# On the Uncertainty and Changeability of the Estimates of Seasonal Maximum Flows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Case Study

^{3}/s), and for winter peak flows it was 213 (m

^{3}/s), while the variation coefficient Cv was 0.718 and 0.569, respectively.

## 3. Materials and Methods

#### 3.1. Model Selection Procedures

- 1.
- The AIC criterion (Akaike information criterion) is based on the likelihood function of distribution maximized with respect to the parameter values, also taking into account the number of distribution parameters $k$. The ${\mathrm{AIC}}_{i}$ selection statistic is defined by Equation (1) for the i-th distribution of the probability density function ${f}_{i}$ from among m alternative distributions and for $i=1,..,m$ [41]:$${\mathrm{AIC}}_{i}=2k-2\xb7\underset{\widehat{\theta}}{\mathrm{max}}\left[{{\displaystyle \sum}}_{j=1}^{N}\mathrm{ln}{f}_{i}\left({\widehat{\theta}}_{i},{x}_{j}\right)\right]$$

- 2.
- The QK (Quesenberry–Kent) procedure is based on the density function modified to the form given by Equation (2), which is invariant under a scale transformation of the data [45]:$${\mathrm{S}}_{i}=-\mathrm{ln}\left\{{{\displaystyle \int}}_{0}^{\infty}{f}_{i}\left(\lambda {x}_{1},\dots ,\lambda {x}_{N}\right){\lambda}^{N-1}d\lambda \right\}$$

- 3.
- The KS (Kolmogorov–Smirnov) goodness of fit test is based on the Kolmogorov–Smirnov statistics defined by Equation (3) [46,47]:$${\mathrm{D}}_{i}^{\mathrm{max}}=\underset{j=1,\dots ,N}{\mathrm{max}}\left|{p}_{i}\left({x}_{j:N}\right)-{\widehat{p}}_{j:N}\right|$$

- 4.
- The R (range) procedure has two variants. The first variant, R
^{1}, is based on the distance between the estimate of the 1% quantile from the maximum likelihood method and its value from the method of moments, and it is calculated using Equation (4) [49]:$${\mathrm{R}}_{i}^{1}=\left|{\widehat{x}}_{1\%\left(i\right)}^{\mathrm{MLM}}-{\widehat{x}}_{1\%\left(i\right)}^{\mathrm{MOM}}\right|$$

#### 3.2. Models Aggregation

## 4. Results

#### 4.1. Models for Seasonal Maximum Flows of Polish Data

#### 4.2. Distribution Identification and Its Parameter Estimation: Comparative Study

#### 4.2.1. Estimation of Quantiles of the Maximum Flow Distribution

#### 4.2.2. Uncertainty of Distribution Selection

^{1}procedure. The GE distribution also turns out to be the best fit to the data among all competitive PDFs, after which the LN and LL are placed. The GE distribution also turns out to be the best fit to the data among all competitive PDFs, when QK procedure or KS test along with MOM method of estimation are applied. In both cases, the second and third place are occupied by Ga and LN distributions, respectively. For the KS test with the LMM estimation method, the best distribution was IG, followed by GE and LN, while with the MLM estimation method, surprisingly, the best was the LL distribution, followed by LN and GE, respectively. Finally, according to the R

^{2}selection procedure, first place was taken by the Ga distribution, followed by We and GE.

^{1}procedures. The IG and LL distributions occupied first place when the KS test was applied, together with the LMM and MLM estimation methods, respectively. Except for the case of the R

^{1}procedure, in all variants of the model selection procedure with an estimation method, the LN distribution was in second place.

#### 4.3. Aggregation of Quantiles

#### 4.3.1. Instability of High Quantile Estimates with Increasing Length of Data Series

#### 4.3.2. Case Study of Model Aggregation

#### 4.3.3. Impact of Competitive Distributions on Aggregated Quantiles

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Basic characteristics of the seasonal peak flows of 37 gauging stations at Polish rivers analyzed in paper.

