# Improved Rainfall Prediction Using Combined Pre-Processing Methods and Feed-Forward Neural Networks

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Artificial Neural Network (ANN)

#### 2.2. Seasonal Artificial Neural Network (SANN)

_{t+l}(l = 1, 2,…, m) represents the predictions for the future s periods; Y

_{t−i}(i = 1, 2, …, m) are the observations of the previous s periods; IW

_{ij}(i = 1, 2,…, m; j = 1, 2,…, n) are the weights of connections from an input layer’s neuron to a hidden layer’s neuron; LW

_{jl}(j = 1, 2, …, n; l = 1, 2, …,m) are the weights of connections from a hidden layer’s neuron to an output layer’s neuron; b

_{l}(j = 1, 2,…, n) and b

_{j}(j = 1, 2, …, n) are the weights of bias connections and f is the activation function.

#### 2.3. ARIMA and GA-SA Models

_{i}(i = 1, 2, …, p) and θ

_{j}(j = 0, 1, 2, …, q) are model parameters; p and q are integers and referred as orders of the model; y

_{t}and ε

_{t}are the actual value and random error at the time period t, respectively; random errors, ε

_{t}, are presumed to be identically distributed with a mean of zero and a constant variance of σ

^{2}and independent of each other’s values.

## 3. Data Analysis

#### 3.1. Data Selection

_{t}) time series on a monthly scale were collected over 39 years (1971–2010).

_{d}), skewness coefficient (C

_{s}), and autocorrelations from 1-day lag to 3-day lag (R

_{1}, R

_{2}, and R

_{3}). It is important to note that ANN, and other data-driven methods, best performs when there is no extrapolation outside the data range that is used in training the models. Therefore, the extreme values of the whole dataset should present in the training dataset. In Table 2, the extreme values of R were within the training set range. When high skewness coefficients may reflect the substantially low performance of the models [54], the skewness coefficients in our models were low. The table also shows comparable statistical characteristics between the datasets, most obviously between the autocorrelation coefficients of the validating and testing sets.

#### 3.2. Data Pre-Processing

#### 3.2.1. Seasonal Decomposition (SD)

_{t}) into a multiplication of the three components as follows:

#### 3.2.2. Wavelet Transform (WT)

## 4. Model Application

#### 4.1. Combination of Models

_{d1}(t), R

_{d2}(t), …, R

_{di}(t), R

_{a}(t), where R

_{d1}(t), R

_{d2}(t), …, R

_{di}(t), and R

_{a}(t), which are the details and approximation of rainfall time series, respectively, must first be completed. The variable di is the ith level of the decomposed time series and signifies the approximate time series. In this paper, the observed R time series were decomposed using three different mother wavelets in four levels. These three wavelet mother functions are depicted in Figure 4. The R signal decomposed to level 4 yields 5 sub-signals (the approximation at level 4 and detail at levels 1, 2, 3 and 4) by the Daubechies-2, 4 (db2 and db4) and Meyer wavelets. Figure 6 shows these sub-signals for the Meyer mother wavelet. It is important to note that the focus of the current study is to evaluate the effectiveness and accuracy of the proposed hybrid models, and not assessing the effects of different decomposition levels and sensitivity of the mother wavelet types in pre-processing by DWT.

#### 4.2. Model Evaluation

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Predicted rainfall using Feed-forward Neural Network (ANN) for the testing period; (

**a**) no pre-processing, (

**b**) pre-processed by seasonal decomposition, (

**c**) pre-processed by discrete wavelet transform (Meyer).

**Figure 9.**Predicted rainfall using Seasonal Artificial Neural Network (SANN) for the testing period; (

**a**) no pre-processing, (

**b**) pre-processed by seasonal decomposition, (

**c**) pre-processed by discrete wavelet transform (Meyer).

**Figure 10.**Predicted rainfall using (

**a**) Autoregressive Integrated Moving Average (ARIMA), and (

**b**) GA-SA models for the testing period.

**Table 1.**Simulated annealing (

**left**) and genetic frame-work (

**right**) used in this GA-SA algorithm [15].

Algorithm 1: Simulation of Annealing | Algorithm 2: Genetic Frame-Work |

Select an initial solution Select an initial temperature t = t _{o} > 0Select number of phases maxphase Select a temperature reduction coefficient α While phase < maxphaseWhile iteration_count < nrep/* s is a neighbor solution of s0 */ Randomly select s $\in $ N(s0); /* compute the change in cost function */ δ = f(s) − f(s0) if δ < 0 then s_{o} = selsegenerate random x $\in $ [0, 1] if x < exp(−δ/t) then s_{o} = st = t * α | Initialize population with random candidate solutions Evaluate each candidate repeatrepeatSelect parents Recombine pairs of parents Mutate the resulting children until iteration_count = num_mateEvaluate children Select individuals for the next generation until Termination-Condition is satisfied |

