# ARMA models

2019, Jérémie DECOCK

## Lag operator

Definition of the *backward shift operator* $B$:

$$B y_t = y_{t-1}$$

$$B^i y_t = y_{t-i}$$

Definition of the *forward shift operator* $F$:

$$  F y_t = B^{-1} y_t = y_{t+1}$$

$$F^i y_t = B^{-i} y_t = y_{t+i}$$

## Lag polynomials

Polynomials of the lag operator are commonly used to simplify ARMA models notation.
These polynomials are called *lag polynomials*.

### Example

$a_{(p)} (B)$ is a *lag polynomials*:
$$
\begin{align}
a_{(p)} (B) &= 1 + \sum_{i=1}^p a_i B^i \\
            &= 1 + a_1 B + a_2 B^2 + \dots + a_p B^p
\end{align}
$$

### Usage example in time series analysis

$$
\begin{align}
a_{(p)} (B)y_t &= \left( 1 + \sum_{i=1}^p a_i B^i \right) ~ y_t \\
               &= y_t + a_1 B y_t + a_2 B^2 y_t + \dots + a_p B^p y_t \\
               &= y_t + a_1 y_{t-1} + a_2 y_{t-2} + \dots + a_p y_{t-p}
\end{align}
$$

## Difference operator

Definition of the *difference operator*:
$$\nabla^i y_t = (1-B)^i ~ y_t$$

### Example: first order difference operator (i.e. $i=1$)

$$
\begin{align}
\nabla y_t &= (1-B) ~ y_t \\
           &= y_t - B ~ y_t \\
           &= y_t - y_{t-1}
\end{align}
$$

### Example: second order difference operator (i.e. $i=2$)

$$
\begin{align}
\nabla^2 y_t &= (1-B)^2 ~ y_t \\
           &= (1-B)(1-B) ~ y_t \\
           &= \left( 1 - 2B + B^2 \right) ~ y_t \\
           &= y_t - 2y_{t-1} + y_{t-2}
\end{align}
$$

### Example: first order difference operator with a lag of 12

$$
\begin{align}
\left( 1-B^{12} \right) ~ y_t &= y_t - B^{12} ~ y_t \\
                              &= y_t - y_{t-12}
\end{align}
$$

## Autoregressive Model (AR)

Using this model, the observation $y_t$ of a time series at time $t$ is defined as a linear combination of its $p$ past observations $y_{t-1}, \dots, y_{t-p}$

$$
\begin{align}
AR(\color{\red}{p}): \quad\quad\quad \color{\red}{\underbrace{\varphi_{(p)} (B) y_t}_{\text{AR}}} 
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{ \left( 1 + \sum_{i=1}^p \varphi_i B^i \right) y_t }
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^p \varphi_i B^i y_t }
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^p \varphi_i y_{t-i} }
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{y_t} 
&= c + \color{\green}{\varepsilon_t} + \color{\red}{\sum_{i=1}^p \phi_i y_{t-i}}
\end{align}
$$

with $\varphi_i = -\phi_i$.

$c$ is a constant used to ensure the series is centered to 0 ($c$ is the average of the series).

$\color{\green}{\varepsilon_t}$ is the noise component of $y_t$.

### Example: Autoregressive order 3 process (i.e. $p=3$) noted AR(3)

$$
\begin{align}
AR(\color{\red}{3}): \quad\quad\quad \color{\red}{\underbrace{\varphi_{(3)} (B) y_t}_{\text{AR}}} 
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{ \left( 1 + \sum_{i=1}^3 \varphi_i B^i \right) y_t }
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^3 \varphi_i B^i y_t }
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^3 \varphi_i y_{t-i} }
&= c + \color{\green}{\varepsilon_t} \\
\color{\red}{y_t} 
&= c + \color{\green}{\varepsilon_t} + \color{\red}{\sum_{i=1}^3 \phi_i y_{t-i}} \\
\color{\red}{y_t} 
&= c + \color{\green}{\varepsilon_t} + \color{\red}{\phi_1 y_{t-1}} + \color{\red}{\phi_2 y_{t-2}} + \color{\red}{\phi_3 y_{t-3}}
\end{align}
$$

with $\varphi_i = -\phi_i$.

## Moving Average Model (MA)

Combinaison linéaire des $q$ erreurs passées
Using this model, the observation $y_t$ of a time series at time $t$ is defined as the output of a linear filter with transfer function $\theta_{(q)}(B)$ when the input is white noise $\varepsilon_t$.

$$
\begin{align}
MA(\color{\green}{q}): \quad\quad\quad
\color{\red}{y_t} &= c + \color{\green}{\underbrace{\theta_{(q)} (B) \varepsilon_t}_{MA}} \\
\color{\red}{y_t} &= c + \color{\green}{\left( 1 + \sum_{j=1}^q \theta_j B^j \right) ~ \varepsilon_t} \\
\color{\red}{y_t} &= c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^q \theta_j B^j \varepsilon_t} \\
\color{\red}{y_t} &= c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^q \theta_j ~ \varepsilon_{t-j}}
\end{align}
$$

### Example: Moving average order 3 process (i.e. $q=3$) noted MA(3)

$$
\begin{align}
MA(\color{\green}{3}): \quad\quad\quad
\color{\red}{y_t} &= c + \color{\green}{\underbrace{\theta_{(3)} (B) \varepsilon_t}_{MA}} \\
\color{\red}{y_t} &= c + \color{\green}{\left( 1 + \sum_{j=1}^3 \theta_j B^j \right) ~ \varepsilon_t} \\
\color{\red}{y_t} &= c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^3 \theta_j B^j \varepsilon_t} \\
\color{\red}{y_t} &= c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^3 \theta_j ~ \varepsilon_{t-j}} \\
\color{\red}{y_t} &= c + \color{\green}{\varepsilon_t} + \color{\green}{\theta_1 ~ \varepsilon_{t-1}} + \color{\green}{\theta_2 ~ \varepsilon_{t-2}} + \color{\green}{\theta_3 ~ \varepsilon_{t-3}}
\end{align}
$$

### AutoRegressive Moving Average (ARMA)

An ARMA(p,q) process of order $p$ and $q$ is a combination of an AR(p) and an MA(q) process.

