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Evaluation of Various Generalized Pareto Probability Distributions for Flood Frequency Analysis

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28 March 2023

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29 March 2023

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Abstract
This article analyzes 6 probability distributions from the Generalized Pareto family, with 3, 4 and 5 parameters, with main purpose to identify other distributions from this family with applicability in flood frequency analysis compared to the distribution already used in the literature from this family such as Generalized Pareto Type II and Wakeby. This analysis is part of a larger and more complex research carried out in the Faculty of Hydrotechnics regarding the elaboration of a norm for flood frequency analysis using the linear moments method. In Romania, the standard method of parameter estimation is the method of ordinary moments, thus the transition from this method to the method of linear moments is desired. All the necessary elements for the distributions use are presented like, the probability density functions, the complementary cumulative distribution functions, the quantile functions, the exact and approximate relations for estimating parameters, for both methods of parameters estimation. All these elements are necessary for a proper transition between the two methods, especially since the use of the method of ordinary moments is done by choosing the skewness of the observed data depending on the origin of the maximum flows. A flood frequency analysis case study, using annual maximum and annual exceedance series, was carried out for the Prigor river, to numerically present the analyzed distributions. The performance of this distributions is evaluated using relative mean error, relative absolute error and linear moments diagram.
Keywords: 
Subject: Environmental and Earth Sciences  -   Water Science and Technology

1. Introduction

The frequency analysis of extreme events in hydrology is of particular importance, with the aim of determining the probability of occurrence of extreme events of a given magnitude.
Flood frequency analysis is important because it determines the maximum flow with certain exceeding probabilities, they have a defining role in the design of dams [1] and in water management [2], and can have significant impacts on human lives, infrastructure, and the environment.
Together with the distributions from the Gamma family and Generalized Extreme Values, the Generalized Pareto Type 2 distribution (PGII) represents one of the most used distributions in flood frequency analysis [3,4,5], especially in the analysis of partial series using the Annual Exceedance Series (AES) or Peak Over Thresholds (POT).
Among the Generalized Pareto distributions analyzed in this article, the ones that received considerable attention in flood frequency analysis are the Generalized Pareto distribution Type II (PGII) using AES or POT [3,4,5], respectively the Generalized Pareto distribution Type III (PGIII) and the Wakeby (WK5) distribution [4,8,9], in the analysis with the Annual Maximum Series (AMS). The PGIII is also known in literature as the Log-Logistic distribution or Generalized Logistic distribution [4,6,7].
The PGII distribution has a broad use in the analysis of extreme events such as, precipitation frequency analysis [10,11,12,13,14,15], in low flow frequency analysis [16] and in flood frequency analysis [4,5,17,18,19,20].
Based on Rao and Hamed [4], respectively Hosking and Wallis [21] the Generalized Pareto distribution ‘’ is the logical choice for modeling flood magnitudes that exceed a fixed threshold when it is reasonable to assume that successive floods follow a Poisson process and have independent magnitudes’’.
In the case of Wakeby distribution, it is a quantile distribution (a distribution expressible in inverse form) which in certain situations [4,9,22] turns into the Generalized Pareto Type II distribution. Its application, using the close form equations for the first four central moments, was realized for the first time by Anghel and Ilinca [22,23], both for the four and five parameters Wakeby distributions.
Other distributions from the Generalized Pareto family have received little to none attention, such as the four parameters Generalized Pareto Type IV (PGIV4), the three parameters Generalized Pareto Type IV (PGIV3) and the Generalized Pareto Type I (PGI) also known as Pearson XI distribution.
Taking into account this, one of the objectives of the article consists in the analysis of the applicability of other distributions belonging to the same family of Generalized Pareto distribution in flood frequency analysis. For a comprehensive analysis, in this article all the distributions belonging to this family are comparatively analyzed. Although some of these distributions have been used in the frequency analysis of extreme events in hydrology, this article brings new elements for these distributions that help to better understand and apply them in hydrology and beyond.
The research from this article is part of a larger and more complex research carried out in the Faculty of Hydrotechnics, to identify the distributions from different families of distributions, which have applicability in frequency analysis of extreme values, partial results presented in other materials [22,23,24].
In this article, the estimation methods of the parameters of these distributions are method of ordinary moments (MOM) and the method of linear moments (L-moments), for some of them being necessary to solve nonlinear systems of equations, which leads to some difficulties in using these distributions. Thus, for the ease of applications of these distributions, parameter approximation relations are presented, using polynomial, exponential or rational functions.
Only these two methods of estimating parameters are analyzed in this article, because MOM is the ‘’parent’’ method in Romania, and the L-moments method is the method that is intended to be used in the new regulations regarding the analysis of extreme phenomena in hydrology, being a much more stable and less sensitive to short lengths of data, as is the general case of hydrometry in Romania.
All the mathematical elements necessary to use these distributions in the flood frequency analysis are presented.
New elements are presented, such as: the first six raw and central moments for PGII, PGI; relations for estimating the parameters with L-moments for the PGIV4, PGIV3 and PGI distributions; new approximate parameter estimation relations for the PGII, PGI and PGIII distributions; the frequency factors for all analyzed distributions, both for MOM and L-moments; approximation relations of the frequency factors for the PGI, PGII and PGIII distributions.
Thus, all these novelty elements for these distributions presented in Table 1 will help hydrology researchers to better understand and easily apply these distributions.
The raw and central moments of the analyzed distributions were determined using the methodology presented in the Supplementary file, based on the probability density functions. It is for the first time that, for the Generalized Pareto distributions Type II and Type III, the raw and central moments up to order 6 are presented, important, along with the frequency factor, in establishing the confidence interval (for MOM) using the Kite approximation [4]. Also for the first time, the frequency factor of these distributions is presented based on the L-moments method, an important aspect in determining the confidence interval using the Chow approximation [4], the latter was used in hydrology only based on the estimation of parameters with MOM.
In order to verify the performances of the proposed distributions, a flood frequency analysis is carried out, using the Annual Maximum Series (AMS) and the Annual Exceedance Series (AES) for the Prigor river, as a case study. All results are presented in comparison with the Pearson III distribution, which is the ‘’parent’’ distribution in flood frequency analysis in Romania [22]. The purpose of the article was to identify other distributions from this family with applicability in flood frequency analysis compared to the distributions already used in the literature from these families such as PGIII and PGII. The article does not exclude the applicability of other distributions from other families (Gamma, GEV, Beta) in flood frequency analysis, especially since these families were also analyzed within the research carried out in the Faculty of Hydrotechnics and presented in other materials [22,23,24].
Comparing the results and choosing the best distribution is based on the performance indicators: relative mean error (RME), relative absolute error (RAE) and L-skewness ( τ 3 )-L-kurtosis ( τ 4 ) diagram.
The article is organized as follows. The description of the statistical distributions by presenting the density function, the complementary cumulative function and the quantile function, in Section 2.1. The presentation of the relations for exact calculation and the approximate relations for determining the parameters of the distributions, in Section 2.2. Case studies by applying these distributions in flood frequency analysis for the Prigor river, in Section 3. Results, discussions and conclusions, in Section 3, Section 4 and Section 5.

