# Hybridizing Artificial Intelligence Algorithms for Forecasting of Sediment Load with Multi-Objective Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Study Area

^{2}or approximately 4.3% of India’s total land area [46]. Odisha receives 53% of the river’s basin area, while Chhattisgarh receives 46% and Maharashtra, Madhya Pradesh, and Jharkhand share the remaining 1% [46]. Until it enters the Bay of Bengal, the river flows for a total of 851 km. Thr MR was located between 19°20′ and 23°35′ north, and 80°30′ and 86°50′ east. The MR contains the Hirakud dam which is the world’s largest earthen dam. In terms of current sediment load, the MR is second among Indian peninsular rivers. Figure 1 shows the MR basin elevation map and the locations of all 11 hydro-climatological sites. The average annual RF was between 1200 and 1400 mm [47]. Approximately 90% of the yearly RF that the MR basin receives occurs during the monsoons. The MR basin has a dispersed pattern of RF strength. In the MR basin, the warmest months are April and May, with summer temperatures of 39 to 45 °C, and the coldest months are December and January, with winter temperatures of 4 to 12 °C [47]. The two largest bodies of water in the MR are Lake Chilka and the Hirakud Dam.

## 3. Methodology and Data

^{9}[24]. In the fifth section, the biases and weights of the connections of ANN models are shown. The length of the chromosome changes because the number of hidden neurons and the number of inputs change. The ANN-MOGA forecasting models are designed with GA parameters such as the number of generations, the size of the population, the rate of mutation, and the probability of a crossover. In this study, a uniform crossover with a high probability value (0.6) and a low probability of constant mutation (0.05) was used. The values of each chromosome’s fitness are estimated using the fitness function (RMSE) of the training dataset. The maximum generations (50) were considered as stopping criteria.

_{0}of size $N$ is produced by the mutation, selection, and crossover operators at iteration 0. The overall number of chromosomal solutions for any iteration t following the genetic operations is R

_{t}= P

_{t}∪ Q

_{t}becomes twice (2N). P and Q represent the parent and child populations, respectively. R represents the total number of chromosomal solutions after the genetic operations. The objective functions of each solution (R

_{t}) were determined by calculating, and the solutions have been ranked using the previously discussed Non-Dominated Sorting criteria and crowding distance. The rest solutions were eliminated from the solution space, and the top $N$ solutions determined by their whole rank were chosen (referred to as elitism) for the following generation. The maximum number of individuals permitted from the i

^{th}non-dominated front as shown in Equation (5) is provided diversity in the new population, based on the geometric distribution [36].

_{i}($i$ = 0, 1, 2,....., $n$) represents the regression model’s coefficients. The MAR was designed using training datasets and the linear combination of the Autoregressive of multiple variables (WD, T, RF, and SL and spatial variables (R, CA, and RT). The MAR formula is shown below

_{i}, b

_{i}, c

_{i}, d

_{i}, e, f, and g (i = 1,2, 3,…, n) represent the coefficients of the MAR model. The a

_{i}, b

_{i}, c

_{i,}and d

_{i}represent the coefficients of WD, RF, T and SL, respectively. The coefficients of the RT, R, and CA are represented by the values e, f, and g, respectively. The maximum lag (n) in AR and MAR model for the SL forecasting is 12, after which the cyclicity begins due to seasonal behaviour of the data. There are four temporal variables: WD, SL, RF, and T. The RT, CA and R are the spatial variables.

## 4. Results and Discussion

#### 4.1. Data Analysis

#### 4.2. ANN-MOGA Forecasting Model

^{°}line, except for the four gauging stations that were mentioned earlier (Figure 8). The scatter plots and hydrographs show that the magnitudes and medium, high, and low SL forecasted values generated by the best ANN-MOGA-51 forecasting model are also fairly close to the corresponding actual SL values. The ANN-MOGA-51 model displayed a positive value SL at each of the 11 gauge stations, although the SL output was either 0 or very near zero (Figure 7 and Figure 8). Based on these findings, it was determined that the application of ANN in conjunction with GA is the method that yields the most accurate results for calculating SL in the MR basin system. The forecasting model provides the highest level of accuracy at the Tikarapara gauging station compared to any other station gauging station. This may be due to the location of Tikarapara, which is at the most downstream portion of the MR basin and possesses the highest WD, CA, RF, and SL of any of the gauging stations [24,47].