$$
\begin{align}
ARMA(\color{\red}{p}, \color{\green}{q}): \quad\quad\quad
\color{\red}{\underbrace{\varphi_{(p)} (B) y_t}_{\text{AR}}} 
&=
c + \color{\green}{\underbrace{\theta_{(q)} (B) \varepsilon_t}_{MA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^p \varphi_i B^i \right) y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^q \theta_j B^j \right) ~ \varepsilon_t}
\\
\color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^p \varphi_i B^i y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^q \theta_j B^j \varepsilon_t}
\\
\color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^p \varphi_i y_{t-i} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^q \theta_j ~ \varepsilon_{t-j}}
\\
\color{\red}{y_t}
&=
c
+ \color{\red}{\underbrace{\sum_{i=1}^p \phi_i y_{t-i}}_{\text{AR model}}}
+ \color{\green}{\varepsilon_t}
+ \color{\green}{\underbrace{\sum_{j=1}^q \theta_j \varepsilon_{t-j}}_{\text{MA model}}}
\end{align}
$$

with $\varphi_i = -\phi_i$

#### Example: ARMA(3,2)

$$
\begin{align}
ARMA(\color{\red}{3}, \color{\green}{2}): \quad\quad\quad
\color{\red}{\underbrace{\varphi_{(3)} (B) y_t}_{\text{AR}}} 
&=
c + \color{\green}{\underbrace{\theta_{(2)} (B) \varepsilon_t}_{MA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^3 \varphi_i B^i \right) y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^2 \theta_j B^j \right) ~ \varepsilon_t}
\\
\color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^3 \varphi_i B^i y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \theta_j B^j \varepsilon_t}
\\
\color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^3 \varphi_i y_{t-i} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \theta_j ~ \varepsilon_{t-j}}
\\
\color{\red}{y_t}
&=
c
+ \color{\red}{\underbrace{\sum_{i=1}^3 \phi_i y_{t-i}}_{\text{AR model}}}
+ \color{\green}{\varepsilon_t}
+ \color{\green}{\underbrace{\sum_{j=1}^2 \theta_j \varepsilon_{t-j}}_{\text{MA model}}}
\\
\color{\red}{y_t}
&=
c
+ \underbrace{ \color{\red}{\phi_1 y_{t-1}} + \color{\red}{\phi_2 y_{t-2}} + \color{\red}{\phi_3 y_{t-3}} }_{\text{AR model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{ \color{\green}{\theta_1 \varepsilon_{t-1}} + \color{\green}{\theta_2 \varepsilon_{t-2}} }_{\text{MA model}}
\end{align}
$$

with $\varphi_i = -\phi_i$

## AutoRegressive Integrated Moving Average (ARIMA)

An ARIMA process is an *integrated* ARMA process.
It's used on non-stationary time series.
An initial transformation step (the "integrated" part of the model) is applied to eliminate the non-stationarity.
This transformation is usually operated by the *difference operator* described above.

An $ARIMA(p,d,q)$ process produce series that can be modeled as a stationary $ARMA(p,q)$ process after being differenced $d$ times.

$$
\begin{align}
ARIMA(\color{\red}{p}, \color{\orange}{d}, \color{\green}{q}): \quad\quad\quad
\color{\red}{\underbrace{\varphi_{(p)} (B) \color{\orange}{\nabla^d} y_t}_{\text{AR}}} 
&=
c + \color{\green}{\underbrace{\theta_{(q)} (B) \varepsilon_t}_{MA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^p \varphi_i B^i \right)} \color{\orange}{\nabla^d} \color{\red}{ y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^q \theta_j B^j \right) ~ \varepsilon_t}
\\
\color{\orange}{\nabla^d} \color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^p \varphi_i B^i } \color{\orange}{\nabla^d} \color{\red}{ y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^q \theta_j B^j \varepsilon_t}
\\
\color{\orange}{\nabla^d} \color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^p \varphi_i } \color{\orange}{\nabla^d} \color{\red}{ y_{t-i} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^q \theta_j ~ \varepsilon_{t-j}}
\\
\color{\orange}{\nabla^d} \color{\red}{y_t}
&=
c
+ \underbrace{
    \color{\red}{\sum_{i=1}^p \phi_i}
    \color{\orange}{\nabla^d}
    \color{\red}{y_{t-i}}
  }_{\text{ARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\sum_{j=1}^q \theta_j \varepsilon_{t-j}}}_{\text{MA model}}
\end{align}
$$

with

$$\nabla = (1-B)$$
$$\nabla^2 = (1-B)^2$$
$$\dots$$
$$\nabla^d = (1-B)^d$$

#### Examples for $d=1$, $d=2$ and $d=3$

If $d=0$:
$$
\begin{align}
\nabla^0 y_t &:= (1-B)^0 y_t \\
              &= y_t
\end{align}
$$

If $d=1$:
$$
\begin{align}
\nabla y_t &:= (1-B) y_t \\
              &= y_t - B y_t \\
              &= y_t - y_{t-1}
\end{align}
$$

If $d=2$:
$$
\begin{align}
\nabla^2 y_t &:= (1-B)^2 y_t \\
              &= (1-B)(1-B) ~ y_t \\
              &= (1 - 2B + B^2) y_t \\
              &= y_t - 2B y_t + B^2 y_t \\
              &= y_t - 2 y_{t-1} + y_{t-2}
\end{align}
$$

The above approach generalises to the i-th difference operator $\nabla^i y_t = (1-B)^i ~ y_t$

