# Prediction for the Sluice Deformation Based on SOA-LSTM-Weighted Markov Model

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## Abstract

**:**

^{2}and the smallest values of RMSE and absolute relative errors for the monitoring data of four monitoring points. Consequently, it concluded that the proposed method shows better prediction ability and accuracy than the SOA-LSTM model and the stepwise regression model.

## 1. Introduction

## 2. Statistical Model of Sluice Deformation

## 3. SOA-LSTM Network Model

#### 3.1. Principles of (Long Short-Term Memory) LSTM Neural Network

_{i}, b

_{c}are the bias terms of the sigmoid layer and the tanh layer, respectively; tanh is the hyperbolic tangent function, $\mathrm{tanh}x=\left(1-{e}^{-x}\right)/\left(1+{e}^{-x}\right)$.

#### 3.2. Principles of Seagull Optimization Algorithm (SOA)

- (1)
- Migration behavior (global search)During the seagull migration process, a seagull moves from one position to another. The above migration behavior should satisfy the following three conditions:
- (1)
- Avoiding collision. To avoid collisions between adjacent seagulls, an additional variable A is introduced to update the position of the seagull during the iterative calculation, as shown in Figure 3a.$$\overrightarrow{{N}_{s}}=A\xb7\overrightarrow{{P}_{s}}\left(i\right)$$$$A={f}_{c}-i\xb7\left(\frac{{f}_{c}}{{\mathrm{Max}}_{iteration}}\right)$$
- (2)
- Moving towards the best neighbor. After avoiding collisions between adjacent seagulls, a seagull moves towards the direction of the best neighbor, as illustrated in Figure 3b.$$\overrightarrow{{B}_{S}}=B\xb7\left(\overrightarrow{{P}_{gS}}\left(i\right)-\overrightarrow{{P}_{s}}\left(i\right)\right)$$$$B=2\xb7{A}^{2}\xb7rd$$
- (3)
- Moving towards the best position. Finally, the seagull updates its own position based on the best position, as depicted in Figure 3c.$$\overrightarrow{{D}_{s}}=|\overrightarrow{{N}_{s}}\left(i\right)+\overrightarrow{{B}_{s}}\left(i\right)|$$

- (2)
- Aggressive behavior (local search)When a seagull needs to attack its prey during flight, it forms a spiral formation in the air, as shown in Figure 3d. This behavior can be described in the xyz three-dimensional plane as follows.$$x=r\xb7\mathrm{sin}k$$$$y=r\xb7\mathrm{cos}k$$$$z=r\xb7k$$$$r=u\xb7{e}^{k\xb7v}$$Considering both seagull migration and aggressive behaviors, the calculation formula of seagull location updating can be obtained based on Equation (12) to Equation (16) as follows.$$\overrightarrow{{P}_{s}}\left(i\right)=x\xb7y\xb7z\xb7\overrightarrow{{D}_{s}}+\overrightarrow{{P}_{gs}}\left(i\right)$$

#### 3.3. SOA Optimized LSTM

- (1)
- Data processing. First, the data set is normalized, and 80% of the data in the monitoring sequence is selected as the training set, and 20% of the data in the monitoring sequence is selected as the test set.
- (2)
- Single LSTM model structure design. The LSTM model results are designed in terms of the number of network layers, the number of neurons in the input and hidden layers, the optimizer, and the loss function of the LSTM, and the multivariate multidimensional single-step LSTM network model with an initial input of nine dimensions, a time step of one, and an output of one dimension is constructed.
- (3)
- Construction of SOA LSTM model. The LSTM hyper-parameters to be optimized, such as the number of neurons in the hidden layer, the initial learning rate, the maximum number of iterations, the minimum number of batches, the time step, and the regularization parameter, are used as the solution objectives of the seagull optimization algorithm, and the root mean square error (RMSE) on the training set is taken as the fitness function to obtain the SOA-LSTM model. The computational formula is given by:$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}}$$
- (4)
- Model training and prediction. The training data are applied to train the SOA-LSTM model, the test data are input into the trained model, and finally obtain the predicted values.
- (5)
- Model comparison. The prediction results are utilized to calculate the prediction accuracy evaluation index to test the model effect.

