# A Comparison of BPNN, GMDH, and ARIMA for Monthly Rainfall Forecasting Based on Wavelet Packet Decomposition

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Region

^{2}. The mainstream Luo River is 446.9 km long, and the tributary Yi River is 264.8 km long. Luoning and Zuoyu Stations are located in the middle stream of Luo River and the middle and upper stream of Yi River, respectively. The average annual rainfall of the two stations are 635.2 mm and 834.3 mm, respectively. The inter-seasonal fluctuations of rainfall in two stations are very strong. For Luoning station, the average annual rainfall in December and January are 7.9 mm and 7.5 mm, respectively, and the average annual rainfall in July and August are 115.6 mm and 96.6 mm, respectively, indicating high difficulty of modeling. The location of the study area is shown in Figure 1.

#### 2.2. Data Sets and Pre-Processing

#### 2.3. Methods

#### 2.3.1. ARIMA Model

#### 2.3.2. BPNN Model

_{j}is input neuron and j ∈ (0, m), m is the number of input neurons, w

_{ij}is weight of the ith neuron in the input layer corresponding to the $j\mathrm{th}$ neuron in the hidden layer, ${\beta}_{j}$ is bias-related weight of hidden neurons, y

_{i}is input of the hidden layer node ($i=0,1,\dots ,n$), and n is the number of neurons in the hidden layer. Tan-sigmoid is the transfer function between the layer output and the hidden layer, and its form is as follows:

_{k}and $O$ represent input and output values of the output layer, respectively.

#### 2.3.3. GMDH Model

#### 2.3.4. Wavelet Packet Decomposition (WPD)

_{3}and DDD

_{3}represent the lowest frequency and highest frequency, respectively.

#### 2.4. Evaluation Indices

#### 2.5. Hybrid Forecasting Models

## 3. Results

#### 3.1. Decomposition Results Using WPD and Input Variables Determination

^{th}variable before the target output variable.

