# Application of a Novel Hybrid Wavelet-ANFIS/Fuzzy C-Means Clustering Model to Predict Groundwater Fluctuations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}. The long-term annual rainfall and temperature are 294 mm and 14 °C, respectively. The climate of the region is arid and cold (see Figure 1).

#### 2.2. Data Description

#### 2.3. Completion of Groundwater Level Missing Values

#### 2.4. Training and Testing of GWL Prediction Models

_{t}

_{−1}and P

_{t}

_{−j}was applied:

_{t}

_{−j}. Although Equation (1) is basically just a simple transfer model, it has some physical basis, as groundwater data usually exhibit some persistence, as illustrated in Figure 3, given that groundwater is recharged by rainfall with a delay [40,41].

_{t}is mostly correlated with GWL

_{t−1,t−2,t−3}and P

_{t−1,t−2,t−3,t−4}. Therefore, the final input-output Wavelet-ANFIS and ANFIS and SVM model Equation (1) can be written as:

#### 2.5. Adaptive Neuro Fuzzy Inference System (ANFIS)

_{1,i}is the output of the ith node of the layer 1.

_{1,2}is the input node i and A

_{i}(or B

_{i}) is a linguistic label associated with this node. Therefore O

_{1,i}is the membership grade of a fuzzy set (A

_{1}, A

_{2}, B

_{1}, B

_{2}). The second layer consists of rule nodes with AND and/or OR operators. The output (O

_{2,i}) is the product of all the incoming signals.

_{3,i}) are called normalized firing strengths.

_{i}, q

_{i}, r

_{i}) is the parameter set of this node; These are referred to as the consequent parameters. Finally, the fifth layer is called the output layer, which computes the overall output as the summation of all incoming signals.

#### 2.6. Fuzzy C-Means (FCM) Clustering Method

_{m}) [48]:

_{ij}]

_{c}

_{×n}is the fuzzy partition matrix of c clusters and n data, V = (v

_{1}, v

_{2}, …, v

_{i}) are the centroids of the clusters, u

_{ij}is the partial membership degree of data x

_{j}in cluster i ($0\le {u}_{ij}\le 1{\sum}_{i=1}^{c}{u}_{ij}=1,2,\dots ,n$), m is the weighting exponent on each fuzzy membership (which is equal to 2 in this study), x

_{j}= (x

_{1}, x

_{2}, …, x

_{n}) is the dataset, v

_{i}is the initial value for the cluster center, and d(x

_{j}, v

_{i}) is the Euclidean distance between the x

_{j}data and the cluster center of the i

^{th}cluster v

_{i}, i.e., x

_{j}−v

_{i}. Equation (11) describes a non-linear optimization problem which can be solved by iterative minimization. The centroid of each cluster is calculated by the partial derivative of Equation (11) with respect to V, then the partial membership degree of data is updated in each iteration by differentiation of the above equation with respect to U:

^{−8}), i.e., a minimum improvement in the FCM objective function (convergence) does not occur in five iterations and/or a maximum number of iterations (1000 iterations in this study) is exceeded. In the present study, the FCM algorithm was incorporated into (i.e., coded in) the ANFIS model. The procedure is sketched in Figure 6. The optimal value of the number of clusters was obtained based on trial and error. The RMSE values were calculated for different numbers in the training and test phases (Figure 7). As shown in Figure 7, using two clusters resulted in the least error.

#### 2.7. Wavelet Transformation

_{0}is the location parameter that must be greater than 0, and S

_{0}is a specified fixed dilation step greater than 1. The DWT performs two functions viewed as high-pass and low-pass filters, through which the original time series are passed and then the original time series data are decomposed and divided into two parts, namely “approximation” and “details” [54]. These components explain behavior better and reveal more information about the process than the original time series. Therefore, they can help forecasting models predict with greater accuracy [55,56]. There are many wavelets that can be used as mother wavelets, For the choice of the discrete father-scaling/mother-wavelet filter functions in the DWT/MRA, popular ones are the Haar, the Daubechies wavelet db4 and the irregular wavelet Symlet (sym4), and these are those have been used here [38,57].

