# Novel Hybrid Data-Intelligence Model for Forecasting Monthly Rainfall with Uncertainty Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Adaptive Neuro-Fuzzy Inference System

_{1}and x

_{2}) in Figure 1a.

#### 2.2. Particle Swarm Optimization (PSO)

_{is}(k) position (${P}_{i}\u03f5P\left(k\right)$) in the hyperspace equals k = 0, which is the first step. This is followed by the second step where each particle’s F function performance is evaluated using the position of the particle (x

_{i}(k)):

_{1}and r

_{2}represent random parameters which lie within 0 and 1; and the parameters (C1 and C2 = 2) [27]. In Equation (6), w represents the inertia weight parameter. A balance between the local and global swarm performance can be achieved by a careful selection of these parameters to reduce the number of iterations. As per [34], the value of w can be determined as follows:

_{min}and w

_{max}, respectively, while the maximum iteration value and iteration number are represented by itr

_{max}and itr, respectively. The particles are transformed to their new locations in the fifth stage using the following equation:

#### 2.3. Genetic Algorithm (GA) Optimization

#### 2.4. Differential Evolution (DE) Optimization

^{μ}

`→`I

^{μ}) involves the production of a mutated vector (μ) using the equation:

_{1}, r

_{2}, r

_{3}$\in $ [1, 2, ..., μ] are randomly designated. F $\in $ [0,2] is a fixed parameter which affects the vector’s differential variation. The capacity of the global search algorithm is usually increased by larger F or population size (μ) quantities. In DE, the crossover operator (CR: I

^{μ}→ I

^{μ}) modifies the vectors $({\overrightarrow{v}}_{i}=[{\overrightarrow{v}}_{1i},{\overrightarrow{v}}_{2i},\dots ,{\overrightarrow{v}}_{di}])$ with a target function $({\overrightarrow{a}}_{i}=[{\overrightarrow{a}}_{1i},{\overrightarrow{a}}_{2i},\dots ,{\overrightarrow{a}}_{di}])$ to generate a trial combination of vectors using the following formula:

#### 2.5. Hybridization of ANFIS Model

**Model 1:**t−1

**Model 2:**t−1, t−2

**Model 3:**t−1, t−3

**Model 4:**t−1, t−6

**Model 5:**t−1, t−12

**Model 6:**t−1, t−2, t−3

**Model 7:**t−1, t−2, t−6

**Model 8:**t−1, t−2, t−12

**Model 9:**t−1, t−3, t−6

**Model 10:**t−1, t−3, t−12

**Model 11:**t−1, t−2, t−3, t−6

**Model 12:**t−1, t−2, t−3, t−12

**Model 13:**t−1, t−2, t−6, t−12

**Model 14:**t−1, t−2, t−3, t−6, t−12

**Model 15:**t−1, t−2, t−3, t−12, t−24

**Model 16:**t−1, t−2, t−3, t−6, t−12, t−24

#### 2.6. Modeling Performance Indicators

#### 2.7. Uncertainty Analysis

## 3. Case Study and Hydrological Data Description

^{2}(Figure 2). The climate of the area is dominated by the northeast monsoon rainfall influence. The average annual rainfall ranges between 1609 and 2132 mm in the basin. In general, the extreme rainfall events occur between November and March. In this study, the monthly rainfall data for the period 2000–2014 were used for the development of hybrid models. The rainfall time series was obtained from the Department of Irrigation and Drainage (DID). It is worth mentioning that the modeling carried out in this research was based on the univariate concept that included only the rainfall information to forecast the rainfall itself. This is highly significant, especially in a watershed that comprises several metrological information sources. The highly stochastic behavior of rainfall patterns often produces floods, and thus the main motivation for this research was to develop an accurate model for rainfall forecasting, which could be used for anticipating the possible occurrence of floods. For this purpose, ANFIS combinations with a couple of evolutionary algorithms were developed.

## 4. Application and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Adaptive neuro-fuzzy inference system model architecture for 2 inputs and 1 output, (

**b**) General flow chart of a hybrid ANFIS model for different scenarios.

**Figure 3.**Time series and scatterplot of observed and forecasted rainfall by three hybrid models, namely, ANFIS-particle swarm optimization (PSO), ANFIS-genetic algorithm (GA), and ANFIS-differential evolution (DE), and the classical ANFIS model for different input combinations over the testing phase of the models.

**Figure 4.**Taylor diagrams for the graphical presentation of the performance of three hybrid models, namely, ANFIS-PSO, ANFIS-GA, and ANFIS-DE, and the classical ANFIS model for different input combinations during the testing phase of the models.

**Figure 5.**Comparative plots of observed and forecasted rainfall patterns obtained using ANFIS-based, ANFIS-PSO, ANFIS-GA, and ANFIS-DE models.