#### 3.2. Model Development

#### 3.3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

WPD | wavelet packet decomposition |

BPNN | back-propagation neural network |

GMDH | group method of data handing |

ARIMA | autoregressive integrated moving average |

ANN | artificial neural network |

GP | genetic programming |

SVM | support vector machines |

ANFIS | adaptive neuro-fuzzy inference system |

AR | auto-regressive |

MA | moving average |

ARMA | autoregressive moving average |

LM | Levenberg–Marquardt |

EMD | empirical mode decomposition |

EEMD | ensemble empirical mode decomposition |

FT | Fourier transform |

SVR | support vector regression |

QPSO | quantum-behaved particle swarm optimization |

VMD | variational mode decomposition |

LSWA | least-squares wavelet analysis |

WD | wavelet decomposition |

DWT | discrete wavelet transform |

WR | wavelet representation |

LPF | low-pass filter |

HPF | high-pass filter |

RMSE | root mean-squared error |

MAE | mean absolute error |

R | coefficient of correlation |

NSEC | Nash–Sutcliffe efficiency coefficient |

ACF | autocorrelation function |

PACF | partial autocorrelation function |

ADF | augmented Dickey–Fuller |

BIC | Bayes information criteria |

SCS-CN | soil conservation service-curve number |

## References

- Adnan, R.M.; Liang, Z.; Heddam, S.; Zounemat-Kermani, M.; Kisi, O.; Li, B. Least square support vector machine and multivariate adaptive regression splines for streamflow prediction in mountainous basin using hydro-meteorological data as inputs. J. Hydrol.
**2020**, 586, 124371. [Google Scholar] [CrossRef] - Ghaderpour, E.; Vujadinovic, T.; Hassan, Q.K. Application of the Least-Squares Wavelet software in hydrology: Athabasca River Basin. J. Hydrol. Reg. Stud.
**2021**, 36, 100847. [Google Scholar] [CrossRef] - Kisi, O.; Cimen, M. A wavelet-support vector machine conjunction model for monthly streamflow forecasting. J. Hydrol.
**2011**, 399, 132–140. [Google Scholar] [CrossRef] - Niu, W.-J.; Feng, Z.-K.; Chen, Y.-B.; Zhang, H.; Cheng, C.-T. Annual Streamflow Time Series Prediction Using Extreme Learning Machine Based on Gravitational Search Algorithm and Variational Mode Decomposition. J. Hydrol. Eng.
**2020**, 25, 04020008. [Google Scholar] [CrossRef] - Ali, M.; Prasad, R.; Xiang, Y.; Yaseen, Z.M. Complete ensemble empirical mode decomposition hybridized with random forest and kernel ridge regression model for monthly rainfall forecasts. J. Hydrol.
**2020**, 584, 124647. [Google Scholar] [CrossRef] - Yang, T.; Asanjan, A.A.; Welles, E.; Gao, X.; Sorooshian, S.; Liu, X. Developing reservoir monthly inflow forecasts using artificial intelligence and climate phenomenon information. Water Resour. Res.
**2017**, 53, 2786–2812. [Google Scholar] [CrossRef] - Wang, W.-C.; Xu, D.-M.; Chau, K.-W.; Chen, S. Improved annual rainfall-runoff forecasting using PSO-SVM model based on EEMD. J. Hydroinform.
**2013**, 15, 1377–1390. [Google Scholar] [CrossRef] - Wang, W.-C.; Cheng, C.-T.; Chau, K.-W.; Xu, D.-M. Calibration of Xinanjiang model parameters using hybrid genetic algorithm based fuzzy optimal model. J. Hydroinform.
**2012**, 14, 784–799. [Google Scholar] [CrossRef][Green Version] - Abbaszadeh, P.; Alipour, A. Development of a coupled wavelet transform and evolutionary Levenberg-Marquardt neural networks for hydrological process modeling. Comput. Intell.
**2018**, 34, 175–199. [Google Scholar] [CrossRef] - Wang, W.-C.; Chau, K.-W.; Cheng, C.-T.; Qiu, L. A comparison of performance of several artificial intelligence methods for forecasting monthly discharge time series. J. Hydrol.
**2009**, 374, 294–306. [Google Scholar] [CrossRef][Green Version] - Aksoy, H.; Dahamsheh, A. Artificial neural network models for forecasting monthly precipitation in Jordan. Stoch. Environ. Res. Risk Assess.
**2009**, 23, 917–931. [Google Scholar] [CrossRef] - Chadalawada, J.; Herath, H.; Babovic, V. Hydrologically Informed Machine Learning for Rainfall-Runoff Modeling: A Genetic Programming-Based Toolkit for Automatic Model Induction. Water Resour. Res.
**2020**, 56, e2019WR026933. [Google Scholar] [CrossRef] - Sain, S.R. The Nature of Statistical Learning Theory. Technometrics
**1996**, 38, 409. [Google Scholar] [CrossRef] - Jang, J.R. ANFIS: Adaptive-network-based fuzzy inference system. IEEE Trans. Syst. Man Cybern.
**1993**, 23, 665–685. [Google Scholar] [CrossRef] - Box, G.; Jenkins, G. Time Series Analysis-Forecast and Control; Prentice-Hall: Englewood Cliffs, NJ, USA, 1976. [Google Scholar]
- Lai, Y.; Dzombak, D.A. Use of the Autoregressive Integrated Moving Average (ARIMA) Model to Forecast Near-Term Regional Temperature and Precipitation. Weather Forecast.
