# The Use of GAMLSS Framework for a Non-Stationary Frequency Analysis of Annual Runoff Data over a Mediterranean Area

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. The Generalized Additive Models in Location, Scale and Shape (GAMLSS) Framework

_{i}, for i = 1, 2, …, n, are distributed according to a probability density function, f(y

_{i}|θ

_{i}), conditional on θ

_{i}= (θ

_{1i}, θ

_{2i}, θ

_{3i}, θ

_{4i}) = (µ

_{i}, σ

_{i}, ν

_{i}, τ

_{i}), which represents the ensemble of four distribution parameters, each of which can be a function of the explanatory variables.

_{i,}σ

_{i}, ν

_{i}, τi) as the distribution parameters. The first two of them, µ

_{i}and σ

_{i}, are usually mentioned as location and scale parameters, while the remaining parameter(s), if any, are characterized as shape parameters, e.g., skewness and kurtosis parameters.

_{1}, y

_{2}, …, y

_{n}) the n-length vector of the response variable and let g

_{k}(·) (for k = 1, 2, 3, 4) be the monotonic functions linking the distribution parameters to the explanatory variables:

_{k}and x

_{jk}, (for j = 1, 2, …, J

_{k}) and k = 1, 2, 3, 4 are vectors of length n. In many practical situations, four distribution parameters are required at most. The function h

_{jk}is a non-parametric additive function of the explanatory variable X

_{jk}evaluated at x

_{jk}. The explanatory vectors x

_{jk}are assumed to be fixed and known. In addition, X

_{k}, for k = 1, 2, 3, 4, are fixed design matrices (fixed effects design matrices of explanatory variables, i.e., covariates) while β

_{k}are the vectors of the parameters of the distribution. Usually, in typical applications, a constant or other simple model is often suitable for each of the two shape parameters (ν and τ).

_{k}and k = 1, 2, 3, 4 are estimated within the GAMLSS framework by maximizing a penalized likelihood function.

#### 2.3. Statistical Distributions in GAMLSS

#### 2.4. The AIC Criterion

- 1.
- The number of independent variables used to build the model;
- 2.
- The maximum likelihood estimates of the model.