2. Methods

The frequency analysis consists in determining the flows with certain exceedance probabilities using AMS, respectively AES, with the Prigor river as a case study.
The series of maximum annual flows consists in choosing the maximum value corresponding to each year. In many cases, the lower maximum values of the annual data series do not always represent "floods". Thus, the use of frequency analysis using AES is required, which allows secondary events, which exceed certain annual maximum values, to be considered as "floods".
The AES was established by descending sorting of all independent maximum values and choosing the first "n" values corresponding to the "n" number of years of analysis [14,15].
Using this criterion, it is important to verify that two or more maximum flow values do not come from the same flood. The independence of flows was verified and established based on the Cunnane criteria, respectively USWRC 1976 [25].
The determination of the maximum flows was carried out in stages according to Figure 1. The verification of the character of outliers (Grubbs, Pilon, Quartile method), homogeneity and independence of flows, were carried out in the data curation phase. No outliers have been detected.
The estimation of the parameters of the analyzed distributions was done with MOM and L-moments. The MOM estimation has the disadvantage that for high-order moments and short data series in many cases it generates unrealistic values because the high-order moments require correction [4,5,26,27]. For skewness, the correction can be made using the Bobee relation [26] or, as is the practice in Romania [22,28,29,30,31], being established according to the origin of the maximum flows, by multiplying the coefficient of variation ( C v ) by a coefficient reflecting this origin. In many cases the choice is subjective and without rigor [22,32].
The L-moments method is a more stable and robust method, having the advantage that it is less sensitive to short data lengths, compared to other parameter estimation methods such as Method of Ordinary Moments, Method of Maximum Likelihood Estimation or the Method of Least Squares [3,22,33].
Considering that in many cases to estimate the parameters of the analyzed distributions, it is necessary to solve some systems of non-linear equations, approximate relations for estimating the parameters were determined in the case of distributions where skewness and L-skewness depend on a single parameter. Also, for a simplified and fast calculation taking into account the fact that the inverse function can be expressed with the frequency factor for both MOM [4,23,34] and L moments [23], the approximation relations of the factor of frequency (and the coefficients of these relations) for the most frequent exceeding probabilities in the analysis of maximum flows. The estimation errors of both the parameters and the frequency factors are between 10-3-10-4.
The quantile results are compared with those of the Pearson III distribution, which is the ‘’parent’’ distribution in Romania in the analysis of extreme events in hydrology, especially in the flood frequency analysis [22,28,29].

2.1. Probability Distributions

In Table 2 are presented the probability density function, f x ; the complementary cumulative distribution function, F x , and quantile function, x p , for analyzed distributions [4,5,6,14,15,16,17]. All F x and x p of the analysed distributions were determined using the methodology presented in the Supplementary file, using only f x

2.2. Parameter Estimation

The parameters estimation of the analyzed statistical distributions is presented for MOM and L-moments, some of the most used methods in hydrology for parameter estimation [3,4,5,7,22].

2.2.1. Generalized Pareto Type IV (PGIV4)

The equations needed to estimate the parameters with MOM have the following expressions:
μ = γ + β ⋅ α ⋅ Γ α ⋅ Γ λ − α λ ⋅ Γ λ
σ 2 = β 2 λ ⋅ Γ λ ⋅ Γ 2 ⋅ α + 1 ⋅ Γ λ − 2 ⋅ α − Γ α + 1 2 ⋅ Γ λ − α 2 λ ⋅ Γ λ
Because they are too long, the relations for estimating skewness and kurtosis are presented in Appendix F.
The equations needed to estimate the parameters with L-moments have the following expressions:
L 1 = γ + β ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ
L 2 = β ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ − Γ 2 ⋅ λ − α ⋅ Γ 1 + α Γ 2 ⋅ λ
L 3 = β ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ − 3 ⋅ Γ 2 ⋅ λ − α ⋅ Γ 1 + α Γ 2 ⋅ λ + 2 ⋅ Γ 3 ⋅ λ − α ⋅ Γ 1 + α Γ 3 ⋅ λ
L 4 = β ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ − 6 ⋅ Γ 2 ⋅ λ − α ⋅ Γ 1 + α Γ 2 ⋅ λ + 10 ⋅ Γ 3 ⋅ λ − α ⋅ Γ 1 + α Γ 3 ⋅ λ − 5 ⋅ Γ 4 ⋅ λ − α ⋅ Γ 1 + α Γ 3 ⋅ λ
where L 1 , L 2 , L 3 , L 4 represent the first four linear moments.

2.2.2. Generalized Pareto Type IV (PGIV3)

The distribution represents a particular case of the Pareto IV distribution when the position parameter γ = 0 . It is also known as the beta_p or Singh-Maddala distribution [6,33].
The equations needed to estimate the parameters with MOM have the following expressions:
μ = β ⋅ Γ α + 1 ⋅ Γ λ − α Γ λ
σ 2 = β 2 Γ λ ⋅ Γ 2 ⋅ α + 1 ⋅ Γ λ − 2 ⋅ α − Γ α + 1 2 ⋅ Γ λ − α 2 Γ λ
C s = Γ λ 2 ⋅ Γ λ − 3 ⋅ α ⋅ Γ 1 + 3 ⋅ α + 2 ⋅ Γ λ − α 3 ⋅ Γ 1 + α 3 − 3 ⋅ Γ λ ⋅ Γ λ − 2 ⋅ α ⋅ Γ 1 + 2 ⋅ α ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ ⋅ Γ λ − 2 ⋅ α ⋅ Γ 1 + 2 ⋅ α − Γ λ − α 2 ⋅ Γ 1 + α 2 1.5
The equations needed to estimate the parameters with L-moments have the following expressions [33]:
L 1 = β ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ
L 2 = β ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ − Γ 2 ⋅ λ − α ⋅ Γ 1 + α Γ 2 ⋅ λ
L 3 = β ⋅ Γ λ − α ⋅ Γ 1 + α Γ λ − 3 ⋅ Γ 2 ⋅ λ − α ⋅ Γ 1 + α Γ 2 ⋅ λ + 2 ⋅ Γ 3 ⋅ λ − α ⋅ Γ 1 + α Γ 3 ⋅ λ

2.2.3. Generalized Pareto Type III (PGIII)

The equations needed to estimate the parameters with MOM have the following expressions [3,4]:
μ = γ + β ⋅ Γ 1 + 1 α ⋅ Γ 1 − 1 α
σ 2 = β 2 ⋅ Γ 1 + 2 α ⋅ Γ 1 − 2 α − Γ 1 + 1 α 2 ⋅ Γ 1 − 1 α 2
C s = Γ 1 + 3 α ⋅ Γ 1 − 3 α + 2 ⋅ Γ 1 + 1 α 3 ⋅ Γ 1 − 1 α 3 − 3 ⋅ Γ 1 + 2 α ⋅ Γ 1 − 2 α ⋅ Γ 1 + 1 α ⋅ Γ 1 − 1 α Γ 1 + 2 α ⋅ Γ 1 − 2 α − Γ 1 + 1 α 2 ⋅ Γ 1 − 1 α 2 1.5
The shape parameter can be obtained approximately depending on the skewness coefficient, using the following function:
if 0.1 ≤ C s ≤ 7 :
α = exp 2.2463456 − 0.8505372 ⋅ ln C s + 0.1230871 ⋅ ln C s 2 + 0.0511417 ⋅ ln C s 3 + 0.0024655 ⋅ ln C s 4 − 0.006948 ⋅ ln C s 5 − 0.0017015 ⋅ ln C s 6 + 0.0004746 ⋅ ln C s 7 + 0.0001676 ⋅ ln C s 8
β = σ Γ 1 + 2 α ⋅ Γ 1 − 2 α − Γ 1 + 1 α 2 ⋅ Γ 1 − 1 α 2
γ = μ − β α ⋅ Γ 1 α ⋅ Γ 1 − 1 α
The equations needed to estimate the parameters with L-moments have the following expressions:
α = 1 τ 3 = L 2 L 3
β = α 2 ⋅ L 2 Γ 1 α ⋅ Γ 1 − 1 α
γ = L 1 − β α ⋅ Γ 1 α ⋅ Γ 1 − 1 α