**2020**, 35, 959–976. [Google Scholar] [CrossRef] - Mishra, A.K.; Desai, V.R. Drought forecasting using stochastic models. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 326–339. [Google Scholar] [CrossRef] - Sanikhani, H.; Kisi, O.; Maroufpoor, E.; Yaseen, Z.M. Temperature-based modeling of reference evapotranspiration using several artificial intelligence models: Application of different modeling scenarios. Theor. Appl. Climatol.
**2019**, 135, 449–462. [Google Scholar] [CrossRef] - Rahman, M.A.; Lou, Y.S.; Sultana, N. Analysis and prediction of rainfall trends over Bangladesh using Mann-Kendall, Spearman′s rho tests and ARIMA model. Meteorol. Atmos. Phys.
**2017**, 129, 409–424. [Google Scholar] [CrossRef] - Mishra, S.; Saravanan, C.; Dwivedi, V.K.; Shukla, J.P. Rainfall-Runoff Modeling using Clustering and Regression Analysis for the River Brahmaputra Basin. J. Geol. Soc. India
**2018**, 92, 305–312. [Google Scholar] [CrossRef] - Rizeei, H.M.; Pradhan, B.; Saharkhiz, M.A. Surface runoff prediction regarding LULC and climate dynamics using coupled LTM, optimized ARIMA, and GIS-based SCS-CN models in tropical region. Arab. J. Geosci.
**2018**, 11, 53. [Google Scholar] [CrossRef] - Wang, Z.-Y.; Qiu, J.; Li, F.-F. Hybrid Models Combining EMD/EEMD and ARIMA for Long-Term Streamflow Forecasting. Water
**2018**, 10, 853. [Google Scholar] [CrossRef][Green Version] - Tan, Q.-F.; Lei, X.-H.; Wang, X.; Wang, H.; Wen, X.; Ji, Y.; Kang, A.-Q. An adaptive middle and long-term runoff forecast model using EEMD-ANN hybrid approach. J. Hydrol.
**2018**, 567, 767–780. [Google Scholar] [CrossRef] - Kashani, M.H.; Ghorbani, M.A.; Shahabi, M.; Naganna, S.R.; Diop, L. Multiple AI model integration strategy—Application to saturated hydraulic conductivity prediction from easily available soil properties. Soil Tillage Res.
**2020**, 196, 104449. [Google Scholar] [CrossRef] - Pradhan, P.; Tingsanchali, T.; Shrestha, S. Evaluation of Soil and Water Assessment Tool and Artificial Neural Network models for hydrologic simulation in different climatic regions of Asia. Sci. Total Environ.
**2020**, 701, 134308. [Google Scholar] [CrossRef] - Dubey, S.K.; Sharma, D.; Babel, M.S.; Mundetia, N. Application of hydrological model for assessment of water security using multi-model ensemble of CORDEX-South Asia experiments in a semi-arid river basin of India. Ecol. Eng.
**2020**, 143, 105641. [Google Scholar] [CrossRef] - Gokbulak, F.; Sengonul, K.; Serengil, Y.; Yurtseven, I.; Ozhan, S.; Cigizoglu, H.K.; Uygur, B. Comparison of Rainfall-Runoff Relationship Modeling using Different Methods in a Forested Watershed. Water Resour. Manag.
**2015**, 29, 4229–4239. [Google Scholar] [CrossRef] - Nourani, V. An Emotional ANN (EANN) approach to modeling rainfall-runoff process. J. Hydrol.
**2017**, 544, 267–277. [Google Scholar] [CrossRef] - Malekzadeh, M.; Kardar, S.; Saeb, K.; Shabanlou, S.; Taghavi, L. A Novel Approach for Prediction of Monthly Ground Water Level Using a Hybrid Wavelet and Non-Tuned Self-Adaptive Machine Learning Model. Water Resour. Manag.
**2019**, 33, 1609–1628. [Google Scholar] [CrossRef] - Mukherjee, A.; Ramachandran, P. Prediction of GWL with the help of GRACE TWS for unevenly spaced time series data in India: Analysis of comparative performances of SVR, ANN and LRM. J. Hydrol.
**2018**, 558, 647–658. [Google Scholar] [CrossRef] - Choong, C.E.; Ibrahim, S.; El-Shafie, A. Artificial Neural Network (ANN) model development for predicting just suspension speed in solid-liquid mixing system. Flow Meas. Instrum.
**2020**, 71, 101689. [Google Scholar] [CrossRef] - Mokhtarzad, M.; Eskandari, F.; Jamshidi Vanjani, N.; Arabasadi, A. Drought forecasting by ANN, ANFIS, and SVM and comparison of the models. Environ. Earth Sci.
**2017**, 76, 729. [Google Scholar] [CrossRef] - Vidyarthi, V.K.; Jain, A. Knowledge extraction from trained ANN drought classification model. J. Hydrol.
**2020**, 585, 124804. [Google Scholar] [CrossRef] - Liu, Y.; Zhao, Q.; Yao, W.; Ma, X.; Yao, Y.; Liu, L. Short-term rainfall forecast model based on the improved BP–NN algorithm. Sci. Rep.
**2019**, 9, 19751. [Google Scholar] [CrossRef] [PubMed] - Danladi, A.; Stephen, M.; Aliyu, B.M.; Gaya, G.K.; Silikwa, N.W.; Machael, Y. Assessing the influence of weather parameters on rainfall to forecast river discharge based on short-term. Alex. Eng. J.
**2018**, 57, 1157–1162. [Google Scholar] [CrossRef] - Cui, Y.K.; Long, D.; Hong, Y.; Zeng, C.; Zhou, J.; Han, Z.Y.; Liu, R.H.; Wan, W. Validation and reconstruction of FY-3B/MWRI soil moisture using an artificial neural network based on reconstructed MODIS optical products over the Tibetan Plateau. J. Hydrol.
**2016**, 543, 242–254. [Google Scholar] [CrossRef] - Yuan, Q.; Xu, H.; Li, T.; Shen, H.; Zhang, L. Estimating surface soil moisture from satellite observations using a generalized regression neural network trained on sparse ground-based measurements in the continental U.S. J. Hydrol.
**2020**, 580, 124351. [Google Scholar] [CrossRef] - Amanifard, N.; Nariman-Zadeh, N.; Farahani, M.H.; Khalkhali, A. Modelling of multiple short-length-scale stall cells in an axial compressor using evolved GMDH neural networks. Energy Convers. Manag.
**2008**, 49, 2588–2594. [Google Scholar] [CrossRef] - Li, Y.; Shi, H.; Liu, H. A hybrid model for river water level forecasting: Cases of Xiangjiang River and Yuanjiang River, China. J. Hydrol.
**2020**, 587, 124934. [Google Scholar] [CrossRef] - Adnan, R.M.; Liang, Z.; Parmar, K.S.; Soni, K.; Kisi, O. Modeling monthly streamflow in mountainous basin by MARS, GMDH-NN and DENFIS using hydroclimatic data. Neural Comput. Appl.