**Table 1.**Statistical performance of the predictive models in forecasting monthly rainfall during the training and testing phases.

(a) ANFIS-based model | ||||||||

Models | RMSE (mm) | MAE (mm) | CC | WI | RMSE (mm) | MAE (mm) | CC | WI |

Training phase | Testing phase | |||||||

Model 1 | 4.23 | 3.16 | 0.492 | 0.564 | 5.32 | 3.69 | 0.439 | 0.478 |

Model 2 | 4.02 | 2.85 | 0.575 | 0.698 | 4.04 | 2.73 | 0.749 | 0.788 |

Model 3 | 2.96 | 2.22 | 0.799 | 0.882 | 3.00 | 2.31 | 0.864 | 0.913 |

Model 4 | 3.41 | 2.43 | 0.723 | 0.810 | 4.69 | 2.91 | 0.627 | 0.714 |

Model 5 | 2.87 | 1.95 | 0.820 | 0.888 | 4.54 | 2.82 | 0.664 | 0.759 |

Model 6 | 3.08 | 2.30 | 0.780 | 0.865 | 2.63 | 2.00 | 0.897 | 0.938 |

Model 7 | 3.40 | 2.26 | 0.724 | 0.806 | 3.17 | 2.28 | 0.848 | 0.911 |

Model 8 | 3.05 | 2.07 | 0.793 | 0.871 | 3.03 | 2.13 | 0.866 | 0.922 |

Model 9 | 3.20 | 2.13 | 0.760 | 0.850 | 4.16 | 2.14 | 0.723 | 0.801 |

Model 10 | 3.06 | 2.20 | 0.796 | 0.880 | 3.72 | 2.42 | 0.807 | 0.850 |

Model 11 | 2.18 | 1.50 | 0.896 | 0.941 | 2.20 | 1.49 | 0.930 | 0.962 |

Model 12 | 2.73 | 1.83 | 0.842 | 0.911 | 2.46 | 1.87 | 0.924 | 0.947 |

Model 13 | 2.78 | 1.94 | 0.832 | 0.903 | 2.51 | 1.66 | 0.915 | 0.946 |

Model 14 | 1.92 | 1.20 | 0.923 | 0.960 | 1.73 | 1.12 | 0.960 | 0.978 |

Model 15 | 2.10 | 1.36 | 0.897 | 0.943 | 1.13 | 0.75 | 0.984 | 0.991 |

Model 16 | 1.40 | 0.91 | 0.956 | 0.976 | 0.99 | 0.65 | 0.987 | 0.994 |

(b) ANFIS-PSO | ||||||||

Models | RMSE (mm) | MAE (mm) | CC | WI | RMSE (mm) | MAE (mm) | CC | WI |

Training phase | Testing phase | |||||||

Model 1 | 4.24 | 3.17 | 0.488 | 0.561 | 5.31 | 3.67 | 0.444 | 0.483 |

Model 2 | 3.63 | 2.66 | 0.671 | 0.782 | 4.31 | 2.92 | 0.692 | 0.756 |

Model 3 | 2.66 | 1.93 | 0.841 | 0.908 | 2.63 | 2.02 | 0.895 | 0.939 |

Model 4 | 3.25 | 2.31 | 0.751 | 0.840 | 4.46 | 2.69 | 0.671 | 0.755 |

Model 5 | 2.76 | 1.95 | 0.835 | 0.898 | 4.40 | 2.74 | 0.689 | 0.786 |

Model 6 | 2.84 | 1.96 | 0.816 | 0.888 | 2.21 | 1.53 | 0.927 | 0.960 |

Model 7 | 3.32 | 2.33 | 0.740 | 0.835 | 2.67 | 1.92 | 0.898 | 0.937 |

Model 8 | 2.68 | 1.75 | 0.846 | 0.907 | 2.75 | 1.81 | 0.892 | 0.939 |

Model 9 | 3.11 | 2.13 | 0.775 | 0.861 | 4.02 | 2.05 | 0.745 | 0.819 |

Model 10 | 2.53 | 1.67 | 0.864 | 0.924 | 3.07 | 2.18 | 0.871 | 0.912 |

Model 11 | 2.04 | 1.33 | 0.910 | 0.950 | 1.81 | 1.20 | 0.953 | 0.975 |

Model 12 | 1.77 | 1.20 | 0.936 | 0.965 | 1.58 | 1.11 | 0.967 | 0.981 |

Model 13 | 2.30 | 1.53 | 0.891 | 0.940 | 2.24 | 1.36 | 0.931 | 0.961 |

Model 14 | 1.16 | 0.75 | 0.973 | 0.985 | 1.14 | 0.62 | 0.982 | 0.991 |

Model 15 | 1.44 | 0.83 | 0.953 | 0.975 | 0.73 | 0.44 | 0.993 | 0.996 |

Model 16 | 0.86 | 0.51 | 0.984 | 0.991 | 0.47 | 0.28 | 0.997 | 0.998 |