## 4. SOA-LSTM-Weighted Markov Model for Sluice Deformation Prediction

#### 4.1. Weighted Markov Model-Based Sluice Deformation Prediction

- (1)
- State classification. The mean square deviation method is proposed to classify the values of random sequence indicator into five state intervals.
- (2)
- Markovianity test. For discrete sluice deformation sequences, the chi-square statistic can be constructed to test for Markovianity.
- (3)
- State transition probability matrix. The one-step state transition frequency matrix ${f}_{ij}^{\left(k\right)}$ with different lags (k is the step size) and the one-step state transition probability matrix ${p}_{ij}^{\left(k\right)}$ should be calculated, separately.
- (4)
- Autocorrelation coefficients and transfer weights of each order. After selecting the lag time, the autocorrelation coefficient ${r}_{k}$ and transfer weight ${w}_{k}$ of each order should be calculated, separately.
- (5)
- Prediction table preparation. The sluice deformation of several years before the prediction year should be taken as the initial state, and then the sluice deformation should be combined with the transfer probability matrix of the corresponding lag time to obtain the probability of each order state in the prediction year. Afterwards, the predicted state probabilities of each year of the same state are weighted and summed, which is the average probability of being in each state in the prediction year. The above calculations can be compiled into a prediction table, and the state corresponding to the maximum probability value in the prediction table is taken and predicted by using the weighted Markov model.

#### 4.2. Sluice Deformation Prediction Procedure of SOA-LSTM-Weighted Markov Model

- (1)
- Based on the training set, the SOA-LSTM network model is established.
- (2)
- After comparing the measured data with the fitted value, the error sequence can be obtained.
- (3)
- The weighted Markov model is constructed according to the steps in Section 3.1.
- (4)
- The sluice deformation prediction model based on the SOA-LSTM-weighted Markov model can be established.

## 5. Project Overview

## 6. Results and Discussions

#### 6.1. Fitted and Predicted Results of the Stepwise Regression Model

#### 6.2. Fitted and Predicted Results of the SOA-LSTM Network Model

#### 6.3. Fitted and Predicted Results of the SOA-LSTM-Weighted Markov Model

- (1)
- State classification. The relative error series has a mean of 75% and a root mean square error of 7.64%, which is classified into five states corresponding to the intervals of (−22.44%, −9.16%), (−9.16%, −4.58%), (−4.58%, 3.07%), (3.07%, 7.65%) and (7.65%, 24.20%), respectively.
- (2)
- Markovianity test. The one-step transfer frequency matrix ${f}_{ij}$, one-step transfer probability matrix ${p}^{\left(1\right)}$, and marginal probability and chi-square statistics were calculated, and the calculation results are described in Table 5. The value of the calculated statistic ${\chi}^{2}=2{\displaystyle \sum _{i=1}^{m}{\displaystyle \sum _{i=1}^{m}{f}_{ij}\left|\mathrm{ln}\frac{{P}_{ij}}{{P}_{.j}}\right|}}$ is 109.9, and given the significance level, $\alpha =0.01$. Then ${\chi}_{\alpha}^{2}=\left[{\left(m-1\right)}^{2}\right]={\chi}_{0.01}^{2}\left(16\right)=32.0$. Because ${\chi}^{2}>{\chi}_{0.01}^{2}\left(16\right)$, the random sequence is highly significant in terms of Markovianity, and can be predicted by the weighted Markov model.
- (3)
- The state transition probability matrix determination. The relative errors of the fitted values of the SOA-LSTM model were taken as a random sequence, and the one-step state transfer probability matrixes with the step size of 2, 3, 4, and 5, respectively.
- (4)
- Calculation of the autocorrelation coefficients and transfer weights of each order. The autocorrelation coefficients (r
_{k}) and transmission weights (w_{k}) of each order are shown in Table 6. - (5)
- Prediction table preparation. According to the relative errors of the SOA-LSTM model, the corresponding state transfer probability is weighted and calculated to predict the relative error state of test set. Table 7 illustrates a subset of the weighted prediction results of the relative error state of the test set. The state S with the maximum probability value in the prediction table was determined, E was taken as the median value of the corresponding state interval. x
_{SL}was selected as the predicted value of the SOA-LSTM network model. x_{SlM}was selected as the predicted value of the SOA-LSTM-weighted Markov model. Therefore, the corrected predicted value can be expressed as:$${x}_{BM}={x}_{BP}/\left(1+E\right)$$

