Table 1 Five procedures were chosen to impute missing forecasts

Mean

\(I^{-1}\sum _{i}{q_{i,k}}\)

Take the mean of all present quantiles where the set I is an index for present forecasts

Median

\(\min _{x} \left\{ F(x)-1/2 \right\}\)

Take the median of all present quantiles where F is the empirical cdf over all I quantiles

Bayesian Ridge regression

\({\mathbb {E}}(X) \text { where } X\sim {\mathcal {N}}(Q_{-k}\beta ,\sigma ^{2})\)  \(\beta \sim {\mathcal {N}}(0,\uplambda ^{-1} I) \;\sigma ^{2} \sim \Gamma (\alpha ,\gamma )\)  

The matrix \(Q_{-k}\) has two columns: a column of ones and a second column of quantiles from present forecasts.

Decision Tree regression

–

The missing quantile value is imputed by the mean of quantiles in the same partition.

Extremely Randomized Trees

–

Multiple decision trees \((D_{i})\) are fit to random subsets of quantiles and the missing forecast is imputed as the average over \(D_{i}\).