No | Gauging Station | River | Observation Period | Mean of Summer Peak Flows (m^{3}/s) | Mean of Winter Peak Flows (m^{3}/s) |
---|---|---|---|---|---|

1 | Jawiszowice | Wisła | 1951–2016 | 153 | 68 |

2 | Tyniec | 1951–1990 | 642 | 378 | |

3 | Jagodniki | 1921–2016 | 990 | 621 | |

4 | Szczucin | 1921–2016 | 1700 | 1018 | |

5 | Sandomierz | 1921–2016 | 2010 | 1443 | |

6 | Zawichost | 1951–2016 | 2727 | 2074 | |

7 | Puławy | 1951–2016 | 2432 | 1960 | |

8 | Warszawa | 1921–2016 | 2225 | 2302 | |

9 | Kępa | 1921–2016 | 2545 | 3305 | |

10 | Toruń | 1921–2016 | 2532 | 3334 | |

11 | Tczew | 1921–2016 | 2400 | 3383 | |

12 | Żywiec | Sola | 1956–2016 | 330 | 160 |

13 | Sucha | Skawa | 1951–2016 | 148 | 66 |

14 | Wadowice | 1951–2016 | 253 | 117 | |

15 | Rudze | Wieprzówka | 1961–2016 | 61 | 22 |

16 | Stróża | Raba | 1956–2016 | 202 | 93 |

17 | Proszówki | 1951–2016 | 433 | 213 | |

18 | Kowaniec | Dunajec | 1951–2016 | 240 | 87 |

19 | Krościenko | 1951–2016 | 427 | 161 | |

20 | Nowy Sącz | 1946–2016 | 887 | 366 | |

21 | Żabno | 1956–2016 | 1010 | 444 | |

22 | Nowy Targ | Czarny Dunajec | 1961–2016 | 160 | 56 |

23 | Zakopane | Biały Dunajec | 1961–2016 | 40 | 9 |

24 | Muszyna | Poprad | 1951–2016 | 214 | 120 |

25 | Stary Sącz | 1951–2016 | 278 | 171 | |

26 | Koszyce Wlk. | Biała | 1951–2016 | 240 | 121 |

27 | Jarosław | San | 1951–2016 | 469 | 465 |

28 | Radomyśl | 1951–2016 | 435 | 657 | |

29 | Tryńcza | Wisłok | 1951–2016 | 158 | 178 |

30 | Żółków | Wisłoka | 1951–2016 | 152 | 99 |

31 | Mielec | 1951–2016 | 419 | 331 | |

32 | Klęczany | Ropa | 1951–2016 | 129 | 55 |

33 | Wyszków | Bug | 1921–2016 | 272 | 587 |

34 | Konin | Warta | 1921–1991 | 122 | 238 |

35 | Poznań | 1822–2016 | 156 | 404 | |

36 | Skwierzyna | 1921–2016 | 184 | 362 | |

37 | Gorzów | 1921–2016 | 276 | 494 |

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**Figure 2.**Regulated (red color) and unregulated (blue color) peak flows at the Proszówki gauging station along with the trends in their average values for the (

**a**) summer season; (

**b**) winter season.

**Figure 3.**Skewness coefficient versus variation coefficient for two-parameter distributions and Polish data of summer (red color) and winter (blue color) peak flows for 37 gauging stations: (

**a**) conventional moment ratio diagrams of Cs vs. Cv; (

**b**) linear moment ratio diagrams of LCs vs. LCv. Key to the distributions: Ga—gamma, We—Weibull, GE—generalized exponential, IG—inverse Gaussian, LN—log-normal, LL—log-logistic, LG—log-Gumbel, Exp—exponential.

**Figure 4.**Estimates of the 1% quantile of maximum flows at Proszówki Station along with the best matching distributions for: (

**a**) summer season; (

**b**) winter season. Distributions: Ga—gamma, We—Weibull, GE—generalized exponential, IG—inverse Gaussian, LN—log-normal.

**Figure 5.**Estimates of the 1% quantile of maximum flows along with the best matching distributions by the Akaike information criterion (AIC) for the stations: (

**a**) Jagodniki [3]*; (

**b**) Koszyce Wielkie [26]; (

**c**) Mielec [31]; (

**d**) Wyszków [33]. Distributions: Ga—gamma, We—Weibull, GE—generalized exponential, IG—inverse Gaussian, LN—log-normal, LG—log-Gumbel. * The number of the station in Table A1.

