# Best-Fit Probability Distributions and Return Periods for Maximum Monthly Rainfall in Bangladesh

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Study Area

## 3. Methodology

#### 3.1. Commonly Used Probability Distributions

_{i}is the observation data.

_{x}(x) is the value of the distribution function for the point where x corresponds to an observed data value of X.

#### 3.1.1. Normal

^{2}, are the parameters of the normal distribution. The probability density function (pdf), f(x) and cumulative distribution function (cdf), F(x) for a normal random variable x are expressed as,

#### 3.1.2. Log-Normal

_{Y}is the standard deviation and μ

_{Y}is the mean for the LN2 distribution. The LN2 should be better suited than the simple N distribution as it allows for a “heavy tail” on one side, which can better fit extreme rainfall data.

#### 3.1.3. Pearson Type 3

^{2}

#### 3.1.4. Log-Pearson Type 3

#### 3.1.5. Exponential

_{1}− α

_{2}

_{1}is the L-location or mean and λ

_{2}is the L-scale of the distribution (for more details see [19]).

#### 3.1.6. Gumbel

#### 3.1.7. Generalized Extreme Value

_{3}is the L-skewness of the distribution [19].

#### 3.1.8. Weibull

#### 3.1.9. Generalized Pareto

#### 3.2. Goodness-of-Fit Test

#### 3.2.1. Kolmogorov-Smirnov (K-S) Test

_{n}(x)) with the cdf of an assumed theoretical distribution (F

_{X}(x)). The maximum difference between S

_{n}(x) and F

_{X}(x) is the K-S test statistic. For a sample size n, the data is rearranged in increasing order, X

_{1}< X

_{2}< … < X

_{n}and the K-S statistic is assessed for each ordered value:

#### 3.2.2. Anderson-Darling (A-D) Test

^{2}as follows:

_{x}(x

_{i}) is the cdf of the proposed distribution at x

_{i}, for i = 1, 2, …, n. The observed data must be arranged in increasing order, as x

_{1}< x

_{2}< … < x

_{n}.

#### 3.2.3. Root Mean Square Error (RMSE)

_{i}denotes the estimated value and X denotes the observed value. The RMSE gives a relatively larger weight to large errors by squaring them.

#### 3.2.4. Graphical Test

_{i}

_{:n}various plotting position formulas are used for the specific distributions. Most of the plotting position formulae currently in use have the following form:

_{(i)}is ascending order), n = 1, 2, 3, …, N. N is the total number of observed values and a is a constant.

_{(i)}, versus the estimated values, x(F), are plotted, where x(F) is the quantile function, with F determined by the p

_{i}

_{:n}for the certain probability distribution.

#### 3.3. Return Period

_{T}, occurs once in T years, then the probability of occurrence P(X ≥ x) in a given year of the variable is:

^{-year}]

^{−1}= 50 years and [1–0.99

^{-year}]

^{−1}=100 years) [15].

## 4. Results and Discussion

#### 4.1. Selecting the Best-Fit Results

#### 4.2. Return Period Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Return period results of all stations (serial number from x axis corresponds to the respective serial number of Table 1). The y axis represents maximum monthly rainfall in mm. From top to bottom the sections represent 10-year, 25-year, 50-year, 100-year return periods. Only the return period of the best-fit distributions is shown here.

**Figure 4.**Rainfall return level estimations (solid lines) and 95% confidence intervals (dashed lines) of the top three fitted distributions—GEV, P3 and LP3 of three stations, as an example.

Sl.no | Station Name | Mean ($\overline{\mathit{X}}$) | SD (σ) | Coeff. of Skew. (γ) | Best-Fit Test Statistic Results | Highest Ranked Distribution (Sum of Ranks) | ||
---|---|---|---|---|---|---|---|---|

By K-S Test | By A-D Test | By RMSE Test | ||||||

1 | Ambagan | 875.2 | 209.2 | 0.118 | LN2(0.115) | P3(0.305) | N(38.25) | N(5) |

2 | Barisal | 555.0 | 167.6 | 1.618 | GEV(0.095) | GEV(0.382) | GEV(37.95) | GEV(3) |

3 | Bhola | 600.6 | 174.8 | 0.675 | GUM(0.081) | LP3(0.280) | P3(28.71) | P3(7) |

4 | Bogra | 483.9 | 151.1 | 0.187 | LP3(0.066) | LP3(0.213) | LP3(21.05) | LP3(3) |

5 | Chandpur | 549.8 | 215.3 | 2.102 | GEV(0.088) | GEV(0.136) | LP3(27.51) | GEV(4) |

6 | Chittagong | 897.4 | 238.0 | 0.335 | GP(0.068) | GP(0.249) | GP(28.14) | GP(3) |

7 | Chuadanga | 433.2 | 147.8 | 0.923 | P3(0.07) | LN2(0.18) | GUM(16.18) | GEV(7) |

8 | Comilla | 531.2 | 152.0 | 0.656 | GP(0.082) | P3(0.243) | W2(18.53) | P3(5) |

