter-specific definitions should take precedence.