(c) ANFIS-GA | ||||||||

Models | RMSE (mm) | MAE (mm) | CC | WI | RMSE (mm) | MAE (mm) | CC | WI |

Training phase | Testing phase | |||||||

Model 1 | 4.24 | 3.17 | 0.488 | 0.561 | 5.31 | 3.67 | 0.442 | 0.481 |

Model 2 | 3.59 | 2.62 | 0.679 | 0.791 | 4.28 | 2.85 | 0.697 | 0.758 |

Model 3 | 2.91 | 2.10 | 0.805 | 0.881 | 2.84 | 2.28 | 0.876 | 0.928 |

Model 4 | 3.43 | 2.37 | 0.720 | 0.822 | 4.12 | 2.47 | 0.738 | 0.794 |

Model 5 | 2.60 | 1.78 | 0.856 | 0.911 | 3.55 | 2.28 | 0.812 | 0.890 |

Model 6 | 2.77 | 2.02 | 0.826 | 0.895 | 2.40 | 1.72 | 0.915 | 0.951 |

Model 7 | 3.42 | 2.15 | 0.718 | 0.812 | 2.98 | 2.02 | 0.868 | 0.922 |

Model 8 | 2.90 | 2.01 | 0.816 | 0.884 | 3.36 | 2.28 | 0.833 | 0.900 |

Model 9 | 3.11 | 2.07 | 0.774 | 0.860 | 4.12 | 2.12 | 0.728 | 0.806 |

Model 10 | 2.50 | 1.63 | 0.872 | 0.930 | 3.26 | 2.27 | 0.851 | 0.898 |

Model 11 | 2.37 | 1.60 | 0.876 | 0.930 | 2.40 | 1.53 | 0.917 | 0.954 |

Model 12 | 2.33 | 1.59 | 0.885 | 0.935 | 1.83 | 1.35 | 0.954 | 0.975 |

Model 13 | 2.40 | 1.57 | 0.880 | 0.932 | 2.06 | 1.30 | 0.942 | 0.967 |

Model 14 | 1.50 | 0.93 | 0.956 | 0.976 | 1.38 | 0.78 | 0.974 | 0.986 |

Model 15 | 1.55 | 0.92 | 0.945 | 0.970 | 0.92 | 0.56 | 0.989 | 0.994 |

Model 16 | 1.21 | 0.69 | 0.967 | 0.982 | 0.83 | 0.51 | 0.991 | 0.995 |

(d) ANFIS-DE | ||||||||

Models | RMSE (mm) | MAE (mm) | CC | WI | RMSE (mm) | MAE (mm) | CC | WI |

Training phase | Testing phase | |||||||

Model 1 | 4.17 | 3.20 | 0.513 | 0.600 | 5.29 | 3.60 | 0.449 | 0.486 |

Model 2 | 3.66 | 2.61 | 0.666 | 0.781 | 4.07 | 2.61 | 0.736 | 0.791 |

Model 3 | 2.71 | 2.00 | 0.834 | 0.900 | 2.75 | 2.17 | 0.885 | 0.935 |

Model 4 | 3.42 | 2.41 | 0.720 | 0.819 | 4.05 | 2.61 | 0.741 | 0.813 |

Model 5 | 2.62 | 1.86 | 0.853 | 0.913 | 2.73 | 2.03 | 0.894 | 0.940 |

Model 6 | 2.74 | 1.89 | 0.830 | 0.895 | 2.35 | 1.66 | 0.917 | 0.955 |

Model 7 | 3.32 | 2.09 | 0.740 | 0.824 | 3.04 | 2.07 | 0.862 | 0.919 |

Model 8 | 2.85 | 1.91 | 0.824 | 0.889 | 2.88 | 1.96 | 0.881 | 0.934 |

Model 9 | 3.17 | 2.18 | 0.763 | 0.850 | 4.20 | 2.16 | 0.715 | 0.797 |

Model 10 | 2.51 | 1.70 | 0.867 | 0.930 | 3.12 | 2.23 | 0.865 | 0.908 |

Model 11 | 2.22 | 1.42 | 0.893 | 0.940 | 2.12 | 1.40 | 0.936 | 0.965 |

Model 12 | 2.43 | 1.68 | 0.875 | 0.930 | 2.09 | 1.65 | 0.941 | 0.966 |

Model 13 | 2.50 | 1.68 | 0.866 | 0.923 | 2.32 | 1.51 | 0.925 | 0.956 |

Model 14 | 1.36 | 0.92 | 0.963 | 0.980 | 1.37 | 0.76 | 0.974 | 0.987 |

Model 15 | 1.58 | 0.99 | 0.943 | 0.970 | 0.70 | 0.46 | 0.994 | 0.997 |

Model 16 | 1.14 | 0.67 | 0.971 | 0.985 | 0.73 | 0.38 | 0.993 | 0.996 |

Indicators | Models | ANFIS-PSO | ANFIS-GA | ANFIS-DE | ANFIS |
---|---|---|---|---|---|