9 | Cox’s Bazar | 1116.9 | 243.1 | 0.840 | P3(0.108) | P3(0.731) | LN2(67.98) | P3 & LP3(7) |

10 | Dhaka | 511.3 | 147.1 | 0.296 | LN2(0.098) | LP3(0.326) | W2(20.04) | LP3(6) |

11 | Dinajpur | 574.1 | 194.6 | 1.592 | GEV(0.102) | GEV(0.432) | GEV(28.82) | GEV(3) |

12 | Faridpur | 450.3 | 156.2 | 1.106 | GEV(0.083) | GEV(0.233) | P3(27.34) | GEV(7) |

13 | Feni | 791.6 | 182.8 | 0.383 | GUM(0.113) | P3(0.453) | P3(36.07) | P3(6) |

14 | Hatiya | 858.5 | 249.7 | −0.112 | GUM(0.062) | P3(0.226) | N(36.49) | GEV(6) |

15 | Ishurdi | 402.1 | 134.2 | 0.433 | LN2(0.07) | LP3(0.19) | GP(16.19) | P3 & LP3(9) |

16 | Jessore | 460.4 | 170.9 | 1.103 | GEV(0.071) | LP3(0.181) | P3(22.27) | LP3(7) |

17 | Khepupara | 766.9 | 150.7 | 0.158 | GEV(0.075) | GEV(0.263) | GP(19.3) | GEV(5) |

18 | Khulna | 483.2 | 159.2 | 0.778 | P3(0.066) | P3(0.193) | P3(23.71) | P3(3) |

19 | Kutubdia | 930.9 | 293.4 | 1.514 | GEV(0.08) | GEV(0.382) | LP3(48.937) | GEV(4) |

20 | Madaripur | 518.6 | 154.1 | 0.580 | P3(0.07) | LP3(0.18) | W2(18.32) | P3(5) |

21 | Maijdi Court | 807.5 | 202.2 | 1.139 | GEV(0.066) | GUM(0.25) | GEV(31.24) | GEV(4) |

22 | Mongla | 484.9 | 143.7 | 1.929 | LP3(0.08) | LP3(0.219) | GEV(32.543) | LP3 & GEV(5) |

23 | Mymensingh | 558.8 | 164.3 | 0.142 | GP(0.105) | GP(0.36) | GP(25.12) | GP(3) |

24 | Patuakhali | 707.8 | 175.6 | 0.156 | P3(0.082) | P3(0.236) | P3(22.59) | P3(3) |

25 | Rajshahi | 388.0 | 123.6 | 1.107 | LN2(0.084) | P3(0.198) | LP3(18.78) | LP3(7) |

26 | Rangamati | 705.1 | 206.8 | 0.450 | GUM(0.064) | GEV(0.262) | LP3(34.64) | GEV(7) |

27 | Rangpur | 634.0 | 202.6 | 1.483 | GP(0.067) | P3(0.192) | LP3(22.08) | LP3(7) |

28 | Sandwip | 1096.9 | 496.5 | 2.399 | GEV(0.054) | GEV(0.125) | LP3(66.31) | LP3(5) |

29 | Satkhira | 449.6 | 117.6 | 0.453 | W2(0.091) | P3(0.306) | W2(18.2) | P3(7) |

30 | Sitakunda | 890.1 | 240.6 | 0.244 | P3(0.076) | GEV(0.274) | GP(29.48) | GEV(6) |

31 | Srimangal | 596.2 | 176.6 | 0.937 | P3(0.05) | LP3(0.135) | GUM(17.37) | GEV & LP3(7) |

32 | Sydpur | 625.8 | 171.4 | −0.103 | LN2(0.104) | N(0.212) | P3(21.5) | N(5) |

33 | Sylhet | 979.8 | 206.6 | 0.246 | GP(0.06) | GEV(0.215) | GP(18.68) | GEV(6) |

34 | Tangail | 444.8 | 125.1 | 0.930 | P3(0.066) | GEV(0.13) | P3(12.77) | P3(7) |

35 | Teknaf | 1332.4 | 238.6 | 1.022 | GUM(0.063) | LP3(0.161) | GUM(29.63) | GUM(6) |

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**MDPI and ACS Style**

Alam, M.A.; Emura, K.; Farnham, C.; Yuan, J.
Best-Fit Probability Distributions and Return Periods for Maximum Monthly Rainfall in Bangladesh. *Climate* **2018**, *6*, 9.
https://doi.org/10.3390/cli6010009

**AMA Style**

Alam MA, Emura K, Farnham C, Yuan J.
Best-Fit Probability Distributions and Return Periods for Maximum Monthly Rainfall in Bangladesh. *Climate*. 2018; 6(1):9.
https://doi.org/10.3390/cli6010009

**Chicago/Turabian Style**

Alam, Md Ashraful, Kazuo Emura, Craig Farnham, and Jihui Yuan.
2018. "Best-Fit Probability Distributions and Return Periods for Maximum Monthly Rainfall in Bangladesh" *Climate* 6, no. 1: 9.
https://doi.org/10.3390/cli6010009