ETC3550 A1

Ari Gestetner - 33966753

Question 1

retail |> autoplot(Turnover, color=plot_colour) +
  labs(title = "Time Series Plot of Retail Data")

Looking at the plot we can notice a somewhat exponential positive trend.
Additionally, we can note some multiplicitive seasonal patterns.
We can also see some indications of cyclical patterns where there appears to be a drop in the late 1980’s. Additionally, we can see what appears to be a drop in the late 2000’s (likely around the 2008 GFC) and another fall in the early 2020’s (likely due to Covid).

Question 2

retail |> gg_season(Turnover, labels = "left")

By plotting the data with the gg_season function we can note some features of the data.
Specifically, we are able to notice that as the years increment, so does the data increase (indicating a positive trend). Furthermore, we can notice how there is an increase in sales by December and January, however, there is then a drop in sales during the month of Febuary.
Additionally, we can note that there are some years with a significant drop in sales, however, the plot is too clutered to investigate specific years. Therefore, we will split the data into years, before and after the year

retail |> 
  filter(Month < yearmonth("Jan 2000")) |>
  gg_season(Turnover) +
  labs(title = "Seasonal plots from before January 2000")

retail |> 
  filter(Month >= yearmonth("Jan 2000")) |>
  gg_season(Turnover) +
  labs(title = "Seasonal plots from after January 2000")

In the first plot we can see a significant drop in the late 1980’s (Green line) around the end of the year. Furthermore, in the second plot we can see another significant dip in 2020 (Purple line) which is also around the time of the covid outbreak.
Nevertheless, there doesn’t appear to be any significant drops in the late 2000’s despite what it looked like in the autoplot plot.
Ultimatly, these drops and recoveries are part of the cyclical behaviour of this data.

Question 3

retail |> 
  gg_subseries(Turnover, color=plot_colour) +
  labs(title = "Subseries plot for retail data")

Taking a look at the gg_subseries plot we can note some features which were hidden in the other 2 plots.
First, we can notice how there is a dip in the late 2000’s over every month which is possibly the reason it wasn’t so obvious in the gg_season plot.
Furthermore, we are also able to notice the how the drop due to covid is extreme druing the months of March to June. However, after June the drop seems minimal compared to the earlier months.
Nevertheless, we can also see some common features with the other 2 plots, mainly the seasonal pattern with the increase in December and January and the slight dip in Febuary. As well, we can see the strong trend in the data which exists in every month.

Question 4

To find a good value for the Box Cox transformation, we will first plot the data with (log transformation) and (original data)

lambda <- 0
retail |> autoplot(box_cox(Turnover, lambda), color = plot_colour) +
  labs(title = glue::glue("Retail Turnover with Box Cox with lambda = {round(lambda, 2)}"), y = glue::glue("box_cox(Turnover, lambda = {round(lambda, 2)})"))

lambda <- 1
retail |> autoplot(box_cox(Turnover, lambda), color = plot_colour) +
  labs(title = glue::glue("Retail Turnover with Box Cox with lambda = {round(lambda, 2)}"), y = glue::glue("box_cox(Turnover, lambda = {round(lambda, 2)})"))

Firstly, we are able to see that the variation clearly looks multiplicative with the identity transformation. Furthermore, We can see here that the transformation is pretty good. however, there is a bit of variation in the seasonality in the middle of the data period.
Now we will use the Guerrero feature and see if it beats the log function.

# Attempt guerro first
lambda <- retail |> 
  features(Turnover, features = guerrero) |>
  pull(lambda_guerrero)
retail |> autoplot(box_cox(Turnover, lambda), color = plot_colour) +
  labs(title = glue::glue("Retail Turnover with Box Cox with lambda = {round(lambda, 2)} using Guerrero"), y = glue::glue("box_cox(Turnover, lambda = {round(lambda, 2)})"))

We can see here that the Guerrero feature doesn’t improve on , thus we will just use as it is the simple log transformation.

Question 5

lambda = 0
retail |>
  model(STL(box_cox(Turnover, lambda) ~ season(window = 9) +
    trend(window = 7), robust = TRUE)) |>
  components() |>
  autoplot(color=plot_colour) + labs(title = "STL Decomposition: US retail employment")

While some seasonality remains in the remainder, we can see some regular features of the data, such as positive trend and some cyclical patterns.
There is one particular feature which stands out in the STL Decomposition, that is how the seasonality changes as the years progress. As in in up untill around 2010, we can see that the December/January increase in sales was much more significant than since 2010. This can be seen by how the peaks seem to be “fatter” in the later years, whereas, they are a lot narrower in the earlier years. This may be due to internet shopping which helps consumers make easy purchases at all times of the year.