**2021**, 33, 2853–2871. [Google Scholar] [CrossRef] - Partal, T.; Kişi, Ö. Wavelet and neuro-fuzzy conjunction model for precipitation forecasting. J. Hydrol.
**2007**, 342, 199–212. [Google Scholar] [CrossRef] - Wang, W.-C.; Chau, K.-W.; Qiu, L.; Chen, Y.-B. Improving forecasting accuracy of medium and long-term runoff using artificial neural network based on EEMD decomposition. Environ. Res.
**2015**, 139, 46–54. [Google Scholar] [CrossRef] [PubMed] - Yu, X.; Zhang, X.; Qin, H. A data-driven model based on Fourier transform and support vector regression for monthly reservoir inflow forecasting. J. Hydro-Environ. Res.
**2018**, 18, 12–24. [Google Scholar] [CrossRef] - Feng, Z.-K.; Niu, W.-J.; Tang, Z.-Y.; Jiang, Z.-Q.; Xu, Y.; Liu, Y.; Zhang, H.-R. Monthly runoff time series prediction by variational mode decomposition and support vector machine based on quantum-behaved particle swarm optimization. J. Hydrol.
**2020**, 583, 124627. [Google Scholar] [CrossRef] - Deering, R.; Kaiser, J.F. The use of a masking signal to improve empirical mode decomposition. In Proceedings of the (ICASSP’05): IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, PA, USA, 23 March 2005; Volume 484, pp. iv/485–iv/488. [Google Scholar]
- Chong, K.L.; Lai, S.H.; El-Shafie, A. Wavelet Transform Based Method for River Stream Flow Time Series Frequency Analysis and Assessment in Tropical Environment. Water Resour. Manag.
**2019**, 33, 2015–2032. [Google Scholar] [CrossRef] - Bayazit, M. Nonstationarity of Hydrological Records and Recent Trends in Trend Analysis: A State-of-the-art Review. Environ. Process.
**2015**, 2, 527–542. [Google Scholar] [CrossRef] - Coifman, R.; Meyer, Y.; Wickerhauser, V.M. Wavelet analysis and signal processing. In Wavelets and Their Applications; Jones Bartlett: Burlington, MA, USA, 1992; pp. 153–178. [Google Scholar]
- Cohen, A.; Daubechies, I.; Feauveau, J.C. Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math.
**1992**, 45, 485–560. [Google Scholar] [CrossRef] - Zhao, L.; Jia, Y. Transcale control for a class of discrete stochastic systems based on wavelet packet decomposition. Inf. Sci.
**2015**, 296, 25–41. [Google Scholar] [CrossRef] - Unnikrishnan, P.; Jothiprakash, V. Hybrid SSA-ARIMA-ANN Model for Forecasting Daily Rainfall. Water Resour. Manag.
**2020**, 34, 3609–3623. [Google Scholar] [CrossRef] - Lu, Y.; AbouRizk, S.M. Automated Box–Jenkins forecasting modelling. Autom. Constr.
**2009**, 18, 547–558. [Google Scholar] [CrossRef] - Fausett, L. Fundamentals of Neural Networks: Architectures, Algorithms, and Applications; Prentice-Hall Inc.: Upper Saddle River, NJ, USA, 1994. [Google Scholar]
- Yang, Z.P.; Lu, W.X.; Long, Y.Q.; Li, P. Application and comparison of two prediction models for groundwater levels: A case study in Western Jilin Province, China. J. Arid Environ.
**2009**, 73, 487–492. [Google Scholar] [CrossRef] - Ivakhnenko, A. Polynomial theory of complex systems. IEEE Trans. Syst. Man Cybern.
**1971**, 1, 364–378. [Google Scholar] [CrossRef][Green Version] - Qaderi, K.; Bakhtiari, B.; Madadi, M.R.; Afzali-Gorouh, Z. Evaluating GMDH-based models to predict daily dew point temperature (case study of Kerman province). Meteorol. Atmos. Phys.
**2020**, 132, 667–682. [Google Scholar] [CrossRef] - Volterra, V. Theory of Functionals and of Integrals and Integro-Differential Equations; Dover Publications: New York, NY, USA, 2005. [Google Scholar]
- Mallat, S.; Mallat, S.G. A Theory of Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Trans. Pattern Anal. Mach. Intell.
**1989**, 11, 674–693. [Google Scholar] [CrossRef][Green Version] - Gentilucci, M.; Materazzi, M.; Pambianchi, G.; Burt, P.; Guerriero, G. Assessment of Variations in the Temperature-Rainfall Trend in the Province of Macerata (Central Italy), Comparing the Last Three Climatological Standard Normals (1961–1990; 1971–2000; 1981–2010) for Biosustainability Studies. Environ. Process.
**2019**, 6, 391–412. [Google Scholar] [CrossRef] - Wang, W.; Lu, Y. Analysis of the Mean Absolute Error (MAE) and the Root Mean Square Error (RMSE) in Assessing Rounding Model. IOP Conf. Ser. Mater. Sci. Eng.
**2018**, 324, 012049. [Google Scholar] [CrossRef] - Kim, H.I.; Keum, H.J.; Han, K.Y. Real-Time Urban Inundation Prediction Combining Hydraulic and Probabilistic Methods. Water
**2019**, 11, 293. [Google Scholar] [CrossRef][Green Version] - Feng, Q.; Wen, X.; Li, J. Wavelet Analysis-Support Vector Machine Coupled Models for Monthly Rainfall Forecasting in Arid Regions. Water Resour. Manag.
**2015**, 29, 1049–1065. [Google Scholar] [CrossRef] - Lin, J.-Y.; Cheng, C.-T.; Chau, K.-W. Using support vector machines for long-term discharge prediction. Hydrol. Sci. J.
**2006**, 51, 599–612. [Google Scholar] [CrossRef] - Yin, Y.; Bai, Y.; Ge, F.; Yu, H.; Liu, Y. Long-term robust identification potential of a wavelet packet decomposition based recursive drift correction of E-nose data for Chinese spirits. Measurement
**2019**, 139, 284–292. [Google Scholar] [CrossRef]