Statistical Parameters | Training Set | Validation Set | Testing Set | Whole Data |
---|---|---|---|---|

Min | 0 | 0 | 0 | 0 |

Max | 782.1 | 748.7 | 656 | 782.1 |

Mean | 198.29 | 223.798 | 196.655 | 202.43 |

S_{d} | 170.266 | 176.683 | 166.183 | 170.53 |

C_{s} | 0.568 | 0.559 | 0.437 | 0.543 |

R_{1} | 0.568 | 0.480 | 0.629 | 0.565 |

R_{2} | 0.297 | 0.239 | 0.339 | 0.298 |

R_{3} | −0.003 | 0.057 | 0.049 | 0.023 |

No. of Model | Pre-Processing Method | Model | Statistical Performance | Number of Neurons | ||||
---|---|---|---|---|---|---|---|---|

3 | 5 | 8 | 10 | 15 | ||||

1 | - | ANN | R | 0.7948 | 0.8092 | 0.8061 | 0.7770 | 0.7248 |

- | RMSE | 103.9385 | 101.4842 | 98.3109 | 104.9657 | 114.1420 | ||

- | MAE | 84.3679 | 80.4982 | 74.0538 | 78.6600 | 82.7604 | ||

2 | - | R | 0.8432 | 0.8601 | 0.8408 | 0.8118 | 0.8300 | |

SD | RMSE | 89.3413 | 85.5113 | 90.2806 | 101.5413 | 93.3553 | ||

- | MAE | 66.1147 | 66.0452 | 69.2322 | 80.5990 | 70.6810 | ||

3 | DWT (Meyer) | R | 0.9802 | 0.9819 | 0.9723 | 0.9776 | 0.9612 | |

RMSE | 33.0155 | 31.5850 | 38.7293 | 34.7847 | 45.9508 | |||

MAE | 25.4845 | 24.6434 | 30.0314 | 27.5218 | 37.7463 | |||

4 | DWT (db2) | R | 0.9248 | 0.9237 | 0.9107 | 0.9299 | 0.8146 | |

RMSE | 65.9907 | 64.0403 | 70.8502 | 62.1080 | 95.9548 | |||

MAE | 52.6840 | 52.0136 | 54.0858 | 48.0775 | 71.8549 | |||

5 | DWT (db4) | R | 0.9625 | 0.9567 | 0.9617 | 0.9564 | 0.8889 | |

RMSE | 45.1868 | 55.3045 | 46.7715 | 48.7092 | 76.6024 | |||

MAE | 33.6868 | 44.3377 | 36.2730 | 38.0701 | 61.1264 | |||

6 | - | SANN | R | 0.8300 | 0.8287 | 0.8049 | 0.8171 | 0.8112 |

RMSE | 94.8723 | 92.8862 | 99.5541 | 96.0473 | 97.0712 | |||

MAE | 78.8377 | 74.2247 | 79.1828 | 78.0073 | 76.7192 | |||

7 | SD | R | 0.9507 | 0.9102 | 0.9284 | 0.9268 | 0.9211 | |

RMSE | 52.1441 | 69.3786 | 61.9278 | 63.6088 | 65.1701 | |||

MAE | 42.6135 | 54.1883 | 49.4063 | 49.5467 | 51.9533 | |||

8 | DWT (Meyer) | R | 0.9927 | 0.9973 | 0.9968 | 0.9955 | 0.9951 | |

RMSE | 20.4926 | 12.1045 | 15.0972 | 16.0551 | 16.5294 | |||

MAE | 15.5346 | 9.3213 | 11.7802 | 12.4678 | 11.8652 | |||

9 | DWT (db2) | R | 0.9624 | 0.9458 | 0.9479 | 0.9338 | 0.9352 | |

RMSE | 45.8911 | 55.3706 | 53.3337 | 59.1869 | 58.8830 | |||

MAE | 37.6343 | 43.2339 | 41.6543 | 47.8722 | 45.4037 | |||

10 | DWT (db4) | R | 0.9570 | 0.9487 | 0.9531 | 0.9612 | 0.9452 | |

RMSE | 49.9123 | 52.6379 | 50.2059 | 46.1096 | 55.2336 | |||

MAE | 38.7917 | 39.2650 | 37.7869 | 36.9486 | 44.3607 |

No. of Model | Pre-Processing Method | Model | Statistical Performance | Number of Neurons | ||||
---|---|---|---|---|---|---|---|---|