2.2.4. Generalized Pareto Type II (PGII)

The equations needed to estimate the parameters with MOM have the following expressions [3,4,5]:
μ = γ + β α + 1
σ 2 = β 2 α + 1 2 ⋅ 2 ⋅ α + 1
C s = α α ⋅ 2 ⋅ α − 1 ⋅ 2 ⋅ α + 1 0.5 3 ⋅ α + 1
The shape parameter can be obtained approximately depending on the skewness coefficient, using the following functions:
if C s < 2 :
α = 0.999946019 − 1.154837085 ⋅ C s + 0.559584297 ⋅ C s 2 − 0.152516332 ⋅ C s 3 − 0.018234805 ⋅ C s 4
if C s ≥ 2 :
α = 1.680766638 − 0.91819165 ⋅ C s + 0.043608198 ⋅ C s 2 − 0.002324901 ⋅ C s 3 1 + 1.805341085 ⋅ C s
β = σ ⋅ α + 1 ⋅ 2 ⋅ α + 1
γ = μ − β α + 1
The frequency factor (for MOM), presented in Appendix B, can be obtained approximately, using a polynomial function of skewness and probability, whose coefficients are presented in table D1 from Appendix D.
The equations needed to estimate the parameters with L-moments have the following expressions:
L 1 = γ + β α + 1
L 2 = β α + 1 ⋅ α + 2
L 3 = − β ⋅ α − 1 α + 1 ⋅ α + 2 ⋅ α + 3
Based on these equations, the parameters have the following expressions:
α = 1 − 3 ⋅ τ 3 τ 3 + 1
β = L 2 ⋅ α + 1 ⋅ α + 2
γ = L 1 − L 2 ⋅ α + 2
The frequency factor (for L-moments) can be obtained approximately, using a polynomial function depending on skewness and probability, whose coefficients are presented in table D2 from Appendix D.

2.2.5. Generalized Pareto Type I (PGI)

The equations needed to estimate the parameters with MOM have the following expressions:
μ = γ + β ⋅ α α − 1
σ 2 = α ⋅ β 2 α − 1 2 ⋅ α − 2
C s = α α ⋅ 2 ⋅ α + 1 ⋅ 2 ⋅ α − 2 0.5 α − 3 ⋅ α
The shape parameter can be obtained approximately depending on the skewness coefficient, using the following functions:
if 0.1 ≤ C s ≤ 1.5 :
α = − 1.00404609 − 2.127093299 ⋅ C s − 1.230376348 ⋅ C s 2 − 0.100951553 ⋅ C s 3 1 + 1.163661138 ⋅ C s − 0.532187406 ⋅ C s 2
if 1.5 < C s < 2.8 :
α = 25953.828411268 − 70057.489315297 ⋅ C s + 66202.507287974 ⋅ C s 2 − 34489.86058943 ⋅ C s 3 + 8475.284356191 ⋅ C s 4 − 823.56972663 ⋅ C s 5 1 + 2982.326081943 ⋅ C s − 1054.442301104 ⋅ C s 2
β = σ ⋅ α − 1 ⋅ α − 2 α
γ = μ − β ⋅ α α − 1
The equations needed to estimate the parameters with MOM have the following expressions:
L 1 = γ + β ⋅ α α − 1
L 2 = α ⋅ β α − 1 ⋅ 2 ⋅ α − 1
L 3 = α ⋅ β ⋅ α + 1 α − 1 ⋅ 2 ⋅ α − 1 ⋅ 3 ⋅ α − 1
Based on these equations, the parameters have the following expressions:
α = τ 3 + 1 3 ⋅ τ 3 − 1
β = 2 ⋅ L 2 3 − 4 ⋅ L 2 ⋅ 7 ⋅ τ 3 − 5 3 ⋅ τ 3 + 1 ⋅ 3 ⋅ τ 3 − 1
γ = L 2 + 3 ⋅ L 1 3 − 8 ⋅ L 2 9 ⋅ τ 3 − 3

2.2.6. The Five-Parameter Wakeby Distribution (WK5)

The equations needed to estimate the parameters with MOM have the following expressions:
μ = ξ + α β + 1 − γ δ − 1
σ 2 = α 2 β + 1 2 ⋅ 2 ⋅ β + 1 − 2 ⋅ α ⋅ γ β + 1 ⋅ β + 1 − δ ⋅ δ − 1 − γ 2 δ − 1 2 ⋅ 2 ⋅ δ − 1
The equations for skewness and kurtosis are presented in Appendix E.
The equations needed to estimate the parameters with L-moments have the following expressions [9]:
L 1 = ξ + α β + 1 + γ 1 − δ
L 2 = α β + 1 ⋅ β + 2 + γ 2 − δ ⋅ 1 − δ
L 3 = γ ⋅ δ + 1 1 − δ ⋅ 2 − δ ⋅ 3 − δ + α ⋅ 1 − β β + 1 ⋅ β + 2 ⋅ β + 3
L 4 = γ ⋅ δ + 1 ⋅ δ + 2 4 − δ ⋅ 3 − δ ⋅ 2 − δ ⋅ 1 − δ + α ⋅ 2 − β ⋅ 1 − β β + 1 ⋅ β + 2 ⋅ β + 3 ⋅ β + 4

3. Case study

The presented case study consists in the determination of maximum annual flows, on the Prigor River, Romania, using the proposed probability distributions.
The Prigor River is the left tributary of the Nera River, and it is located in the south-western part of Romania, as shown in Figure 2. The geographical coordinates of the location are 44°55'25.5"N 22°07'21.7"E.
The main morphometric characteristics of the river are presented in Table 3 [35].
In the section of the hydrometric station, the watershed area is 153 km2 and the average altitude is 713 m. The river has a length of 33 km, with an average slope of 22 ‰ and a sinuosity coefficient of 1.83.
There are 31 annual maximum flows, with the values presented in Table 4.
For the analysis with AES, the maximum flows resulting from the selection are presented in Table 5.
Table 4. The AMS from the Prigor hydrometric station.
Table 4. The AMS from the Prigor hydrometric station.
AMS
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Flow [m3/s] 9.96 15 10.1 14.8 7.30 21.2 18.2 21.4 13.1 14.5 35
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
Flow [m3/s] 19.9 22.1 11.8 80.3 88 51.6 72.2 16.2 42.6 28.5 12.8
2012 2013 2014 2015 2016 2017 2018 2019 2020
Flow [m3/s] 31.2 24.1 52.2 21.1 18.9 6.40 24.9 15.1 36.6
Table 5. The AES from the Prigor hydrometric station.
Table 5. The AES from the Prigor hydrometric station.
AES
1995 1996 1997 2000 2001 2002 2004 2004 2004 2005 2005
Flow [m3/s] 21.2 18.2 21.4 35 19.9 22.1 80.3 22.2 19.2 88 38.9
2005 2005 2006 2007 2007 2007 2008 2009 2009 2010 2012
Flow [m3/s] 24 17.5 51.6 72.2 33.8 15.9 16.2 42.6 23.1 28.5 31.2
2012 2012 2013 2014 2015 2016 2016 2018 2020
Flow [m3/s] 27.3 18.7 24.1 52.2 21.1 18.9 16.8 24.9 36.6
The main statistical indicators of the data series, are presented in Table 6.
Table 6. The statistical indicators of the data series.
Table 6. The statistical indicators of the data series.
Prigor River Statistical Indicators
μ σ C v C s C k L 1 L 2 L 3 L 4 τ 2 τ 3 τ 4
[m3/s] [m3/s] [-] [-] [-] [m3/s] [m3/s] [m3/s] [m3/s] [-] [-] [-]
AMS 27.6 21.1 0.762 1.66 5.16 27.6 10.7 4.26 2.43 0.386 0.399 0.228
AES 31.7 18.9 0.595 1.83 5.77 31.7 9.30 4.22 2.14 0.293 0.454 0.230
where μ , σ , C v , C s , C k , L 1 , L 2 , L 3 , L 4 , τ 2 , τ 3 , τ 4 represent the mean, the standard deviation, the coefficient of variation, the skewness, the kurtosis, the four L-moments, the L-coefficient of variation, the L-skewness, and the L-kurtosis, respectively. For parameter estimation with L-moments, the data series must be in ascending order for the calculation of natural estimators, respectively L-moments.

