SSARIMA stands for “State-space ARIMA” or “Several Seasonalities
ARIMA”. Both names show what happens in the heart of the function: it
constructs ARIMA in a state-space form and allows to model several
(actually more than several) seasonalities. `ssarima()`

is a
function included in smooth package. This
vignette covers `ssarima()`

and `auto.ssarima()`

functions. For more details about the underlying model, read (Svetunkov and Boylan 2019).

As usual, we will use data from `Mcomp`

package, so it is
advised to install it.

Let’s load the necessary packages:

`require(smooth)`

The default call constructs ARIMA(0,1,1):

`ssarima(AirPassengers, h=12, silent=FALSE)`

```
## Time elapsed: 0.01 seconds
## Model estimated: ARIMA(0,1,1)
## Matrix of MA terms:
## Lag 1
## MA(1) 0.4066
## Initial values were produced using backcasting.
##
## Loss function type: likelihood; Loss function value: 700.9086
## Error standard deviation: 31.6734
## Sample size: 144
## Number of estimated parameters: 2
## Number of degrees of freedom: 142
## Information criteria:
## AIC AICc BIC BICc
## 1405.817 1405.902 1411.757 1411.968
```

Some more complicated model can be defined using parameter
`orders`

the following way:

`ssarima(AirPassengers, orders=list(ar=c(0,1),i=c(1,0),ma=c(1,1)), lags=c(1,12), h=12)`

```
## Time elapsed: 0.05 seconds
## Model estimated: SARIMA(0,1,1)[1](1,0,1)[12]
## Matrix of AR terms:
## Lag 12
## AR(1) 1
## Matrix of MA terms:
## Lag 1 Lag 12
## MA(1) -0.3269 0.1017
## Initial values were produced using backcasting.
##
## Loss function type: likelihood; Loss function value: 554.9947
## Error standard deviation: 11.58
## Sample size: 144
## Number of estimated parameters: 4
## Number of degrees of freedom: 140
## Information criteria:
## AIC AICc BIC BICc
## 1117.989 1118.277 1129.869 1130.584
```

This would construct seasonal ARIMA(0,1,1)(1,0,1)\(_{12}\).

We could try selecting orders manually, but this can also be done
automatically via `auto.ssarima()`

function:

`auto.ssarima(AirPassengers, h=12)`

```
## Time elapsed: 2.4 seconds
## Model estimated: SARIMA(0,1,3)[1](0,1,0)[12]
## Matrix of MA terms:
## Lag 1
## MA(1) -0.3587
## MA(2) 0.1056
## MA(3) -0.2230
## Initial values were produced using backcasting.
##
## Loss function type: likelihood; Loss function value: 549.9204
## Error standard deviation: 11.179
## Sample size: 144
## Number of estimated parameters: 4
## Number of degrees of freedom: 140
## Information criteria:
## AIC AICc BIC BICc
## 1107.841 1108.129 1119.720 1120.435
```

Automatic order selection in SSARIMA with optimised initials does not work well and in general is not recommended. This is partially because of the possible high number of parameters in some models and partially because of potential overfitting of first observations when non-zero order of AR is selected:

`auto.ssarima(AirPassengers, h=12, initial="backcasting")`

```
## Time elapsed: 2.4 seconds
## Model estimated: SARIMA(0,1,3)[1](0,1,0)[12]
## Matrix of MA terms:
## Lag 1
## MA(1) -0.3587
## MA(2) 0.1056
## MA(3) -0.2230
## Initial values were produced using backcasting.
##
## Loss function type: likelihood; Loss function value: 549.9204
## Error standard deviation: 11.179
## Sample size: 144
## Number of estimated parameters: 4
## Number of degrees of freedom: 140
## Information criteria:
## AIC AICc BIC BICc
## 1107.841 1108.129 1119.720 1120.435
```

`auto.ssarima(AirPassengers, h=12, initial="optimal")`

```
## Time elapsed: 22.85 seconds
## Model estimated: SARIMA(0,1,3)[1](0,1,0)[12] with drift
## Matrix of MA terms:
## Lag 1
## MA(1) -0.3645
## MA(2) 0.1120
## MA(3) -0.2297
## Constant value is: 0.2186
## Initial values were optimised.
##
## Loss function type: likelihood; Loss function value: 549.269
## Error standard deviation: 11.7305
## Sample size: 144
## Number of estimated parameters: 18
## Number of degrees of freedom: 126
## Information criteria:
## AIC AICc BIC BICc
## 1134.538 1140.010 1187.995 1201.592
```

As can be seen from the example above the model with optimal initials takes more time and we end up with a different model than in the case of backcasting.

