In coatless/ITS: A Tour of Time Series Analysis with R

SARIMA Models

An introductory seasonal ARIMA model would be a seasonal AR(1) model.

```{definition, name = "Seasonal Autoregressive Model of Order 1"} A sesaonal autoregressive model of order 1 is defined to be:

$$X_t = \Phi_1 X_{t-12} + W_{t}$$

The model would have the following properties:

$$\e{X_t} = 0$$
$$\gamma(0) = \var{X_t} = \frac{\sigma^2}{1-\Phi^2_1}$$
$$\rho(h) = \begin{cases} 1, &\text{ if } h = 0\ \Phi^{\left|h\right|}, &\text{ if } h = \pm 12k, k = 1, 2, \cdots\ 0, &\text{ Otherwise } \end{cases}$$

These properties are similar to that of an AR(1).

```{definition, name = "Seasonal Moving Average of Order 1"}
A sesaonal moving average model of order 1 is defined to be:

$$X_t = W_t + \theta W_{t-12} \Leftrightarrow X_t = (1 - \theta B^{12}) W_t$$

$$\gamma(h) = \begin{cases}
\left({1+\theta^2}\right)\sigma^2, &\text{ if } h = 0 \\
\theta \sigma^2, &\text{ if } h = \pm 12k, k = 1, 2, \cdots \\
0, &\text{ Otherwise } \\
\end{cases}$$

```{definition, name = "Seasonal Autoregressive Operator"} Similarly, to the regular autoregressive operator, there exists a seasonal variant known as:

$$\Phi_p(B^S) = 1 - \Phi_1 B^S - \Phi_2B^{2S} - \cdots - \Phi_PB^{PS}$$

```{definition, name = "Seasonal Moving Average Operator"}
The seasonal moving average operator is defined to be:

$$\Theta_p(B^S) = 1 + \Theta_1 B^S + \Theta_2B^{2S} + \cdots + \Theta_PB^{PS}$$

```{example, name = "Mixed Seasonal Model"} Consider the following time series model that contains both a seasonality term and a traditional time series component:

$$X_t = \Phi X_{t-12} + W_t + \theta W_{t-1} \, \left| \Theta \right| < 1, \left| \theta \right| < 1$$

The properties of this model can be derived as follows:

\begin{align} \var \left( {{X_t}} \right) &= {\Phi ^2}\var \left( {{X_{t - 12}}} \right) + {\sigma ^2} + {\theta ^2}{\sigma ^2} \notag \ \Rightarrow \gamma \left( 0 \right) &= \frac{{{\sigma ^2}\left( {1 + {\theta ^2}} \right)}}{{1 - {\Phi ^2}}} \ \gamma \left( 1 \right) &= \cov\left( {{X_t},{X_{t - 1}}} \right) = \cov\left( {\Phi {X_{t - 12}} + {W_t} + \theta {W_{t - 1}},{X_{t - 1}}} \right) \notag \ &= \Phi \cov\left( {{X_{t - 12}},{X_{t - 1}}} \right) + \underbrace {\cov\left( {{W_t},{X_{t - 1}}} \right)}{ = 0} + \theta \cov\left( {{W{t - 1}},{X_{t - 1}}} \right) \notag \ &= \Phi \gamma \left( {11} \right) + \theta {\sigma ^2} \ \gamma \left( h \right) &= \cov\left( {{X_t},{X_{t - h}}} \right) = \cov\left( {\Phi {X_{t - 12}} + {W_t} + \theta {W_{t - 1}},{X_{t - h}}} \right) \notag \ &\overbrace{=^{h \ge 2}}\Phi \cov\left( {{X_{t - 12}},{X_{t - h}}} \right) \notag \ &= \Phi \gamma \left( {h - 12} \right) \ \end{align}

If the autocovariance is defined within the appropriate seasonal lag, then we have a realized value other than zero \begin{equation} \gamma \left( 1 \right) = \Phi \gamma \left( {11} \right) + \theta {\sigma ^2} = {\Phi ^2}\gamma \left( 1 \right) + \theta {\sigma ^2} = \frac{{\theta {\sigma ^2}}}{{1 - {\Phi ^2}}} \end{equation}

When this is not the case, the autocovariance will be zero: \begin{align} \gamma \left( 2 \right) &= \cov \left( {{X_t},{X_{t - 2}}} \right) = \operatorname{cov} \left( {\Phi {X_{t - 12}} + {W_t} + \theta {W_{t - 1}},{X_{t - 2}}} \right) \notag \ &= \Phi \cov \left( {{X_{t - 12}},{X_{t - 2}}} \right) = \Phi \gamma \left( {10} \right) = {\Phi ^2}\gamma \left( 2 \right) = 0 \end{align}

In this example, this would hold for:

\begin{equation} \gamma \left( 3 \right) = \gamma \left( 4 \right) = \cdots = \gamma \left( 10 \right) = 0 \end{equation}

Therefore, the autocovariance can be denoted as: \begin{align} \gamma \left( {12h} \right) &= {\Phi ^h}\gamma \left( 0 \right), &h = 0, 1, 2, \ldots \ \gamma \left( {12h + 1} \right) &= \gamma \left( {12h - 1} \right) = {\Phi ^h}\gamma \left( 1 \right), &h = 0, 1, 2, \ldots \ \gamma \left( {h} \right) &= 0, &\text{Otherwise} \end{align}

As a result, the autocorrelation is given as:

\begin{align} \rho \left( {12h} \right) &= {\Phi ^h}, & h = 0, 1, 2, \ldots \ \rho \left( {12h - 1} \right) &= \rho \left( {12h + 1} \right) = \frac{\theta }{{1 + {\theta ^2}}}{\Phi ^h}, & h = 0, 1, 2, \ldots \ \rho \left( h \right) &= 0, & \text{Otherwise} \ \end{align}

The correlation structure can be viewed quite straightforwardly.
```r
library("exts")
model = SARIMA(ar=0, i=0,ma=-0.8, sar=0.95, si = 0 , sma = 0, s = 12)
xt = gen_gts(100000, model)
autoplot(emp_acf(xt, lag.max = 40))

```{definition, name = "Seasonal ARMA Model Form"} The form of Seasonal Autoregressive Moving Average models is often written as $ARMA(p, q)\times(P, Q)_{S}$:

$$\Phi_p \left({B^S}\right) \phi\left(B\right) X_t = \Theta_Q \left({ B^S }\right) \theta \left({ B }\right) W_t$$

