# Probabilistic Hydrological Post-Processing at Scale: Why and How to Apply Machine-Learning Quantile Regression Algorithms

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Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical University of Athens, Heroon Polytechneiou 5, 157 80 Zographou, Greece

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Air Force Support Command, Hellenic Air Force, Elefsina Air Base, 192 00 Elefsina, Greece

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Department of Civil Engineering, School of Engineering, University of Patras, University Campus, Rio, 26 504, Patras, Greece

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Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

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Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

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Department of Civil, Chemical, Environmental and Materials Engineering (DICAM), University of Bologna, via del Risorgimento 2, 40136 Bologna, Italy

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Authors to whom correspondence should be addressed.

Received: 11 August 2019 / Revised: 16 September 2019 / Accepted: 30 September 2019 / Published: 14 October 2019

(This article belongs to the Special Issue Techniques for Mapping and Assessing Surface Runoff)

We conduct a large-scale benchmark experiment aiming to advance the use of machine-learning quantile regression algorithms for probabilistic hydrological post-processing “at scale” within operational contexts. The experiment is set up using 34-year-long daily time series of precipitation, temperature, evapotranspiration and streamflow for 511 catchments over the contiguous United States. Point hydrological predictions are obtained using the Génie Rural à 4 paramètres Journalier (GR4J) hydrological model and exploited as predictor variables within quantile regression settings. Six machine-learning quantile regression algorithms and their equal-weight combiner are applied to predict conditional quantiles of the hydrological model errors. The individual algorithms are quantile regression, generalized random forests for quantile regression, generalized random forests for quantile regression emulating quantile regression forests, gradient boosting machine, model-based boosting with linear models as base learners and quantile regression neural networks. The conditional quantiles of the hydrological model errors are transformed to conditional quantiles of daily streamflow, which are finally assessed using proper performance scores and benchmarking. The assessment concerns various levels of predictive quantiles and central prediction intervals, while it is made both independently of the flow magnitude and conditional upon this magnitude. Key aspects of the developed methodological framework are highlighted, and practical recommendations are formulated. In technical hydro-meteorological applications, the algorithms should be applied preferably in a way that maximizes the benefits and reduces the risks from their use. This can be achieved by (i) combining algorithms (e.g., by averaging their predictions) and (ii) integrating algorithms within systematic frameworks (i.e., by using the algorithms according to their identified skills), as our large-scale results point out.