## 3. Results and Discussion

#### 3.1. Stationary Analysis

#### 3.2. Non-Stationary Analysis with Rainfall as a Covariate

#### 3.3. Non-Stationary Analysis with Time as a Covariate

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Cannarozzo, M.; Noto, L.; Viola, F.; la Loggia, G. Annual Runoff Regional Frequency Analysis in Sicily. Phys. Chem. Earth Parts A/B/C
**2009**, 34, 679–687. [Google Scholar] [CrossRef] - Markovic, R.D. Probability Functions of the Best Fit to Distributions of Annual Precipitation and Runoff Hydrology. Doctoral Dissertation, Colorado State University, Fort Collins, CO, USA, 1965. Paper No. 8. [Google Scholar]
- Vogel, R.M.; Wilson, I. Probability distribution of annual maximum, mean, and minimum streamflows in the united states. J. Hydrol. Eng.
**1996**, 1, 69–76. [Google Scholar] [CrossRef] - Salas, J.D. Analysis and Modeling of Hydrological Time Series. Handb. Hydrol.
**1993**, 19, 11–72. [Google Scholar] - Liu, L.; Xu, H.; Wang, Y.; Jiang, T. Impacts of 1.5 and 2 °C global warming on water availability and extreme hydrological events in Yiluo and Beijiang River catchments in China. Clim. Change
**2017**, 145, 145–158. [Google Scholar] [CrossRef] - Donnelly, C.; Andersson, J.C.M.; Arheimer, B. Using flow signatures and catchment similarities to evaluate the E-HYPE multi-basin model across Europe. Hydrol. Sci. J.
**2016**, 61, 255–273. [Google Scholar] [CrossRef] - Li, S.; Qin, Y. Frequency Analysis of the Nonstationary Annual Runoff Series Using the Mechanism-Based Reconstruction Method. Water
**2022**, 14, 76. [Google Scholar] [CrossRef] - Sadri, S.; Kam, J.; Sheffield, J. Nonstationarity of low flows and their timing in the eastern United States. Hydrol. Earth Syst. Sci.
**2016**, 20, 633–649. [Google Scholar] [CrossRef] - Debele, S.E.; Bogdanowicz, E.; Strupczewski, W.G. Around and about an application of the GAMLSS package to non-stationary flood frequency analysis. Acta Geophys.
**2017**, 65, 885–892. [Google Scholar] [CrossRef] - Jiang, C.; Xiong, L.; Xu, C.Y.; Guo, S. Bivariate frequency analysis of nonstationary low-flow series based on the time-varying copula. Hydrol. Processes
**2015**, 29, 1521–1534. [Google Scholar] [CrossRef] - Kang, L.; Jiang, S.; Hu, X.; Li, C. Evaluation of return period and risk in bivariate non-stationary flood frequency analysis. Water
**2019**, 11, 79. [Google Scholar] [CrossRef] - Nasri, B.R.; Bouezmarni, T.; St-Hilaire, A.; Ouarda, T. Non-Stationary Hydrologic Frequency Analysis using B-Spline Quantile Regression. J. Hydrol.
**2017**, 554, 532–544. [Google Scholar] [CrossRef][Green Version] - Nogaj, M.; Parey, S.; Dacunha-Castelle, D. Non-stationary extreme models and a climatic application. Nonlinear Processes Geophys.
**2007**, 14, 305–316. [Google Scholar] [CrossRef] - Villarini, G.; Smith, J.A.; Napolitano, F. Nonstationary modeling of a long record of rainfall and temperature over Rome. Adv. Water Resour.
**2010**, 33, 1256–1267. [Google Scholar] [CrossRef] - Xiong, L.; Jiang, C.; Du, T. Statistical attribution analysis of the nonstationarity of the annual runoff series of the Weihe River. Water Sci. Technol.
**2014**, 70, 939–946. [Google Scholar] [CrossRef] [PubMed] - Yang, L.; Smith, J.A.; Wright, D.B.; Baeck, M.L.; Villarini, G.; Tian, F.; Hu, H. Urbanization and climate change: An examination of nonstationarities in urban flooding. J. Hydrometeorol.
**2013**, 14, 1791–1809. [Google Scholar] [CrossRef] - Koutsoyiannis, D.; Montanari, A. Risks from dismissing stationarity. In Proceedings of the AGU Fall Meeting Abstracts, San Francisco, CA, USA, 15–19 December 2014; p. H54F-01. [Google Scholar]
- Matalas, N.C. Comment on the announced death of stationarity. J. Water Resour. Plan. Manag.
**2012**, 138, 311–312. [Google Scholar] [CrossRef] - Milly, P.C.; Betancourt, J.; Falkenmark, M.; Hirsch, R.M.; Kundzewicz, Z.W.; Lettenmaier, D.P.; Stouffer, R.J. Stationarity is dead: Whither water management? Science
**2008**, 319, 573–574. [Google Scholar] [CrossRef] - Caracciolo, D.; Noto, L.V.; Istanbulluoglu, E.; Fatichi, S.; Zhou, X. Climate change and Ecotone boundaries: Insights from a cellular automata ecohydrology model in a Mediterranean catchment with topography controlled vegetation patterns. Adv. Water Resour.
**2014**, 73, 159–175. [Google Scholar] [CrossRef] - Francipane, A.; Fatichi, S.; Ivanov, V.Y.; Noto, L.V. Stochastic assessment of climate impacts on hydrology and geomorphology of semiarid headwater basins using a physically based model. J. Geophys.Res. Earth Surf.
**2015**, 120, 507–533. [Google Scholar] [CrossRef] - Giuntoli, I.; Renard, B.; Vidal, J.-P.; Bard, A. Low flows in France and their relationship to large-scale climate indices. J. Hydrol.
