From today’s email:
I have just finished reading a copy of ‘Forecasting:Principles and Practice’ and I have found the book really interesting. I have particularly enjoyed the case studies and focus on practical applications.
After finishing the book I have joined a forecasting competition to put what I’ve learnt to the test. I do have a couple of queries about the forecasting outputs required. The output required is a quantile forecast, is this the same as prediction intervals? Is there any R function to produce quantiles from 0 to 99?
If you were able to point me in the right direction regarding the above it would be greatly appreciated.
The future value of a time series is unknown, so you can think of it as a random variable, and its distribution is the “forecast distribution”. A “quantile forecast” is a quantile of the forecast distribution. The usual point forecast is often the mean or the median of the forecast distribution. A prediction interval is a range of specified coverage probability under that distribution. For example, if we assume the forecast distribution is normal, then the 95% prediction interval is defined by the 2.5% and 97.5% quantiles of the forecast distribution.
Still assuming normality, we could generate the forecast quantiles from 1% to 99% in R using
qnorm((1:99)/100, m, s)
sigma are the estimated mean and standard deviation of the forecast distribution. So if you are using the forecast package in R, you can do something like this:
library(forecast) fit <- auto.arima(WWWusage) fc <- forecast(fit, h=20, level=95) qf <- matrix(0, nrow=99, ncol=20) m <- fc$mean s <- (fc$upper-fc$lower)/1.96/2 for(h in 1:20) qf[,h] <- qnorm((1:99)/100, m[h], s[h]) plot(fc) matlines(101:120, t(qf), col=rainbow(120), lty=1)
Of course, assuming a normal distribution is rather restrictive and not very interesting. For a more interesting but much more complicated approach to generating quantiles, see my 2010 paper on Density forecasting for long-term peak electricity demand.