#### Example: ARIMA(3, 1, 2)

$$
\begin{align}
ARIMA(\color{\red}{3}, \color{\orange}{1}, \color{\green}{2}): \quad\quad\quad
\color{\red}{\underbrace{\varphi_{(3)} (B) \color{\orange}{\nabla} y_t}_{\text{AR}}} 
&=
c + \color{\green}{\underbrace{\theta_{(2)} (B) \varepsilon_t}_{MA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^3 \varphi_i B^i \right)} \color{\orange}{\nabla} \color{\red}{ y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^2 \theta_j B^j \right) ~ \varepsilon_t}
\\
\color{\orange}{\nabla} \color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^3 \varphi_i B^i } \color{\orange}{\nabla} \color{\red}{ y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \theta_j B^j \varepsilon_t}
\\
\color{\orange}{\nabla} \color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^3 \varphi_i } \color{\orange}{\nabla} \color{\red}{ y_{t-i} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \theta_j ~ \varepsilon_{t-j}}
\\
\color{\orange}{\nabla} \color{\red}{y_t}
&=
c
+ \underbrace{
    \color{\red}{\sum_{i=1}^3 \phi_i}
    \color{\orange}{\nabla}
    \color{\red}{y_{t-i}}
  }_{\text{ARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\sum_{j=1}^2 \theta_j \varepsilon_{t-j}}}_{\text{MA model}}
\\
\color{\orange}{\nabla} \color{\red}{y_t}
&=
c
+ \underbrace{
      \color{\red}{\phi_1} \color{\orange}{\nabla} \color{\red}{y_{t-1}}
    + \color{\red}{\phi_2} \color{\orange}{\nabla} \color{\red}{y_{t-2}}
    + \color{\red}{\phi_3} \color{\orange}{\nabla} \color{\red}{y_{t-3}}
  }_{\text{ARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\theta_1 \varepsilon_{t-1}} + \color{\green}{\theta_2 \varepsilon_{t-2}}}_{\text{MA model}}
\\
\color{\orange}{(1-B)} \color{\red}{y_t}
&=
c
+ \underbrace{
      \color{\red}{\phi_1} \color{\orange}{(1-B)} \color{\red}{y_{t-1}}
    + \color{\red}{\phi_2} \color{\orange}{(1-B)} \color{\red}{y_{t-2}}
    + \color{\red}{\phi_3} \color{\orange}{(1-B)} \color{\red}{y_{t-3}}
  }_{\text{ARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\theta_1 \varepsilon_{t-1}} + \color{\green}{\theta_2 \varepsilon_{t-2}}}_{\text{MA model}}
\\
\color{\red}{y_t} - \color{\orange}{B y_t}
&=
c
+ \underbrace{
      \color{\red}{\phi_1} (\color{\red}{y_{t-1}} - \color{\orange}{B y_{t-1}})
    + \color{\red}{\phi_2} (\color{\red}{y_{t-2}} - \color{\orange}{B y_{t-2}})
    + \color{\red}{\phi_3} (\color{\red}{y_{t-3}} - \color{\orange}{B y_{t-3}})
  }_{\text{ARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\theta_1 \varepsilon_{t-1}} + \color{\green}{\theta_2 \varepsilon_{t-2}}}_{\text{MA model}}
\\
\color{\red}{y_t} - \color{\orange}{y_{t-1}}
&=
c
+ \underbrace{
      \color{\red}{\phi_1} (\color{\red}{y_{t-1}} - \color{\orange}{y_{t-2}})
    + \color{\red}{\phi_2} (\color{\red}{y_{t-2}} - \color{\orange}{y_{t-3}})
    + \color{\red}{\phi_3} (\color{\red}{y_{t-3}} - \color{\orange}{y_{t-4}})
  }_{\text{ARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\theta_1 \varepsilon_{t-1}} + \color{\green}{\theta_2 \varepsilon_{t-2}}}_{\text{MA model}}
\\
\color{\red}{y_t}
&=
c
+ \color{\orange}{y_{t-1}} 
+ \underbrace{
      \color{\red}{\phi_1 y_{t-1}} - \color{\orange}{\phi_1 y_{t-2}}
    + \color{\red}{\phi_2 y_{t-2}} - \color{\orange}{\phi_2 y_{t-3}}
    + \color{\red}{\phi_3 y_{t-3}} - \color{\orange}{\phi_3 y_{t-4}}
  }_{\text{ARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\theta_1 \varepsilon_{t-1}} + \color{\green}{\theta_2 \varepsilon_{t-2}}}_{\text{MA model}}
\end{align}
$$

## AutoRegressive Moving Average Including Exogenous Covariates (ARMAX)

$$
\begin{align}
\color{\red}{\underbrace{\varphi (B) y_t}_{AR}}
&= c
+ \color{\green}{\underbrace{\theta (B) \varepsilon_t}_{MA}}
+ \color{\purple}{\underbrace{\beta_1 x_{1,~t} + \beta_2 x_{2,~t} + \dots + \beta_r x_{r,~t}}_{X}}
\end{align}
$$

$x_{1,~t}, x_{2,~t}, \dots, x_{r,~t}$ is the value of the $r$ exogenous variables at time $t$, with $\beta_1, \beta_2, \dots, \beta_r$ the $r$ corresponding coefficients.

$$
ARMAX(\color{\red}{p}, \color{\green}{q}): \quad
\color{\red}{y_t} = c
+ \color{\red}{\underbrace{\sum_{i=1}^p \varphi_i y_{t-i}}_{\text{AR model}}}
+ \color{\green}{\varepsilon_t}
+ \color{\green}{\underbrace{\sum_{j=1}^q \theta_j \varepsilon_{t-j}}_{\text{MA model}}}
+ \color{\purple}{\underbrace{\sum_{k=1}^r \beta_k x_{k,~t}}_{\text{Exogenous}}}
$$

## Seasonal AutoRegressive Moving Average (SARMA)

#### Purely seasonal processes

$$
\begin{align}
ARMA(\color{\red}{P}, \color{\green}{Q})_{\color{\red}{s}, \color{\green}{s'}}: \quad\quad\quad
\color{\red}{\underbrace{\Phi_{(P)} (B^s) y_t}_{\text{SAR}}}
&= 
c + \color{\green}{\underbrace{\Theta_{(Q)} (B^{s'}) \varepsilon_t}_{SMA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^P \Phi_i B^{si} \right) y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^Q \Theta_j B^{s'j} \right) ~ \varepsilon_t}
\\
\color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^P \Phi_i B^{si} y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^Q \Theta_j B^{s'j} \varepsilon_t}
\\
\color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^P \Phi_i ~ y_{t-si} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^Q \Theta_j ~ \varepsilon_{t-s'j}}
\\
\color{\red}{y_t} &= c
+ \color{\red}{\underbrace{\sum_{i=1}^P \varPhi_i ~ y_{t-si}}_{\text{SAR model}}}
+ \color{\green}{\varepsilon_t}
+ \color{\green}{\underbrace{\sum_{j=1}^Q \Theta_j ~ \varepsilon_{t-s'j}}_{\text{SMA model}}}
\end{align}
$$

with $\varPhi = -\Phi$ and :

- $s$  is the number of time steps for a single seasonal period for the AR process
- $s'$ is the number of time steps for a single seasonal period for the MA process
- $P$ is the seasonal AR process order
- $Q$ is the seasonal MA process order

Remark: most of the time, $s = s'$. In this case, the process is simply noted $ARMA(\color{\red}{P}, \color{\green}{Q})_{s}$