#### 2.8. Hybrid Wavelet-ANFIS Model

## 3. Results and Discussion

^{2}and RMSE statistical parameters. Shown in Table 3, as expected, the performance of the ANFIS model is lower than that of the Wavelet-ANFIS hybrid model, because the Wavelet-ANFIS hybrid model takes advantage of both ANFIS and WT, simultaneously. In particular, WT, or specifically, the DWTMRA, has the ability to consider various aspects of a time series such as trends, discontinuities and breakpoints, and this time-scale localization feature was used to provide decomposition of the input time series up to the 4 levels in the present application, allowing for a separation of the approximation and for providing details of the non-stationary noisy time series, thus enhancing the ANFIS, which uses a combination of fuzzification of the input through membership functions with the network-based algorithm and the use of the hybrid (back propagation tries and gradient descent) optimization method. Table 3 shows the minimum RMSE and the maximum R

^{2}in the training phase with 0.05 and 0.997 and in the test phase as 0.08 and 0.974, respectively, which are obtained with the hybrid wavelet ANFIS model and using the Sym4-mother wavelet (second input combination, Level Decomposition = 3). The best results of the Haar mother wavelets in the test phase has RMSE and R

^{2}values of 0.13 and 0.929, respectively. The db4 mother wavelets also exhibits the best results in the test phase, with corresponding values of 0.09 and 0.968. Comparison of the results of the db4 and sym4 mother wavelets revealed that, although there was no significant difference between the results of these two mother wavelets, the sym4 mother wavelets outperforms db4 mother wavelets slightly. The less accurate results are related to Haar mother wavelets since these are only orthogonal wavelet with a linear phase that cannot provide nonlinear shifts between the original and decomposed signals. By adding the precipitation parameter to the input data, the output accuracy increased in most simulations (on average, 10.2%), although it was more significant in the Wavelet-ANFIS models. In this research, the effect of the decomposition level on the accuracy of the output results of the Wavelet-ANFIS models was investigated. Among the different levels, the decomposition at level 3 had the highest accuracy. However, the accuracy of decomposition level 2 was also acceptable (with only a small difference compared to level 3).

^{2}, the points are nearly perfectly lying on the “slope = 1” regression line. The GWL-time series observed and simulated with the best combination of the Wavelet-ANFIS model are presented in Figure 9. There is an acceptable correlation between the observed and simulated GWL data, and the effect of the applied delays on the input data, especially the piezometric head data, is clearly visible. All data-driven prediction methods are based on the idea that the random errors are drawn from a normal distribution and the hybrid Wavelet-ANFIS model is not an exception. Figure 10 illustrates that a-posteriori computed errors of the GWLs predicted with this model, follow indeed a normal distribution.

^{2}and RMSE for both the training and test phase are given in Figure 11. According to most R

^{2}values above 0.95 and most RMSE values less than 0.20 m, especially in the test phase, it can be argued that the model has an acceptable performance in almost all wells.

## 4. Conclusions

^{2}were obtained 0.05 m and 0.997, respectively, in the training phase and of 0.08 m and 0.974 the testing (prediction) phase, which indicates that the hybrid Wavelet-ANFIS using the Symlet mother wavelet and decomposition level 3 performs best. According to the results of the hybrid Wavelet-ANFIS model, the model performance was acceptable in estimating GWL fluctuations all over the plain, such that the difference between the observed and simulated values was negligible, meaning the incorporation of the wavelet transforms in ANFIS increased the performance of ANFIS model, especially in the prediction phase. Another noteworthy characteristic of this coupled model in the simulated and predicted values of GWL fluctuations is the use of the FCM clustering method which generates fewer fuzzy rules, thereby overcoming the well-known data dimensionality problem.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Autocorrelation function for GWL (

**a**) and cross correlation function between GL and Precipitation (

**b**).

**Figure 5.**Architecture of ANFIS [11].

**Figure 6.**Flow diagram of the combination of fuzzy c-means clustering (FCM) method and ANFIS [11].

**Figure 8.**Regression plots of Wavelet-ANFIS (Sym4 mother wavelet)-simulated over observed GWL-data for training (left panel) and testing (right) phases.

**Figure 9.**Wavelet-ANFIS/Sym4-simulated and observed GWL-time series for the training (upper panel) and testing (lower panel) phases.

**Figure 11.**Results of GWL-simulation in all 25 wells (see Figure 2).

ID | Location | A(ha) | Elevation(m) | ID | Location | A(ha) | Elevation(m) |
---|---|---|---|---|---|---|---|

1 | Seraju (khoshkbarchi) | 1194 | 1397.2 | 14 | Akhond gheshlagh | 650 | 1280.58 |

2 | Sezavar mousavi | 1659 | 1331 | 15 | Alghou | 2099 | 1336.31 |

3 | Varjouy | 1801 | 1404.8 | 16 | Pahr abad | 2243 | 1413.41 |

4 | Yengi kand khoushe mehr | 3694 | 1327.92 | 17 | Khoushe mehr | 1007 | 1314.25 |

5 | Rousht bozorg istgahe rah ahan | 2096 | 1337.89 | 18 | Rousht kouchak | 921 | 1296.26 |

6 | Khaneh bargh ghadim | 931 | 1287.01 | 19 | Bonab energy atomi | 1527 | 1280.25 |

7 | Gharah chopogh behdasht | 1553 | 1285.09 | 20 | Bonab mantagheh abiyari | 1089 | 1282.14 |

