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.