^{2}) of the training set and the root mean square error (RMSE) of the test set of each model. The calculation formula of R

^{2}can be given by:

^{2}and RMSE of the stepwise regression model vary from 0.8001 to 0.9114 and from 1.6705 to 3.8577, respectively. The R

^{2}and RMSE of the stepwise regression model vary from 0.7801 to 0.8514 and from 2.0705 to 4.9561, respectively. The R

^{2}and RMSE of the stepwise regression model vary from 0.9301 to 0.9951 and from 0.725 to 1.517, respectively. Among all four monitoring points, the proposed SOA-LSTM-weighted Markov model has the smallest RMSE, which indicates that the proposed model is feasible and has better prediction accuracy overall.

## 7. Conclusions

- (1)
- The stepwise regression model is more suitable for dealing with linear problems; it has some limitations in fitting and predicting nonlinear and fluctuating monitoring sequences. By contrast, the proposed model shows obvious superiority in dealing with monitoring data with fluctuation.
- (2)
- The SOA improves the training efficiency of the neural network hyper-parameters of the LSTM model. The SOA-LSTM model can more accurately reflect the nonlinear change rule of the sluice settlement, and the weighted Markov model takes into account the strength of the dependency relationship between different lags, which can fully utilize the information of the training set, significantly reduce the model prediction error, and improve the model prediction accuracy. As listed in Table 1, the maximum settlement variation ranges from 1.73 mm to 31.59 mm, which indicates the apparent fluctuation of sluice settlement monitoring data. On the basis of the fitted results, predicted results and absolute relative errors, the proposed model demonstrates the largest values of R
^{2}and the smallest values of RMSE for the monitoring data of four monitoring points. Therefore, the SOA-LSTM-weighted Markov model is especially suitable for fitting and predicting the larger fluctuation of the sluice settlement, it is also especially suitable for dealing with the more volatile settlement monitoring data. - (3)
- Due to the better regularity of the actual data series selected in this paper, the prediction accuracy of the SOA-LSTM model and the SOA-LSTM-weighted Markov model is higher. When more monitoring data are obtained, the prediction model should be trained and adjusted in time, and then the subsequent dynamic prediction of the sluice settlement can be carried out, which will be able to better describe the settlement law of the sluice. To our knowledge, the proposed model expresses the stable and accurate results in predicting long sequenced monitoring data. In view of the practical application performance in the case study, the proposed model is supposed to applied extensively in deformation prediction of more sluice projects to improve safety monitoring efficiency, which provides a new aid for structural engineers involved in the real-time health assessment and safety monitoring in sluices. The study in this paper focuses mainly on the data-driven sluice deformation prediction model. A further study should pay more attention to investigating the physics-data double-driven intelligent prediction model, and the proposed method will have prospective applications in sluice structural behavior safety monitoring.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Schematic diagram of migration and aggressive behavior of SOA: (

**a**) avoiding collision, (

**b**) moving towards the best neighbor, (

**c**) moving towards the best position, (

**d**) the spiral aggressive behavior.

**Figure 6.**(

**a**) Engineering scene of the regulating sluice with 12 holes at Bengbu sluice; (

**b**) elevation map of the regulating sluice with 12 holes at Bengbu sluice; (

**c**) layout of settlement monitoring points of the regulating sluice with 12 holes at Bengbu sluice.

**Figure 14.**Comparison results of the fitting and prediction results of different models based on measured data of four monitoring points.