#### Example: $ARMA(3,2)_{4, 5}$

$$
\begin{align}
ARMA(\color{\red}{3}, \color{\green}{2})_{4, 5}: \quad\quad\quad
\color{\red}{\underbrace{\Phi_{(3)} (B^4) y_t}_{\text{SAR}}} 
&=
c + \color{\green}{\underbrace{\Theta_{(2)} (B^5) \varepsilon_t}_{SMA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^3 \Phi_i B^{4i} \right) y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^2 \Theta_j B^{5j} \right) ~ \varepsilon_t}
\\
\color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^3 \Phi_i B^{4i} y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \Theta_j B^{5j} \varepsilon_t}
\\
\color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^3 \Phi_i y_{t-4i} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \Theta_j ~ \varepsilon_{t-5j}}
\\
\color{\red}{y_t}
&=
c
+ \color{\red}{\underbrace{\sum_{i=1}^3 \varPhi_i y_{t-4i}}_{\text{SAR model}}}
+ \color{\green}{\varepsilon_t}
+ \color{\green}{\underbrace{\sum_{j=1}^2 \Theta_j \varepsilon_{t-5j}}_{\text{SMA model}}}
\\
\color{\red}{y_t}
&=
c
+ \underbrace{ \color{\red}{\varPhi_1 y_{t-4}} + \color{\red}{\varPhi_2 y_{t-8}} + \color{\red}{\varPhi_3 y_{t-12}} }_{\text{SAR model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{ \color{\green}{\Theta_1 \varepsilon_{t-5}} + \color{\green}{\Theta_2 \varepsilon_{t-10}} }_{\text{SMA model}}
\end{align}
$$

with $\varPhi_i = -\Phi_i$

### Seasonal and non seasonal processes : additive process $ARMA(p,q)+ARMA(P,Q)_{s, s'}$

These processes are rarely used $ARMA(p,q)+ARMA(P,Q)_{s, s'}$.

$$
\begin{align}
ARMA(p,q)+ARMA(P,Q)_{s, s'}: \quad 
\left(
\underbrace{\varphi_p \left(B\right)}_{AR} +
\underbrace{\Phi_P \left(B^s\right)}_{SAR} -
1
\right) ~  y_t
&=
c +
\left(
\underbrace{\theta_q \left(B\right)}_{MA} +
\underbrace{\Theta_Q \left(B^{s'}\right)}_{SMA}
- 1 \right) ~ \varepsilon_t \\
\underbrace{\varphi_p (B) ~ y_t}_{AR} +
\underbrace{\Phi_P \left(B^s\right) ~ y_t}_{SAR} -
y_t
&=
c +
\underbrace{\theta_q \left(B\right) ~ \varepsilon_t}_{MA} +
\underbrace{\Theta_Q \left(B^{s'}\right) ~ \varepsilon_t}_{SMA} -
\varepsilon_t \\
\underbrace{\left( 1 + \sum_{i=1}^p \varphi_i B^i \right) ~ y_t}_{AR} +
\underbrace{\left( 1 + \sum_{k=1}^P \Phi_k B^{sk} \right) ~ y_t}_{SAR} -
y_t
&=
c +
\underbrace{\left( 1 + \sum_{j=1}^q \theta_j B^j \right) ~ \varepsilon_t}_{MA} +
\underbrace{\left( 1 + \sum_{l=1}^Q \Theta_l B^{s'l} \right) ~ \varepsilon_t}_{SMA} -
\varepsilon_t \\
\underbrace{y_t + \sum_{i=1}^p \varphi_i B^i y_t}_{AR} +
\underbrace{y_t + \sum_{k=1}^P \Phi_k B^{sk} y_t}_{SAR} -
y_t
&=
c +
\underbrace{\varepsilon_t + \sum_{j=1}^q \theta_j B^j \varepsilon_t}_{MA} +
\underbrace{\varepsilon_t + \sum_{l=1}^Q \Theta_l B^{s'l} \varepsilon_t}_{SMA} -
\varepsilon_t \\
y_t +
\sum_{i=1}^p \varphi_i B^i y_t +
\sum_{k=1}^P \Phi_k B^{sk} y_t
&=
c +
\varepsilon_t + \sum_{j=1}^q \theta_j B^j \varepsilon_t +
\sum_{l=1}^Q \Theta_l B^{s'l} \varepsilon_t \\
y_t +
\sum_{i=1}^p \varphi_i y_{t-i} +
\sum_{k=1}^P \Phi_k y_{t-sk}
&=
c +
\varepsilon_t +
\sum_{j=1}^q \theta_j \varepsilon_{t-j} +
\sum_{l=1}^Q \Theta_l \varepsilon_{t-s'l} \\
y_t
&=
c
+ \underbrace{\sum_{i=1}^p \phi_i y_{t-i}}_{\text{AR model}}
+ \underbrace{\sum_{k=1}^{P} \varPhi_k y_{t-sk}}_{\text{SAR model}}
+ \varepsilon_t
+ \underbrace{\sum_{j=1}^q \theta_j \varepsilon_{t-j}}_{\text{MA model}}
+ \underbrace{\sum_{l=1}^{Q} \Theta_l \varepsilon_{t-s'l}}_{\text{SMA model}}
\end{align}
$$

with $\varPhi = -\Phi$ and $\varphi = -\phi$

Remark: most of the time, $s = s'$. In this case, the process is simply noted $ARMA(\color{\red}{p}, \color{\green}{q}) + ARMA(\color{\red}{P}, \color{\green}{Q})_{s}$

### Seasonal and non seasonal processes : multiplicative process $ARMA(p,q) \times ARMA(P,Q)_{s, s'}$

These process are more often used than previously described additive processes.

Seasonal series are characterized by a strong serial correlation at the seasonal lag (and possibly multiples thereof).

$$
\begin{align}
ARMA(\color{\red}{p},\color{\green}{q}) \times ARMA(\color{\red}{P},\color{\green}{Q})_{\color{\red}{s}, \color{\green}{s'}}: \quad 
\color{\red}{
\underbrace{\varphi_{(p)} (B)}_{AR} ~
\underbrace{\Phi_{(P)} (B^s)}_{SAR} ~ 
y_t
}
&=
c +
\color{\green}{
\underbrace{\theta_{(q)} (B)}_{MA} ~
\underbrace{\Theta_{(Q)} (B^{s'})}_{SMA} ~
\varepsilon_t
}
\\
\color{\red}{
\underbrace{\left( 1 + \sum_{i=1}^p \varphi_i B^i \right)}_{AR} ~
\underbrace{\left( 1 + \sum_{k=1}^P \Phi_k B^{sk} \right)}_{SAR} ~ 
y_t
}
&=
c +
\color{\green}{
\underbrace{\left( 1 + \sum_{j=1}^q \theta_j B^j \right)}_{MA} ~
\underbrace{\left( 1 + \sum_{l=1}^Q \Theta_l B^{s'l} \right)}_{SMA} ~
\varepsilon_t
}
\\
\color{\red}{
\underbrace{
\left( 1 + \varphi_1 B + \varphi_2 B^2 + \dots + \varphi_p B^p \right)
\left( 1 + \Phi_1 B^{s} + \Phi_2 B^{s2} + \dots + \Phi_P B^{sP} \right)
y_t
}_{\text{Order of the equivalent AR process} ~=~ p + Ps}
}
&=
c +
\color{\green}{
\underbrace{
\left( 1 + \theta_1 B + \theta_2 B^2 + \dots + \theta_q B^q \right)
\left( 1 + \Theta_1 B^{s'} + \Theta_2 B^{s'2} + \dots + \Theta_Q B^{s'Q} \right)
\varepsilon_t
}_{\text{Order of the equivalent MA process} ~=~ q + Qs'}
}
\\
\end{align}
$$