**2013**, 482, 105–118. [Google Scholar] [CrossRef] - Giuntoli, I.; Villarini, G.; Prudhomme, C.; Hannah, D.M. Uncertainties in projected runoff over the conterminous United States. Clim. Change
**2018**, 150, 149–162. [Google Scholar] [CrossRef][Green Version] - Kormos, P.R.; Luce, C.H.; Wenger, S.J.; Berghuijs, W.R. Trends and sensitivities of low streamflow extremes to discharge timing and magnitude in Pacific Northwest mountain streams. Water Resour. Res.
**2016**, 52, 4990–5007. [Google Scholar] [CrossRef] - Jiang, C.; Xiong, L.; Yan, L.; Dong, J.; Xu, C.-Y. Multivariate hydrologic design methods under nonstationary conditions and application to engineering practice. Hydrol. Earth Syst. Sci.
**2019**, 23, 1683–1704. [Google Scholar] [CrossRef] - Li, Y.; Chang, J.; Luo, L.; Wang, Y.; Guo, A.; Ma, F.; Fan, J. Spatiotemporal impacts of land use land cover changes on hydrology from the mechanism perspective using SWAT model with time-varying parameters. Hydrol. Res.
**2019**, 50, 244–261. [Google Scholar] [CrossRef] - Katz, R.W.; Parlange, M.B.; Naveau, P. Statistics of extremes in hydrology. Adv. Water Resour.
**2002**, 25, 1287–1304. [Google Scholar] [CrossRef] - Villarini, G.; Smith, J.A.; Serinaldi, F.; Bales, J.; Bates, P.D.; Krajewski, W.F. Flood frequency analysis for nonstationary annual peak records in an urban drainage basin. Adv. Water Resour.
**2009**, 32, 1255–1266. [Google Scholar] [CrossRef] - Rigby, R.A.; Stasinopoulos, D.M. Generalized additive models for location, scale and shape. J. R. Stat.Soc. Ser. C (Appl.Stat.)
**2005**, 54, 507–554. [Google Scholar] [CrossRef] - Jiang, C.; Xiong, L.; Wang, D.; Liu, P.; Guo, S.; Xu, C.-Y. Separating the impacts of climate change and human activities on runoff using the Budyko-type equations with time-varying parameters. J. Hydrol.
**2015**, 522, 326–338. [Google Scholar] [CrossRef] - Li, J.; Tan, S. Nonstationary flood frequency analysis for annual flood peak series, adopting climate indices and check dam index as covariates. Water Resour. Manag.
**2015**, 29, 5533–5550. [Google Scholar] [CrossRef] - López, J.; Francés, F. Non-stationary flood frequency analysis in continental Spanish rivers, using climate and reservoir indices as external covariates. Hydrol. Earth Syst. Sci.
**2013**, 17, 3189–3203. [Google Scholar] [CrossRef] - Villarini, G.; Strong, A. Roles of climate and agricultural practices in discharge changes in an agricultural watershed in Iowa. Agric. Ecosyst. Environ.
**2014**, 188, 204–211. [Google Scholar] [CrossRef] - Li, J.; Gao, Z.; Guo, Y.; Zhang, T.; Ren, P.; Feng, P. Water supply risk analysis of Panjiakou reservoir in Luanhe River basin of China and drought impacts under environmental change. Theor. Appl. Climatol.
**2019**, 137, 2393–2408. [Google Scholar] [CrossRef] - Stasinopoulos, M.; Rigby, B.; Akantziliotou, C. Instructions on how to use the gamlss package in R Second Edition. 2008. Available online: http://gamlss.com/wp-content/uploads/2013/01/gamlss-manual.pdf (accessed on 1 August 2022).
- Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control.
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Akaike, H. On the likelihood of a time series model. J. R. Stat.Soc. Ser. D (Stat.)
**1978**, 27, 217–235. [Google Scholar] [CrossRef] - Nelson, D.B. Stationarity and persistence in the GARCH (1, 1) model. Econom. Theory
**1990**, 6, 318–334. [Google Scholar] [CrossRef] - Shumway, R.; Stoffer, D. Time Series Analysis and Its Applications with R Examples; Springer: New York, NY, USA, 2011; Volume 9. [Google Scholar]
- Chen, H.-L.; Rao, A.R. Testing hydrologic time series for stationarity. J. Hydrol. Eng.
**2002**, 7, 129–136. [Google Scholar] [CrossRef] - Buuren, S.v.; Fredriks, M. Worm plot: A simple diagnostic device for modelling growth reference curves. Stat. Med.
**2001**, 20, 1259–1277. [Google Scholar] [CrossRef] - Stasinopoulos, M.D.; Rigby, R.A.; Bastiani, F.D. GAMLSS: A distributional regression approach. Stat. Model.
**2018**, 18, 248–273. [Google Scholar] [CrossRef] - Rigby, R.A.; Stasinopoulos, D.M. A semi-parametric additive model for variance heterogeneity. Stat. Comput.
**1996**, 6, 57–65. [Google Scholar] [CrossRef] - Rigby, R.A.; Stasinopoulos, M.D. Mean and Dispersion Additive Models. In Statistical Theory and Computational Aspects of Smoothing; Physica-Verlag HD: Heidelberg, Germany, 1996; pp. 215–230. [Google Scholar]
- Zhang, T.; Wang, Y.; Wang, B.; Tan, S.; Feng, P. Nonstationary Flood Frequency Analysis Using Univariate and Bivariate Time-Varying Models Based on GAMLSS. Water
**2018**, 10, 819. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Locations of the gauges considered in this study and the related catchments overlaid on the perimeter of Sicily.