3 | 5 | 8 | 10 | 15 | ||||

1 | - | ANN | R | 0.7185 | 0.7389 | 0.7496 | 0.7323 | 0.7340 |

- | RMSE | 119.3717 | 116.1545 | 112.7167 | 116.2497 | 115.5481 | ||

- | MAE | 92.9199 | 90.6317 | 82.9759 | 85.3620 | 85.2603 | ||

2 | - | R | 0.8305 | 0.8569 | 0.8344 | 0.8627 | 0.8497 | |

SD | RMSE | 94.8184 | 88.0335 | 94.2725 | 87.7624 | 91.2737 | ||

- | MAE | 68.5089 | 60.6942 | 69.5760 | 62.6814 | 64.6820 | ||

3 | DWT (Meyer) | R | 0.9740 | 0.9804 | 0.9773 | 0.9791 | 0.9721 | |

RMSE | 38.5218 | 33.5312 | 36.1287 | 34.6227 | 39.9093 | |||

MAE | 30.5493 | 26.3544 | 27.7637 | 27.0257 | 30.1284 | |||

4 | DWT (db2) | R | 0.9174 | 0.9295 | 0.9290 | 0.9498 | 0.8475 | |

RMSE | 69.0054 | 63.0776 | 64.3189 | 54.4427 | 90.2813 | |||

MAE | 54.3111 | 48.8261 | 47.7219 | 39.6865 | 67.9992 | |||

5 | DWT (db4) | R | 0.9639 | 0.9596 | 0.9611 | 0.9597 | 0.9196 | |

RMSE | 45.4499 | 53.0644 | 47.5084 | 47.8804 | 66.8622 | |||

MAE | 34.4683 | 40.9454 | 36.4628 | 30.7056 | 47.5880 | |||

6 | - | SANN | R | 0.8010 | 0.8164 | 0.8070 | 0.8200 | 0.8213 |

RMSE | 102.2606 | 97.1078 | 100.3804 | 96.7091 | 96.1686 | |||

MAE | 79.9284 | 70.2974 | 73.5328 | 73.3155 | 71.8607 | |||

7 | SD | R | 0.9690 | 0.9393 | 0.9607 | 0.9521 | 0.9275 | |

RMSE | 41.5860 | 57.7945 | 46.8358 | 52.4555 | 63.0528 | |||

MAE | 31.3174 | 41.0834 | 31.3976 | 38.5993 | 47.0755 | |||

8 | DWT (Meyer) | R | 0.9919 | 0.9985 | 0.9965 | 0.9967 | 0.9972 | |

RMSE | 22.0805 | 9.4251 | 15.4858 | 13.7718 | 12.5305 | |||

MAE | 14.8010 | 6.6855 | 9.8296 | 10.0559 | 7.2495 | |||

9 | DWT (db2) | R | 0.9674 | 0.9593 | 0.9632 | 0.9615 | 0.9618 | |

RMSE | 42.9408 | 47.5008 | 45.2887 | 46.2991 | 46.1271 | |||

MAE | 31.6252 | 34.3351 | 32.4117 | 33.7197 | 32.1644 | |||

10 | DWT (db4) | R | 0.9747 | 0.9716 | 0.9744 | 0.9765 | 0.9639 | |

RMSE | 37.9720 | 40.6086 | 37.8868 | 37.4568 | 44.8296 | |||

MAE | 24.7977 | 29.0884 | 24.3499 | 26.9415 | 27.2420 |

**Table 5.**Comparison of proposed methods and ARIMA and Genetic Algorithm and Simulated Annealing algorithm (GA-SA) methods for the testing period.

Methods | R | RMSE | MAE |
---|---|---|---|

ARIMA | 0.7628 | 108.070 | 83.235 |

GA-SA | 0.8190 | 96.000 | 76.595 |

Raw data + ANN | 0.8061 | 98.311 | 74.054 |

Raw data + SANN | 0.8287 | 92.886 | 74.225 |

SD + ANN | 0.8601 | 85.511 | 66.045 |

SD + SANN | 0.9507 | 52.144 | 42.614 |

Meyer Wavelet + ANN | 0.9819 | 31.585 | 24.643 |

Meyer Wavelet + SANN | 0.9973 | 12.105 | 9.3213 |

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## Share and Cite

**MDPI and ACS Style**

Tran Anh, D.; Duc Dang, T.; Pham Van, S.
Improved Rainfall Prediction Using Combined Pre-Processing Methods and Feed-Forward Neural Networks. *J* **2019**, *2*, 65-83.
https://doi.org/10.3390/j2010006

**AMA Style**

Tran Anh D, Duc Dang T, Pham Van S.
Improved Rainfall Prediction Using Combined Pre-Processing Methods and Feed-Forward Neural Networks. *J*. 2019; 2(1):65-83.
https://doi.org/10.3390/j2010006

**Chicago/Turabian Style**

Tran Anh, Duong, Thanh Duc Dang, and Song Pham Van.
2019. "Improved Rainfall Prediction Using Combined Pre-Processing Methods and Feed-Forward Neural Networks" *J* 2, no. 1: 65-83.
https://doi.org/10.3390/j2010006