4. Results

The proposed distributions from the Generalized Pareto family, were applied to perform a flood frequency analysis using the annual maximum series (AMS) and annual exceedance series (AES) analysis, on the Prigor river.
MOM and L-moments were used to estimate the parameters of the distributions. For the MOM, the skewness coefficient was chosen depending on the origin of the flows according to Romanian regulations [22,28], based on some multiplication coefficients for the coefficent of variation ( C v ). For the Prigor river, the multiplication coefficient 3 applied to the coefficient of variation resulting in a skewness of 2.29 for AMS and 1.786 for AES.
In Table 7 and 8 are presented the results values of quantile distributions, for some of the most common exceedance probabilities in flood frequency analysis.
Figure 3 and Figure 4, show the fitting distributions for AMS and AES, for the Prigor river. For plotting positions, the Alexeev formula was used [2].
In Figure S1, from the supplementary material, the confidence interval for each analyzed distribution is presented, both for MOM and L-moments, using Chow's relation [4,34], for 95% confidence intervals.
Table 9 shows the values of the distributions parameters for the two methods of estimating and for both AMS and AES.
The performance of the analyzed distribution was evaluated using the relative mean error (RME) criterion, the relative absolute error (RAE) criterion [36,37,38].
R M E = 1 n ⋅ ∑ i = 1 n x i − x p x i 2
R A E = 1 n ⋅ ∑ i = 1 n x i − x p x i
where n , x i , x p represent sample size, observed value, and estimated value for a given probability, respectively. For the RME and RAE performance indicators, the best model is the one with the minimum values. For the L-moments method, the variation diagram is additionally used for choosing the best fit model.
The distributions performance values are presented in Table 10 and Table 11. The values for the best model are highlighted in bold.

5. Discussion

In this article, the applicability of the distributions from the Generalized Pareto family in flood frequency analysis was analyzed, with the Prigor river as a case study.
The analysis was performed using AMS and AES. As can be seen both graphically (Figure 4) and tabularly (Table 7), the analysis with AMS is more conservative than the analysis with AES.
The main advantage of the AMS analysis is the ease of data selection, these being chosen as the maximum flow corresponding to each year. The disadvantage of the analysis is the use of maximum flows characteristic of each year, which do not always represent floods. These values located in the right-hand (high probabilities), lead to a steeper graph with higher values of quantiles in the field of low probabilities (left-hand).
The advantage of the analysis with AES is that the flows of the data series always represent floods. The disadvantage is the greater effort in data selection, through additional analyzes regarding data independence, respecting the condition that these maximum flows do not come from the same flood.
The estimation methods of the analyzed distribution parameters were performed for MOM (standard method in Romania) and L-moments, two of the most used estimation methods in hydrology. In general, the L-moments method is a much more stable and robust method, being less influenced by the length of the observed data.
For the MOM analysis, the skewness was chosen depending on the origin of the flows, as is the hydrological practice in Romania.
All analyzed distributions represent particular cases of the Generalized Pareto distribution, being distributions of 3 and 4 parameters. The Wakeby distribution, which is a five-parameter distribution, was analyzed because it has as its particular case the PGII distribution, being a quantile function whose structure is made up of two quantile functions of the PGII distribution.
All the results obtained in the case study are presented compared to the Pearson III distribution, which is considered the ‘’parent’’ distribution in Romania, for the most used exceedance probabilities in hydrology.
According to Tabel 10 and Tabel 11, the best fitted distributions with the smallest RME and RAE values are WK5 and PGII distributions. But based only on these indicators, the best fit distribution cannot always be selected, because these statistical indicators only properly evaluate the probability area of the observed values. Outside of this domain (left hand), they lose their relevance, because, in general, the data sets in Romania are not large enough (n>80). Thus, considering that it is desired to implement the L-moments method in Romania, it is necessary to use an additional criterion for choosing the distributions in the analysis with L-moments, namely the τ 3 - τ 4 variation diagram. It is recommended to use distributions that have the values of these indicators very close to those of the observed data, especially in the case of 3-parameter distributions.
From the obtained results with L-moments, it can be seen that the best results are with the PGIV4 and WK5 distributions, which best approximate the statistical indicators of the data set, τ 3 and τ 4 . PGII and PGI distributions give satisfactory results because the natural values τ 3 and τ 4 of the distributions are close to those of the data sets.
For PE3, PGIV3 and PGIII, both for MOM and L-moments, the resulting values are characterized by a high degree of uncertainty, especially in the area of small exceedance probabilities (left-hand), due to the fact that a proper calibration of the higher moments cannot be done.
As observed in other materials [24], the apparent stability of the Pearson III distribution is due to the fact that the variation of the shape parameter for the two estimation methods does not differ much, except in the upper area of C s and τ 3 . The same cannot be said about the PGI, PGII and PGIII distributions, in which the variation is extremely large.
Figure 5 shows the variation graph of the shape parameter for the PE3, PGIII, PGII and PGI distributions. As it could be observed in Section 2.1, both skewness and L-skewness depend only on the shape coefficient α .
The results of the quantiles obtained with the L-moments method, for the PGII and PGI distributions, both for AES and for AMS, and presented in Table 4 and Table 5, are the same, the two distributions being mutual special cases. The distributions WK5 and PGIV4 best approximate, as expected, the values of the indicators obtained with L moments, being distributions of 5 and 4 parameters, respectively.
Figure 6 shows the τ 3 − τ 4 variation diagram of the distributions, as well as their relation to the values of the two indicators of the data sets.
Regarding the results obtained with MOM, for AES it can also be observed that the PGII and WK5 distributions have extremely close quantile values, due to the fact that for the same C s they correspond to the same value of C k , the WK5 distribution becoming the PGII distribution, case as it was highlighted in other materials [4,8,9,22].
The WK5 distribution, although it is a distribution that was introduced in flood frequency analysis to achieve the so-called "separation effect" described by Matalas [9,22], it can be seen that it is extremely sensitive depending on the analysis used (AMS or AES), and this is due to the particular cases in which it can take.
Figure 7 shows the skeness ( C s )-kurtozis ( C k ) variation diagram of the distributions.

