# Modeling Average Grain Velocity for Rectangular Channel Using Soft Computing Techniques

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup and Data Observation

#### 2.1.1. Methodology

#### 2.1.2. Input Parameters

_{e}) to the base area (A

_{b}) of the particle, which changes with the change in the discharge and slope of the channel for the same particle and is calculated as

#### 2.2. Multiple Linear Regression—MLR

#### 2.3. Artificial Neural Network—ANN

_{h}) for layer j is given as:

_{h}represents the neuron threshold value for h, O

_{pi}is the i-th output of the previous layer, and W

_{ih}is the weight between the layers i and h. The Levenberg-Marquardt (LM) training algorithm was considered to adjust the weights for the current study.

#### 2.4. Support Vector Machine-SVM

_{1}, y

_{1}), (x

_{2}, y

_{2}),…, (x

_{m}, y

_{m})

^{n}represents the training inputs, and y Y ⸦ ℝ

^{n}represents the training outputs. that a nonlinear function f(x) which is non-linear, is given by:

^{T}Φ(x

_{i}) + b

_{i}) denotes the high-dimensional feature space. Furthermore, data set T and, Equation (6) is transformed into Equation (7) as a constrained complex optimization problem stated as

^{T}w

#### 2.5. Performance Evaluation

## 3. Results and Discussion

#### 3.1. Statistical Parameters

_{p}of all data, training, and testing sets included various statistical parameters such as mean, median, minimum and maximum value, standard deviation, CV and skewness. These statistical parameters showed data heterogeneity over the whole time series. Cross-validating is essential for the same statistical population if the data is divided into training and test subsets. Due to the high skewness, the model’s efficiency was adversely affected. The standard deviation values suggest that the values are farther from zero, indicating that the data heterogeneity is more significant. The mean value variance is greater (Table 1).

#### 3.2. Trial Selection

#### 3.3. Quantitative Performance Evaluation

^{−6}for ANN, P = 6.34 × 10

^{−5}for MLR, and P = 9.4 × 10

^{−5}for SVM), which is less than 0.05 and suggested that there is a significant difference between observed and predicted mean values of grain velocity in all three models. The observed mean values of grain velocity were 0.8806 m/s. They predicted that the mean grain velocity was 0.7565 m/s, 0.970815 m/s, and 0.9529 m/s for the ANN, MLR, and SVM models. Thus, the mean absolute difference between observed and predicted grain velocities was 0.1240 for ANN, 0.09019 for MLR, and 0.07236 for SVM. Therefore, it is confirmed that the SVM-based model predicted close to the corresponding observed values than MLR and ANN-based models. The Friedman test also verified a significant (p < 0.05) difference between observed and predicted grain velocity in MLR, ANN, and SVM models.

#### 3.4. Qualitative Performance Evaluation

^{2}) were highest (0.8608) for the SVM model, and the lowest value (0.6729) was obtained for the ANN-based model. The value of (R

^{2}) was 0.7967 in the case of the MLR-based model.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations and Nomenclature

## References

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**Figure 1.**Experimental setup of the hydraulic flume; schematic presentation (

**top part**) and lab view (

**lower part**).

**Figure 3.**Soft margin loss setting for a linear SVM and ε-insentive loss function adapted after Lin et al. [41].

**Figure 4.**Plots of the dependent variable V

_{p}(m/s) and input variable: (

**a**) EATBAR, (

**b**) weight (gm), (

**c**) shear velocity (m/s), (

**d**) relative depth.

**Figure 5.**Observed versus predicted grain velocity by the (

**a**) MLR, (

**b**) ANN, and (

**c**) SVM models during the testing phase.

Variables | Mean | Median | Minimum | Maximum | Std. Dev. | C.V. | Skewness |
---|---|---|---|---|---|---|---|

