# Improving Drought Modeling Using Hybrid Random Vector Functional Link Methods

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## Abstract

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^{2}). The HGS algorithm provided a better performance than the alternative algorithms, and it considerably improved the accuracy of the RVFL method in drought forecasting; the improvement in RMSE for the SPI3, SP6, SPI9, and SPI12 was by 6.14%, 11.89%, 14.14%, 24.5% in station 1, by 6.02%, 17.42%, 13.49%, 24.86% in station 2 and by 7.55%, 26.45%, 15.27%, 13.21% in station 3, respectively. The outcomes of the study recommend the use of a HGS-based RVFL in drought modeling.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study

^{2}[39]. Bangladesh has a subtropical monsoon climate with large spatial and temporal variability. It has four different seasons (e.g., pre-monsoon, monsoon, post-monsoon, and winter). In 85% of the country, annual rainfall occurs in the monsoon period. Less than 6% of rainfall occurs in the winter period. In the northwest, the annual mean total rainfall is 1329 mm [40]. In the winter season, the mean temperature varies between 17 and 20.6 °C while in the post-monsoon, it ranges from 26.9 to 31.1 °C. The Bangladesh Meteorological Department (BMD) reported that the temperature in pre-monsoon increases to 45 °C and the temperature in winter falls at 5 °C in the northwest part of the study area [41]. The northwest region experiences extreme events such as drought more frequently than the remaining parts of the country.

#### 2.2. Standard Precipitation Index (SPI)

_{i}is the precipitation data sample; $\overline{x}$ is the average precipitation value. G(x) is obtained by integrating the probability density function g(x) of I then completing the gamma distribution. The accumulated value under the time scale is:

_{0}= 2.515517; c

_{1}= 0.802853; c

_{2}= 0.010328; d

_{1}= 1.432788; d

_{2}= 0.189269; d

_{3}= 0.001308 (McKee et al., 1993). Classification of drought based on SPI can be obtained from the past studies [47,48].

#### 2.3. Random Vector Functional Link Networks (RVFL)

_{s}sample data (X), characterized as a pair (x

_{i}, y

_{i}), where y

_{i}is the goal variable. The input data are processed through the middle (hidden) nodes afterward. These middle nodes are called nodes of enhancement. Each middle node’s output is computed as below:

_{j}indicates the weights between the input layer nodes and the enhancement (middle) nodes, and α

_{j}refers to the bias. S refers to the scale factor and is evaluated throughout the optimization procedure. The formula below computes the outcomes of RVFL:

_{1}and the outcome of middle layer F

_{2}:

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

_{i,t}, and x

_{i,t}represent the next and the current iterations, the velocity, and the position of the particle i at iteration t, respectively. pbest

_{i,t}and gbest

_{i,t}then are the individual’s best previous and global position i. Moreover, c

_{1}and c

_{2}are the learning rates that indicate the influence of the social and cognitive components and r

_{1}and r

_{2}are arbitrarily selected numbers that vary between 0 and 1. Eventually, X is a convergence factor and is usually considered to be about 0.729 [51].

#### 2.5. Genetic Algorithm (GA)

#### 2.6. Grey Wolves Optimization (GWO)

_{p}(t) is the target vector which is likely to be the position of the wolf α. D and A are calculated as below:

_{1}− α

_{2}, X (t) shows the grey wolf situation; α is usually reduced linearly from 2 to 0; r

_{1}and r

_{2}are random vectors from 0 to 1. Considering the positions of wolves α, β, and δ, the group ω changes its location according to the below equation:

_{1}, X

_{2}, X

_{3}are calculated as follows:

_{α}, X

_{β}, and X

_{δ}correspond to α, β, and δ positions, respectively. Moreover, D

_{α}, D

_{β}, and D

_{δ}are evaluated as below:

_{1}, C

_{2}, and C

_{3}are random vectors, while X(t) depicts the current solution. When the abovementioned steps are completed, the corresponding fitness value is calculated, and the best solution is assumed to be α. This step keeps going to satisfy the stopping criterion, where GWO is ended.

#### 2.7. Social Spider Optimization (SSO)

_{c}and S

_{b}show the best closest neighbor and the fittest spider in the population, correspondingly. The procedure allowing the male spider to update in position varies from the females. The males here are divided into dominated (D) and non-dominated (ND). The mathematical expression of the before-mentioned categories, i.e., ND and D-type males, is as below:

_{f}is the closest female to i

_{th}male, and α, δ, and rare random values in [0, 1]. The mating operator, which modifies the search agents, is the last tool of SSO. ND males need to meet females in a distinct area around them, called the mating radius. There may be more than one pair of males and females in the mating radius, and the fitness of pairs, therefore, is essential in their selection. Producing the new generation is the next step after the mating process. The new generation fitness, then, is checked. If it is better than the worst spider in population, the new generation will be saved, and the worst will be removed. The abovementioned process is iterated until the constraints are satisfied.

