This is an example of ETS, see the blog for Theory.

International Tourist Visitor nights to Australia

library(fpp2)

## Warning: package 'fpp2' was built under R version 3.6.1

## Loading required package: ggplot2

## Registered S3 methods overwritten by 'ggplot2':
##   method         from 
##   [.quosures     rlang
##   c.quosures     rlang
##   print.quosures rlang

## Loading required package: forecast

## Warning: package 'forecast' was built under R version 3.6.1

## Registered S3 method overwritten by 'xts':
##   method     from
##   as.zoo.xts zoo

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

## Registered S3 methods overwritten by 'forecast':
##   method             from    
##   fitted.fracdiff    fracdiff
##   residuals.fracdiff fracdiff

## Loading required package: fma

## Warning: package 'fma' was built under R version 3.6.1

## Loading required package: expsmooth

## Warning: package 'expsmooth' was built under R version 3.6.1

Lets use the function

aust <- window(austourists,start=2005)
fit <- ets(aust)
summary(fit)

## ETS(M,A,M) 
## 
## Call:
##  ets(y = aust) 
## 
##   Smoothing parameters:
##     alpha = 0.1908 
##     beta  = 0.0392 
##     gamma = 2e-04 
## 
##   Initial states:
##     l = 32.3679 
##     b = 0.9281 
##     s = 1.0218 0.9628 0.7683 1.2471
## 
##   sigma:  0.0383
## 
##      AIC     AICc      BIC 
## 224.8628 230.1569 240.9205 
## 
## Training set error measures:
##                      ME     RMSE     MAE        MPE     MAPE     MASE
## Training set 0.04836907 1.670893 1.24954 -0.1845609 2.692849 0.409454
##                   ACF1
## Training set 0.2005962

The Model selected is MAM Lets plot it

autoplot(fit)

Lets get the residuals and forecast errors

cbind('Residuals' = residuals(fit),
'Forecast errors' = residuals(fit,type='response')) %>%
autoplot(facet=TRUE) + xlab("Year") + ylab("")

Lets Forecast with ETS function

fit %>% forecast(h=8) %>%
autoplot() +
ylab("International visitor night in Australia (millions)")