#### 4.3. AR Forecasting Model

#### 4.4. The Multivariate Autoregressive (MAR) Forecasting Model

#### 4.5. Comparison Results of Forecasting Models

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The elevation map of the Mahanadi River basin with the geographical location of 11 gauging stations [48].

**Figure 3.**Spatial variation of monthly average hydro-climatic and geomorphological data such as water discharge (cms) (m

^{3}/s), temperature (°C), rainfall (mm) and sediment load (t/ month), relief (m), and catchment area (km

^{2}).

**Figure 4.**Comparative analysis of long-term variation over the past two decades of (

**a**) WD, (

**b**) RF, (

**c**) SL, and (

**d**) T.

**Figure 5.**The autocorrelation function (ACF) for monthly hydro-climatical data (

**a**) SL, (

**b**) WD, (

**c**) RF, (

**d**) T.

**Figure 6.**(

**a**) Dilemma between the bias and variance objectives. (

**b**) Variation of individuals and distance. (

**c**) Variation of individuals and corresponding ranking. (

**d**) Generation-wise variation of average spreads of Pareto.

**Figure 7.**Comparison of the actual and forecasted SL during the testing phase of the ANN-MOGA-51 forecasting model (

**a**–

**k**).

**Figure 8.**Scatter plot of the actual and forecasted SL of the ANN-MOGA-51 forecasting model during the testing phase (

**a**–

**k**).

**Figure 9.**Comparison of the actual and forecasted SL during the testing phase of the AR forecasting model (

**a**–

**k**).

**Figure 10.**Scatter plot of the actual and forecasted SL of the AR forecasting model during the testing phase (

**a**–

**k**).

**Figure 11.**Comparison of the actual and forecasted SL during the testing phase of the MAR forecasting model (

**a**–

**k**).

**Figure 12.**Scatter plot of the actual and forecasted SL of the MAR forecasting model during the testing phase (

**a**–

**k**).

**Table 1.**Spearman rank correlation coefficient

^{®}of hydro-climatolic data from eleven stations for the MR.

Stations | r1 (WD-SL) | r2 (SL-RF) | r3 (SL-T) |
---|---|---|---|

Tikarapara | 0.891951579 | 0.667566099 | 0.167164223 |

Sundargarh | 0.933162643 | 0.719490012 | 0.083275757 |

Simga | 0.912171157 | 0.669144968 | 0.016117111 |

Jondhara | 0.953615062 | 0.634788084 | 0.024002708 |

Andhiyarakhore | 0.930440984 | 0.679483679 | 0.172300271 |

Kurubhata | 0.914790531 | 0.739541786 | 0.09457768 |

Bamnidih | 0.792975574 | 0.673800075 | 0.19294038 |

Rajim | 0.933515452 | 0.652771319 | −0.053703323 |

Kantamal | 0.784858255 | 0.653582343 | 0.045884228 |

Baronda | 0.896128553 | 0.718865603 | 0.170015607 |

Basantpur | 0.900261237 | 0.717722236 | 0.120971863 |

Models | Number of Initial Inputs | Input Parameters |
---|---|---|

ANN-MOGA-51 | 51 | SL, WD, RF, T, RT, R and CA |

ANN-MOGA-48 | 48 | SL, WD, RF and T |

ANN-MOGA-15 | 15 | SL, RT, R and CA |

ANN-MOGA-12 | 12 | SL |

**Table 3.**Performance comparison of the hybrid multi-objective GA-based ANN forecasting models in the testing phase.

Models | RMSE | Initially Inputs No. | MSE | MAE | VAR | r |
---|---|---|---|---|---|---|