#### 2.8. Salp SWARM Algorithm (SSA)

_{i}, lb

_{i,}and ub

_{i}are the first salp, food position, the lower bound, and the upper bound of the i

_{th}dimension, respectively. The coefficient r

_{1}is also calculated as below, while r

_{2}and r

_{3}are random numbers between 0 to 1:

_{th}salp in the i

_{th}dimension, while δ

_{0}and t are initial speed and time:

#### 2.9. Hunger Games Search (HGS)

_{1}and r

_{2}are random numbers in the range of [0, 1]. randn (1) is a random number and satisfies normal distribution. t shows the existing iterations. $\overrightarrow{{W}_{1}}$ and $\overrightarrow{{W}_{2}}$ indicate the weights of hunger. Moreover, $\overrightarrow{{X}_{b}}$ and $\overrightarrow{X(t)}$ show the location of the best separate iteration and each individual’s location, respectively. (1 + randn (1)) represents the random food search by a hungry agent at a current location. The range of individual activity can be modeled $\left|\overrightarrow{{X}_{b}}-\overrightarrow{X(t)}\right|$ at present. When $\left|\overrightarrow{{X}_{b}}-\overrightarrow{X(t)}\right|$ is multiplied by $\overrightarrow{{W}_{2}}$, the result can show the effect of hunger on the activity range. $\overrightarrow{R}$ is a ranging controller that limits the activity range and slowly decreases to 0. E can be formulated as below:

_{3}, r

_{4}, and r

_{5}are random numbers in the range of [0, 1]. hungry (i), therefore, can be evaluated as below:

_{6}is a random number between 0 and 1; WF indicates the worst fitness gained in the current iteration so far; and UB and LB are the upper and lower bounds of the feature space, respectively. The hunger sensation H is limited to a lower bound, LH, to allow the algorithm to obtain a better performance [60].

#### 2.10. Model Performance Evaluation Matrices

## 3. Application and Results

## 4. Discussion

^{2}of 0.780 and 0.908. The study conducted by Jalalkamali et al. [67] examined the accuracy of three methods, MLP, ANN, and ANFIS, in forecasting SPI-3, SPI-6, and SPI-9. They found R

^{2}of 0.801, 0.774, and 0.706 for the optimal models, respectively. The study of Kisi et al. [25] investigated the performances of the ANFIS hybridized with metaheuristic methods in forecasting SPI indices. The best models produced R

^{2}of 0.760, 0.850, 0.771, and 0.832 for the SPI-3, SPI-6, SPI-9, and SPI-12, respectively. Recently, in a study performed by Gorgij et al. [68], the efficiency of four methods, long short-term memory (LSTM), extra-trees (ET), the vector autoregressive approach (VAR), and multivariate adaptive regression spline (MARS), was investigated in forecasting SPI-3, SPI-6, SPI-9, and SPI-12 in four stations. The NSE of the best model (LSTM) ranged from 0.682 to 0.697, 0.780 to 0.815, 0.777 to 0.848 and 0.852 to 0.904 for forecasting SPI-3, SPI-6, SPI-9, and SPI-12, respectively. The outcomes of this study reported in Tables S2–S7 indicate that the RVFL–HGS model can generate accurate drought forecasts based on the standard precipitation index.

## 5. Conclusions

^{2}, NSE, and visual inspections, including scatter plots, and Taylor and violin charts. The metaheuristic algorithms improved the accuracy of the single RFVL in the simulation (training stage) and forecasting (testing stage) drought based on SPI3, SPI6, SPI9, and SPI12. Among the algorithms, the HGS provided the best performance by improving the RMSE accuracy of RFVL by 6.14, 11.89, 14.14 and 24.5% in forecasting the SPI3, SP6, SPI9, and SPI12 in station 1, by 6.02, 17.42, 13.49 and 24.86 in station two and by 7.55, 26.45, 15.27 and 13.21 in station 3, respectively. The HGS also increased the forecasting accuracy of the RVFL-based PSO, GA, GWO, SSO, SSA in forecasting SPI3, SPI6, SPI9, and SPI12.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Scatterplots of the observed and predicted SPI3 drought index by different RFVL based models in the test period —Station 2.

**Figure 4.**Taylor and violin diagrams of the SPI3 drought index by different RFVL-based models in the test period–Station 2.

