# Assessing the Predictability of an Improved ANFIS Model for Monthly Streamflow Using Lagged Climate Indices as Predictors

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{3}/s, so that the stream flow for summer, based on climate indexes, is more than that in other seasons.

## 1. Introduction

_{3}(5° S–5° N, 150°–90° W), NINO3.4 (5° S–5° N, 170°–120° W), NINO4 (5° S–5° N, 150°–160° W) and NINO1 + 2 (0–10° S, 90°–80° W) (Figure 1).

## 2. Background

## 3. Materials and Methods

#### 3.1. ANFIS

_{1}and f

_{2}are output functions and ${p}_{1}$, ${q}_{1}$, ${r}_{1}$, ${p}_{2}$, ${q}_{2}$ and ${r}_{2}$ are linear parameters (Figure 2).

_{i}and B

_{i}are fuzzy sets, ${\mu}_{Ai}\left(x\right)$ and ${\mu}_{{B}_{i-2}}\left(y\right)$ are the degrees of MF, and x and y are the inputs for node i.

_{i}, q

_{i}, r

_{i}):

#### 3.2. Bat Algorithm (BA)

- (1)
- All bats use echolocation to identify the food location.
- (2)
- The bats fly at the random velocity (v
_{l}) at the location y_{l}with the frequency f_{min}and the wavelength ${\lambda}_{l}$. The loudness parameter for the bats is given by A_{0}. - (3)
- The volume can vary from A
_{0}to A_{min}.

_{l}) are updated for each level. The pulsation rate increases and volume decrease when the bats find the food. The volume and pulsation rate are updated based on following equation:

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

_{is}(k) position $\left({P}_{i}\in {P}_{k}\right)$ equals (k = 0, k: number of levels), which is known as the first step. Each particle’s F function is computed based on the following equation:

_{1}and r

_{2}are random parameters, w is the inertia weight and c

_{1}and c

_{2}are the acceleration coefficients.

#### 3.4. Genetic Algorithm (GA)

#### 3.5. Principal Component Analysis (PCA)

- (1)
- PCA is considered to be a statistical nonparametric method and thus, it is necessary to evaluate the Kaiser–Meyer–Olkin (KMO) test. This index is computed based on simple and partial correlation coefficients. If the value of the KMO coefficient is more than 0.5, the PCA method can be applied to the data [36,37,38].$$KMO=\frac{{\displaystyle \sum _{i=1}^{p}{\displaystyle \sum _{j=1}^{p}{r}_{ij}^{2}}}}{{\displaystyle \sum _{i=1}^{p}{\displaystyle \sum _{j=1}^{p}{r}_{ij}^{2}+{\displaystyle \sum _{i=1}^{p}{\displaystyle \sum _{j=1}^{p}{r}_{ij}^{2}}}}}}i\ne j$$
_{ij}are the simple correlation coefficient and partial correlation coefficient, respectively, between variables i,j. - (2)
- The second level is used for the conversion of data to the standard format:$$Z=\frac{X-\mu}{\sigma}$$
- (3)
- The correlation matrix is computed to show the variations in the samples and the correlations of different variables with each other. The members of the main diagonal of the matrix are considered as variance of the input variables, and other arrays are considered as covariances of the input variables [34].
- (4)
- The Eigen vectors and Eigen values are computed based on the following equation:$$\left|R-\lambda {I}_{p}\right|=0$$

#### 3.6. Data Splitting

## 4. Case Study

^{6}m

^{3}per year. The data from 1987–2007 were available. The river water regulation and the irrigation demand supply for the Aidoughmoush basin were considered and thus, the construction of the Aydoughmoush dam was also considered. The network area for this basin is 13,500 ha, with 1341.5 ha at a water level above sea level for this dam. Table 1 shows the predicator as input variables and the source of the data collection. There are seven stations for the precipitation measurement, and the Koppen index was used to classify the region’s climate [39]. The Koppen classification includes five main climate groups, with the groups classified based on precipitation and temperature (Appendix A). The central part of the basin is shown by the symbol Bwk, which refers to the cold desert climate based on the Koppen classification. The upstream part of the basin is shown by the Cwb symbol, where there is a dry winter and a warm summer, and the downstream part of the basinis shown by Bsk, referring to a cold semi-arid climate. There are some stations that measure precipitation, including Maktu, Ghezel gheye, Tlkhab, Kangavr, Tunnel 7, Poldokhtar and Tazekand. Furthermore, the effect of some stations that are out of the basin were considered because they are located close to the edges and may affect the basin. The inverse distance weighting method was used to obtain the precipitation values for the different points in the maps. The power parameter for this method was obtained based on the optimization algorithm. In fact, the RMSE between the observed and simulated precipitation was used as an objective function and then, the power parameter, as an initial population, was inserted into the BA so that the optimization algorithm gave the optimal value for the power parameter. This is because minimizing the RMSE is suitable for the decision maker [40].