**Figure 12.**Variation of statistical indicators with the number of hidden layer nodes for Luoning station.

**Figure 13.**Variation of statistical indicators with the number of hidden layer nodes for Zuoyu station.

**Figure 14.**(

**a**) Forecasting results of Luoning station in the training period; (

**b**) forecasting errors of Luoning station in the training period.

**Figure 15.**(

**a**) Forecasting results of Luoning station in the testing period; (

**b**) forecasting errors of Luoning station in the testing period.

**Figure 16.**(

**a**) Forecasting results of Zuoyu station in training period; (

**b**) forecasting errors of Zuoyu station in the training period.

**Figure 17.**(

**a**) Forecasting results of Zuoyu station in testing period; (

**b**) forecasting errors of Luoning station in the testing period.

Station | Max (mm) | Min (mm) | Mean (mm) | Std (mm) | |
---|---|---|---|---|---|

Luoning | All | 313.8 | 0 | 47.18 | 50.93 |

Training | 313.8 | 0 | 47.05 | 50.86 | |

Testing | 261.7 | 0.4 | 48.66 | 52.39 | |

Zuoyu | All | 430.2 | 0 | 69.53 | 74.39 |

Training | 430.2 | 0 | 69.92 | 75.33 | |

Testing | 316.2 | 0 | 65.05 | 63.44 |

**Table 2.**Number of input variables for different data series from Luoning and Zuoyu stations based on ACF and PACF analysis.