RFVL | Activation function | radial basis |

Hidden neurons | 200 | |

PSO | $\mathrm{Cognitive}\text{}\mathrm{component}\text{}({c}_{1}$) | 2 |

$\mathrm{Social}\text{}\mathrm{component}\text{}({c}_{2}$) | 2 | |

inertia weight | 0.2–0.9 | |

GA | Crossover percentage | 0.9 |

Mutation percentage | 0.5 | |

Mutation rate | 0.1 | |

GWO | $a$ | decreased from 2 to 0 |

HGS | $L$ | 0.03 |

$LH$ | 1000 | |

SSO | $PF$ | 0.7 |

SSA | ${v}_{0}$ | 0 |

All algorithms | Population | 40 |

Number of iterations | 100 | |

Number of runs for each algorithm | 20 |

**Table 2.**The comparison of different RFVL-based models in forecasting SPI3 and SPI6 drought indices—Station 2.

Drought Indices | Input Comb. | Models | Training | Testing | ||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |||

SPI3 | SPI3t-1 | RFVL | 0.506 | 0.392 | 0.803 | 0.803 | 0.570 | 0.409 | 0.705 | 0.647 |

PSO | 0.495 | 0.381 | 0.814 | 0.814 | 0.547 | 0.374 | 0.740 | 0.688 | ||

GA | 0.493 | 0.380 | 0.817 | 0.817 | 0.537 | 0.363 | 0.748 | 0.706 | ||

GWO | 0.482 | 0.377 | 0.837 | 0.837 | 0.535 | 0.357 | 0.757 | 0.710 | ||

SSO | 0.472 | 0.368 | 0.865 | 0.865 | 0.523 | 0.351 | 0.765 | 0.730 | ||

SSA | 0.470 | 0.366 | 0.870 | 0.870 | 0.523 | 0.344 | 0.767 | 0.731 | ||

HGS | 0.454 | 0.351 | 0.883 | 0.883 | 0.511 | 0.340 | 0.769 | 0.734 | ||

SPI3t-1,SPI3t-2 | RFVL | 0.492 | 0.378 | 0.820 | 0.820 | 0.543 | 0.368 | 0.736 | 0.696 | |

PSO | 0.479 | 0.374 | 0.842 | 0.842 | 0.541 | 0.363 | 0.746 | 0.700 | ||

GA | 0.474 | 0.372 | 0.851 | 0.851 | 0.525 | 0.349 | 0.760 | 0.727 | ||

GWO | 0.465 | 0.359 | 0.865 | 0.865 | 0.522 | 0.346 | 0.764 | 0.733 | ||

SSO | 0.463 | 0.360 | 0.868 | 0.868 | 0.516 | 0.339 | 0.773 | 0.743 | ||

SSA | 0.444 | 0.344 | 0.898 | 0.898 | 0.507 | 0.331 | 0.781 | 0.748 | ||

HGS | 0.440 | 0.335 | 0.905 | 0.905 | 0.502 | 0.328 | 0.784 | 0.759 | ||

SPI6 | SPI6t-1,SP6t-2 | RFVL | 0.375 | 0.268 | 0.843 | 0.843 | 0.489 | 0.260 | 0.751 | 0.705 |

PSO | 0.365 | 0.253 | 0.858 | 0.858 | 0.446 | 0.229 | 0.788 | 0.740 | ||

GA | 0.364 | 0.258 | 0.858 | 0.858 | 0.440 | 0.218 | 0.791 | 0.749 | ||

GWO | 0.353 | 0.249 | 0.875 | 0.875 | 0.404 | 0.223 | 0.832 | 0.755 | ||

SSO | 0.329 | 0.229 | 0.858 | 0.858 | 0.389 | 0.212 | 0.824 | 0.778 | ||

SSA | 0.304 | 0.218 | 0.891 | 0.891 | 0.378 | 0.190 | 0.823 | 0.794 | ||

HGS | 0.304 | 0.218 | 0.891 | 0.891 | 0.372 | 0.190 | 0.831 | 0.802 | ||

SPI6t-1,SPI6t-2,SPI6t-3,SPI6t-4 | RFVL | 0.370 | 0.252 | 0.850 | 0.850 | 0.465 | 0.219 | 0.774 | 0.709 | |