_{i}is the measured distance between the prediction and observation point, and q is the power parameter. The spatial correlation based on the IDW was 0.94 and Figure 5 shows the spatial distribution of precipitation.

## 5. Discussion and Results

#### 5.1. Results of PCA

#### 5.2. Study of Sensitivity Analysis by Vayring Parameter Values

_{1}and c

_{2}) and inertia weight (w), were calculated as shown in Table 5.

#### 5.3. Results for Comparison of ANFIS-BA, ANFIS, PSO and ANFIS GA

_{f}is the cumulative probability for the forecasted variable and p

_{v}is the cumulative probability for the observed variable. The probability value for each parameter was computed based on historical probable distribution. The sum of the S values was computed based on the following equation:

_{f}= p

_{v}) is computed when ${S}^{\u2033}$ has a positive value. If it has a negative value, ${{S}^{\u2033}}_{mj}$ is computed based on the assumption of the worst prediction. The value of LEPS is between the worst value (−100) and the best value (100). Figure 7 shows the relative error based on percentage, where the percentage relative error between the predicated and observed streamflow based on the ANFIS-BA varied from 0 to 4, while for the GA, it varied from 20 to 42%, and furthermore, the ANFIS-BA performed better than the ANIFS-PSO based on computed indexes. The average LEPS for different months of the 1987–2007 period is shown in Figure 7, where the average LEPS value for the most months varied from 60 to 75 for the ANFIS-BA, which was more than the ANFIS-PSO and ANFIS-GA. However, the different indexes showed the superiority of the ANFIS-BA compared to the other models.

#### 5.4. Discussion

_{C}is the relative observed agreement, and P

_{c}is the hypothetical probability. The probabilities for each observer were computed based on the observed data. Kappa equals 1 if the rates have complete agreement. Figure 8 shows that the spatial streamflow for the different methods and the Kappa for the ANFIS-BA was 0.91, while it was 0.85 and 0.78 for the ANFIS-BA and ANFIS-GA, respectively. Thus, the ANFIS-BA has the most accurate streamflow for the different parts of the basin.

^{3}/s, which is more than in other seasons. This is related to the snowmelt during summer, which increases the runoff, streamflow and variation of streamflow, and is greater during summer than other seasons. However, the ENSO for the summer can also increase the runoff and streamflow significantly so that the food probability can increase for the summer compared to the other seasons.

## 6. Conclusions

_{3}, NINO3.4, NINO

_{4}and PDO. The results indicated that the ANFIS-BA could decrease the error index more than other methods. For example, the RMSE for the ANFIS-BA was 25 and 30% less than that for the ANFIS-PSO and ANFIS-GA, respectively. In addition, the average LEPS value for the most months varied from 60 to 75 for the ANFIS-BA and was more than the ANFIS-PSO and ANFIS-GA. Also, a weight method was used to obtain the spatial map for the streamflow and Kappa coefficient, which had the greatest value for the ANFIS-BA. The results indicated that the ANFIS-BA could have these results because of less uncertainty and the increased summer streamflow due to the snow melt. The current paper showed a large signal climate index could increase or decrease the streamflow and thus, events such as a floods can form through the variations of these indexes. Future studies should consider predicting the streamflow based on satellite images. The results of soft computing methods were compared with such images to determine which tools could produce results with a greater level of agreement compared to the observed data.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

First Part | Second Part | Third Part |

B (arid) | W (desert) | - |

S (steppe) | - | |

h (hot) | ||

k (cold) | ||

C (temperate) | S (dry summer) | - |

W (dry winter) | - | |

F (without dry season) | - | |

- | a (hot summer) | |

- | b (warm summer) | |

- | c (cold summer) |

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**Figure 4.**ANFIS and Evolutionary Algorithm [26].

**Figure 5.**(

**a**) The location of the case study, (

**b**) The spatial of climate change and (

**c**) the spatial distribution based on rainfall (mm).

**Figure 6.**The upload of coefficient values for the different climate indexes and different PCA components.

**Figure 8.**(

**a**) spatial streamflow based on ANIFS-BA, (

**b**) observed spatial streamflow based on ANFIS-PSO, (

**c**) spatial streamflow based on ANFIS-GA, (

**d**) spatial streamflow based on observed map.

**Figure 9.**(

**a**) spatial streamflow based on summer, (

**b**) observed spatial streamflow based on autumn, (

**c**) spatial streamflow based on winter, (

**d**) spatial streamflow based on spring.

Predicators | Predicator Definition | Origin | Data Period | Data Source |
---|---|---|---|---|