with $s > p$ and $s' > q$.

$ARMA(p,q) \times ARMA(P,Q)_{s, s'}$ can be defined as an $ARMA(p + Ps, ~ q + Qs')$ with some parameters fixed to 0.

Remark: most of the time, $s = s'$. In this case, the process is simply noted $ARMA(\color{\red}{p}, \color{\green}{q}) \times ARMA(\color{\red}{P}, \color{\green}{Q})_{s}$

#### Example: $ARMA(3,0) \times ARMA(2,0)_{6, 0}$

$$
\begin{align}
ARMA(\color{\red}{3},0) \times ARMA(\color{\pink}{2},0)_{6,0}: \quad
\color{\red}{
\underbrace{\varphi_{(3)} (B)}_{AR} ~
}
\color{\pink}{
\underbrace{\Phi_{(2)} (B^6)}_{SAR} ~ 
}
y_t
=~&
c + \varepsilon_t
\\
\color{\red}{
\underbrace{\left( 1 + \sum_{i=1}^3 \varphi_i B^i \right)}_{AR} ~
}
\color{\pink}{
\underbrace{\left( 1 + \sum_{k=1}^2 \Phi_k B^{6k} \right)}_{SAR} ~ 
}
y_t
=~&
c + \varepsilon_t
\\
\underbrace{
    \color{\red}{
    \left( 1 + \varphi_1 B + \varphi_2 B^2 + \varphi_3 B^3 \right)
    }
\color{\pink}{
    \left( 1 + \Phi_1 B^{6} + \Phi_2 B^{12} \right)
    }
y_t
}_{\text{Order of the equivalent AR process} ~=~ 3 + 2 \times 6 = 15}
=~&
c + \varepsilon_t
\\
\left(
1 + \Phi_1 B^{6} + \Phi_2 B^{12}
+ \color{\red}{\varphi_1} B    + \color{\red}{\varphi_1} \color{\pink}{\Phi_1} B^{7} + \color{\red}{\varphi_1} \color{\pink}{\Phi_2} B^{13}
+ \color{\red}{\varphi_2} B^2  + \color{\red}{\varphi_2} \color{\pink}{\Phi_1} B^{8} + \color{\red}{\varphi_2} \color{\pink}{\Phi_2} B^{14}
+ \color{\red}{\varphi_3} B^3  + \color{\red}{\varphi_3} \color{\pink}{\Phi_1} B^{9} + \color{\red}{\varphi_3} \color{\pink}{\Phi_2} B^{15}
\right)
y_t
=~&
c + \varepsilon_t
\\
y_t =
  & ~ c \\
  & + \color{\red}{\phi_1} y_{t-1} + \color{\red}{\phi_2} y_{t-2} + \color{\red}{\phi_3} y_{t-3} \\
  & + \color{\pink}{\varPhi_1}   y_{t-6} + \color{\red}{\phi_1} \color{\pink}{\varPhi_1} y_{t-7}  + \color{\red}{\phi_2} \color{\pink}{\varPhi_1}  y_{t-8} + \color{\red}{\phi_3} \color{\pink}{\varPhi_1} y_{t-9} \\
  & + \color{\pink}{\varPhi_2}  y_{t-12} + \color{\red}{\phi_1} \color{\pink}{\varPhi_2} y_{t-13} + \color{\red}{\phi_2} \color{\pink}{\varPhi_2} y_{t-14} + \color{\red}{\phi_3} \color{\pink}{\varPhi_2} y_{t-15} \\
  & + \varepsilon_t
\end{align}
$$