**Figure 6.**The estimates of the 1% quantile of (

**a**) summer and (

**b**) winter peak flows for the Proszówki gauging station on the Raba River obtained by the method of selecting the best fit distribution and by the aggregation method with variants of the candidate distributions.

Distribution | Probability Density Function (PDF) |
---|---|

Gamma (Ga) | $f\left(x\right)=\frac{{\alpha}^{k}}{\mathsf{\Gamma}\left(k\right)}{x}^{k-1}{e}^{-\alpha x};k0$ |

Weibull (We) | $f\left(x\right)=\frac{k}{\alpha}{\left(\frac{x}{\alpha}\right)}^{k-1}\mathrm{exp}\left[-{\left(\frac{x}{\alpha}\right)}^{k}\right];k0$ |

Inverse Gaussian (IG) | $f\left(x\right)=\frac{\alpha}{\sqrt{\pi {x}^{3}}}\mathrm{exp}\left[-{\left(\alpha -\frac{k}{\alpha}x\right)}^{2}/x\right];k0$ |

Generalized exponential (GE) | $f\left(x\right)=\alpha k{\left(1-{e}^{-\alpha x}\right)}^{\left(k-1\right)}{e}^{-\alpha x};k0$ |

Log-normal (LN) | $f\left(x\right)=\frac{1}{xk\sqrt{2\pi}}\mathrm{exp}\left[-\frac{{\left(\mathrm{ln}\left(x\right)-\alpha \right)}^{2}}{2{k}^{2}}\right];k0$ |

Log-logistic (LL) | $f\left(x\right)=\frac{1}{\alpha}{\left[-\frac{kx}{\alpha}\right]}^{\frac{1}{k}-1}{\left[1+{\left\{-\frac{kx}{\alpha}\right\}}^{\frac{1}{k}}\right]}^{-2};k0$ |

Log-Gumbel (LG) | $f\left(x\right)=\frac{1}{\alpha}{\left[-\frac{kx}{\alpha}\right]}^{\frac{1}{k}-1}\mathrm{exp}\left\{-{\left[-\frac{kx}{\alpha}\right]}^{\frac{1}{k}}\right\};k0$ |

**Table 2.**Estimates of the 1% quantile of seasonal maximum flows (m

^{3}/s) from the period 1951–2016 at the Proszówki gauging station and assuming selected two-parameter distributions along with the standard deviation $\left(\sigma \right)$ of estimates.

Season | Estimation Method | Probability Distribution | σ | ||||||
---|---|---|---|---|---|---|---|---|---|

Ga | We | GE | IG | LN | LL | LG | |||

Summer | MOM | 1457 | 1403 | 1482 | 1570 | 1577 | 1533 | 1530 | 63.26 |

LMM | 1477 | 1391 | 1515 | 1718 | 1760 | 1965 | 2247 | 302.36 | |

MLM | 1435 | 1348 | 1468 | 1941 | 1983 | 2528 | 10760 | 3417.87 | |

Winter | MOM | 590 | 554 | 609 | 630 | 676 | 642 | 656 | 41.54 |

LMM | 585 | 535 | 613 | 653 | 755 | 739 | 858 | 112.54 | |

MLM | 566 | 539 | 596 | 671 | 780 | 803 | 2063 | 540.33 |

**Table 3.**The results of the model selection procedures for the two-parameter distributions being fitted to the seasonal maximum flows in 1951–2016 at the Proszówki gauging station.