No. | Series | Input Variables | |
---|---|---|---|

Luoning Station | Zuoyu Station | ||

1 | Original | x_{t−1}~x_{t−12} | x_{t−1}~x_{t−13} |

2 | WPD_{1} | x_{t−1}~x_{t−12} | x_{t−1}~x_{t−12} |

3 | WPD_{2} | x_{t−1}~x_{t−12} | x_{t−1}~x_{t−12} |

4 | WPD_{3} | x_{t−1}~x_{t−12} | x_{t−1}~x_{t−13} |

5 | WPD_{4} | x_{t−1}~x_{t−11} | x_{t−1}~x_{t−11} |

6 | WPD_{5} | x_{t−1}~x_{t−12} | x_{t−1}~x_{t−12} |

7 | WPD_{6} | x_{t−1}~x_{t−12} | x_{t−1}~x_{t−12} |

8 | WPD_{7} | x_{t−1}~x_{t−13} | x_{t−1}~x_{t−13} |

9 | WPD_{8} | x_{t−1}~x_{t−11} | x_{t−1}~x_{t−13} |

Name | Sample Data Set | t-Statistic Value | Critical Value |
---|---|---|---|

Luoning | Original | −5.85207 | −3.42041 |

WPD_{1} | −6.70412 | −3.42041 | |

WPD_{2} | −14.33 | −3.42041 | |

WPD_{3} | −12.6215 | −3.42041 | |

WPD_{4} | −16.2217 | −3.42041 | |

WPD_{5} | −9.58776 | −3.42041 | |

WPD_{6} | −18.1806 | −3.42041 | |

WPD_{7} | −14.919 | −3.42041 | |

WPD_{8} | −24.0394 | −3.42041 | |

Zuoyu | Original | −4.86539 | −3.42041 |

WPD_{1} | −6.63558 | −3.42041 | |

WPD_{2} | −14.6596 | −3.42041 | |

WPD_{3} | −12.2977 | −3.42041 | |

WPD_{4} | −16.5301 | −3.42041 | |

WPD_{5} | −10.0685 | −3.42041 | |

WPD_{6} | −17.6399 | −3.42041 | |

WPD_{7} | −14.3801 | −3.42041 | |

WPD_{8} | −25.9766 | −3.42041 |

Name | Sample Data Set | ARIMA (p, d, q)/SARIMA (p, d, q) (P, D, Q) | BIC |
---|---|---|---|