6. Conclusions

Generalized Pareto distribution represents a usual distribution used in the analysis of extreme events in hydrology. In flood frequency analysis this is especially applied using the partial series of maximum flows.
This article presents 5 distributions, of 3, 4 and 5 parameters that represent different forms of the Generalized Pareto distribution, some of them received limited attention in flood frequency analysis.
These distributions were analyzed (besides other families of distributions) in the research carried out in the Faculty of Hydrotechnics regarding the elaboration of a norm in Romania for the frequency analysis using L-moments method.
The main purpose of the article was to identify other distributions from the same family that have applicability in flood frequency analysis using both AMS and AES.
For the transparency and ease of use of these distributions, all the necessary elements for their use are presented, such as the exact and approximate relations of parameters estimation, and of frequency factors, care eliminates the need for iterative numerical calculation, thus facilitating their applicability.
The performances of these distributions were verified in flood frequency analysis of the Prigor River, using the Annual Maximum Series and the Annual Exceedance Series.
The results were evaluated using relative mean error and relative absolute error, which uses the inverse function of the distribution. In general, performance indicators have the disadvantage that they are only valid for the range of recorded values. Thus, in the case of the analysis with L-moments, the additional selection criterion of the best fit distribution is represented by the values of τ 3 and τ 4 , based on the τ 3 − τ 4 diagram, compared to that of the data sets, being a more robust indicator over the entire range of excess probabilities.
Based on the study results, and also from the research carried out in the Faculty of Hydrotechnics for other sites, for flood frequency analysis and the L-moments estimation method, good candidates, from the Generalized Pareto family, are the PGIV4 and WK5 distributions, being distributions of four and five parameters, having the advantage that it can calibrate all linear moments.
Regarding the Wakeby distribution, this requires an additional analysis, because in some cases it turns into the PGII distribution, which is a three-parameter distribution, not achieving a satisfactory calibration of the linear moments.
In general, the three parameters distributions can be used in the analysis with L-moments, but the selection of their use must be made based on the τ 3 - τ 4 diagram, so that the natural values τ 3 and τ 4 of the distribution to be very close to those of the observed data. Based on the work of Anghel si Ilinca [23], in Appendix A is presented the τ 3 − τ 4 diagram for a wide range of distributions used in hydrology.
Mathematical support in statistical analysis is useful because the use of software (EasyFit, HEC-SSP, etc) without knowledge of mathematical foundations often leads to superficial analyzes with negative consequences. Another important aspect of the presentation of all the mathematical elements necessary for the application of these distributions is the fact that this software dedicated to statistical analysis are limited, and do not offer the possibility of choosing the skewness coefficient depending on the origin of the maximum flows, as is the practice in Romania.
The research in this article is part of a more complex research carried out within the Faculty of Hydrotechnics, with the main aim of establishing the necessary guidelines for a robust, clear and concise norm regarding the determination of the maximum flow using the L-moment estimation method.

Supplementary Materials

The following supporting information can be downloaded at the website of this paper posted on Preprints.org.

Author Contributions

Conceptualization, C.I. and C.G.A.; methodology, C.I. and C.G.A.; software, C.I. and C.G.A.; validation, C.I. and C.G.A.; formal analysis, C.I. and C.G.A.; investigation, C.I. and C.G.A.; resources, C.I. and C.G.A.; data curation, C.I. and C.G.A.; writing—original draft preparation, C.I. and C.G.A.; writing—review and editing, C.I. and C.G.A.; visualization, C.I. and C.G.A.; supervision, C.I. and C.G.A.; project administration, C.I. and C.G.A.; funding acquisition, C.I. and C.G.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

MOM the method of ordinary moments
L-moments the method of linear moments
μ expected value; arithmetic mean
σ standard deviation
C v coefficient of variation
C s coefficient of skewness; skewness
C k coefficient of kurtosis; kurtosis
L 1 , L 2 , L 3 linear moments
τ 2 , L C v coefficient of variation based on the L-moments method
τ 3 , L C s coefficient of skewness based on the L-moments method
τ 4 , L C k coefficient of kurtosis based on the L-moments method
Distr. Distributions
RME relative mean error
RAE relative absolute error
xi observed values

Appendix A. The Variation of L-kurtosis - L-skewness

In the next section are presented the variation of L-kurtosis depending on the positive L-skewness, obtained with the L-moments method, for certain theoretical distributions often used in hydrology.
Preprints 70262 g0a1
Figure A1. The variation diagram of L C s − L C k .
Figure A1. The variation diagram of L C s − L C k .
Preprints 70262 g0a1

Appendix B. The Frequency Factors for the Analyzed Distributions

Table A1 shows the expressions of the frequency factors for MOM and L-moments.
Table A1. Frequency factors.
Table A1. Frequency factors.
Distribution Frequency factor, K p p
Quantile function (inverse function)
Method of ordinary moments (MOM) L-moments
x p = μ + σ ⋅ K p p x p = L 1 + L 2 ⋅ K p p
PGIV4 Γ λ ⋅ p − 1 λ − 1 − Γ α + 1 ⋅ Γ λ − α Γ 2 ⋅ α + 1 ⋅ Γ λ − 2 ⋅ α ⋅ Γ λ − Γ α + 1 2 ⋅ Γ λ − α 2 p − 1 λ − 1 α − Γ α + 1 ⋅ Γ λ − α Γ λ Γ α + 1 ⋅ Γ λ − α Γ λ − Γ 2 ⋅ λ − α Γ 2 ⋅ λ
PGIV3 Γ λ ⋅ Γ p − 1 λ − 1 α − Γ α + 1 ⋅ Γ λ − α Γ 2 ⋅ α + 1 ⋅ Γ λ − 2 ⋅ α ⋅ Γ λ − Γ α + 1 2 ⋅ Γ λ − α 2 Γ p − 1 λ − 1 α − Γ α + 1 ⋅ Γ λ − α Γ λ Γ α + 1 ⋅ Γ λ − α Γ λ − Γ 2 ⋅ λ − α Γ 2 ⋅ λ
PGIII 1 p − 1 1 α − π α ⋅ sin π α 2 ⋅ π α ⋅ sin 2 ⋅ π α − π 2 α 2 ⋅ sin π α 2 1 − p p 1 α ⋅ α 2 ⋅ sin π α π − α
PGII 1 − 2 ⋅ α α ⋅ 1 − 1 + α ⋅ p α 1 + 2 α ⋅ 1 − p α − α + 3 ⋅ p α
PGI α α ⋅ α − 1 ⋅ α − 2 ⋅ 1 − p − 1 α − α α − 1 α α α ⋅ α − 1 ⋅ 2 ⋅ α − 1 ⋅ 1 − p − 1 α − α α − 1 α
WK5 α β ⋅ 1 − p β − γ δ ⋅ 1 − p − δ − α β + 1 + γ δ − 1 α 2 β + 1 2 ⋅ 2 ⋅ β + 1 − 2 ⋅ α ⋅ γ β + 1 ⋅ β + 1 − δ ⋅ δ − 1 − γ 2 δ − 1 2 ⋅ 2 ⋅ δ − 1 α β ⋅ 1 − p β − γ δ ⋅ 1 − p − δ − α β + 1 − γ 1 − δ α β + 1 ⋅ β + 2 + γ 2 − δ ⋅ 1 − δ