ANN-MOGA-51 | 0.011639 | 51 | 0.000135 | 0.003802 | 0.000136 | 0.643313 |

ANN-MOGA-48 | 0.013343 | 48 | 0.000178 | 0.00381194 | 0.0001783 | 0.5674853 |

ANN-MOGA-15 | 0.013637 | 15 | 0.000186 | 0.0044 | 0.000186 | 0.513217 |

ANN-MOGA-12 | 0.01181 | 12 | 0.000139 | 0.003626 | 0.00014 | 0.623344 |

Models | Transfer Function | Neurons | Inputs | µ |
---|---|---|---|---|

ANN-MOGA-51 | Log-sigmoid, pure linear | 29 | 22 | 2 |

ANN-MOGA-48 | Tan-sigmoid, pure linear | 19 | 21 | 9 |

ANN-MOGA-15 | Pure linear, tan-sigmoid | 15 | 10 | 10 |

ANN-MOGA-12 | Tan-sigmoid, pure linear | 4 | 6 | 6 |

**Table 5.**Error statistics of ANN-MOGA-51 forecasting model during validation, testing, and training phase.

ANN-MOGA-51 | MSE | RMSE | r | Error Variance | MAE |
---|---|---|---|---|---|

Training | 0.000241 | 0.015526 | 0.668938 | 0.000241 | 0.004677 |

Validation | 5.25 × 10^{−5} | 0.007243 | 0.7730 | 5.26 × 10^{−5} | 0.002867 |

Testing | 0.000135 | 0.011639 | 0.643313 | 0.000136 | 0.003802 |

Tikarapara | 0.000396 | 0.019905 | 0.731051 | 0.000407 | 0.010517 |

Simga | 1.17 × 10^{−5} | 0.003422 | 0.5930 | 9.81 × 10^{−6} | 0.001707 |

Andhiyarakhore | 1.10 × 10^{−5} | 0.003319 | 0.4001 | 9.06 × 10^{−5} | 0.001749 |

Sundargarh | 2.60 × 10^{−5} | 0.005097 | 0.635 | 2.67 × 10^{−5} | 0.002466 |

Bamnidih | 3.33 × 10^{−5} | 0.005769 | 0.695 | 3.27 × 10^{−5} | 0.002765 |

Jondhara | 3.02 × 10^{−5} | 0.005499 | 0.737 | 2.91 × 10^{−5} | 0.002534 |

Kurubhata | 2.06 × 10^{−5} | 0.001434 | 0.914 | 1.97 × 10^{−6} | 0.000865 |

Basantpur | 0.000229 | 0.015121 | 0.478143 | 0.000222 | 0.006185 |

Baronda | 5.66 × 10^{−6} | 0.002379 | 0.495 | 5.23 × 10^{−6} | 0.001319 |

Rajim | 2.10 × 10^{−6} | 0.001448 | 0.669 | 2.15 × 10^{−6} | 0.000722 |

Kantamal | 0.000806 | 0.028395 | 0.659518 | 0.000801 | 0.01198 |

**Table 6.**Error statistics of the autoregressive (AR) forecasting model at different gauging stations.

AR | RMSE | MSE | MAE | VAR | r |
---|---|---|---|---|---|

Training | 0.01640 | 0.00027 | 0.00473 | 0.00027 | 0.61268 |

Testing | 0.01335 | 0.00018 | 0.00365 | 0.00018 | 0.49670 |

Tikarapara | 0.02576 | 0.00066 | 0.01119 | 0.00066 | 0.51087 |

Simga | 0.00185 | 3.42 × 10^{−6} | 0.00069 | 3.49 × 10^{−06} | 0.19322 |

Andhiyarakhore | 0.00020 | 4.07 × 10^{−8} | 9.30×10^{−5} | 4.15 × 10^{−08} | 0.40166 |