All Data | |||||||

λ | 0.9707 | 1.000 | 0.5700 | 1.000 | 0.0823 | 0.0848 | −3.283 |

W | 51.733 | 49.60 | 32.200 | 73.40 | 16.966 | 0.32795 | 0.1874 |

U* | 0.0754 | 0.0731 | 0.0529 | 0.1030 | 0.0120 | 0.1598 | 0.4072 |

y/d | 0.8573 | 0.7142 | 0.2500 | 2.352 | 0.4281 | 0.4994 | 1.220 |

V_{p} | 0.5473 | 0.600 | 0.0 | 1.174 | 0.3352 | 0.6124 | −0.2776 |

Training Data | |||||||

λ | 0.9793 | 1.0 | 0.6 | 1.0 | 0.0706 | 0.0721 | −3.964 |

W | 51.476 | 49.6 | 32.2 | 73.4 | 17.035 | 0.3309 | 0.2091 |

U* | 0.0696 | 0.0688 | 0.0529 | 0.0862 | 0.008 | 0.1150 | 0.0828 |

y/d | 0.7894 | 0.6667 | 0.25 | 2.222 | 0.4031 | 0.5106 | 1.295 |

V_{p} | 0.4070 | 0.474 | 0.0 | 0.8440 | 0.2793 | 0.6863 | −0.3741 |

Testing Data | |||||||

λ | 0.9503 | 1.0 | 0.57 | 1.0 | 0.1035 | 0.1089 | −2.368 |

W | 52.344 | 49.6 | 32.2 | 73.4 | 17.05 | 0.3258 | 0.1369 |

U* | 0.0892 | 0.0903 | 0.0774 | 0.103 | 0.0081 | 0.0914 | 0.1011 |

y/d | 1.0185 | 0.8012 | 0.5714 | 2.352 | 0.4489 | 0.4407 | 1.1701 |

V_{p} | 0.8806 | 0.8855 | 0.333 | 1.174 | 0.1901 | 0.2159 | −0.6570 |

_{p}= grain velocity).

**Table 2.**During the training and testing phase, performance evaluations of MLR, ANN, and SVM models.

Model | Training | Testing | ||||
---|---|---|---|---|---|---|

RMSE (m/s) | PCC | WI | RMSE (m/s) | PCC | WI | |

MLR | 0.1340 | 0.8756 | 0.7532 | 0.1459 | 0.8375 | 0.6789 |

ANN | ||||||

Trial-1 | 0.1266 | 0.8911 | 0.8106 | 0.3109 | 0.1509 | 0.3420 |

Trial-2 | 0.0873 | 0.9502 | 0.8756 | 0.2154 | 0.3636 | 0.4579 |

Trial-3 | 0.0689 | 0.9692 | 0.8799 | 0.1721 | 0.5176 | 0.5012 |

Trial-4 | 0.0663 | 0.9728 | 0.8999 | 0.2302 | 0.1945 | 0.3666 |

Trial-5 | 0.0699 | 0.9678 | 0.8916 | 0.1906 | 0.4365 | 0.4751 |

Trial-6 | 0.0759 | 0.9625 | 0.9439 | 0.1821 | 0.4900 | 0.5058 |

SVM | ||||||

Trial-1 | 0.1423 | 0.8595 | 0.7475 | 0.1208 | 0.8852 | 0.7231 |

Trial-2 | 0.1381 | 0.8675 | 0.7531 | 0.1341 | 0.8688 | 0.7022 |

Trial-3 | 0.1431 | 0.8577 | 0.7479 | 0.1195 | 0.8877 | 0.7243 |

Trial-4 | 0.1408 | 0.8622 | 0.7513 | 0.1247 | 0.8795 | 0.7150 |

Model/Trial | Architecture |
---|---|

ANN | |

Trial-1 | 4-1-1 |

Trial-2 | 4-4-1 |

Trial-3 | 4-5-1 |

Trial-4 | 4-7-1 |

Trial-5 | 4-5-5-1 |

Trial-6 | 4-4-4-4-1 |

SVM | |

Trial-1 | C = 10, $\mathsf{\gamma}$ = 0.25, ε = 0.01 |

Trial-2 | C = 10, $\mathsf{\gamma}$ = 0.25, ε = 0.1 |

Trial-3 | C = 10, $\mathsf{\gamma}$ = 0.45, ε = 0.01 |

Trial-4 | C = 10, $\mathsf{\gamma}$ = 0.45, ε = 0.05 |

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**MDPI and ACS Style**

Kumari, A.; Kumar, A.; Kumar, M.; Kuriqi, A.
Modeling Average Grain Velocity for Rectangular Channel Using Soft Computing Techniques. *Water* **2022**, *14*, 1325.
https://doi.org/10.3390/w14091325

**AMA Style**

Kumari A, Kumar A, Kumar M, Kuriqi A.
Modeling Average Grain Velocity for Rectangular Channel Using Soft Computing Techniques. *Water*. 2022; 14(9):1325.
https://doi.org/10.3390/w14091325

**Chicago/Turabian Style**

Kumari, Anuradha, Akhilesh Kumar, Manish Kumar, and Alban Kuriqi.
2022. "Modeling Average Grain Velocity for Rectangular Channel Using Soft Computing Techniques" *Water* 14, no. 9: 1325.
https://doi.org/10.3390/w14091325