Quantile Regression Averaging (QRA) is a
forecast combination approach to the computation of
prediction interval
In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are o ...
s. It involves applying
quantile regression
Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional ''mean'' of the response variable across values of the predictor variables, quantile regressi ...
to the point forecasts of a small number of individual forecasting models or experts. It has been introduced in 2014 by Jakub Nowotarski and Rafał Weron and originally used for
probabilistic forecasting Probabilistic forecasting summarizes what is known about, or opinions about, future events. In contrast to single-valued forecasts (such as forecasting that the maximum temperature at a given site on a given day will be 23 degrees Celsius, or that t ...
of electricity prices
and loads. Despite its simplicity it has been found to perform extremely well in practice - the top two performing teams in the ''price track'' of the
Global Energy Forecasting Competition The Global Energy Forecasting Competition (GEFCom) is a competition conducted by a team led by Dr. Tao Hong that invites submissions around the world for forecasting energy demand. GEFCom was first held in 2012 on Kaggle, and the second GEFCom was h ...
(GEFCom2014) used variants of QRA.
Introduction
The individual point forecasts are used as
independent variables
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
and the corresponding observed target variable as the
dependent variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
in a standard
quantile regression
Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional ''mean'' of the response variable across values of the predictor variables, quantile regressi ...
setting. The Quantile Regression Averaging method yields an interval forecast of the target variable, but does not use the prediction intervals of the individual methods. One of the reasons for using point forecasts (and not interval forecasts) is their availability. For years, forecasters have focused on obtaining accurate point predictions. Computing
probabilistic forecasts, on the other hand, is generally a much more complex task and has not been discussed in the literature nor developed by practitioners so extensively. Therefore, QRA may be found particularly attractive from a practical point of view as it allows to leverage existing development of point forecasting.
Computation
The
quantile regression
Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional ''mean'' of the response variable across values of the predictor variables, quantile regressi ...
problem can be written as follows:
,
where
is the conditional ''q''-th
quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...
of the dependent variable (
),