d-factor | Model 1 | 0.14 | 0.14 | 0.14 | 0.15 |

Model 2 | 0.68 | 0.69 | 0.69 | 0.72 | |

Model 3 | 0.68 | 0.67 | 0.67 | 0.65 | |

Model 4 | 0.69 | 0.69 | 0.68 | 0.67 | |

Model 5 | 0.67 | 0.65 | 0.65 | 0.61 | |

Model 6 | 1.17 | 1.18 | 1.18 | 1.19 | |

Model 7 | 0.95 | 0.96 | 0.96 | 0.98 | |

Model 8 | 1.02 | 1.02 | 1.02 | 1.03 | |

Model 9 | 0.99 | 0.99 | 0.99 | 0.98 | |

Model 10 | 0.88 | 0.88 | 0.88 | 0.88 | |

Model 11 | 1.28 | 1.28 | 1.29 | 1.30 | |

Model 12 | 1.30 | 1.31 | 1.31 | 1.32 | |

Model 13 | 1.29 | 1.29 | 1.30 | 1.31 | |

Model 14 | 1.43 | 1.43 | 1.44 | 1.44 | |

Model 15 | 1.38 | 1.39 | 1.40 | 1.42 | |

Model 16 | 1.41 | 1.41 | 1.42 | 1.43 | |

95PPU | Model 1 | 12.29 | 12.29 | 12.29 | 12.29 |

Model 2 | 60.67 | 60.67 | 60.11 | 60.67 | |

Model 3 | 57.63 | 56.50 | 55.37 | 50.28 | |

Model 4 | 58.62 | 58.05 | 57.47 | 55.75 | |

Model 5 | 65.48 | 64.88 | 64.29 | 61.31 | |

Model 6 | 76.84 | 76.84 | 76.84 | 75.71 | |

Model 7 | 66.67 | 67.24 | 64.94 | 65.52 | |

Model 8 | 76.19 | 75.60 | 74.40 | 72.62 | |

Model 9 | 70.11 | 69.54 | 68.39 | 65.52 | |

Model 10 | 73.81 | 73.21 | 72.02 | 70.83 | |

Model 11 | 79.31 | 78.74 | 78.16 | 77.59 | |

Model 12 | 85.71 | 85.71 | 85.12 | 85.12 | |

Model 13 | 82.74 | 82.14 | 81.55 | 81.55 | |

Model 14 | 85.12 | 84.52 | 84.52 | 83.33 | |

Model 15 | 89.10 | 89.10 | 88.46 | 85.26 | |

Model 16 | 91.67 | 91.03 | 89.74 | 88.46 |

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## Share and Cite

**MDPI and ACS Style**

Yaseen, Z.M.; Ebtehaj, I.; Kim, S.; Sanikhani, H.; Asadi, H.; Ghareb, M.I.; Bonakdari, H.; Wan Mohtar, W.H.M.; Al-Ansari, N.; Shahid, S.
Novel Hybrid Data-Intelligence Model for Forecasting Monthly Rainfall with Uncertainty Analysis. *Water* **2019**, *11*, 502.
https://doi.org/10.3390/w11030502

**AMA Style**

Yaseen ZM, Ebtehaj I, Kim S, Sanikhani H, Asadi H, Ghareb MI, Bonakdari H, Wan Mohtar WHM, Al-Ansari N, Shahid S.
Novel Hybrid Data-Intelligence Model for Forecasting Monthly Rainfall with Uncertainty Analysis. *Water*. 2019; 11(3):502.
https://doi.org/10.3390/w11030502

**Chicago/Turabian Style**

Yaseen, Zaher Mundher, Isa Ebtehaj, Sungwon Kim, Hadi Sanikhani, H. Asadi, Mazen Ismaeel Ghareb, Hossein Bonakdari, Wan Hanna Melini Wan Mohtar, Nadhir Al-Ansari, and Shamsuddin Shahid.
2019. "Novel Hybrid Data-Intelligence Model for Forecasting Monthly Rainfall with Uncertainty Analysis" *Water* 11, no. 3: 502.
https://doi.org/10.3390/w11030502