# 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

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**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 |

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**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