NINO4 | Average SST anomaly over centre Pacific Ocean | Pacific Ocean | 1987–2007 | https://library.noaa.gov http://sdwebx.worldbank.org/climateporta |

NINO3 | Average SST anomaly over centre Pacific Ocean | Pacific Ocean | 1987–2007 | https://library.noaa.gov http://sdwebx.worldbank.org/climateporta |

NINO3.4 | Average SST anomaly over centre Pacific Ocean | Pacific Ocean | 1987–2007 | https://library.noaa.gov http://sdwebx.worldbank.org/climateporta |

PDO | Average SST anomaly over centre Pacific Ocean | Pacific Ocean | 1987–2007 | http://research.jisao.washington.edu/pdo/PDO.latest.txt |

Components | Value of Each Component from 16 | Varince Prenatage of Data | Comulative Variantagece Prenatage |
---|---|---|---|

PCA1 | 6.72 | 42.000 | 42.000 |

PCA2 | 3.68 | 23.000 | 65.000 |

PCA3 | 2.08 | 13.000 | 78.000 |

PCA4 | 1.12 | 7.000 | 85.000 |

PCA5 | 0.88 | 5.500 | 90.500 |

PCA6 | 0.80 | 5.000 | 95.500 |

PCA7 | 0.496 | 3.100 | 98.600 |

PCA8 | 0.179 | 1.12 | 99.720 |

PCA9 | 0.0128 | 0.08 | 99.80 |

PCA10 | 0.0128 | 0.08 | 99.88 |

PCA11 | 0.0064 | 0.04 | 99.92 |

PCA12 | 0.0064 | 0.04 | 99.96 |

PCA13 | 0.0016 | 0.01 | 99.97 |

PCA14 | 0.0016 | 0.01 | 99.98 |

PCA15 | 0.0016 | 0.01 | 99.99 |

PCA16 | 0.0016 | 0.01 | 100 |

Components | PCA1 | PCA2 | PCA3 | PCA4 | PCA5 |
---|---|---|---|---|---|

NINO3 (t) | 0.12 | 0.11 | 0.09 | 0.07 | 0.06 |

NINO3 (t − 3) | 0.67 | 0.64 | 0.61 | 0.55 | 0.53 |

NINO3 (t − 6) | 0.94 | 0.92 | 0.91 | 0.90 | 0.87 |

NINO3 (t − 9) | 0.61 | 0.60 | 0.59 | 0.52 | 0.50 |

NINO4 (t) | 0.11 | 0.10 | 0.08 | 0.05 | 0.03 |

NINO4 (t − 3) | 0.62 | 0.60 | 0.57 | 0.55 | 0.51 |

NINO4 (t − 6) | 0.91 | 0.90 | 0.88 | 0.86 | 0.85 |

NINO4 (t − 9) | 0.60 | 0.55 | 0.52 | 0.50 | 0.49 |

NINO3.4 (t) | 0.10 | 0.09 | 0.08 | 0.07 | 0.05 |

NINO3.4 (t − 3) | 0.61 | 0.56 | 0.51 | 0.49 | 0.42 |

NINO3.4 (t − 6) | 0.89 | 0.82 | 0.80 | 0.79 | 0.77 |

NINO3.4 (t − 9) | 0.60 | 0.52 | 0.49 | 0.45 | 0.40 |

PDO (t) | 0.11 | 0.10 | 0.09 | 0.08 | 0.07 |

PDO (t − 3) | 0.59 | 0.57 | 0.55 | 0.51 | 0.45 |

PDO (t − 6) | 0.90 | 0.89 | 0.82 | 0.80 | 0.79 |

PDO (t − 9) | 0.62 | 0.60 | 0.57 | 0.44 | 0.42 |

Components | PCA1 | PCA2 | PCA3 | PCA4 | PCA5 |
---|---|---|---|---|---|

NINO3 (t) | 0.12 | 0.10 | 0.08 | 0.06 | 0.05 |

NINO3 (t − 3) | 0.56 | 0.55 | 0.52 | 0.49 | 0.48 |

NINO3 (t − 6) | 0.91 | 0.89 | 0.87 | 0.86 | 0.83 |

NINO3 (t − 9) | 0.55 | 0.53 | 0.50 | 0.49 | 0.48 |

NINO4 (t) | 0.11 | 0.12 | 0.10 | 0.09 | 0.08 |

NINO4 (t − 3) | 0.52 | 0.49 | 0.47 | 0.45 | 0.44 |

NINO4 (t − 6) | 0.90 | 0.88 | 0.85 | 0.83 | 0.82 |

NINO4 (t − 9) | 0.50 | 0.45 | 0.45 | 0.42 | 0.41 |

NINO3.4 (t) | 0.09 | 0.08 | 0.06 | 0.05 | 0.05 |

NINO3.4 (t − 3) | 0.51 | 0.50 | 0.49 | 0.47 | 0.45 |

NINO3.4 (t − 6) | 0.30 | 0.29 | 0.27 | 0.26 | 0.24 |