Project Site | Maximum Average Variation (mm) | Minimum Settlement (mm) | Maximum Settlement (mm) | Maximum Settlement Variation (mm) | ||
---|---|---|---|---|---|---|

Point | Settlement | Point | Settlement | |||

Upper left-wing wall | −14.99 | Ul-6 | −12.71 | Ul-2 | −16.33 | 3.62 |

Lower left-wing wall | −14.95 | Ll-6 | −9.80 | Ll-4 | −17.82 | 8.02 |

Upper right-wing Wall | −16.80 | Ur-1 | −14.14 | Ur-2 | −18.37 | 4.23 |

Lower right-wing wall | −18.11 | Lr-2 | −8.03 | Lr-5 | −28.71 | 20.68 |

Upper sluice pier | −13.52 | 1-3 | −0.21 | 3-4 | −31.80 | 31.59 |

Lower sluice pier | −14.30 | 2-2 | −4.19 | 5-1 | −35.75 | 31.56 |

Abutment pier | −8.45 | Right pier | −7.59 | Left pier | −9.32 | 1.73 |

Time (d) | Monitoring Data (mm) | Time (d) | Monitoring Data (mm) | Time (d) | Monitoring Data (mm) |
---|---|---|---|---|---|

0 | 1.43 | 210 | 29.14 | 420 | 34.71 |

30 | 11.72 | 240 | 30.83 | 450 | 33.20 |

60 | 17.74 | 270 | 29.32 | 480 | 33.43 |

90 | 23.85 | 300 | 29.72 | 510 | 36.25 |

120 | 26.79 | 330 | 32.21 | 540 | 35.75 |

150 | 28.95 | 360 | 34.41 | ||

180 | 28.56 | 390 | 33.06 |

Time (d) | Monitoring Data | Predicted Value (mm) | Relative Error (%) | Time | Monitoring Data | Predicted Value (mm) | Relative Error (%) |
---|---|---|---|---|---|---|---|

0 | 1.43 | 1.431 | 0 | 300 | 29.72 | 29.17 | −1.85 |

30 | 11.72 | 10.87 | −7.23 | 330 | 32.21 | 29.66 | −7.91 |

60 | 17.74 | 17.99 | 1.46 | 360 | 34.41 | 30.10 | −12.53 |

90 | 23.85 | 21.93 | −8.01 | 390 | 33.06 | 30.51 | −7.69 |

120 | 26.79 | 24.48 | −8.61 | 420 | 34.71 | 30.89 | −11.00 |

150 | 28.95 | 25.63 | −11.48 | 450 | 33.20 | 31.24 | −5.89 |

180 | 28.56 | 26.56 | −7.02 | 480 | 33.43 | 31.57 | −5.56 |

210 | 29.14 | 27.35 | −6.16 | 510 | 36.25 | 31.88 | −12.04 |

240 | 30.83 | 28.03 | −9.09 | 540 | 35.75 | 32.18 | −10.01 |

270 | 29.32 | 28.63 | −2.33 |

**Table 4.**Upper bound and lower bound of the parameters set in SOA, and the optimized results of the LSTM model.

Time (d) | Hidden Layers | Nodes in First Layer | Nodes in Second Layer | Learning Rate |
---|---|---|---|---|