8 | Khalilvand rouberouye ajorpazi | 998 | 1286.82 | 21 | Jadeh sarj baghal kanal | 1454 | 1345.71 |

9 | Maragheh foroudgah | 1094 | 1325.23 | 22 | Khanghah ghabrestan | 2126 | 1359.17 |

10 | Zavasht ghabrestan | 1416 | 1297.9 | 23 | Maragheh khajeh nasir | 3251 | 1474.24 |

11 | Varjouy maleki | 1465 | 1371.99 | 24 | Bonab aval rah darya | 1529 | 1284.25 |

12 | Khezerlou shourgol | 1645 | 1284.4 | 25 | Chelghay masjed | 2285 | 1305.22 |

13 | Zavaregh bimarestan | 1706 | 1292.36 | - | - | - | - |

Well ID | 3 | 8 | 10 | 12 | 18 | 20 | 24 |

Number of Missing Values | 17 | 16 | 27 | 17 | 30 | 7 | 5 |

**Table 3.**Results of the various model combinations of input data with the optimum number of clusters (c = 2) for the training and testing phases of ANFIS.

Method | Decomposition Level | ANFIS Inputs | Training | Testing | ||
---|---|---|---|---|---|---|

RMSE(m) | R^{2} | RMSE(m) | R^{2} | |||

ANFIS | - | GLt−1,GLt−2,GLt−3 | 0.16 | 0.962 | 0.19 | 0.870 |

Wavelet-ANFIS/haar | 1 | - | 0.11 | 0.983 | 0.14 | 0.923 |

2 | - | 0.11 | 0.983 | 0.14 | 0.921 | |

3 | - | 0.10 | 0.984 | 0.19 | 0.878 | |

4 | - | 0.11 | 0.983 | 0.16 | 0.891 | |

Wavelet-ANFIS/db4 | 1 | - | 0.09 | 0.987 | 0.13 | 0.949 |

2 | - | 0.07 | 0.992 | 0.11 | 0.964 | |

3 | - | 0.07 | 0.995 | 0.08 | 0.972 | |

4 | - | 0.06 | 0.996 | 0.14 | 0.921 | |

Wavelet-ANFIS/sym4 | 1 | - | 0.10 | 0.951 | 0.14 | 0.933 |

2 | - | 0.08 | 0.99 | 0.12 | 0.955 | |

3 | - | 0.06 | 0.994 | 0.09 | 0.969 | |

4 | - | 0.06 | 0.994 | 0.09 | 0.966 | |

ANFIS | - | GLt−1,GLt−2,GLt−3,Pt−1,Pt−2,Pt−3,Pt−4 | 0.13 | 0.975 | 0.18 | 0.878 |

- | - | - | - | - | ||

Wavelet-ANFIS/haar | 1 | - | 0.09 | 0.988 | 0.16 | 0.891 |

2 | - | 0.09 | 0.988 | 0.13 | 0.927 | |

3 | - | 0.09 | 0.988 | 0.13 | 0.929 | |

4 | - | 0.09 | 0.988 | 0.13 | 0.925 | |

Wavelet-ANFIS/db4 | 1 | - | 0.08 | 0.991 | 0.12 | 0.949 |

2 | - | 0.07 | 0.994 | 0.09 | 0.967 | |

3 | - | 0.06 | 0.996 | 0.09 | 0.968 | |

4 | - | 0.06 | 0.997 | 0.1 | 0.959 | |

Wavelet-ANFIS/sym4 | 1 | - | 0.08 | 0.99 | 0.13 | 0.931 |

2 | - | 0.07 | 0.993 | 0.11 | 0.954 | |

3 | - | 0.06 | 0.995 | 0.08 | 0.974 | |

4 | - | 0.05 | 0.996 | 0.08 | 0.974 |

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Jafari, M.M.; Ojaghlou, H.; Zare, M.; Schumann, G.J.-P.
Application of a Novel Hybrid Wavelet-ANFIS/Fuzzy C-Means Clustering Model to Predict Groundwater Fluctuations. *Atmosphere* **2021**, *12*, 9.
https://doi.org/10.3390/atmos12010009

**AMA Style**

Jafari MM, Ojaghlou H, Zare M, Schumann GJ-P.
Application of a Novel Hybrid Wavelet-ANFIS/Fuzzy C-Means Clustering Model to Predict Groundwater Fluctuations. *Atmosphere*. 2021; 12(1):9.
https://doi.org/10.3390/atmos12010009

**Chicago/Turabian Style**

Jafari, Mohammad Mahdi, Hassan Ojaghlou, Mohammad Zare, and Guy Jean-Pierre Schumann.
2021. "Application of a Novel Hybrid Wavelet-ANFIS/Fuzzy C-Means Clustering Model to Predict Groundwater Fluctuations" *Atmosphere* 12, no. 1: 9.
https://doi.org/10.3390/atmos12010009