Published on Fri Jan 14 2022

Expectile-based hydrological modelling for uncertainty estimation: Life after mean

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Predictions of hydrological models should be probabilistic in nature. Our aim
is to introduce a method that estimates directly the uncertainty of
hydrological simulations using expectiles, thus complementing previous
quantile-based direct approaches as well as generalizing mean-based approaches.
Expectiles are new risk measures in hydrology. Compared to quantiles that use
information of the frequency of process realizations over a specified value,
expectiles use additional information of the magnitude of the exceedances over
the specified value. Expectiles are least square analogues of quantiles and can
characterize the probability distribution in much the same way as quantiles do.
Moreover, the mean of the probability distribution is the special case of the
expectile at level 0.5. To this end, we propose calibrating hydrological models
using the expectile loss function, which is strictly consistent for expectiles.
We apply our method to 511 basins in contiguous US and deliver predictive
expectiles of hydrological simulations with the GR4J, GR5J and GR6J
hydrological models at expectile levels 0.5, 0.9, 0.95 and 0.975. An honest
assessment empirically proves that the GR6J model outperforms the other two
models at all expectile levels. Great opportunities are offered for moving
beyond the mean in hydrological modelling by simply adjusting the objective
function.

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