Forecasting within limits
It is common to want forecasts to be positive, or to require them to be within some specified range \([a,b]\). 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 \(\lambda=0\). 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 \(a=50\) and \(b=400\). Then we can transform the data using a scaled logit transform which maps \((a,b)\) to the whole real line: \[ y = \log\left(\frac{x-a}{b-x}\right) \] where \(x\) is on the original scale and \(y\) 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.