**Figure 3.**Scatterplots of annual runoff vs. rainfall (

**first row**) and annual runoff vs. time (

**second row**) for IM-DRA, BE-SPA, SL-MON and VA-SER.

**Figure 4.**Empirical and theoretical cumulative distribution functions (cdfs) and worm plots for the distributions with the lowest AIC values for all the stations.

**Figure 5.**Variation of µ and σ parameters with annual rainfall for the IM-DRA LOGNO distribution (

**left panel**) and the centiles plot (

**right panel**) for the same station, along with the plot for annual runoff vs. annual rainfall.

**Figure 6.**Summary of results for the P1 model for all the stations with generalized additive models in location, scale and shape parameters and the corresponding worm plots for runoff series. The legend for the (

**first line**) is the same as in Figure 5.

**Figure 7.**Best suitable distribution centile curves (

**first row**) and the corresponding worm plots (

**second row**) for the P2 model. The legend for the first line is the same as in Figure 5.

**Figure 8.**Best suitable distribution centile curves (

**first row**) and the corresponding worm plots (

**second row**) for the µ~cs(p), σ~p model. The legend for the first line is the same as in Figure 5.

**Figure 9.**Centiles plots (

**first row**) of the best model for the four considered stations. In the (

**second row**) the corresponding worm plots are displayed. The legend for the first line is the same as in Figure 5.

Distributions | Probability Density Function | Distribution Moments |
---|---|---|

Normal (NO) | $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathsf{\sigma}\sqrt{2\mathsf{\pi}}}{\mathrm{e}}^{-\frac{1}{2}{\left(\frac{\mathrm{x}-\mathsf{\mu}}{\mathsf{\sigma}}\right)}^{2}}$ | $\mathrm{E}\left[\mathrm{x}\right]=\text{}\mathsf{\mu}\text{}$; $\mathrm{Var}\left[\mathrm{x}\right]={\text{}\mathsf{\sigma}}^{2}\text{}$; |

Gamma (GA) | $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathsf{\sigma}}^{\mathsf{\mu}}\mathsf{\Gamma}\left(\mathsf{\mu}\right)}{\mathrm{x}}^{\mathsf{\mu}-1}{\mathrm{e}}^{-\frac{\mathrm{x}}{\mathsf{\sigma}}}$ | $\mathrm{E}\left[\mathrm{x}\right]=\text{}\mathsf{\mu}\text{}$; $\mathrm{Var}\left[\mathrm{x}\right]={\text{}\mathsf{\sigma}}^{2}{\text{}\mathsf{\mu}}^{2}$ |

Log-normal 2 parameters (LOGNO) | $\mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{e}}^{\frac{-{\left(\mathrm{ln}\left(\mathrm{x}\right)-\text{}\mathsf{\mu}\right)}^{2}}{2{\mathsf{\sigma}}^{2}}}\text{}}{\mathrm{x}\sqrt{2\mathsf{\pi}}\text{}\mathsf{\sigma}}$ | $\mathrm{E}\left[\mathrm{x}\right]={\mathsf{\phi}}^{\frac{1}{2}}{\text{}\mathrm{e}}^{\mathsf{\mu}}\text{}$; $\mathrm{Var}\left[\mathrm{x}\right]=\text{}\mathsf{\phi}\text{}\left(\mathsf{\phi}-1\right){\text{}\mathrm{e}}^{2\text{}\mathsf{\mu}}$; where $\mathsf{\phi}={\mathrm{e}}^{{\mathsf{\sigma}}^{2}}$ |