Season | Model Selection Procedure | Estimation Method | Probability Distribution | ||||||
---|---|---|---|---|---|---|---|---|---|

Ga | We | GE | IG | LN | LL | LG | |||

Summer | AIC criterion | MOM | 921.1 | 923.1 | 920.7 | 934.3 | 927.9 | 936.6 | 2933 |

LMM | 921.3 | 923.0 | 920.9 | 928.9 | 923.8 | 925.2 | 1128 | ||

MLM | 921.0 | 922.8 | 920.7 | 926.5 | 923.0 | 923.6 | 942.5 | ||

QK procedure | MLM | 460.0 | 461.0 | 459.9 | 462.7 | 460.9 | 461.3 | 470.6 | |

KS test | MOM | 0.063 | 0.074 | 0.058 | 0.069 | 0.066 | 0.122 | 0.224 | |

LMM | 0.061 | 0.075 | 0.055 | 0.054 | 0.056 | 0.084 | 0.117 | ||

MLM | 0.073 | 0.084 | 0.059 | 0.088 | 0.051 | 0.048 | 0.110 | ||

R procedure | R^{1} | 22 | 56 | 14 | 371 | 407 | 995 | 9230 | |

R^{2} | 42 | 44 | 47 | 222 | 223 | 562 | 8512 | ||

Winter | AIC criterion | MOM | 804.2 | 808.7 | 802.9 | 804.8 | 803.9 | 807.0 | 1171 |

LMM | 804.1 | 808.5 | 803.0 | 804.1 | 803.3 | 803.7 | 878.6 | ||

MLM | 803.8 | 808.5 | 802.8 | 804.0 | 803.3 | 803.5 | 818.2 | ||

QK procedure | MLM | 401.7 | 404.1 | 401.2 | 401.8 | 401.4 | 401.5 | 454.6 | |

KS test | MOM | 0.089 | 0.111 | 0.077 | 0.058 | 0.060 | 0.083 | 0.147 | |

LMM | 0.089 | 0.115 | 0.077 | 0.055 | 0.056 | 0.058 | 0.095 | ||

MLM | 0.091 | 0.119 | 0.076 | 0.066 | 0.058 | 0.053 | 0.113 | ||

R procedure | R^{1} | 24 | 15 | 13 | 41 | 45 | 162 | 1407 | |

R^{2} | 19 | 4 | 17 | 19 | 16 | 64 | 1206 |

**Table 4.**Data for the aggregation of the 1% quantile for individual candidate distributions of seasonal maximum flows from 1951–2016 at Proszówki Station.

Season | Probability Distribution | AIC | ${\mathit{w}}_{\mathit{i}}$ | ${\widehat{\mathit{Q}}}_{\mathbf{max}1\mathit{\%}}\phantom{\rule{0ex}{0ex}}({\mathbf{m}}^{3}/\mathbf{s})$ | ${\overline{\mathit{Q}}}_{\mathbf{max}1\mathit{\%}}\phantom{\rule{0ex}{0ex}}({\mathbf{m}}^{3}/\mathbf{s})$ |
---|---|---|---|---|---|

Summer | Ga | 921.0 | 0.310 | 1420 | 1591 |

We | 922.8 | 0.125 | 1348 | ||

GE | 920.7 | 0.352 | 1467 | ||

IG | 926.5 | 0.019 | 1941 | ||

LN | 923.0 | 0.112 | 1983 | ||

LL | 923.6 | 0.082 | 2528 | ||

LG | 942.5 | 0.000 | 10758 | ||

Winter | Ga | 803.8 | 0.184 | 566 | 642 |

We | 808.5 | 0.018 | 539 | ||

GE | 802.8 | 0.305 | 596 | ||

IG | 804.0 | 0.171 | 671 | ||

LN | 803.2 | 0.247 | 679 | ||

LL | 805.6 | 0.075 | 853 | ||

LG | 818.2 | 0.000 | 2063 |

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Markiewicz, I.; Bogdanowicz, E.; Kochanek, K. On the Uncertainty and Changeability of the Estimates of Seasonal Maximum Flows. *Water* **2020**, *12*, 704.
https://doi.org/10.3390/w12030704

**AMA Style**

Markiewicz I, Bogdanowicz E, Kochanek K. On the Uncertainty and Changeability of the Estimates of Seasonal Maximum Flows. *Water*. 2020; 12(3):704.
https://doi.org/10.3390/w12030704

**Chicago/Turabian Style**

Markiewicz, Iwona, Ewa Bogdanowicz, and Krzysztof Kochanek. 2020. "On the Uncertainty and Changeability of the Estimates of Seasonal Maximum Flows" *Water* 12, no. 3: 704.
https://doi.org/10.3390/w12030704