PSO | 0.340 | 0.245 | 0.859 | 0.859 | 0.417 | 0.205 | 0.792 | 0.746 | ||

GA | 0.339 | 0.241 | 0.864 | 0.864 | 0.406 | 0.196 | 0.798 | 0.780 | ||

GWO | 0.332 | 0.238 | 0.871 | 0.871 | 0.377 | 0.192 | 0.810 | 0.785 | ||

SSO | 0.320 | 0.231 | 0.880 | 0.880 | 0.384 | 0.188 | 0.812 | 0.786 | ||

SSA | 0.299 | 0.214 | 0.898 | 0.898 | 0.355 | 0.175 | 0.840 | 0.798 | ||

HGS | 0.284 | 0.195 | 0.921 | 0.921 | 0.342 | 0.159 | 0.848 | 0.803 |

**Table 3.**The comparison of different RFVL-based models in forecasting SPI9 and SPI12 drought indices—Station 2.

Drought Indices | Input Comb. | Models | Training | Testing | ||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |||

SPI9 | SPI9t-1,SPI9t-2, SPI9t-3, | RFVL | 0.395 | 0.242 | 0.796 | 0.796 | 0.430 | 0.291 | 0.788 | 0.760 |

PSO | 0.390 | 0.242 | 0.801 | 0.801 | 0.400 | 0.288 | 0.815 | 0.801 | ||

GA | 0.389 | 0.241 | 0.803 | 0.803 | 0.397 | 0.277 | 0.825 | 0.806 | ||

GWO | 0.379 | 0.239 | 0.816 | 0.815 | 0.386 | 0.267 | 0.830 | 0.810 | ||

SSO | 0.346 | 0.218 | 0.853 | 0.853 | 0.379 | 0.265 | 0.833 | 0.812 | ||

SSA | 0.313 | 0.190 | 0.878 | 0.878 | 0.374 | 0.262 | 0.835 | 0.816 | ||

HGS | 0.307 | 0.189 | 0.884 | 0.884 | 0.372 | 0.260 | 0.838 | 0.823 | ||

SPI9t-1,SPI9t-2,SPI9t-3,SPI9t-4,SPI9t-5 | RFVL | 0.416 | 0.273 | 0.770 | 0.769 | 0.490 | 0.315 | 0.733 | 0.709 | |

PSO | 0.396 | 0.244 | 0.794 | 0.794 | 0.427 | 0.293 | 0.793 | 0.765 | ||

GA | 0.395 | 0.241 | 0.795 | 0.795 | 0.420 | 0.288 | 0.806 | 0.778 | ||

GWO | 0.366 | 0.226 | 0.828 | 0.828 | 0.408 | 0.275 | 0.820 | 0.790 | ||

SSO | 0.355 | 0.220 | 0.842 | 0.842 | 0.401 | 0.270 | 0.827 | 0.804 | ||

SSA | 0.349 | 0.218 | 0.849 | 0.849 | 0.397 | 0.267 | 0.831 | 0.806 | ||

HGS | 0.340 | 0.210 | 0.860 | 0.860 | 0.395 | 0.264 | 0.834 | 0.810 | ||

SPI12 | SPI12t-1,SPI12t-2, SPI12t-3, | RFVL | 0.300 | 0.165 | 0.897 | 0.897 | 0.346 | 0.208 | 0.793 | 0.778 |

PSO | 0.298 | 0.162 | 0.900 | 0.900 | 0.337 | 0.213 | 0.804 | 0.789 | ||

GA | 0.295 | 0.163 | 0.902 | 0.902 | 0.327 | 0.203 | 0.814 | 0.801 | ||

GWO | 0.284 | 0.158 | 0.914 | 0.913 | 0.302 | 0.181 | 0.837 | 0.820 | ||

SSO | 0.247 | 0.140 | 0.947 | 0.947 | 0.287 | 0.163 | 0.874 | 0.851 | ||

SSA | 0.240 | 0.134 | 0.953 | 0.953 | 0.271 | 0.156 | 0.883 | 0.864 | ||

HGS | 0.225 | 0.118 | 0.966 | 0.966 | 0.260 | 0.151 | 0.888 | 0.875 | ||

SPI12t-1,SPI12t-2,SPI12t-3,SPI12t-4,SPI12t-5 | RFVL | 0.312 | 0.174 | 0.885 | 0.885 | 0.409 | 0.276 | 0.722 | 0.708 | |

PSO | 0.309 | 0.164 | 0.895 | 0.895 | 0.355 | 0.233 | 0.790 | 0.766 | ||

GA | 0.301 | 0.161 | 0.896 | 0.896 | 0.353 | 0.224 | 0.798 | 0.769 | ||

GWO | 0.293 | 0.157 | 0.907 | 0.907 | 0.329 | 0.194 | 0.826 | 0.816 | ||

SSO | 0.284 | 0.152 | 0.913 | 0.913 | 0.291 | 0.169 | 0.869 | 0.845 | ||

SSA | 0.272 | 0.145 | 0.925 | 0.925 | 0.287 | 0.159 | 0.875 | 0.856 | ||

HGS | 0.249 | 0.137 | 0.946 | 0.946 | 0.269 | 0.157 | 0.884 | 0.867 |

**Table 4.**The comparison of different RFVL-based models in SPI3 drought estimation for the test period using periodicity with optimal inputs.