NINO3.4 (t − 9) | 0.50 | 0.44 | 0.47 | 0.45 | 0.43 |

PDO (t) | 0.08 | 0.07 | 0.06 | 0.06 | 0.05 |

PDO (t − 3) | 0.50 | 0.47 | 0.46 | 0.44 | 0.42 |

PDO (t − 6) | 0.29 | 0.27 | 0.25 | 0.22 | 0.21 |

PDO (t − 9) | 0.49 | 0.45 | 0.44 | 0.40 | 0.38 |

BA | |||||||

Objective Function | Maximum Load Ness | Objective Function | Minimum Frequency | Objective Function | Maximum Frequency | Objective Function | Population Size |

2.6 | 0.3 | 2.9 | 1 | 3.1 | 3 | 2.7 | 20 |

2.5 | 0.5 | 2.7 | 2 | 2.9 | 5 | 2.3 | 40 |

2.2 | 0.7 | 2.2 | 3 | 2.2 | 7 | 2.2 | 60 |

2.7 | 0.90 | 2.8 | 4 | 3.2 | 9 | 2.4 | 80 |

PSO | |||||||

Objective Function | w | Objective Function | C_{2} | Objective Function | C_{1} | Objective Function | Population Size |

3.5 | 0.3 | 4.4 | 1.6 | 4.1 | 1.6 | 4.12 | 20 |

2.89 | 0.5 | 3.1 | 1.8 | 3.90 | 1.8 | 3.89 | 40 |

2.93 | 0.7 | 2.2 | 2.0 | 3.82 | 2.0 | 3.82 | 60 |

3.23 | 0.90 | 2.8 | 2.2 | 3.89 | 2.2 | 3.94 | 80 |

GA | |||||||

Objective Function | Crossover Rate | Objective Function | Mutation Probability | Objective Function | Population Size | ||

7.01 | 0.30 | 7.12 | 0.20 | 7.25 | 20 | ||

6.14 | 0.50 | 6.91 | 0.40 | 6.92 | 40 | ||

6.34 | 0.70 | 6.14 | 0.60 | 6.12 | 60 | ||

6.52 | 0.90 | 6.45 | 0.80 | 6.25 | 80 |

Model | Train | Test | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | WI | NSE | RMSE | MAE | WI | NSE | |

ANFIS-GA | 3.22 | 2.89 | 0.87 | 0.88 | 4.25 | 4.02 | 0.85 | 0.84 |

ANFIS-PSO | 3.02 | 2.85 | 0.89 | 0.90 | 4.01 | 3.85 | 0.88 | 0.86 |

ANFIS-BA | 2.10 | 1.76 | 0.95 | 0.94 | 2.98 | 2.78 | 0.92 | 0.92 |

Model | p Value | d Factor |
---|---|---|

ANFIS-BA | 90% | 0.52 |

ANFIS-PSO | 86% | 0.72 |

ANFIS-GA | 83% | 0.75 |

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## Share and Cite

**MDPI and ACS Style**

Ehteram, M.; Afan, H.A.; Dianatikhah, M.; Ahmed, A.N.; Ming Fai, C.; Hossain, M.S.; Allawi, M.F.; Elshafie, A. Assessing the Predictability of an Improved ANFIS Model for Monthly Streamflow Using Lagged Climate Indices as Predictors. *Water* **2019**, *11*, 1130.
https://doi.org/10.3390/w11061130

**AMA Style**

Ehteram M, Afan HA, Dianatikhah M, Ahmed AN, Ming Fai C, Hossain MS, Allawi MF, Elshafie A. Assessing the Predictability of an Improved ANFIS Model for Monthly Streamflow Using Lagged Climate Indices as Predictors. *Water*. 2019; 11(6):1130.
https://doi.org/10.3390/w11061130

**Chicago/Turabian Style**

Ehteram, Mohammad, Haitham Abdulmohsin Afan, Mojgan Dianatikhah, Ali Najah Ahmed, Chow Ming Fai, Md Shabbir Hossain, Mohammed Falah Allawi, and Ahmed Elshafie. 2019. "Assessing the Predictability of an Improved ANFIS Model for Monthly Streamflow Using Lagged Climate Indices as Predictors" *Water* 11, no. 6: 1130.
https://doi.org/10.3390/w11061130