## 8.5 Examples of application

### 8.5.1 Non-seasonal data

Using the time series from the Box-Jenkins textbook (Box and Jenkins, 1976), we fit the ARIMA model to the data but based on our judgment rather than their approach. Just a reminder, here is how the data looks (series `BJsales`

, Figure 8.17):

It seems to exhibit the trend in the data, so we can consider ARIMA(1,1,2), ARIMA(0,2,2) and ARIMA(1,1,1) models. We do not consider models with drift in this example, because they would imply the same slope over time for the whole series, which does not seem to be the case here. We use `msarima()`

function from the `smooth`

package, which is a wrapper for the `adam()`

function:

```
<- vector("list",3)
adamARIMABJ # ARIMA(1,1,2)
1]] <- msarima(BJsales, orders=c(1,1,2),
adamARIMABJ[[h=10, holdout=TRUE)
# ARIMA(0,2,2)
2]] <- msarima(BJsales, orders=c(0,2,2),
adamARIMABJ[[h=10, holdout=TRUE)
# ARIMA(1,1,1)
3]] <- msarima(BJsales, orders=c(1,1,1),
adamARIMABJ[[h=10, holdout=TRUE)
names(adamARIMABJ) <- c("ARIMA(1,1,2)", "ARIMA(0,2,2)",
"ARIMA(1,1,1)")
```

Comparing information criteria (we will use AICc) of the three models, we can select the most appropriate one:

`sapply(adamARIMABJ, AICc)`

```
## ARIMA(1,1,2) ARIMA(0,2,2) ARIMA(1,1,1)
## 503.8630 497.0121 505.7502
```

*Remark*. Note that this comparison is possible in `adam()`

, `ssarima()`

and `msarima()`

because the implemented ARIMA is formulated in state space form, sidestepping the issue of the conventional ARIMA (where taking differences reduces the sample size).

Based on this comparison, it looks like the ARIMA(0,2,2) is the most appropriate model (among the three) for the data. Here how the fit and the forecast from the model looks (Figure 8.18):

`plot(adamARIMABJ[[2]], which=7)`

Comparing this model with the ETS(A,A,N), we will see a slight difference because the two models are initialised and estimated differently:

`adam(BJsales, "AAN", h=10, holdout=TRUE)`

```
## Time elapsed: 0.02 seconds
## Model estimated using adam() function: ETS(AAN)
## Distribution assumed in the model: Normal
## Loss function type: likelihood; Loss function value: 243.4073
## Persistence vector g:
## alpha beta
## 1.0000 0.2392
##
## Sample size: 140
## Number of estimated parameters: 5
## Number of degrees of freedom: 135
## Information criteria:
## AIC AICc BIC BICc
## 496.8146 497.2624 511.5228 512.6292
##
## Forecast errors:
## ME: 3.229; MAE: 3.341; RMSE: 3.797
## sCE: 14.178%; Asymmetry: 91.7%; sMAE: 1.467%; sMSE: 0.028%
## MASE: 2.826; RMSSE: 2.491; rMAE: 0.928; rRMSE: 0.924
```

If we are interested in a more classical Box-Jenkins approach, we can always analyse the residuals of the constructed model and try improving it further. Here is an example of ACF and PACF of the residuals of the ARIMA(0,2,2):

```
par(mfcol=c(1,2))
plot(adamARIMABJ[[2]], which=c(10,11), main="")
```

As we see from the plot in Figure 8.19, all autocorrelation coefficients lie inside the confidence interval, implying that there are no significant AR / MA lags to include in the model.

### 8.5.2 Seasonal data

Similarly to the previous cases, we use Box-Jenkins AirPassengers data, which exhibits a multiplicative seasonality and a trend. We will model this using SARIMA(0,2,2)(0,1,1)\(_{12}\), SARIMA(0,2,2)(1,1,1)\(_{12}\) and SARIMA(0,2,2)(1,1,0)\(_{12}\) models, which are selected to see what type of seasonal ARIMA is more appropriate to the data:

```
<- vector("list",3)
adamSARIMAAir # SARIMA(0,2,2)(0,1,1)[12]
1]] <- msarima(AirPassengers, lags=c(1,12),
adamSARIMAAir[[orders=list(ar=c(0,0), i=c(2,1),
ma=c(2,1)),
h=12, holdout=TRUE)
# SARIMA(0,2,2)(1,1,1)[12]
2]] <- msarima(AirPassengers, lags=c(1,12),
adamSARIMAAir[[orders=list(ar=c(0,1), i=c(2,1),
ma=c(2,1)),
h=12, holdout=TRUE)
# SARIMA(0,2,2)(1,1,0)[12]
3]] <- msarima(AirPassengers, lags=c(1,12),
adamSARIMAAir[[orders=list(ar=c(0,1), i=c(2,1),
ma=c(2,0)),
h=12, holdout=TRUE)
names(adamSARIMAAir) <- c("SARIMA(0,2,2)(0,1,1)[12]",
"SARIMA(0,2,2)(1,1,1)[12]",
"SARIMA(0,2,2)(1,1,0)[12]")
```

Note that now that we have a seasonal component, we need to provide the SARIMA lags: 1 and \(m=12\) and specify `orders`

differently – as a list with values for AR, I and MA orders separately. This is done because the SARIMA implemented in `msarima()`

and `adam()`

supports multiple seasonalities (e.g. you can have `lags=c(1,24,24*7)`

if you want). The resulting information criteria of models are:

`sapply(adamSARIMAAir, AICc)`

```
## SARIMA(0,2,2)(0,1,1)[12] SARIMA(0,2,2)(1,1,1)[12] SARIMA(0,2,2)(1,1,0)[12]
## 1028.828 1078.016 1086.773
```

It looks like the second model is slightly better than the other two, so we will use it in order to produce forecasts (see Figure 8.20):

```
forecast(adamSARIMAAir[[2]], h=12,
interval="prediction") |>
plot(main="")
```

This model is directly comparable with ETS models, so here is, for example, AICc of ETS(M,A,M) on the same data:

```
AICc(adamETSAir <- adam(AirPassengers, "MAM",
h=12, holdout=TRUE))
```

`## [1] 973.9646`

It is lower than for the SARIMA model, which means that ETS(M,A,M) is more appropriate for the data in terms of information criteria than SARIMA(0,2,2)(1,1,1)\(_{12}\). We can also investigate if there is a way to improve ETS(M,A,M) by adding some ARMA components (Figure 8.21):

```
par(mfcol=c(1,2))
plot(adamETSAir, which=c(10,11), main="")
```

Judging by the plots in Figure 8.21, significant correlation coefficients exist for some lags. Still, it is not clear whether they appear due to randomness or not. Just to check, we will see if adding SARIMA(0,0,0)(0,0,1)\(_{12}\) helps (reduces AICc) in this case:

```
adam(AirPassengers, "MAM", h=12, holdout=TRUE,
order=list(ma=c(0,1)), lags=c(1,12)) |>
AICc()
```

`## [1] 1009.721`

As we see, the increased complexity does not decrease the AICc (probably because now we need to estimate 13 parameters more than in just ETS(M,A,M)), so we should not add the SARIMA component. We could try adding other SARIMA elements to see if they improve the model, but we do not aim to find the best model here. The interested reader is encouraged to do that as an additional exercise.