Appendix C. Estimation of the Frequency Factor for the PGIII Distribution

The frequency factor, for MOM, can be estimated using a polynomial function:
K p p = a + b ⋅ C s + c ⋅ C s 2 + d ⋅ C s 3 + e ⋅ C s 4 + f ⋅ C s 5 + g ⋅ C s 6 + h ⋅ C s 7
Table A2. The frequency factor for estimation with MOM.
Table A2. The frequency factor for estimation with MOM.
P
[%]		 a b c d e f g h
0.01 5.111737 2.313409 1.34999E+00 -7.76028E-01 1.82704E-01 -2.32091E-02 1.5563E-03 -4.330E-05
0.1 3.808941 1.377403 3.45331E-01 -3.18752E-01 9.03167E-02 -1.30843E-02 9.7470E-04 -2.960E-05
0.5 2.912620 0.812083 1.70251E-02 -1.14635E-01 3.94326E-02 -6.29710E-03 4.9970E-04 -1.590E-05
1 2.527259 0.600436 -5.35487E-02 -5.65675E-02 2.33015E-02 3.98010E-03 3.2800E-04 1.070E-05
2 2.140031 0.411098 -9.11407E-02 -1.47256E-02 1.07835E-02 -2.10400E-03 1.8480E-04 -6.200E-06
3 1.911447 0.311395 -1.00372E-01 2.88960E-03 5.08580E-03 -1.21600E-03 1.1520E-04 -4.100E-06
5 1.619324 0.197967 -1.00797E-01 1.85338E-02 -4.64200E-04 -3.15600E-04 4.3000E-05 -1.700E-06
10 1.208950 0.066749 -8.47851E-02 2.91844E-02 -5.24780E-03 5.25900E-04 -2.7400E-05 6.000E-07
20 0.763538 -0.0362722 -5.31111E-02 2.86703E-02 -6.99220E-03 9.32800E-04 -6.5700E-05 1.900E-06
40 0.224364 -0.103222 -6.80020E-03 1.62353E-02 -5.34190E-03 8.40400E-04 -6.63000E-05 2.100E-06
50 0.001255 -0.111838 1.19116E-02 8.67100E-03 -3.76100E-03 6.53200E-04 -5.44000E-05 1.8000E-06
80 -0.762791 -0.0547579 5.93627E-02 -2.07334E-02 3.89270E-03 -4.16000E-04 2.37000E-05 -6.000E-07
90 -1.210684 0.0415455 6.68030E-02 -3.60124E-02 8.85020E-03 -1.19170E-03 8.48000E-05 -2.500E-06
Table A3. The frequency factor for estimation with L-moments.
Table A3. The frequency factor for estimation with L-moments.
P
[%]		 a b c d e f g h
0.01 9.0865E+00 -9.0870E+00 -5.6824E+00 1.4136E+01 -1.9532E+01 1.5682E+01 -6.8872E+00 1.2851E+00
0.1 6.8300E+00 -6.8367E+00 -4.3709E+00 8.5137E+00 -9.3692E+00 6.1000E+00 -2.2178E+00 3.5120E-01
0.5 5.3161E+00 -5.3440E+00 -3.2817E+00 4.4099E+00 -2.2009E+00 -1.0766E+00 1.7928E+00 6.1653E-01
1 4.6040E+00 -4.6529E+00 -2.9168E+00 3.6177E+00 -1.7699E+00 -6.9899E-01 1.2546E+00 -4.3830E-01
2 3.8961E+00 -3.9826E+00 -2.5218E+00 2.8156E+00 -1.2339E+00 -6.2320E-01 9.9972E-01 -3.5026E-01
3 3.4784E+00 -3.5994E+00 -2.2840E+00 2.4127E+00 -1.0762E+00 -3.9996E-01 7.3125E-01 -2.6304E-01
5 2.9457E+00 -3.1313E+00 -1.9655E+00 1.9096E+00 -8.1060E-01 -2.9906E-01 5.5319E-01 -2.0219E-01
10 2.1976E+00 -2.5343E+00 -1.4975E+00 1.3114E+00 -5.3308E-01 -1.5403E-01 3.3379E-01 -1.2411E-01
20 1.3864E+00 -2.0186E+00 -9.5952E-01 8.1735E-01 -2.7803E-01 -7.4878E-02 1.8663E-01 -5.9390E-02
40 4.0582E-01 -4.0582E-01 -2.5879E-01 3.8534E-01 1.6327E-01 -4.2751E-02 -6.2078E-02 8.8377E-02
50 1.9341E-04 -1.6511E+00 3.3144E-02 3.7384E-01 1.9857E-01 1.1770E-02 -3.1091E-02 6.4863E-02
80 -1.3869E+00 -5.2845E-01 -9.5705E-02 1.2009E+00 -4.2838E-01 4.7547E-02 4.0683E-01 -2.1635E-01
90 -2.1977E+00 1.0582E+00 -1.2265E-01 7.0118E-01 -4.1310E-01 -1.1862E-01 1.7678E-01 -8.4297E-02

Appendix D. Estimation of the Frequency Factor for the PGII Distribution

The frequency factor, for MOM, can be estimated using a polynomial function:
K p p = a + b ⋅ C s + c ⋅ C s 2 + d ⋅ C s 3 + e ⋅ C s 4 + f ⋅ C s 5 + g ⋅ C s 6 + h ⋅ C s 7
Table A4. The frequency factor for estimation with MOM.
Table A4. The frequency factor for estimation with MOM.
P
[%]		 a b c d e f g h
0.01 1.932014 0.061904 2.87112E+00 -7.95586E-01 4.95418E-02 1.03747E-02 -1.7871E-03 7.910E-05
0.1 1.809609 0.670411 2.12699E+00 -1.15732E+00 2.84537E-01 -3.77979E-02 2.6299E-03 -7.530E-05
0.5 1.704417 1.213220 7.62347E-01 -6.51934E-01 1.98072E-01 -3.05909E-02 2.3988E-03 -7.580E-05
1 1.664419 1.314780 1.78438E-01 -3.55564E-01 1.25632E-01 -2.09008E-02 1.7176E-03 -5.610E-05
2 1.622183 1.265729 -2.90832E-01 -7.64258E-02 5.00117E-02 -9.92700E-03 8.9160E-04 -3.080E-05
3 1.590286 1.151049 -4.79305E-01 5.83412E-02 1.03575E-02 -3.85500E-03 4.1630E-04 -1.570E-05
5 1.530888 0.909001 -5.98849E-01 1.79061E-01 -2.88591E-02 2.47620E-03 -9.6700E-05 9.000E-07
10 1.377562 0.424384 -5.22494E-01 2.28357E-01 -5.36328E-02 7.14020E-03 -5.0690E-04 1.490E-05
20 1.047703 -0.1346753 -1.97916E-01 1.36073E-01 -3.93659E-02 5.98720E-03 -4.6780E-04 1.480E-05
40 0.355375 -0.4660863 1.82363E-01 -3.47085E-02 2.20900E-03 2.60000E-04 -4.8700E-05 2.100E-06
50 0.0051002 -0.428129 2.42055E-01 -7.58534E-02 1.42094E-02 -1.57970E-03 9.61000E-05 -2.5000E-06
80 -1.042946 0.1095022 7.44175E-02 -5.46187E-02 1.56946E-02 -2.36020E-03 1.82600E-04 -5.700E-06
90 -1.389758 0.3823067 -6.62687E-02 -9.67300E-03 6.70430E-03 -1.27020E-03 1.09700E-04 -3.700E-06
The frequency factor, for L-moments, can be estimated using a polynomial function:
K p p = a + b ⋅ τ 3 + c ⋅ τ 3 2 + d ⋅ τ 3 3 + e ⋅ τ 3 4 + f ⋅ τ 3 5 + g ⋅ τ 3 6 + h ⋅ τ 3 7 + i ⋅ τ 3 8 + j ⋅ τ 3 9
Table A5. The frequency factor for estimation with L-moments.
Table A5. The frequency factor for estimation with L-moments.
P
[%]		 a b c d e f g h i j
0.01 2.8976E+00 1.6049E+01 -1.6633E+02 2.1072E+03 -1.1711E+04 4.1260E+04 -8.4364E+04 1.0671E+05 -6.8465E+04 1.4604E+04
0.1 2.9921E+00 7.9122E+00 2.1160E+01 4.6340E+01 2.0703E+02 -3.9480E+02 1.4671E+03 -2.0265E+03 6.6772E+02 -
0.5 2.9751E+00 7.0733E+00 2.1320E+01 5.3452E+00 9.7924E+01 -9.9393E+01 -9.5564E+01 -5.9326E+01 - -
1 2.9389E+00 6.8741E+00 1.4223E+01 1.0895E+01 3.3199E+01 -1.1624E+02 4.7113E+01 - - -
2 2.8667E+00 6.5598E+00 3.8790E+00 2.8740E+01 -6.6614E+01 2.3564E+01 - - - -
3 2.8214E+00 5.3368E+00 6.0252E+00 1.9543E+00 -3.1305E+01 1.4174E+01 - - - -
5 2.7060E+00 3.9951E+00 4.1471E+00 -1.3220E+01 -2.6956E+00 4.0738E+00 - - - -
10 2.4017E+00 2.0082E+00 -1.6983E+00 -9.9252E+00 7.8868E+00 -1.6725E+00 - - - -
20 1.7989E+00 -4.8676E-01 -4.2616E+00 1.0072E+00 1.7490E+00 -8.0770E-01 - - - -
40 6.0025E-01 -2.4049E+00 -3.1267E-01 2.3727E+00 -1.6966E+00 4.4147E-01 - - - -
50 3.8891E-04 -2.3323E+00 1.3905E+00 4.8092E-01 -8.1390E-01 2.7475E-01 - - - -
80 -1.8003E+00 5.2569E-01 1.0120E+00 -1.3741E+00 8.6642E-01 -2.2985E-01 - - - -
90 -2.4001E+00 2.1286E+00 -9.6924E-01 2.3252E-01 4.1754E-02 -3.3573E-02 - - - -

