STA457 Notation Index#
Standard notation used across all STA457 notes.
Time Series#
| Symbol | Meaning | Notes |
|---|
| $\{X_t\}$ or $\{X_t\}_{t \in T}$ | Stochastic process | Capital = random variable |
| $x_t$ | Observed value at time $t$ | Lowercase = realization |
| $n$ | Number of observations | |
| $h$ | Time lag | |
Components and Decomposition#
| Symbol | Meaning | Notes |
|---|
| $m_t$ | Deterministic trend | Classical Decomposition |
| $s_t$ | Seasonal component | Period $d$: $s_t = s_{t+d}$ |
| $Y_t$ | Random (noise) component | Zero-mean |
Operators#
| Symbol | Meaning | Definition |
|---|
| $B$ | Backshift Operator | $BX_t = X_{t-1}$ |
| $B^k$ | $k$-step backshift | $B^k X_t = X_{t-k}$ |
| $\nabla$ | Difference operator | $\nabla = 1 - B$ |
| $\nabla^p$ | $p$-th order difference | $(1-B)^p$ |
| $\nabla_d$ | Lag-$d$ difference | $1 - B^d$ |
| $\phi(B)$ | AR polynomial in $B$ | $1 - \phi_1 B - \cdots - \phi_p B^p$ |
| $\theta(B)$ | MA polynomial in $B$ | $1 + \theta_1 B + \cdots + \theta_q B^q$ |
Statistical Functions#
| Symbol | Meaning | Notes |
|---|
| $\mu_X$ or $\mu$ | Mean of $\{X_t\}$ | Constant under Stationarity |
| $\sigma_X^2$ or $\sigma^2$ | Variance of $\{X_t\}$ | $= \gamma_X(0)$ |
| $\gamma_X(h)$ | ACVF at lag $h$ | $\text{Cov}(X_t, X_{t+h})$ |
| $\rho_X(h)$ | ACF at lag $h$ | $\gamma_X(h)/\gamma_X(0)$ |
| $\hat{\gamma}(h)$ | Sample ACVF | $n^{-1}\sum(x_t - \bar{x})(x_{t+h}-\bar{x})$ |
| $\hat{\rho}(h)$ | Sample ACF | $\hat{\gamma}(h)/\hat{\gamma}(0)$ |
Model Parameters#
| Symbol | Meaning | Context |
|---|
| $\phi_1, \dots, \phi_p$ | AR coefficients | Autoregressive Process |
| $\theta_1, \dots, \theta_q$ | MA coefficients | Moving Average Process |
| $\sigma^2$ | White noise variance | $\{Z_t\} \sim \text{WN}(0, \sigma^2)$ |
| $\{Z_t\}$ | White noise process | Input/innovation |
| $\{a_j\}$ | Filter coefficients | Linear Filter |
Prediction#
| Symbol | Meaning | Notes |
|---|
| $P_n X_{n+h}$ | Best Linear Predictor of $X_{n+h}$ given $X_1, \dots, X_n$ | Minimizes MSE |
| $\text{MSE}$ | Mean squared error | $\mathbb{E}[(X_{n+h} - P_n X_{n+h})^2]$ |
| $\Gamma_n$ | Covariance matrix | $(\gamma_X(i-j))_{i,j=1}^n$ |
Common Abbreviations#
| Abbreviation | Meaning |
|---|
| WN | White Noise Process |
| AR($p$) | Autoregressive of order $p$ |
| MA($q$) | Moving average of order $q$ |
| ARMA($p,q$) | Autoregressive moving average |
| ACVF | Autocovariance function |
| ACF | Autocorrelation function |
| BLP | Best linear predictor |