<center></center>

#### Example: $ARMA(3,2) \times ARMA(2,1)_{6, 12}$

$$
\begin{align}
ARMA(\color{\red}{3},\color{\green}{2}) \times ARMA(\color{\red}{2},\color{\green}{1})_{\color{\red}{6},\color{\green}{12}}: \quad
\color{\red}{
\underbrace{\varphi_{(3)} (B)}_{AR} ~
\underbrace{\Phi_{(2)} (B^6)}_{SAR} ~ 
y_t
}
=~&
c +
\color{\green}{
\underbrace{\theta_{(2)} (B)}_{MA} ~
\underbrace{\Theta_{(1)} (B^{12})}_{SMA} ~
\varepsilon_t
}
\\
\color{\red}{
\underbrace{\left( 1 + \sum_{i=1}^3 \varphi_i B^i \right)}_{AR} ~
\underbrace{\left( 1 + \sum_{k=1}^2 \Phi_k B^{6k} \right)}_{SAR} ~ 
y_t
}
=~&
c +
\color{\green}{
\underbrace{\left( 1 + \sum_{j=1}^2 \theta_j B^j \right)}_{MA} ~
\underbrace{\left( 1 + \sum_{l=1}^1 \Theta_l B^{12l} \right)}_{SMA} ~
\varepsilon_t
}
\\
\color{\red}{
\underbrace{
\left( 1 + \varphi_1 B + \varphi_2 B^2 + \varphi_3 B^3 \right)
\left( 1 + \Phi_1 B^{6} + \Phi_2 B^{12} \right)
y_t
}_{\text{Order of the equivalent AR process} ~=~ 3 + 2 \times 6 = 15}
}
=~&
c +
\color{\green}{
\underbrace{
\left( 1 + \theta_1 B + \theta_2 B^2 \right)
\left( 1 + \Theta_1 B^{12} \right)
\varepsilon_t
}_{\text{Order of the equivalent MA process} ~=~ 2 + 1 \times 12 = 14}
}
%
%\\
%\left(
%1 + \Phi_1 B^{6} + \Phi_2 B^{12}
%+ \varphi_1 B    + \varphi_1 B   \times \Phi_1 B^{6} + \varphi_1 B   \times \Phi_2 B^{12}
%+ \varphi_2 B^2  + \varphi_2 B^2 \times \Phi_1 B^{6} + \varphi_2 B^2 \times \Phi_2 B^{12}
%+ \varphi_3 B^3  + \varphi_3 B^3 \times \Phi_1 B^{6} + \varphi_3 B^3 \times \Phi_2 B^{12}
%\right)
%y_t
%=~&
%c +
%\left(
%1 + \Theta_1 B^{12}
%+ \theta_1 B   + \theta_1 B   \times \Theta_1 B^{12}
%+ \theta_2 B^2 + \theta_2 B^2 \times \Theta_1 B^{12}
%\right)
%\varepsilon_t
%
\\
\color{\red}{
\left(
1 + \Phi_1 B^{6} + \Phi_2 B^{12}
+ \varphi_1 B    + \varphi_1 \Phi_1 B^{7} + \varphi_1 \Phi_2 B^{13}
+ \varphi_2 B^2  + \varphi_2 \Phi_1 B^{8} + \varphi_2 \Phi_2 B^{14}
+ \varphi_3 B^3  + \varphi_3 \Phi_1 B^{9} + \varphi_3 \Phi_2 B^{15}
\right)
y_t
}
=~&
c +
\color{\green}{
\left(
1 + \Theta_1 B^{12}
+ \theta_1 B   + \theta_1 \Theta_1 B^{13}
+ \theta_2 B^2 + \theta_2 \Theta_1 B^{14}
\right)
\varepsilon_t
}
\\
\color{\red}{y_t} =
  & ~ c \\
  & \color{\red}{+ \varPhi_1 y_{t-1} + \varPhi_2 y_{t-2} + \varPhi_3 y_{t-3}} \\
  & \color{\red}{+ \phi_1 y_{t-6} + \varPhi_1 \phi_1 y_{t-7} + \varPhi_2 \phi_1 y_{t-8} + \varPhi_3 \phi_1 y_{t-9}} \\
  & \color{\red}{+ \phi_2 y_{t-12} + \varPhi_1 \phi_2 y_{t-13} + \varPhi_2 \phi_2 y_{t-14} + \varPhi_3 \phi_2 y_{t-15}} \\
  & \color{\green}{+ \varepsilon_t} \\
  & \color{\green}{+ \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2}} \\
  & \color{\green}{+ \Theta_1 \varepsilon_{t-12} + \theta_1 \Theta_1 \varepsilon_{t-13} + \theta_2 \Theta_1 \varepsilon_{t-14}}
\end{align}
$$

### Seasonal AutoRegressive Moving Average (SARIMA) : $ARIMA(P,D,Q)_{s,s'',s'}$ with seasonal differencing

$$
\begin{align}
ARIMA(\color{\red}{P}, \color{\orange}{D}, \color{\green}{Q})_{\color{\red}{s}, \color{\orange}{s''}, \color{\green}{s'}}: \quad\quad\quad
\color{\red}{\underbrace{\Phi_{(P)} (B^s) \color{\orange}{\nabla^D_{s''}} y_t}_{\text{SARI}}}
&= 
c + \color{\green}{\underbrace{\Theta_{(Q)} (B^{s'}) \varepsilon_t}_{SMA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^P \Phi_i B^{si} \right) \color{\orange}{\nabla^D_{s''}} y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^Q \Theta_j B^{s'j} \right) ~ \varepsilon_t}
\\
\color{\orange}{\nabla^D_{s''}} \color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^P \Phi_i B^{si} \color{\orange}{\nabla^D_{s''}} y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^Q \Theta_j B^{s'j} \varepsilon_t}
\\
\color{\orange}{\nabla^D_{s''}} \color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^P \Phi_i ~ \color{\orange}{\nabla^D_{s''}} y_{t-si} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^Q \Theta_j ~ \varepsilon_{t-s'j}}
\\
\color{\orange}{\nabla^D_{s''}} \color{\red}{y_t} &= c
+ \color{\red}{\underbrace{\sum_{i=1}^P \varPhi_i ~ \color{\orange}{\nabla^D_{s''}} y_{t-si}}_{\text{SARI model}}}
+ \color{\green}{\varepsilon_t}
+ \color{\green}{\underbrace{\sum_{j=1}^Q \Theta_j ~ \varepsilon_{t-s'j}}_{\text{SMA model}}}
\end{align}
$$

with

$$\nabla_{s''} = (1-B^{s''})$$
$$\nabla_{s''}^2 = (1-B^{s''})^2$$
$$\dots$$
$$\nabla_{s''}^d = (1-B^{s''})^d$$

Remark: most of the time, $s = s' = s''$. In this case, the process is simply noted $ARIMA(\color{\red}{P}, \color{\orange}{D}, \color{\green}{Q})_{s}$

