Table of Notations - Time Series Analysis for Data Scientists

The following table gives a brief overview of the notations used in this book along with their definitions and examples of how they might appear in the text.

Symbol	Meaning	Examples
Lower case unbolded letter	Scalar or realization/observation of random variable	$x$ , $\alpha$
Upper case unbolded letter	Random variable	$X$ , $Y$
Lower case bolded letter	Vector	$\mathbf{x}$ , $\boldsymbol{\alpha}$
Upper case bolded letter	Matrix	$\boldsymbol{\Phi}$ , $\mathbf{R}$
$\stackrel{\triangle}{=}$	Defined as	$\boldsymbol{\theta}\stackrel{\triangle}{=}(\theta_0, \theta_1,\ldots,\theta_{p-1})$
$\hat{}\,$ over letter	Statistical estimator	$\hat{\mu}$ , $\hat{\sigma}^2$ , $\hat{S}_t$
$\mathbb{E}[\cdot]$	Expectation	$\mathbb{E}[x]$ , $\mathbb{E}[(x_t-\mu_x)(y_t-\mu_y)]$
$\mathbb{P}(\cdot)$	Probability	$\mathbb{P}(x)$ , $\mathbb{P}(x_i\|\theta)$
$P(\cdot)$	Probability function (discrete) or probability density function (continuous)	$\sum_{x_i} P(x_i)$
$\mathbb{B}$	Backshift operator	$\mathbb{B}x_t=x_{t-1}$
$\mathbb{B}^{-1}$	Forward-shift operator	$\mathbb{B}^{-1}x_{t-1}=x_t$
Variable with subscript	Value of discrete variable at given index or series indexed by given variable	$x_{t-2}$ , $\theta_q$
Variable followed by $(\cdot)$	Value of continuous variable at given value or function of given variable	$x(t-2)$
Variable with subscript and superscript	Value of discrete variable at lower index conditioned on (i.e. given) known values up to upper index	$x_{t}^{t-1}$
$\mathbb{V}[\cdot]$ or $\sigma_{\cdot}^2$	Variance	$\mathbb{V}[x_t]$ , $\sigma_x^2$
$\text{Cov}(\cdot,\cdot)$ or $\sigma_{\cdot,\cdot}$	Covariance	$\text{Cov}(\mathbf{X}, \mathbf{Y})$ , $\sigma_{x,y}$
$\text{Cor}(\cdot,\cdot)$	Correlation	$\text{Cor}(\mathbf{X}, \mathbf{Y})$
$\gamma(h)$ or $\gamma_x(h)$	Autocovariance of series $x$ at lag $h$	$\gamma(2)$ , $\gamma_x(0)$
$\rho(h)$ or $\rho_x(h)$	Autocorrelation of series $x$ at lag $h$	$\rho(2)$ , $\rho_x(0)$
$\kappa(h)$ or $\kappa_x(h)$	Partial autocorrelation of series $x$ at lag $h$	$\kappa(2)$
$\gamma_{x,y}(h)$	Cross-covariance of series $x$ and $y$ at lag $h$	$\gamma_{x,y}(2)$ , $\gamma_{x,y}(0)$
$\rho_{x,y}(h)$	Cross-correlation of series $x$ and $y$ at lag $h$	$\rho_{x,y}(2)$ , $\rho_{x,y}(0)$
$\|\|\cdot\|\|$	Vector norm	$\|\|\mathbf{u}\|\|$
$\lfloor\cdot\rfloor$	Floor function	$\lfloor\pi\rfloor=3$ , $\lfloor1\rfloor=1$
$\lceil\cdot\rceil$	Ceiling function	$\lceil\pi\rceil=4$ , $\lceil1\rceil=1$
$\sim$	Distributed as	$x_t\sim \chi_1^2$
$wn(\cdot,\cdot)$	White noise process or distribution	$w_t\sim wn(0,\sigma_w^2)$
$\mathcal{N}(\cdot,\cdot)$	Gaussian process or distribution	$w_t\sim \mathcal{N}(0,\sigma_w^2)$
$\Re(\cdot)$	Real portion of complex number	$\Re(z)$
$\Im(\cdot)$	Imaginary portion of complex number	$\Im(z)$