It is common to want forecasts to be positive, or to require them to be within some specified range . Both of these situations are relatively easy to handle using transformations.

Positive forecasts

To impose a positivity constraint, simply work on the log scale. With the forecast package in R, this can be handled by specifying the Box-Cox parameter . For example, consider the real price of a dozen eggs (1900–1993; in cents):

library(fpp)
fit <- ets(eggs, lambda=0)
plot(forecast(fit, h=50))

Forecasts constrained to an interval

To see how to handle data constrained to an interval, imagine that the egg prices were constrained to lie within and . Then we can transform the data using a scaled logit transform which maps to the whole real line:

where is on the original scale and is the transformed data.

# Bounds
a <- 50
b <- 400
# Transform data
y <- log((eggs-a)/(b-eggs))
fit <- ets(y)
fc <- forecast(fit, h=50)
# Back-transform forecasts
fc$mean <- (b-a)*exp(fc$mean)/(1+exp(fc$mean)) + a
fc$lower <- (b-a)*exp(fc$lower)/(1+exp(fc$lower)) + a
fc$upper <- (b-a)*exp(fc$upper)/(1+exp(fc$upper)) + a
fc$x <- eggs
# Plot result on original scale
plot(fc)

The prediction intervals from these transformations have the same coverage probability as on the transformed scale, because quantiles are preserved under monotonically increasing transformations.