Appendix E. The Skewness and Kurtosis for the WK5 Distribution

C s = 6 ⋅ α ⋅ γ 2 ⋅ δ 2 − δ ⋅ β + δ − 1 β + 1 ⋅ β + 1 − 2 ⋅ δ ⋅ β + 1 − δ ⋅ δ − 1 2 ⋅ 2 ⋅ δ − 1 + 6 ⋅ α 2 ⋅ γ ⋅ β 2 − 1 − β ⋅ 1 + δ β + 1 2 ⋅ 2 ⋅ β + 1 ⋅ β + 1 − δ ⋅ δ − 1 ⋅ 2 ⋅ β + 1 − δ − 2 ⋅ γ 3 ⋅ 1 + δ δ − 1 3 ⋅ 1 + δ ⋅ 6 ⋅ δ − 5 − 2 ⋅ α 3 ⋅ β + 1 β + 1 3 ⋅ 1 + β ⋅ 5 + 6 ⋅ β α 4 β + 1 2 ⋅ 2 ⋅ β + 1 − γ 2 δ − 1 2 ⋅ 2 ⋅ δ − 1 + 2 ⋅ α ⋅ λ β + 1 ⋅ δ − 1 ⋅ δ − β − 1 1.5
C k = 3 ⋅ α 4 ⋅ 2 ⋅ β 2 − β + 3 β + 1 4 ⋅ 24 ⋅ β 3 + 26 ⋅ β 2 + 9 ⋅ β + 1 − γ 4 ⋅ 3 + δ + 2 ⋅ δ 2 δ − 1 4 ⋅ 4 ⋅ δ − 1 ⋅ 1 + δ ⋅ 6 ⋅ δ − 5 + 2 ⋅ α 2 ⋅ γ 2 ⋅ − 9 − β ⋅ 30 + β ⋅ 29 + 4 ⋅ β ⋅ 2 + β + β ⋅ δ ⋅ 71 + β ⋅ 43 + 2 ⋅ β ⋅ 7 + 4 ⋅ β − δ 2 ⋅ 29 + β ⋅ 43 + 4 ⋅ 6 + 7 ⋅ β + 2 ⋅ δ 3 ⋅ 4 + 7 ⋅ β ⋅ 1 + 2 ⋅ β − 4 ⋅ δ 4 ⋅ 1 + 2 ⋅ β + 30 ⋅ δ β + 1 2 ⋅ 2 ⋅ β + 1 ⋅ 1 + β − 2 ⋅ δ ⋅ 1 + 2 ⋅ β − 2 ⋅ δ ⋅ 1 + β − δ ⋅ 1 + 2 ⋅ β − δ ⋅ δ − 1 2 ⋅ 2 ⋅ δ − 1 + 4 ⋅ α ⋅ γ 4 ⋅ 3 + 2 ⋅ β + β 2 − δ ⋅ 12 + β ⋅ 3 + β − δ 2 11 + β + 2 ⋅ β 2 − 2 ⋅ δ 3 ⋅ 1 + 3 ⋅ β − 4 ⋅ δ 4 1 + β ⋅ 1 + β − 3 ⋅ δ ⋅ 1 + β − 2 ⋅ δ ⋅ 1 + β − δ ⋅ δ − 1 3 ⋅ 1 + δ ⋅ 6 ⋅ δ − 5 + 4 ⋅ α 3 ⋅ γ ⋅ 3 + 4 ⋅ β 4 + β 3 ⋅ 2 − 6 ⋅ δ + δ ⋅ δ − 2 + β ⋅ 12 + δ ⋅ δ − 3 + β 2 ⋅ 11 + δ ⋅ 2 ⋅ δ − 1 1 + β 3 ⋅ 1 + β ⋅ 6 ⋅ β + 5 ⋅ 1 + β − δ ⋅ 1 + 2 ⋅ β − δ ⋅ 1 + 3 ⋅ β − δ ⋅ δ − 1 γ 2 δ − 1 2 ⋅ 2 ⋅ δ − 1 − α 4 β + 1 2 ⋅ 2 ⋅ β + 1 + 2 ⋅ α ⋅ λ β + 1 ⋅ δ − 1 ⋅ δ − β − 1 2

Appendix F. The Skewness and Kurtosis for the PGIV Distribution

C s = Γ 1 + 3 ⋅ α ⋅ Γ λ − 3 ⋅ α λ ⋅ Γ λ + 2 ⋅ Γ 1 + α 3 ⋅ Γ λ − α 3 Γ λ + 1 3 − 3 ⋅ α ⋅ Γ 1 + 2 ⋅ α ⋅ Γ λ − 2 ⋅ α ⋅ Γ α ⋅ Γ λ − α Γ λ + 1 2 Γ 1 + 2 ⋅ α ⋅ Γ λ − 2 ⋅ α λ ⋅ Γ λ − Γ 1 + α 2 ⋅ Γ λ − α 2 Γ λ + 1 2 3 / 2 C k = Γ 1 + 4 ⋅ α ⋅ Γ λ − 4 ⋅ α Γ λ + 1 − 4 ⋅ Γ 1 + 3 ⋅ α ⋅ Γ λ − 3 ⋅ α ⋅ Γ 1 + α ⋅ Γ λ − α Γ λ + 1 2 + 6 ⋅ Γ 1 + 2 ⋅ α ⋅ Γ λ − 2 ⋅ α ⋅ Γ 1 + α 2 ⋅ Γ λ − α 2 Γ λ + 1 3 − 3 ⋅ Γ 1 + α 4 ⋅ Γ λ − α 4 Γ λ + 1 4 Γ 1 + 2 ⋅ α ⋅ Γ λ − 2 ⋅ α Γ λ + 1 − Γ 1 + α 2 ⋅ Γ λ − α 2 Γ λ + 1 2 2

