## 12.5 Prediction intervals for aggregates

A common problem is to forecast the aggregate of several time periods of data, using a model fitted to the disaggregated data. For example, we may have monthly data but wish to forecast the total for the next year. Or we may have weekly data, and want to forecast the total for the next four weeks.

If the point forecasts are means, then adding them up will give a good estimate of the total. But prediction intervals are more tricky due to the correlations between forecast errors.

A general solution is to use simulations. Here is an example using ETS models applied to Australian monthly gas production data, assuming we wish to forecast the aggregate gas demand in the next six months.

```
# First fit a model to the data
fit <- ets(gas/1000)
# Forecast six months ahead
fc <- forecast(fit, h=6)
# Simulate 10000 future sample paths
nsim <- 10000
h <- 6
sim <- numeric(nsim)
for(i in seq_len(nsim))
sim[i] <- sum(simulate(fit, future=TRUE, nsim=h))
meanagg <- mean(sim)
```

The mean of the simulations is very close to the sum of the individual forecasts:

```
sum(fc[["mean"]][1:6])
#> [1] 281.8
meanagg
#> [1] 281.7
```

Prediction intervals are also easy to obtain:

```
#80% interval:
quantile(sim, prob=c(0.1, 0.9))
#> 10% 90%
#> 263 301
#95% interval:
quantile(sim, prob=c(0.025, 0.975))
#> 2.5% 97.5%
#> 254.0 311.4
```