Upper bound | 10 | 200 | 200 | 0.1 |

Lower bound | 1 | 10 | 10 | 0.001 |

Optimized parameters | 2 | 120 | 60 | 0.005 |

Status | Marginal Probability | ${\mathit{f}}_{\mathit{i}1}\left|\mathbf{ln}\frac{{\mathit{P}}_{\mathit{i}1}}{{\mathit{P}}_{,1}}\right|$ | ${\mathit{f}}_{\mathit{i}2}\left|\mathbf{ln}\frac{{\mathit{P}}_{\mathit{i}2}}{{\mathit{P}}_{,2}}\right|$ | ${\mathit{f}}_{\mathit{i}3}\left|\mathbf{ln}\frac{{\mathit{P}}_{\mathit{i}3}}{{\mathit{P}}_{,3}}\right|$ | ${\mathit{f}}_{\mathit{i}4}\left|\mathbf{ln}\frac{{\mathit{P}}_{\mathit{i}4}}{{\mathit{P}}_{,4}}\right|$ | ${\mathit{f}}_{\mathit{i}5}\left|\mathbf{ln}\frac{{\mathit{P}}_{\mathit{i}5}}{{\mathit{P}}_{,5}}\right|$ | Total |
---|---|---|---|---|---|---|---|

1 | 0.127 | 10.768 | 5.410 | 1.94 | 0.961 | 0.000 | 19.079 |

2 | 0.158 | 0.575 | 0.062 | 2.010 | 0.867 | 0.851 | 4.365 |

3 | 0.436 | 2.083 | 0.972 | 7.210 | 0.439 | 1.836 | 12.54 |

4 | 0.157 | 0.000 | 0.087 | 1.521 | 0.247 | 1.957 | 3.842 |

5 | 0.122 | 0.723 | 0.000 | 2.513 | 4.984 | 6.745 | 14.965 |

Total | 1 | 14.149 | 6.531 | 15.194 | 7.498 | 11.389 | 54.761 |

Project | Order | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

r_{k} | 0.568 | 0.354 | 0.248 | 0.126 | 0.139 |

w_{k} | 0.416 | 0.245 | 0.168 | 0.081 | 0.105 |

Initial Time (d) | Status | Lag Time | Weights | Status | ||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||||

60 | 3 | 1 | 0.4106 | 0.0771 | 0.1328 | 0.5587 | 0.1528 | 0.0759 |

120 | 3 | 2 | 0.2514 | 0.0979 | 0.1757 | 0.4892 | 0.1363 | 0.0978 |

180 | 3 | 3 | 0.1659 | 0.1000 | 0.2000 | 0.4785 | 0.1000 | 0.1207 |

240 | 4 | 4 | 0.0824 | 0.1581 | 0.0000 | 0.6296 | 0.1043 | 0.1062 |

300 | 5 | 5 | 0.1021 | 0.0657 | 0.0657 | 0.4000 | 0.2678 | 0.1985 |

P_{j} (Weighted sum) | 0.0922 | 0.1315 | 0.5249 | 0.0901 | 0.1447 |

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## Share and Cite

**MDPI and ACS Style**

Peng, J.; Xie, W.; Wu, Y.; Sun, X.; Zhang, C.; Gu, H.; Zhu, M.; Zheng, S.
Prediction for the Sluice Deformation Based on SOA-LSTM-Weighted Markov Model. *Water* **2023**, *15*, 3724.
https://doi.org/10.3390/w15213724

**AMA Style**

Peng J, Xie W, Wu Y, Sun X, Zhang C, Gu H, Zhu M, Zheng S.
Prediction for the Sluice Deformation Based on SOA-LSTM-Weighted Markov Model. *Water*. 2023; 15(21):3724.
https://doi.org/10.3390/w15213724

**Chicago/Turabian Style**

Peng, Jianhe, Wei Xie, Yan Wu, Xiaoran Sun, Chunlin Zhang, Hao Gu, Mingyuan Zhu, and Sen Zheng.
2023. "Prediction for the Sluice Deformation Based on SOA-LSTM-Weighted Markov Model" *Water* 15, no. 21: 3724.
https://doi.org/10.3390/w15213724