Navigate
Courses Tutorials Practice Pricing Tools Upgrade

Box-Cox and variance-stabilizing transforms

Today let's understand how to pick a transform that keeps a seasonal swing the same size, no matter how tall the series itself has grown.

Victoria's monthly turnover for cafes, restaurants and takeaway food services is a real series the Australian Bureau of Statistics has tracked every month since April 1982. Over 441 months it climbs from \$81.3 million a month to \$1,066.2 million a month.

Here is the whole series, plotted in the order the months actually happened.

Look near the start of that line, back in 1982. November sits at \$97.7 million and December at \$109.3 million, a jump of \$11.6 million. Now look at the far end, 2018. November closes at \$981.8 million and December jumps to \$1,066.2 million, a jump of \$84.4 million.

The line has grown more than ten times taller since 1982, and its December jump has grown right along with it.

Seasonal swings that grow with the level, and why that breaks an additive model

That December jump is not one unlucky year. It shows up every single year in this series, and its size has been climbing for decades.

An additive decomposition explains a series like this one by splitting it into parts that add back together: a trend, the slow-moving level of the series, and a seasonal component, the part of the pattern that repeats every twelve months. For this series, the seasonal component's December value is one fixed number, call it the December effect. Every December in the whole 37-year history gets that same fixed number added on top of whatever the trend says that month should be.

Look at what the actual December jump, December's turnover minus November's, has done across nine separate years: four from the early 1980s and five from the recent past.

RInteractive R
# Compute the December-minus-November jump for nine separate years library(tsibble) library(tsibbledata) library(dplyr) library(feasts) library(fabletools) library(fable) cafe <- tsibbledata::aus_retail |> filter(State == "Victoria", Industry == "Cafes, restaurants and takeaway food services") jump_for <- function(yr) { nov <- cafe |> filter(Month == yearmonth(paste(yr, "Nov"))) |> pull(Turnover) dec <- cafe |> filter(Month == yearmonth(paste(yr, "Dec"))) |> pull(Turnover) dec - nov } early_years <- 1982:1985 late_years <- c(2014, 2015, 2016, 2017, 2018) data.frame( year = c(early_years, late_years), jump = round(c(sapply(early_years, jump_for), sapply(late_years, jump_for)), 1) ) #> year jump #> 1 1982 11.6 #> 2 1983 15.9 #> 3 1984 11.8 #> 4 1985 14.5 #> 5 2014 59.9 #> 6 2015 64.1 #> 7 2016 82.3 #> 8 2017 82.4 #> 9 2018 84.4

  

  





Average the first four years and you get \$13.45 million. Average the last five and you get \$74.62 million, more than five times bigger. Whatever fixed December effect an additive model settles on, it will be too small for the recent years, too big for the early ones, or some uneasy compromise between the two. The swing itself has grown right along with the series, and one constant cannot describe a moving target.