Luoning | Original | SARIMA (5,1,1) (1,1,1) | 7.602 |

WPD_{1} | ARIMA (2,1,3) | 1.005 | |

WPD_{2} | ARIMA (2,0,8) | 4.016 | |

WPD_{3} | ARIMA (2,0,7) | 2.091 | |

WPD_{4} | ARIMA (3,0,5) | 3.333 | |

WPD_{5} | ARIMA (5,0,7) | −0.274 | |

WPD_{6} | ARIMA (2,0,7) | 1.469 | |

WPD_{7} | ARIMA (2,0,7) | 1.098 | |

WPD_{8} | ARIMA (6,0,8) | 1.132 | |

Zuoyu | Original | SARIMA (5,1,1) (1,1,1) | 8.119 |

WPD_{1} | ARIMA (2,0,5) | 1.469 | |

WPD_{2} | ARIMA (3,0,3) | 4.844 | |

WPD_{3} | ARIMA (2,0,8) | 2.209 | |

WPD_{4} | ARIMA (2,0,8) | 2.992 | |

WPD_{5} | ARIMA (10,0,7) | -0.128 | |

WPD_{6} | ARIMA (3,0,8) | 1.745 | |

WPD_{7} | ARIMA (2,0,7) | 1.43 | |

WPD_{8} | ARIMA (6,0,4) | 2.229 |

Model | Training | Testing | ||||||
---|---|---|---|---|---|---|---|---|

R | RMSE | MAE | NSEC | R | RMSE | MAE | NSEC | |

ARIMA | 0.608 | 40.926 | 27.442 | 0.352 | 0.459 | 46.474 | 27.792 | 0.191 |

WPD-ARIMA | 0.984 | 9.210 | 7.298 | 0.967 | 0.988 | 8.224 | 6.060 | 0.975 |

BPNN | 0.667 | 37.896 | 25.913 | 0.445 | 0.484 | 45.775 | 29.204 | 0.215 |

WPD-BPNN | 0.998 | 3.296 | 2.384 | 0.996 | 0.997 | 4.054 | 2.912 | 0.994 |

GMDH | 0.584 | 41.299 | 28.495 | 0.340 | 0.600 | 41.844 | 24.575 | 0.344 |

WPD-GMDH | 0.970 | 12.372 | 9.588 | 0.941 | 0.966 | 13.734 | 11.171 | 0.929 |

Model | Training | Testing | ||||||
---|---|---|---|---|---|---|---|---|

R | RMSE | MAE | NSEC | R | RMSE | MAE | NSEC | |

ARIMA | 0.679 | 55.717 | 38.124 | 0.459 | 0.576 | 53.689 | 31.856 | 0.263 |

WPD-ARIMA | 0.987 | 12.455 | 9.525 | 0.973 | 0.992 | 7.970 | 6.415 | 0.984 |

BPNN | 0.709 | 53.518 | 36.089 | 0.500 | 0.607 | 50.37 | 31.846 | 0.352 |

WPD-BPNN | 0.997 | 5.935 | 4.102 | 0.994 | 0.998 | 3.705 | 2.889 | 0.996 |

GMDH | 0.662 | 56.751 | 39.460 | 0.433 | 0.643 | 48.271 | 30.439 | 0.405 |

WPD-GMDH | 0.973 | 17.771 | 13.970 | 0.945 | 0.980 | 14.797 | 11.623 | 0.944 |

Model | Index | Training (%) | Testing (%) |
---|---|---|---|

WPD-ARIMA & ARIMA | R(↑) | 61.81 | 115.48 |

NSEC(↑) | 174.54 | 732.71 | |

RMSE(↓) | 77.5 | 80.3 | |

MAE(↓) | 73.4 | 78.19 | |

WPD-BPNN & BPNN | R(↑) | 45.52 | 106.8 |

NSEC(↑) | 123.98 | 362.45 | |

RMSE(↓) | 91.3 | 91.14 | |

MAE(↓) | 90.8 | 90.3 | |

WPD-GMDH & GMDH | R(↑) | 66.22 | 61.17 |

NSEC(↑) | 176.38 | 170.22 | |

RMSE(↓) | 70.04 | 74.35 | |

MAE(↓) | 66.35 | 54.54 |

Model | Index | Training (%) | Testing (%) |
---|---|---|---|

WPD-ARIMA&ARIMA | R(↑) | 45.34 | 72.08 |

NSEC(↑) | 105.30 | 273.50 | |

RMSE(↓) | 77.65 | 85.15 | |

MAE(↓) | 75.02 | 79.86 | |

WPD-BPNN&BPNN | R(↑) | 40.53 | 64.51 |

NSEC(↑) | 98.86 | 183.50 | |

RMSE(↓) | 88.91 | 92.64 | |

MAE(↓) | 88.63 | 90.93 | |

WPD-GMDH&GMDH | R(↑) | 46.81 | 52.33 |

NSEC(↑) | 118.32 | 133.37 | |

RMSE(↓) | 68.69 | 69.35 | |

MAE(↓) | 64.60 | 61.82 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, W.; Du, Y.; Chau, K.; Chen, H.; Liu, C.; Ma, Q.
A Comparison of BPNN, GMDH, and ARIMA for Monthly Rainfall Forecasting Based on Wavelet Packet Decomposition. *Water* **2021**, *13*, 2871.
https://doi.org/10.3390/w13202871

**AMA Style**

Wang W, Du Y, Chau K, Chen H, Liu C, Ma Q.
A Comparison of BPNN, GMDH, and ARIMA for Monthly Rainfall Forecasting Based on Wavelet Packet Decomposition. *Water*. 2021; 13(20):2871.
https://doi.org/10.3390/w13202871

**Chicago/Turabian Style**

Wang, Wenchuan, Yujin Du, Kwokwing Chau, Haitao Chen, Changjun Liu, and Qiang Ma.
2021. "A Comparison of BPNN, GMDH, and ARIMA for Monthly Rainfall Forecasting Based on Wavelet Packet Decomposition" *Water* 13, no. 20: 2871.
https://doi.org/10.3390/w13202871