Generating quantile forecasts in R

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.

Many Thanks,

Presumably the competition is GEFCOM2014 which I’ve posted about before.

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)

where mu and 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.

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