# } END

#### Abbreviations

a.k.a. == also known as

w.r.t. == with respect to

s.t. == such that

iff == if and only if

e.g. == for example (Latin: exempli gratia)

i.e. == that is (Latin: id est)

etc. == and so forth (Latin: et cetera)

AI == artificial intelligence

API == application programming interface

GA == genetic algorithm

GLM == generalised linear model

ML == machine learning

NN == neural network

IDE = integrated development environment

#### Notation Convention – Logic and Set Theory

$$\forall$$ – for all

$$\exists$$ – exists

By writing $$x \in \{a, b, c\}$$ we mean that “$$x$$ is in a set that consists of $$a$$, $$b$$ and $$c$$” or “$$x$$ is either $$a$$, $$b$$ or $$c$$

$$A\subseteq B$$ – set $$A$$ is a subset of set $$B$$ (every element in $$A$$ belongs to $$B$$, $$x\in A$$ implies that $$x\in B$$)

$$A\cup B$$ – union (sum) of two sets, $$x\in A\cup B$$ iff $$x\in A$$ or $$x\in B$$

$$A\cap B$$ – intersection (sum) of two sets, $$x\in A\cap B$$ iff $$x\in A$$ and $$x\in B$$

$$A\setminus B$$ – difference of two sets, $$x\in A\setminus B$$ iff $$x\in A$$ and $$x\not\in B$$

$$A\times B$$ – Cartesian product of two sets, $$A\times B = \{ (a,b): a\in A, b\in B \}$$

$$A^p = A\times A \times \dots\times A$$ ($$p$$ times) for any $$p$$

#### Notation Convention – Symbols

$$\mathbf{X,Y,A,I,C}$$ – bold (I use it for denoting vectors and matrices)

$$\mathbb{X,Y,A,I,C}$$ – blackboard bold (I sometimes use it for sets)

$$\mathcal{X,Y,A,I,C}$$ – calligraphic (I use it for set families = sets of sets)

$$X, x, \mathbf{X}, \mathbf{x}$$ – inputs (usually)

$$Y, y, \mathbf{Y}, \mathbf{y}$$ – outputs

$$\hat{Y}, \hat{y}, \hat{\mathbf{Y}}, \hat{\mathbf{y}}$$ – predicted outputs (usually)

• $$X$$ – independent/explanatory/predictor variable

• $$Y$$ – dependent/response/predicted variable

$$\mathbb{R}$$ – the set of real numbers, $$\mathbb{R}=(-\infty, \infty)$$

$$\mathbb{N}$$ – the set of natural numbers, $$\mathbb{N}=\{1,2,3,\dots\}$$

$$\mathbb{N}_0$$ – the set of natural numbers including zero, $$\mathbb{N}_0=\mathbb{N}\cup\{0\}$$

$$\mathbb{Z}$$ – the set of integer numbers, $$\mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\}$$

$$[0,1]$$ – the unit interval

$$(a, b)$$ – an open interval; $$x\in(a,b)$$ iff $$a < x < b$$ for some $$a< b$$

$$[a, b]$$ – a closed interval; $$x\in[a,b]$$ iff $$a \le x \le b$$ for some $$a\le b$$

#### Notation Convention – Vectors and Matrices

$$\boldsymbol{x}=(x_1,\dots,x_n)$$ – a sequence of $$n$$ elements ($$n$$-ary sequence/vector)

if it consists of real numbers, we write $$\boldsymbol{x}\in\mathbb{R}^n$$

$$\mathbf{x}=[x_1\ x_2\ \dots\ x_p]$$ – a row vector, $$\mathbf{x}\in\mathbb{R}^{1\times p}$$ (a matrix with 1 row)

$$\mathbf{x}=[x_1\ x_2\ \dots\ x_n]^T$$ – a column vector, $$\mathbf{x}\in\mathbb{R}^{n\times 1}$$ (a matrix with 1 column)

$$\mathbf{X}\in\mathbb{R}^{n\times p}$$ – matrix with $$n$$ rows and $$p$$ columns

$\mathbf{X}= \left[ \begin{array}{cccc} x_{1,1} & x_{1,2} & \cdots & x_{1,p} \\ x_{2,1} & x_{2,2} & \cdots & x_{2,p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n,1} & x_{n,2} & \cdots & x_{n,p} \\ \end{array} \right]$

$$x_{i,j}$$ – element in the $$i$$-th row, $$j$$-th column

$$\mathbf{x}_{i,\cdot}$$ – the $$i$$-th row of $$\mathbf{X}$$

$$\mathbf{x}_{\cdot,j}$$ – the $$j$$-th column of $$\mathbf{X}$$

$\mathbf{X}=\left[ \begin{array}{c} \mathbf{x}_{1,\cdot} \\ \mathbf{x}_{2,\cdot} \\ \vdots\\ \mathbf{x}_{n,\cdot} \\ \end{array} \right] = \left[ \begin{array}{cccc} \mathbf{x}_{\cdot,1} & \mathbf{x}_{\cdot,2} & \cdots & \mathbf{x}_{\cdot,p} \\ \end{array} \right].$

$\mathbf{x}_{i,\cdot} = \left[ \begin{array}{cccc} x_{i,1} & x_{i,2} & \cdots & x_{i,p} \\ \end{array} \right].$

$\mathbf{x}_{\cdot,j} = \left[ \begin{array}{cccc} x_{1,j} & x_{2,j} & \cdots & x_{n,j} \\ \end{array} \right]^T=\left[ \begin{array}{c} {x}_{1,j} \\ {x}_{2,j} \\ \vdots\\ {x}_{n,j} \\ \end{array} \right],$

$${}^T$$ denotes the matrix transpose; $$\mathbf{B}=\mathbf{A}^T$$ is a matrix such that $$b_{i,j}=a_{j,i}$$.

$$\|\boldsymbol{x}\| = \|\boldsymbol{x}\|_2 = \sqrt{ \sum_{i=1}^n x_i^2 }$$ – the Euclidean norm

#### Notation Convention – Functions

$$f:X\to Y$$ means that $$f$$ is a function mapping inputs from set $$X$$ (the domain of $$f$$) to elements of $$Y$$ (the codomain)

$$x\mapsto x^2$$ denotes a (inline) function mapping $$x$$ to $$x^2$$, equivalent to function(x) x^2 in R

$$\exp x = e^x$$ – exponential function with base $$e\simeq 2.718$$

$$\log x$$ – natural logarithm (base $$e$$)

it holds $$e^x = y$$ iff $$\log y = x$$

$$\log ab = \log a + \log b$$

$$\log a^c = c\log a$$

$$\log a/b = \log a - \log b$$

$$\log 1 = 0$$

$$\log e = 1$$

hence $$\log e^x = x$$

#### Notation Convention – Sums and Products

$$\sum_{i=1}^n x_i=x_1+x_2+\dots+x_n$$

$$\sum_{i=1,\dots,n} x_i$$ – the same

$$\sum_{i\in\{1,\dots,n\}} x_i$$ – the same

note display (stand-alone) $$\displaystyle\sum_{i=1}^n x_i$$ vs text (in-line) $$\textstyle\sum_{i=1}^n x_i$$ style

$$\prod_{i=1}^n x_i = x_1 x_2 \dots x_n$$