A power of `ssarima()`

function is that it can estimate
SARIMA models with multiple seasonalities. For example,
SARIMA(0,1,1)(0,0,1)_6(1,0,1)_12 model can be estimated the following
way:

`ssarima(AirPassengers, orders=list(ar=c(0,0,1),i=c(1,0,0),ma=c(1,1,1)), lags=c(1,6,12), h=12, silent=FALSE)`

It probably does not make much sense for this type of data, it would
make more sense on high frequency data (for example, `taylor`

series from `forecast`

package). However, keep in mind that
multiple seasonal ARIMAs are very slow in estimation and are very
capricious. So it is really hard to obtain an appropriate and efficient
multiple seasonal ARIMA model. To tackle this issue, I’ve developed an
alternative ARIMA model for multiple seasonalities, called
`msarima()`

.

Now let’s introduce some artificial exogenous variables:

`<- cbind(rnorm(length(AirPassengers),50,3),rnorm(length(AirPassengers),100,7)) x `

If we save model:

`<- auto.ssarima(AirPassengers, h=12, holdout=TRUE, xreg=x) ourModel `

we can then reuse it:

`ssarima(AirPassengers, model=ourModel, h=12, holdout=FALSE, xreg=x, interval=TRUE)`

```
## Time elapsed: 0.25 seconds
## Model estimated: SARIMAX(0,1,3)[1](0,1,0)[12] with drift
## Matrix of MA terms:
## Lag 1
## MA(1) -0.2541
## MA(2) 0.1523
## MA(3) -0.2556
## Constant value is: 0.4061
## Initial values were provided by user.
## Xreg coefficients were estimated in a normal style
##
## Loss function type: likelihood; Loss function value: 550.2307
## Error standard deviation: 11.085
## Sample size: 144
## Number of estimated parameters: 1
## Number of provided parameters: 19
## Number of degrees of freedom: 143
## Information criteria:
## AIC AICc BIC BICc
## 1102.461 1102.490 1105.431 1105.501
##
## 95% parametric prediction interval was constructed
```

Finally, we can combine several SARIMA models:

`ssarima(AirPassengers, h=12, holdout=FALSE, interval=TRUE, combine=TRUE)`

```
## Time elapsed: 0.01 seconds
## Model estimated: ARIMA(0,1,1)
## Matrix of MA terms:
## Lag 1
## MA(1) 0.4066
## Initial values were produced using backcasting.
##
## Loss function type: likelihood; Loss function value: 700.9086
## Error standard deviation: 31.6734
## Sample size: 144
## Number of estimated parameters: 2
## Number of degrees of freedom: 142
## Information criteria:
## AIC AICc BIC BICc
## 1405.817 1405.902 1411.757 1411.968
##
## 95% parametric prediction interval was constructed
```

While SSARIMA is flexible, it is not fast. In fact, it cannot handle high frequency data well and most probably will take ages to estimate the parameter and produce forecasts. This is because of the transition matrix, which becomes huge in case of multiple seasonalities. The MSARIMA model (Multiple Seasonal ARIMA) is formulated in a different state-space form, which reduces the size of transition matrix, significantly reducing the computational time for cases with high frequency data.

There are `auto.msarima()`

and `msarima()`

function in the package, that do things similar to
`auto.ssarima()`

and `ssarima()`

. Here’s just one
example of what can be done with it:

`msarima(AirPassengers, orders=list(ar=c(0,0,1),i=c(1,0,0),ma=c(1,1,1)),lags=c(1,6,12),h=12, silent=FALSE)`

```
## Time elapsed: 0.29 seconds
## Model estimated using msarima() function: SARIMA(0,1,1)[1](0,0,1)[6](1,0,1)[12]
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 579.5247
## ARMA parameters of the model:
## AR:
## phi1[12]
## 0.9841
## MA:
## theta1[1] theta1[6] theta1[12]
## 0.2906 -0.2172 -0.2202
##
## Sample size: 144
## Number of estimated parameters: 24
## Number of degrees of freedom: 120
## Information criteria:
## AIC AICc BIC BICc
## 1207.049 1217.133 1278.325 1303.383
```

The forecasts of the two models might differ due to the different state space form. The detailed explanation of MSARIMA is given in Chapter 9 of ADAM textbook.

Svetunkov, Ivan, and John E. Boylan. 2019. “State-space ARIMA for supply-chain
forecasting.” *International Journal of Production
Research* 0 (0): 1–10. https://doi.org/10.1080/00207543.2019.1600764.