Sundargarh | 0.00555 | 3.08 × 10^{−5} | 0.00303 | 3.02 × 10^{−05} | 0.44791 |

Bamnidih | 0.00024 | 5.56 × 10^{−8} | 0.00012 | 5.46 × 10^{−08} | 0.58226 |

Jondhara | 0.00668 | 4.46 × 10^{−5} | 0.00323 | 4.48 × 10^{−05} | 0.48917 |

Kurubhata | 0.00222 | 4.92 × 10^{−6} | 0.00103 | 4.72 × 10^{−06} | 0.74650 |

Basantpur | 0.01059 | 0.00011 | 0.00411 | 0.00011 | 0.32857 |

Baronda | 0.00187 | 3.49 × 10^{−6} | 0.00073 | 3.56 × 10^{−6} | 0.25906 |

Rajim | 0.00177 | 3.14 × 10^{−6} | 0.00079 | 3.18 × 10^{−6} | 0.37205 |

Kantamal | 0.03386 | 0.00115 | 0.01481 | 0.00116 | 0.36473 |

MAR | MAE | VAR | r | MSE | RMSE |
---|---|---|---|---|---|

Training | 0.00640 | 0.00023 | 0.68010 | 0.00023 | 0.01516 |

Testing | 0.00562 | 0.00017 | 0.55620 | 0.00017 | 0.01296 |

Tikarapara | 0.01306 | 0.00054 | 0.62380 | 0.00052 | 0.02284 |

Simga | 0.00395 | 2.65×10^{−5} | 0.49380 | 2.99×10^{−5} | 0.00547 |

Andhiyarakhore | 0.00260 | 1.20×10^{−5} | 0.39130 | 1.16×10^{−5} | 0.00341 |

Sundargarh | 0.00486 | 4.91×10^{−5} | 0.45670 | 4.79×10^{−5} | 0.00692 |

Bamnidih | 0.00272 | 1.53×10^{−5} | 0.61440 | 1.66×10^{−5} | 0.00407 |

Jondhara | 0.00458 | 5.44×10^{−5} | 0.63700 | 5.36×10^{−5} | 0.00732 |

Kurubhata | 0.00228 | 1.01×10^{−5} | 0.74210 | 9.80×10^{−6} | 0.00313 |

Basantpur | 0.00610 | 0.00010 | 0.62500 | 0.00011 | 0.01025 |

Baronda | 0.00306 | 1.75×10^{−5} | 0.44590 | 1.70×10^{−5} | 0.00412 |

Rajim | 0.00309 | 1.64×10^{−5} | 0.43390 | 1.59×10^{−5} | 0.00398 |

Kantamal | 0.01652 | 0.00111 | 0.39780 | 0.00108 | 0.03292 |

**Table 8.**Statistical performance evaluation of ANN-MOGA-51, AR, and MAR models along with all gauging stations in a testing phase.

Models | ANN-MOGA-51 | MAR | AR | |||
---|---|---|---|---|---|---|

Statistics | RMSE | r | RMSE | r | RMSE | r |

Testing | 0.01164 | 0.6433 | 0.01296 | 0.5562 | 0.01335 | 0.4967 |

Tikarapara | 0.01991 | 0.7311 | 0.02284 | 0.6238 | 0.02576 | 0.5109 |

Simga | 0.00342 | 0.5930 | 0.00547 | 0.4938 | 0.00185 | 0.1932 |

Andhiyarakhore | 0.00332 | 0.4001 | 0.00341 | 0.3913 | 0.00020 | 0.4017 |

Sundargarh | 0.00510 | 0.6350 | 0.00692 | 0.4567 | 0.00555 | 0.4479 |

Bamnidih | 0.00577 | 0.6950 | 0.00407 | 0.6144 | 0.00024 | 0.5821 |

Jondhara | 0.00550 | 0.7370 | 0.00732 | 0.6370 | 0.00668 | 0.4892 |

Kurubhata | 0.00143 | 0.9140 | 0.00313 | 0.7421 | 0.00222 | 0.7465 |

Basantpur | 0.01512 | 0.4781 | 0.01025 | 0.6250 | 0.01059 | 0.3286 |

Baronda | 0.00238 | 0.4950 | 0.00412 | 0.4459 | 0.00187 | 0.2591 |

Rajim | 0.00145 | 0.6690 | 0.00398 | 0.4339 | 0.00177 | 0.3721 |

Kantamal | 0.02840 | 0.6595 | 0.03292 | 0.3978 | 0.03386 | 0.3647 |

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**MDPI and ACS Style**

Yadav, A.; Ali Albahar, M.; Chithaluru, P.; Singh, A.; Alammari, A.; Kumar, G.V.; Miro, Y.
Hybridizing Artificial Intelligence Algorithms for Forecasting of Sediment Load with Multi-Objective Optimization. *Water* **2023**, *15*, 522.
https://doi.org/10.3390/w15030522

**AMA Style**

Yadav A, Ali Albahar M, Chithaluru P, Singh A, Alammari A, Kumar GV, Miro Y.
Hybridizing Artificial Intelligence Algorithms for Forecasting of Sediment Load with Multi-Objective Optimization. *Water*. 2023; 15(3):522.
https://doi.org/10.3390/w15030522

**Chicago/Turabian Style**

Yadav, Arvind, Marwan Ali Albahar, Premkumar Chithaluru, Aman Singh, Abdullah Alammari, Gogulamudi Vijay Kumar, and Yini Miro.
2023. "Hybridizing Artificial Intelligence Algorithms for Forecasting of Sediment Load with Multi-Objective Optimization" *Water* 15, no. 3: 522.
https://doi.org/10.3390/w15030522