#### Example: $ARIMA(3, 1, 2)_{24, 24, 24}$

$$
\begin{align}
ARIMA(\color{\red}{3}, \color{\orange}{1}, \color{\green}{2})_{\color{\red}{24}, \color{\orange}{24}, \color{\green}{24}}: \quad\quad\quad
\color{\red}{\underbrace{\Phi_{(3)} (B^{24}) \color{\orange}{\nabla_{24}} y_t}_{\text{SARI}}}
&= 
c + \color{\green}{\underbrace{\Theta_{(2)} (B^{24}) \varepsilon_t}_{SMA}}
\\
\color{\red}{ \left( 1 + \sum_{i=1}^3 \Phi_i B^{24i} \right) \color{\orange}{\nabla_{24}} y_t }
&=
c + \color{\green}{\left( 1 + \sum_{j=1}^2 \Theta_j B^{24j} \right) ~ \varepsilon_t}
\\
\color{\orange}{\nabla_{24}} \color{\red}{ y_t } +  \color{\red}{\sum_{i=1}^3 \Phi_i B^{24i} \color{\orange}{\nabla_{24}} y_t }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \Theta_j B^{24j} \varepsilon_t}
\\
\color{\orange}{\nabla_{24}} \color{\red}{ y_t } + \color{\red}{ \sum_{i=1}^3 \Phi_i ~ \color{\orange}{\nabla_{24}} y_{t-24i} }
&=
c + \color{\green}{\varepsilon_t} + \color{\green}{\sum_{j=1}^2 \Theta_j ~ \varepsilon_{t-24j}}
\\
\color{\orange}{\nabla_{24}} \color{\red}{y_t} &= c
+ \color{\red}{\underbrace{\sum_{i=1}^3 \varPhi_i ~ \color{\orange}{\nabla_{24}} y_{t-24i}}_{\text{SARI model}}}
+ \color{\green}{\varepsilon_t}
+ \color{\green}{\underbrace{\sum_{j=1}^2 \Theta_j ~ \varepsilon_{t-24j}}_{\text{SMA model}}}
\\
\color{\orange}{\nabla_{24}} \color{\red}{y_t}
&=
c
+ \underbrace{
      \color{\red}{\varPhi_1} \color{\orange}{\nabla_{24}} \color{\red}{y_{t-24}}
    + \color{\red}{\varPhi_2} \color{\orange}{\nabla_{24}} \color{\red}{y_{t-48}}
    + \color{\red}{\varPhi_3} \color{\orange}{\nabla_{24}} \color{\red}{y_{t-72}}
  }_{\text{SARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\Theta_1 \varepsilon_{t-24}} + \color{\green}{\Theta_2 \varepsilon_{t-48}}}_{\text{MA model}}
\\
\color{\orange}{(1 - B^{24})} \color{\red}{y_t}
&=
c
+ \underbrace{
      \color{\red}{\varPhi_1} \color{\orange}{(1 - B^{24})} \color{\red}{y_{t-24}}
    + \color{\red}{\varPhi_2} \color{\orange}{(1 - B^{24})} \color{\red}{y_{t-48}}
    + \color{\red}{\varPhi_3} \color{\orange}{(1 - B^{24})} \color{\red}{y_{t-72}}
  }_{\text{SARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\Theta_1 \varepsilon_{t-24}} + \color{\green}{\Theta_2 \varepsilon_{t-48}}}_{\text{MA model}}
\\
\color{\red}{y_t} - \color{\orange}{B^{24} y_t}
&=
c
+ \underbrace{
      \color{\red}{\varPhi_1} (\color{\red}{y_{t-24}} - \color{\orange}{B^{24} y_{t-24}})
    + \color{\red}{\varPhi_2} (\color{\red}{y_{t-48}} - \color{\orange}{B^{24} y_{t-48}})
    + \color{\red}{\varPhi_3} (\color{\red}{y_{t-72}} - \color{\orange}{B^{24} y_{t-72}})
  }_{\text{SARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\Theta_1 \varepsilon_{t-24}} + \color{\green}{\Theta_2 \varepsilon_{t-48}}}_{\text{MA model}}
\\
\color{\red}{y_t} - \color{\orange}{y_{t-24}}
&=
c
+ \underbrace{
      \color{\red}{\varPhi_1} (\color{\red}{y_{t-24}} - \color{\orange}{y_{t-48}})
    + \color{\red}{\varPhi_2} (\color{\red}{y_{t-48}} - \color{\orange}{y_{t-72}})
    + \color{\red}{\varPhi_3} (\color{\red}{y_{t-72}} - \color{\orange}{y_{t-96}})
  }_{\text{SARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\Theta_1 \varepsilon_{t-24}} + \color{\green}{\Theta_2 \varepsilon_{t-48}}}_{\text{MA model}}
\\
\color{\red}{y_t}
&=
c
+ \color{\orange}{y_{t-24}} 
+ \underbrace{
      \color{\red}{\varPhi_1 y_{t-24}} - \color{\orange}{\varPhi_1 y_{t-48}}
    + \color{\red}{\varPhi_2 y_{t-48}} - \color{\orange}{\varPhi_2 y_{t-72}}
    + \color{\red}{\varPhi_3 y_{t-72}} - \color{\orange}{\varPhi_3 y_{t-96}}
  }_{\text{SARI model}}
+ \color{\green}{\varepsilon_t}
+ \underbrace{\color{\green}{\Theta_1 \varepsilon_{t-24}} + \color{\green}{\Theta_2 \varepsilon_{t-48}}}_{\text{MA model}}
\end{align}
$$

### Seasonal AutoRegressive Moving Average (SARIMA) : $ARIMA(p,d,q) \times ARIMA(P,D,Q)_{s,s'',s'}$ with seasonal and non-seasonal differencing

$$
\begin{align}
ARIMA(\color{\red}{p}, \color{\orange}{d}, \color{\green}{q}) \times ARIMA(\color{\red}{P}, \color{\orange}{D}, \color{\green}{Q})_{\color{\red}{s}, \color{\orange}{s'}, \color{\green}{s''}}: \quad\quad\quad
\color{\red}{
\underbrace{\varphi_{(p)} (B)}_{AR} ~
\underbrace{\Phi_{(P)} (B^s)}_{SAR} ~ 
}
\color{\orange}{\nabla^D_{s''}} \color{\orange}{\nabla^d}
\color{\red}{y_t}
&=
c +
\color{\green}{
\underbrace{\theta_{(q)} (B)}_{MA} ~
\underbrace{\Theta_{(Q)} (B^{s'})}_{SMA} ~
\varepsilon_t
}
\\
\color{\red}{
\underbrace{\left( 1 + \sum_{i=1}^p \varphi_i B^i \right)}_{AR} ~
\underbrace{\left( 1 + \sum_{k=1}^P \Phi_k B^{sk} \right)}_{SAR} ~ 
}
\color{\orange}{\nabla^D_{s''}} \color{\orange}{\nabla^d}
\color{\red}{y_t}
&=
c +
\color{\green}{
\underbrace{\left( 1 + \sum_{j=1}^q \theta_j B^j \right)}_{MA} ~
\underbrace{\left( 1 + \sum_{l=1}^Q \Theta_l B^{s'l} \right)}_{SMA} ~
\varepsilon_t
}
\\
\underbrace{
\color{\red}{
\left( 1 + \varphi_1 B + \varphi_2 B^2 + \dots + \varphi_p B^p \right)
\left( 1 + \Phi_1 B^{s} + \Phi_2 B^{s2} + \dots + \Phi_P B^{sP} \right)
}
\color{\orange}{\nabla^D_{s''}} \color{\orange}{\nabla^d}
\color{\red}{y_t}
}
&=
c +
\underbrace{
\color{\green}{
\left( 1 + \theta_1 B + \theta_2 B^2 + \dots + \theta_q B^q \right)
\left( 1 + \Theta_1 B^{s'} + \Theta_2 B^{s'2} + \dots + \Theta_Q B^{s'Q} \right)
\varepsilon_t
}
}
\\
\end{align}
$$

