# Quantile-Based Hydrological Modelling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Post-processing (two-stage) techniques using stochastic or regression-based methods (including machine learning ones). These techniques model residual errors. Several relevant examples can be found in the literature; see [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] and the review by [38], including quantile regression, which is based on the quantile loss function. Further classification of the methods is possible, depending on specific attributes, but this is out of scope here.

- Post-processing approaches: The quantile loss function is directly implemented by the hydrological model. On the other hand, post-processing methods model the residuals of the fitted hydrological model (that have been obtained by implementing a squared error type loss function) at a second stage (i.e., after applying the hydrological models), using a statistical or machine learning method.
- Monte Carlo of Bayesian joint inference approaches: These methods are applied directly to the hydrological model; therefore, there is some resemblance with our approach. In this case, the differences between the two approaches (Bayesian joint inference and our proposed approach) are identical with those identified in the statistical literature regarding the differences between Bayesian statistics and quantile regression modelling. These differences are thoroughly discussed in the Discussion in Section 6.

## 2. Methods

#### 2.1. Quantile Loss Function

^{−1}(a) is defined by:

^{−1}(a):= inf{x: F(x) ≥ a},

^{−1}(a) is referred to as the ath quantile of x, while inf denotes the infimum of a set of real numbers. For instance, F

^{−1}(1/2) is the median or 0.50th quantile. In regression modelling, one minimises the sum of absolute errors to estimate the median of the conditional distribution. The natural question then is “are there analogs for regression of the other quantiles?” [53] (p. 5). The idea, elaborated by [52], is to apply the quantile loss function defined by Equation (3), instead of using the absolute error function:

**1**(x ≤ r) – a),

**1**(∙) denotes the indicator function, x is the materialisation of the variable x, a is the quantile level of interest and r is the respective predictive quantile. For a = 1/2, Equation (3) reduces to:

_{i}L(r

_{i}; x

_{i}, a)/n, i ∈ {1, …, n}.

#### 2.2. Theoretical Properties of the Quantile Loss Function

#### 2.3. Hydrological Models

`R`programming language allows the minimisation of a customised loss function.

## 3. Data

## 4. Implementation and Key Components

`airGR`

`R`package was used to implement the hydrological models after appropriate adaptations. The latter facilitate hydrological model calibration based on the quantile loss function.

- Define the 2-year period 1980–1981 as the warm-up period of the hydrological models.
- Calibrate the hydrological models in the 16-year period 1982–1997 using: (a) the quantile loss function L(r; x, a) at quantile levels a ∈ {0.025, 0.050, 0.100, 0.500, 0.900, 0.950, 0.975}; and (b) the squared error function. That equates to 8 (i.e., 7 + 1; number of loss functions) × 3 (number of hydrological models) = 24 sets of parameters at each river basin. These sets of parameters correspond to different model setups.
- Simulate streamflow in the 16-year period 1998–2013 for each of the 24 model setups.

`R`programming language using the contributed packages mentioned in Appendix A.

## 5. Results

_{rel}measure can be defined by Equation (5) [81]; note the signs of the benchmark’s SCORE

_{bench}and model’s SCORE

_{model}, reflecting that quantile loss functions are negatively oriented; therefore, improvements are obtained if SCORE

_{rel}> 0:

_{rel}: = (SCORE

_{bench}− SCORE

_{model})/SCORE

_{bench},

## 6. Discussion

- When one is interested in events at the “limits of probability”.
- When the conditional distribution does not follow a known distribution.
- The possible presence of many outliers of the dependent variable (recall also that median regression is more robust compared to mean regression in the presence of outliers).
- The presence of heteroscedasticity.

- Estimating the parameters of the model is harder compared to Gaussian regression.
- Inference on the parameters (e.g., the computation of confidence intervals) is complicated.
- The possible presence of quantile crossing, i.e., estimated quantiles at higher levels might be lower than respective quantiles at lower levels.
- The full conditional distribution is not available; although, the computation of multiple quantiles can substitute predictive distributions [61]. In this case, a drawback of the method is that it requires the estimation of a high number of sets of model parameters (one set for each quantile).

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**Illustration of the quantile score (value of the quantile loss function) at the quantile levels a ∈ {0.05, 0.95} when x = 0 materialises and for varying predictive quantiles r (see Equation (3)).

**Figure 3.**Illustration of observed streamflows and quantiles predicted by the GR4J, GR5J and GR6J hydrological models at the levels a ∈ {0.025, 0.500, 0.975} for a two-year period at an arbitrary river basin.

**Figure 4.**Scatterplot of the mean quantile scores at the level a = 0.500, as computed for the testing periods and separately for each of the 511 river basins, in the case that the quantile loss function at the level a = 0.500 is minimised (x-axis), and in the case that the squared error function is minimised (y-axis) for the calibration of the GR4J model.

**Figure 5.**Histograms of the relative improvements (with red colour) and the median values of these relative improvements (with black dashed lines) against the performance of the GR4J hydrological model in terms of quantile score, as computed for all the 511 river basins and the 0.025, 0.050, 0.100, 0.500, 0.900, 0.950, 0.975 quantile levels, for the (

**a**) GR5J and (

**b**) GR6J hydrological models. Truncation at −50% and 50% has been applied for illustration purposes.

**Figure 6.**Heatmap of the median of the relative improvements summarizing the results for the 511 river basins of the performance of the GR5J and GR6J models against the performance of the GR4J hydrological model in terms of quantile scores.

**Figure 7.**Heatmap of the median coverages of the predictions issued by the GR4J, GR5J and GR6J hydrological models at varying quantile levels summarizing the results for the 511 river basins.

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**MDPI and ACS Style**

Tyralis, H.; Papacharalampous, G. Quantile-Based Hydrological Modelling. *Water* **2021**, *13*, 3420.
https://doi.org/10.3390/w13233420

**AMA Style**

Tyralis H, Papacharalampous G. Quantile-Based Hydrological Modelling. *Water*. 2021; 13(23):3420.
https://doi.org/10.3390/w13233420

**Chicago/Turabian Style**

Tyralis, Hristos, and Georgia Papacharalampous. 2021. "Quantile-Based Hydrological Modelling" *Water* 13, no. 23: 3420.
https://doi.org/10.3390/w13233420