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Figure 1. Methodological approach.
Figure 1. Methodological approach.
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Figure 2. The Prigor River location—Prigor hydrometric station.
Figure 2. The Prigor River location—Prigor hydrometric station.
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Figure 3. The fitting distributions for AMS.
Figure 3. The fitting distributions for AMS.
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Figure 4. The fitting distributions for AES.
Figure 4. The fitting distributions for AES.
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Figure 5. The variation of parameter α .
Figure 5. The variation of parameter α .
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Figure 6. The τ 3 − τ 4 diagram for analyzed distributions.
Figure 6. The τ 3 − τ 4 diagram for analyzed distributions.
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Figure 7. The C s − C k diagram for the analyzed distributions.
Figure 7. The C s − C k diagram for the analyzed distributions.
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Table 1. Novelty elements.
Table 1. Novelty elements.
New Elements Distribution
Exact parameter estimation PGIV4, PGI
Approximate estimation of parameters PGIII, PGII, PGI
The frequency factor for MOM PGIV4, PGIV3, PGIII, PGII, PGI, WK5
The frequency factor for L-moments PGIV4, PGIV3, PGIII, PGII, PGI, WK5
Approximate estimation of the frequency factor PGIII, PGII
Raw and central moments* PGIV, PGIII, PGII
*are presented in Supplementary file.
Table 2. The analyzed probability distributions.
Table 2. The analyzed probability distributions.
Distr. f x F x x p
PGIV4 λ ⋅ x − γ β 1 α − 1 ⋅ x − γ β 1 α + 1 − λ − 1 α ⋅ β x − γ β 1 α + 1 − λ γ + β ⋅ 1 p 1 λ − 1 α
PGIV3 λ ⋅ x β 1 α − 1 ⋅ x β 1 α + 1 − λ − 1 α ⋅ β 1 − x β 1 α + 1 − λ β ⋅ 1 p 1 λ − 1 α
PGIII α ⋅ x − γ β α − 1 ⋅ x − γ β α + 1 − 2 β 1 + x − γ β α − 1 γ + β ⋅ 1 p − 1 1 α
PGII 1 β ⋅ 1 − α β ⋅ x − γ 1 α − 1 1 − α β ⋅ x − γ 1 α γ + β α ⋅ 1 − p α
PGI α β ⋅ x − γ β − α − 1 x − γ β − α γ + β ⋅ p − 1 α
WK5 No closed form No closed form ξ + α β ⋅ 1 − p β − γ δ ⋅ 1 − p − δ
Table 3. The morphometric characteristics.
Table 3. The morphometric characteristics.
Length
[km]		 Average
Stream Slope [‰]		 Sinuosity
Coefficient [-]		 Average
Altitude, [m]		 Watershed
Area, [km2]
33 22 1.83 713 153
Table 7. Quantile results of the analyzed distributions for AMS.
Table 7. Quantile results of the analyzed distributions for AMS.
Distribution Annual Maximum Series (AMS)
Exceedance probabilities [%]
MOM L-moments
0.01 0.1 0.5 1 2 3 5 80 0.01 0.1 0.5 1 2 3 5 80
PE3 214 160 123 107 90.7 81.5 69.5 12.0 231 172 130 113 95.4 85.3 72.7 11.4
PGIV4 265 166 118 100 84.7 76.2 66.1 11.3 364 217 145 119 96.9 84.9 70.9 11.7
PGIV3 260 166 118 101 85.5 76.9 66.6 11.3 813 323 169 128 96.8 82.1 66.5 12.6
PGIII 279 169 117 98.7 82.7 74.2 64.3 12.0 800 320 168 128 95.7 82.1 66.6 12.5
PGII 225 162 122 106 89.6 80.4 69.1 11.8 329 207 142 118 96.8 85.1 71.4 11.7
PGI 171 138 112 100 87.6 80 70.2 10.5 329 207 142 118 86.8 85.1 71.4 11.7
WK5 227 163 122 106 89.4 80.3 69.00 11.8 358 216 145 120 97.0 85.0 71.0 11.7
Table 8. Quantile results of the analyzed distributions for AES.
Table 8. Quantile results of the analyzed distributions for AES.
Distribution Annual Exceedance Series (AES)
Exceedance probabilities [%]
MOM L-moments
0.01 0.1 0.5 1 2 3 5 80 0.01 0.1 0.5 1 2 3 5 80
PE3 178 138 110 97.7 85.4 78.2 69.1 16.6 233 172 130 113 95.3 85.3 72.9 18.0
PGIV4 233 150 108 93.5 80.3 73.2 64.9 17.2 277 190 137 116 96.0 85.1 72.0 18.1
PGIV3 219 146 108 94.3 81.6 74.6 66.1 16.6 836 327 170 128 96.4 81.7 66.4 19.2
PGIII 232 150 109 93.7 80.4 73.2 64.7 17.4 940 338 168 126 94.4 80.1 65.3 18.8
PGII 170 136 110 98.0 86.0 78.9 69.7 16.6 452 240 150 121 96.4 83.9 69.9 18.3
PGI 134 117 101 92.6 83.5 77.7 70.0 15.6 452 240 150 121 96.4 83.9 69.9 18.3
WK5 150 128 108 98.1 87.3 80.5 71.4 17.0 202 164 130 114 96.7 86.6 73.7 18.2
Table 9. Parameter values of each distribution for MOM and L-moments.
Table 9. Parameter values of each distribution for MOM and L-moments.
Distri. AMS AES
MOM L-moments MOM L-moments
α β γ λ δ ξ α β γ λ δ ξ α β γ λ δ ξ α β γ λ δ ξ
PE3 0.766 24.1 9.2 - - - 0.694 26.9 9.0 - - - 1.254 16.9 10.6 - - - 0.526 28.2 16.9 - - -
PGIV4 0.505 47.4 -2.51 2.66 - - 0.940 80.0 7.47 5.19 - - 0.222 49.3 -17 1.26 - - 1.29 1649 16.2 42.9 - -
PGIV3 0.58 52.7 - 3.27 - - 0.37 20.3 - 0.92 - - 0.431 46.7 - 2.55 - - 0.15 19.6 - 0.36 - -
PGIII 5.23 52.9 -28.5 - - - 2.51 20.3 0.90 - - - 6.075 57.1 -28 - - - 2.2 14.2 11.2 - - -
PGII -0.042 19.3 7.5 - - - -0.14 17.1 7.78 - - - 0.039 20.4 12.1 - - - -0.25 12.2 15.4 - - -
PGI -15.4 -367 373 - - - 7.14 122 -114 - - - -6.735 -166 176 - - - 4.023 49.2 -33.8 - - -
WK5 3.317 4.014 19 - 0.047 7.057 2.82 2.09 16 - 0.17 7.56 -20.5 1.82 27.5 - -0.133 14.7 -508 0.28 515 - -0.25 16.33
Table 10. Distributions performance values for AMS.
Table 10. Distributions performance values for AMS.
Distr. Statistical measures
Methods of parameters estimation AES values
MOM L-moments
RME RAE RME RAE τ 3 τ 4 τ 3 τ 4
PE3 0.0238 0.0879 0.0224 0.0902 0.399 0.192 0.399 0.228
PGIV4 0.0352 0.1574 0.0184 0.0772 0.228
PGIV3 0.0300 0.1394 0.0183 0.0750 0.303
PGIII 0.0533 0.2081 0.0181 0.0736 0.299
PGII 0.0228 0.1047 0.0190 0.0787 0.221
PGI 0.0470 0.2327 0.0646 0.2671 0.221
WK5 0.0184 0.0807 0.0185 0.0775 0.228
Table 11. Distributions performance values for AES.
Table 11. Distributions performance values for AES.
Distr. Statistical measures
Methods of parameters estimation AES values
MOM L-moments
RME RAE RME RAE τ 3 τ 4 τ 3 τ 4
PE3 0.0219 0.1063 0.0093 0.0405 0.454 0.220 0.454 0.230
PGIV4 0.0395 0.1736 0.0094 0.0376 0.230
PGIV3 0.0347 0.1556 0.0156 0.0680 0.348
PGIII 0.0399 0.1747 0.0153 0.0656 0.338
PGII 0.0195 0.0967 0.0108 0.0394 0.272
PGI 0.0269 0.1313 0.0108 0.0394 0.272
WK5 0.0137 0.0689 0.0086 0.0359 0.230
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