Log-normal 3 parameters (LNO) | $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\sqrt{2\mathsf{\pi}}\mathsf{\nu}\left(\mathrm{x}-\mathsf{\mu}\right)}{\mathrm{e}}^{-\frac{1}{2}{\left(\frac{\mathrm{log}\left(\mathrm{x}-\mathsf{\mu}\right)-\mathsf{\sigma}}{\mathsf{\nu}}\right)}^{2}}$ | $\mathrm{E}\left[\mathrm{x}\right]=\text{}\mathsf{\mu}+{\mathrm{e}}^{\mathsf{\sigma}+\frac{{\mathsf{\nu}}^{2}}{2}}$; $\mathrm{Var}\left[\mathrm{x}\right]=\left({\mathrm{e}}^{{\mathsf{\nu}}^{2}}-1\right){\text{}\mathrm{e}}^{2\text{}\mathsf{\sigma}+{\text{}\mathsf{\nu}}^{2}}\text{}$; ${\mathrm{E}}^{3}\left[\mathrm{x}\right]=\left({\mathrm{e}}^{{\mathsf{\nu}}^{2}}+2\right)\sqrt{{\mathrm{e}}^{{\mathsf{\nu}}^{2}}-1}$; |

**Table 2.**AIC values for all the considered distributions of the stationary analysis. In bold are shown the lowest AIC values among the analyzed distributions for each station.

AIC Values | ||||
---|---|---|---|---|

Distributions | BE-SPA | IM-DRA | SL-MON | VA-SER |

Normal (NO) | 394.90 | 410.29 | 632.50 | 440.18 |

Gamma (GA) | 396.54 | 386.01 | 630.31 | 435.31 |

Log-normal 2 parameters (LOGNO) | 402.17 | 380.99 | 635.66 | 441.47 |

Log-normal 3 parameters (LNO) | 394.64 | 390.44 | 629.31 | 434.29 |

**Table 3.**Comparison of the best stationary and non-stationary models with rainfall as covariate. In bold are shown the distributions that provided the lowest AIC values between the analyzed distributions for each station.

AIC Values | ||||
---|---|---|---|---|

Models | BE-SPA | IM-DRA | SL-MON | VA-SER |

S—Stationary | LNO 394.64 | LOGNO 380.79 | LNO 629.31 | LNO 434.29 |

P1—µ~p, σ~c | LNO 373.41 | LOGNO 365.33 | NO 581.02 | LNO 409.10 |

P2—µ~p, σ~p | NO373.36 | LOGNO361.12 | LNO580.46 | LNO 407.57 |

P3—µ~cs(p), σ~p | NO 374.66 | LOGNO 362.94 | LNO 584.36 | LOGNO405.71 |

**Table 4.**Comparison of the stationary and non-stationary best models with time as covariate. In bold are shown those models which provided the lowest AIC values among the non-stationary models.

AIC Values | ||||
---|---|---|---|---|

Models | BE-SPA | IM-DRA | SL-MON | VA-SER |

S—Stationary | LNO 394.64 | LOGNO 380.79 | LNO 629.31 | LNO 434.29 |

T1—µ~t, σ~c | LNO 395.48 | LOGNO 382.98 | LNO 630.21 | LNO 434.96 |

T2—µ~t, σ~t | LNO 397.42 | LOGNO 380.87 | LNO 630.99 | GA 429.06 |

T3—µ~cs(t), σ~t | LNO 400.30 | LOGNO 383.96 | LNO 631.10 | GA 428.69 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Scala, P.; Cipolla, G.; Treppiedi, D.; Noto, L.V.
The Use of GAMLSS Framework for a Non-Stationary Frequency Analysis of Annual Runoff Data over a Mediterranean Area. *Water* **2022**, *14*, 2848.
https://doi.org/10.3390/w14182848

**AMA Style**

Scala P, Cipolla G, Treppiedi D, Noto LV.
The Use of GAMLSS Framework for a Non-Stationary Frequency Analysis of Annual Runoff Data over a Mediterranean Area. *Water*. 2022; 14(18):2848.
https://doi.org/10.3390/w14182848

**Chicago/Turabian Style**

Scala, Pietro, Giuseppe Cipolla, Dario Treppiedi, and Leonardo Valerio Noto.
2022. "The Use of GAMLSS Framework for a Non-Stationary Frequency Analysis of Annual Runoff Data over a Mediterranean Area" *Water* 14, no. 18: 2848.
https://doi.org/10.3390/w14182848