Symbols and Notation
This page is a work in progress and minor changes will be made over time.
The most common symbols and notation used throughout this book are presented below; in rare cases where different meanings are intended within the book, this will be made clear.
Fonts, matrices, vectors
A lower-case letter in normal font, \(x\), refers to a single, fixed observation. When in bold font, a lower-case letter, \(\mathbf{x}\), refers to a vector of fixed observations, and an upper-case letter, \(\mathbf{X}\), represents a matrix. Calligraphic letters, \(\mathcal{X}\), are used to denote sets.
A matrix will always be defined with its dimensions using the notation, \(\mathbf{X}\in \mathcal{X}^{n \times p}\), or if for example \(\mathcal{X}\) is the set of Reals, it may be written as “\(\mathbf{X}\) is a \(n \times p\) Real-valued matrix”, analogously for integer-valued matrices etc. By default, a ‘vector’ will refer to a column vector, which may be thought of as a matrix with \(n\) rows and one column, and may be represented as:
\[ \mathbf{x}= \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \]
Vectors are usually defined using transpose notation, for example the vector above may instead be written as \(\mathbf{x}^\top= (x_1 \ x_2 \cdots x_n)\) or \(\mathbf{x}= (x_1 \ x_2 \cdots x_n)^\top\). Vectors may also be defined in a shortened format as \(\mathbf{x}\in \mathcal{X}^{n \times 1}\) or more simply \(\mathbf{x}\in \mathcal{X}^n\), which implies a column vector of length \(n\) with elements as represented above.
A letter in normal font with one subscript refers to a single element from a vector. For example, given \(\mathbf{x}\in \mathcal{X}^n\), the \(i\)th element is denoted \(x_i\). Given a matrix \(\mathbf{X}\in \mathcal{X}^{n \times p}\), a bold-face lower-case letter with a single subscript refers to the row of a matrix, for example the \(i\)th row would be \(\mathbf{x}_i = (x_{i;1} \ x_{i;2} \cdots x_{i;p})^\top\). Whereas a column is referenced with a semi-colon before the subscript, for example the \(j\)th column would be \(\mathbf{x}_{;j} = (x_{1;j} \ x_{2;j} \cdots x_{n;j})^\top\). Two subscripts can be used to reference a single element of a matrix, for example \(x_{i;j} \in \mathcal{X}\) would be the element in the \(i\)th row and \(j\)th column of \(\mathbf{X}\).
Functions
Typically, a ‘hat’, \(\hat{x}\), will refer to the prediction or estimation of a variable, \(x\), with bold-face used again to represent vectors. A ‘bar’, \(\bar{x}\), refers to the sample mean of \(\mathbf{x}\). Capital letters in normal font, \(X\), refer to scalar or vector random variables, which will be made clear from context. \(\mathbb{E}(X)\) is the expectation of the random variable \(X\). We write \(A \perp \!\!\! \perp B\), to denote that \(A\) and \(B\) are independent, i.e., that \(P(A \cap B) = P(A)P(B)\).
A function \(f\), will either be written as a formal map of domain to codomain, \(f: \mathcal{X}\rightarrow \mathcal{Y}; (x, y) \mapsto f(x, y)\) (which is most useful for understanding inputs and outputs), or more simply and commonly as \(f(x, y)\). Given a random variable, \(X\), following distribution \(\zeta\) (mathematically written \(X \sim \zeta\)), then \(f_X\) denotes the probability density function, and analogously for other distribution defining functions such as the cumulative distribution function, survival function, etc. In the survival analysis context (4 Survival Analysis), a subscript “\(0\)” refers to a “baseline” function, for example, \(S_0\) is the baseline survival function.
Finally, \(\exp\), refers to the exponential function, \(f(x) = e^x\), and \(\log\) refers to the natural logarithm \(\ln(x) = \log_e(x)\).
Variables and acronyms
Common variables and acronyms used in the book are given in Table 1 and Table 2 respectively.
| Variable | Definition |
|---|---|
| \(\mathbb{R}, \mathbb{R}_{>0}, \mathbb{R}_{\geq 0}, \bar{\mathbb{R}}\) | Set of Reals, positive Reals (excl. zero), non-negative Reals (incl. zero), and Reals including \(\pm\infty\). |
| \(\mathbb{N}_{> 0}\) | Set of Naturals excluding zero. |
| \((\mathbf{X}, \mathbf{t}, \boldsymbol{\delta})\) | Survival data where \(\mathbf{X}\in \mathbb{R}^{n \times p}\) is a real-valued matrix of \(n\) observations (rows) and \(p\) features (columns), \(\mathbf{t}\in \mathbb{R}^n\) is a vector of observed outcome times, and \(\boldsymbol{\delta}\in \mathbb{R}^n\) is a vector of observed outcome indicators. |
| \(\boldsymbol{\beta}\) | Vector of model coefficients/weights, \(\boldsymbol{\beta}\in \mathbb{R}^p\). |
| \(\boldsymbol{\eta}\) | Vector of linear predictors, \(\boldsymbol{\eta} = ({\eta}_1 \ {\eta}_2 \cdots {\eta}_{n})^\top\), where \(\boldsymbol{\eta}= \mathbf{X}\boldsymbol{\beta}\) and \(\eta_i = \mathbf{x}_{i}^\top\boldsymbol{\beta}\). |
| \(\mathcal{D}, \mathcal{D}_{train}, \mathcal{D}_{test}\) | Dataset, training data, and testing data. |
| Acronym | Definition |
|---|---|
| AFT | Accelerated Failure Time |
| cdf | Cumulative Distribution Function |
| chf | Cumulative Hazard Function |
| CPH | Cox Proportional Hazards |
| GBM | Gradient Boosting Machine |
| GLM | Generalised Linear Model |
| IPC(W) | Inverse Probability of Censoring (Weighted) |
| ML | Machine Learning |
| Probability Density Function | |
| PH | Proportional Hazards |
| (S)SVM | (Survival) Support Vector Machine |
| t.v.i. | Taking Values In |