2

The plastics data set consists of the monthly sales (in thousands) of product A for a plastics manufacturer for five years.

a. Plot the time series of sales of product A. Can you identify seasonal fluctuations and/or a trend-cycle?

autoplot(plastics, main = 'Monthly sales (in thousands) of product A for a plastics manufacturer')

There is a visible upward trend throughout 5 years. Additionally, there are seasonal fluctuations, sales start at the lowest level in the beginning of a year, peak mid-year and then slow down for the remainder of each year.

b. Use a classical multiplicative decomposition to calculate the trend-cycle and seasonal indices.

fit <- plastics %>% decompose(type = 'multiplicative')
autoplot(fit) + ggtitle('Classical multiplicative decomposition')

c. Do the results support the graphical interpretation from part a?

Yes, the results of decomposition support the assessment from part ‘a’. There few runs that suggest that some of the trend-cycle component got into remainder component because the trend-cycle has over-smoothed the data.

d. Compute and plot the seasonally adjusted data.

autoplot(plastics, series = 'Unadjusted') +
  autolayer(seasadj(fit),series="Seasonally adjusted") + theme(legend.title=element_blank())

e. Change one observation to be an outlier (e.g., add 500 to one observation), and recompute the seasonally adjusted data. What is the effect of the outlier?

plastics_mid <- replace(plastics,30,350)
fit <- plastics_mid %>% decompose(type = 'multiplicative')
a <- autoplot(fit) + ggtitle('Multiplicative decomposition')
b <- autoplot(plastics_mid, series = 'Unadjusted') + ggtitle('With seasonal adjustment') +
  autolayer(seasadj(fit),series="Seasonally adjusted") + theme(legend.position="bottom") + theme(legend.title=element_blank())
grid.arrange(a, b, nrow = 1)

f. Does it make any difference if the outlier is near the end rather than in the middle of the time series?

plastics_end <- replace(plastics,57,350)
fit <- plastics_end %>% decompose(type = 'multiplicative')
a <- autoplot(fit) + ggtitle('Multiplicative decomposition')
b <- autoplot(plastics_end, series = 'Unadjusted') + ggtitle('With seasonal adjustment') +
  autolayer(seasadj(fit),series="Seasonally adjusted") + theme(legend.position="bottom") + theme(legend.title=element_blank())
grid.arrange(a, b, nrow = 1)

It appears that adding an outlier towards the end of time series doesn’t change seasonally adjusted series by much other than creating a drop to indicate the outlier. Mid-series adjustment, however, seemed to keep some seasonality in as evident from the part ‘e’, seasonally adjusted series have a spike corresponding to each unadjusted series spike for all years but the one with an outlier, and hence affecting the whole series.

3

Recall your retail time series data (from Exercise 3 in Section 2.10). Decompose the series using X11. Does it reveal any outliers, or unusual features that you had not noticed previously?

download.file('https://otexts.com/fpp2/extrafiles/retail.xlsx','retail.xlsx')
retaildata <- readxl::read_excel("retail.xlsx", skip=1)
myts <- ts(retaildata[,"A3349398A"],
  frequency=12, start=c(1982,4))

fit <- myts %>% seas(x11='')
autoplot(fit) + ggtitle('X11 decomposition')

autoplot(myts, series='Data') + 
  autolayer(trendcycle(fit), series='Trend') +
  autolayer(seasadj(fit),series="Seasonally adjusted") + 
  scale_colour_manual(values=c("gray","blue","red"),breaks=c("Data","Seasonally adjusted","Trend")) +
  theme(legend.position="bottom") + theme(legend.title=element_blank())

Yes, few spikes in the remainder indicate the presence of potential outliers and we can see the data points that were not previously noticeable when plotted as part of time series.