Remark: $\color{\orange}{\nabla^D_{s''}} \color{\orange}{\nabla^d} y_t = \color{\orange}{\nabla^d} \color{\orange}{\nabla^D_{s''}} y_t$

#### Example: $ARMA(3,1,2) \times ARMA(2,1,1)_{24, 24, 24}$

$$
\begin{align}
ARIMA(\color{\red}{3}, \color{\orange}{1}, \color{\green}{2}) \times ARIMA(\color{\red}{2}, \color{\orange}{1}, \color{\green}{1})_{\color{\red}{24}, \color{\orange}{24}, \color{\green}{24}}: \quad\quad\quad
\color{\red}{
\underbrace{\varphi_{(3)} (B)}_{AR} ~
\underbrace{\Phi_{(2)} \left( B^{24} \right)}_{SAR} ~ 
}
\color{\orange}{\nabla_{24}} \color{\orange}{\nabla}
\color{\red}{y_t}
&=
c +
\color{\green}{
\underbrace{\theta_{(2)} (B)}_{MA} ~
\underbrace{\Theta_{(1)} \left( B^{24} \right)}_{SMA} ~
\varepsilon_t
}
\\
\color{\red}{
\underbrace{\left( 1 + \sum_{i=1}^3 \varphi_i B^i \right)}_{AR} ~
\underbrace{\left( 1 + \sum_{k=1}^2 \Phi_k B^{24 k} \right)}_{SAR} ~ 
}
\color{\orange}{\nabla_{24}} \color{\orange}{\nabla}
\color{\red}{y_t}
&=
c +
\color{\green}{
\underbrace{\left( 1 + \sum_{j=1}^2 \theta_j B^j \right)}_{MA} ~
\underbrace{\left( 1 + \sum_{l=1}^1 \Theta_l B^{24 l} \right)}_{SMA} ~
\varepsilon_t
}
\\
\color{\red}{
\left( 1 + \varphi_1 B + \varphi_2 B^2 + \varphi_3 B^3 \right)
\left( 1 + \Phi_1 B^{24} + \Phi_2 B^{48} \right)
}
\color{\orange}{\nabla_{24}} \color{\orange}{\nabla}
\color{\red}{y_t}
&=
c +
\color{\green}{
\left( 1 + \theta_1 B + \theta_2 B^2 \right)
\left( 1 + \Theta_1 B^{24} \right)
\varepsilon_t
}
\\
\color{\red}{
\left(
1 + \Phi_1 B^{24} + \Phi_2 B^{48}
+ \varphi_1 B    + \varphi_1 \Phi_1 B^{25} + \varphi_1 \Phi_2 B^{49}
+ \varphi_2 B^2  + \varphi_2 \Phi_1 B^{26} + \varphi_2 \Phi_2 B^{50}
+ \varphi_3 B^3  + \varphi_3 \Phi_1 B^{27} + \varphi_3 \Phi_2 B^{51}
\right)
}
\color{\orange}{\nabla_{24}} \color{\orange}{\nabla}
\color{\red}{y_t}
=~&
c +
\color{\green}{
\left(
1 + \Theta_1 B^{24}
+ \theta_1 B   + \theta_1 \Theta_1 B^{25}
+ \theta_2 B^2 + \theta_2 \Theta_1 B^{26}
\right)
\varepsilon_t
}
\\
\color{\red}{y_t} =
  & ~ c \\
  & \color{\red}{+ \varPhi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-1} + \varPhi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-2} + \varPhi_3 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-3}} \\
  & \color{\red}{+ \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-24}  + \varPhi_1 \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-25} + \varPhi_2 \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-26} + \varPhi_3 \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-27}} \\
  & \color{\red}{+ \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-48} + \varPhi_1 \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-49} + \varPhi_2 \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-50} + \varPhi_3 \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-51}} \\
  & \color{\green}{+ \varepsilon_t} \\
  & \color{\green}{+ \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2}} \\
  & \color{\green}{+ \Theta_1 \varepsilon_{t-24} + \theta_1 \Theta_1 \varepsilon_{t-25} + \theta_2 \Theta_1 \varepsilon_{t-26}}
\\
\end{align}
$$

As we have:

$$
\begin{align}
\nabla                  &= (1-B) \\
\nabla_{24}             &= (1-B^{24}) \\
\nabla \nabla_{24}      &= (1-B)(1-B^{24}) \\
                        &= 1-B^{24}-B+B^{25} \\
\nabla \nabla_{24} y_t  &= (1-B^{24}-B+B^{25}) y_t \\
                        &= y_t - B^{24} y_t - B y_t   + B^{25} y_t \\
                        &= y_t - y_{t-24}   - y_{t-1} + y_{t-25} \\
\end{align}
$$

Thus:

$$
\begin{align}
\color{\red}{y_t} =
  & ~ c \\
  & \color{\red}{+ \varPhi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-1} + \varPhi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-2} + \varPhi_3 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-3}} \\
  & \color{\red}{+ \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-24}  + \varPhi_1 \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-25} + \varPhi_2 \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-26} + \varPhi_3 \phi_1 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-27}} \\
  & \color{\red}{+ \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-48} + \varPhi_1 \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-49} + \varPhi_2 \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-50} + \varPhi_3 \phi_2 \color{\orange}{\nabla_{24}} \color{\orange}{\nabla} y_{t-51}} \\
  & \color{\green}{+ \varepsilon_t} \\
  & \color{\green}{+ \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2}} \\
  & \color{\green}{+ \Theta_1 \varepsilon_{t-24} + \theta_1 \Theta_1 \varepsilon_{t-25} + \theta_2 \Theta_1 \varepsilon_{t-26}}
\\
= & ...
\\
\end{align}
$$

ff8423cee428259428f191ab99da22c825e9c845
ipynb
Jupyter Notebook
nb_sci_ai/ai_arma_en.ipynb
jdhp-docs/python-notebooks
91a97ea5cf374337efa7409e4992ea3f26b99179
MIT
2017-05-03T12:23:36.000Z
2020-10-26T17:30:56.000Z
nb_sci_ai/ai_arma_en.ipynb
jdhp-docs/python-notebooks
91a97ea5cf374337efa7409e4992ea3f26b99179
MIT
nb_sci_ai/ai_arma_en.ipynb
jdhp-docs/python-notebooks
91a97ea5cf374337efa7409e4992ea3f26b99179
MIT
2020-10-26T17:30:57.000Z
2020-10-26T17:30:57.000Z
Qwen/Qwen-72B
1. YES
2. YES