```{example, name = "Classifying a Seasonal ARMA"}
Returning to our previous example, we can see that the time series follows an $ARMA(0,1)\times(1,0)_{12}$ process.

\begin{align*}
  {X_t} &= \Phi {X_{t - 12}} + {W_t} + \theta {W_{t - 1}} \hfill \\
  \underbrace {\left( {{X_t} - \Phi {B^{12}}} \right)}_{{\Phi _1}\left( {{B^{12}}} \right)}\underbrace 1_{\phi \left( B \right)}{X_t} &= \underbrace 1_{{\theta _Q}\left( B \right)}\underbrace {\left( {1 - \theta B} \right)}_{\theta \left( B \right)}{W_t} \hfill \\ 
\end{align*}

```{definition, name = "Seasonal ARIMA Model Form"} The form of a Seasonal Autoregressive Integrated Moving Average models is denoted as $ARIMA(p, d, q)\times(P, D, Q)_S$:

$$\Phi_p \left({B^S}\right) \phi\left(B\right) \nabla^D_S \nabla^d X_t = \delta + \Theta_Q \left({ B^S }\right) \theta \left({ B }\right) W_t$$

where $\nabla^d = \left({1-B}\right)^d$ and $\nabla^D_S = \left({1-B^S}\right)^D$.

```{example, name = "Classifying a SARIMA"}
Consider the following time series:
\begin{align}
  {X_t} &= {X_{t - 1}} + {X_{t - 12}} - {X_{t - 13}} + {W_t} + \phi {W_{t - 1}} + \theta {W_{t - 12}} + \theta \phi {W_{t - 13}} \hfill \\
  {X_t} - {X_{t - 1}} - {X_{t - 12}} + {X_{t - 13}} &= {W_t} + \phi {W_{t - 1}} + \theta {W_{t - 12}} + \theta \phi {W_{t - 13}} \hfill \\
  \left( {1 - B - {B^{12}} + {B^{13}}} \right){X_t} &= \left( {1 + \phi B + \theta {B^{12}} + \phi \theta {B^{13}}} \right){W_t} \hfill \\
  \underbrace {\left( {1 - {B^{12}}} \right)}_{{\nabla _{12}}}\underbrace {\left( {1 - B} \right)}_\nabla {X_t} &= \underbrace {\left( {1 + \theta {B^{12}}} \right)}_{{\theta _Q}\left( {{B^S}} \right)}\underbrace {\left( {1 + \phi B} \right)}_{\theta \left( B \right)}{W_t} \hfill \\ 
\end{align}

The end result indicates that the SARIMA is given by: $ARIMA(0,1,1)\times(0,1,1)_{12}$.