Station | Input Comb. | Models | Training | Testing | ||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |||

Station 1 | OPT SPI3+MN | RFVL | 0.493 | 0.343 | 0.788 | 0.788 | 0.519 | 0.361 | 0.764 | 0.678 |

PSO | 0.476 | 0.328 | 0.816 | 0.816 | 0.500 | 0.343 | 0.812 | 0.718 | ||

GA | 0.475 | 0.323 | 0.818 | 0.817 | 0.498 | 0.344 | 0.791 | 0.723 | ||

GWO | 0.473 | 0.318 | 0.821 | 0.821 | 0.496 | 0.343 | 0.796 | 0.727 | ||

SSO | 0.450 | 0.310 | 0.856 | 0.856 | 0.491 | 0.341 | 0.801 | 0.731 | ||

SSA | 0.439 | 0.303 | 0.873 | 0.873 | 0.487 | 0.338 | 0.810 | 0.746 | ||

HGS | 0.430 | 0.290 | 0.892 | 0.892 | 0.482 | 0.327 | 0.821 | 0.756 | ||

Station 2 | OPT SPI3+MN | RFVL | 0.467 | 0.321 | 0.782 | 0.782 | 0.541 | 0.367 | 0.768 | 0.730 |

PSO | 0.456 | 0.310 | 0.798 | 0.798 | 0.537 | 0.365 | 0.777 | 0.733 | ||

GA | 0.445 | 0.304 | 0.818 | 0.818 | 0.535 | 0.362 | 0.778 | 0.735 | ||

GWO | 0.442 | 0.301 | 0.823 | 0.823 | 0.526 | 0.356 | 0.802 | 0.762 | ||

SSO | 0.435 | 0.286 | 0.838 | 0.838 | 0.512 | 0.345 | 0.814 | 0.784 | ||

SSA | 0.431 | 0.283 | 0.845 | 0.845 | 0.509 | 0.343 | 0.817 | 0.784 | ||

HGS | 0.402 | 0.261 | 0.890 | 0.890 | 0.505 | 0.341 | 0.819 | 0.787 | ||

Station 3 | OPT SPI3+MN | RFVL | 0.489 | 0.376 | 0.823 | 0.823 | 0.541 | 0.365 | 0.739 | 0.700 |

PSO | 0.476 | 0.371 | 0.845 | 0.845 | 0.538 | 0.360 | 0.749 | 0.702 | ||

GA | 0.471 | 0.370 | 0.854 | 0.854 | 0.522 | 0.346 | 0.763 | 0.730 | ||

GWO | 0.463 | 0.357 | 0.869 | 0.869 | 0.517 | 0.343 | 0.767 | 0.735 | ||

SSO | 0.460 | 0.355 | 0.872 | 0.872 | 0.513 | 0.336 | 0.777 | 0.746 | ||

SSA | 0.440 | 0.340 | 0.902 | 0.902 | 0.504 | 0.328 | 0.782 | 0.750 | ||

HGS | 0.434 | 0.332 | 0.908 | 0.908 | 0.500 | 0.325 | 0.787 | 0.765 |

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## Share and Cite

**MDPI and ACS Style**

Adnan, R.M.; Mostafa, R.R.; Islam, A.R.M.T.; Gorgij, A.D.; Kuriqi, A.; Kisi, O.
Improving Drought Modeling Using Hybrid Random Vector Functional Link Methods. *Water* **2021**, *13*, 3379.
https://doi.org/10.3390/w13233379

**AMA Style**

Adnan RM, Mostafa RR, Islam ARMT, Gorgij AD, Kuriqi A, Kisi O.
Improving Drought Modeling Using Hybrid Random Vector Functional Link Methods. *Water*. 2021; 13(23):3379.
https://doi.org/10.3390/w13233379

**Chicago/Turabian Style**

Adnan, Rana Muhammad, Reham R. Mostafa, Abu Reza Md. Towfiqul Islam, Alireza Docheshmeh Gorgij, Alban Kuriqi, and Ozgur Kisi.
2021. "Improving Drought Modeling Using Hybrid Random Vector Functional Link Methods" *Water* 13, no. 23: 3379.
https://doi.org/10.3390/w13233379