Transformations for variance stabilization

Forecasting in R

Rob J. Hyndman

Professor of Statistics at Monash University

Variance stabilization

  • If the data show increasing variation as the level of the series increases, then a transformation can be useful

  • $y_1,...,y_n$: original observations, $w_1,...,w_n$: transformed observations
Square root $w_t = \sqrt{y_t}$ $\downarrow$
Cube root $w_t = \sqrt[3]{y_t}$ Increasing
Logarithm $w_t = \text{log}(y_t)$ Strength
Inverse $w_t = -1/y_t$ $\downarrow$
Forecasting in R

Variance stabilization

autoplot(usmelec) +
  xlab("Year") + ylab("") +
  ggtitle("US monthly net electricity generation")

ch4_vid1_vs_0.png

Forecasting in R

Variance stabilization

autoplot(usmelec^0.5) +
  xlab("Year") + ylab("") +
  ggtitle("Square root electricity generation")

ch4_vid1_vs_1.png

Forecasting in R

Variance stabilization

autoplot(usmelec^0.33333) +
  xlab("Year") + ylab("") +
  ggtitle("Cube root electricity generation")

ch4_vid1_vs_2.png

Forecasting in R

Variance stabilization

autoplot(log(usmelec)) +
  xlab("Year") + ylab("") +
  ggtitle("Log electricity generation")

ch4_vid1_vs_3.png

Forecasting in R

Variance stabilization

autoplot(-1/usmelec) +
  xlab("Year") + ylab("") +
  ggtitle("Inverse electricity generation")

ch4_vid1_vs_4.png

Forecasting in R

Box-Cox transformations

  • Each of these transformations is close to a member of the family of Box-Cox transformations

$$w_t = \begin{cases} log(y_t) & \lambda = 0 \\ (y_t^\lambda - 1)/\lambda & \lambda \neq 0 \end{cases}$$

  • $\lambda = 1 \ $: No substantive transformation
  • $\lambda = \frac{1}{2} \ $: Square root plus linear transformation
  • $\lambda = \frac{1}{3} \ $: Cube root plus linear transformation
  • $\lambda = 0 \ $: Natural logarithm transformation
  • $\lambda = -1 \ $: Inverse transformation
Forecasting in R

Box-Cox transformations

BoxCox.lambda(usmelec)

-0.5738331
Forecasting in R

Back-transformation

usmelec %>%
  ets(lambda = -0.57) %>%
  forecast(h = 60) %>%
  autoplot()

ch4_vid1_back.png

Forecasting in R

Let's practice!

Forecasting in R

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