Abstract

Dam deformation monitoring and prediction are crucial for evaluating the safety of reservoirs. There are several elements that influence dam deformation. However, the mixed effects of these elements are not always linear. Oppose to a single-kernel extreme learning machine, which suffers from poor generalization performance and instability, in this study, we proposed an improved bat algorithm for dam deformation prediction based on a hybrid-kernel extreme learning machine. To improve the learning ability of the global kernel and the generalization ability of the local kernel, we combined the global kernel function (polynomial kernel function) and local kernel function (Gaussian kernel function). Moreover, a Lévy flight bat optimization algorithm (LBA) was proposed to overcome the shortages of bat algorithms. The results showed that our model outperformed other models. This proves that our proposed algorithm and methods can be used in dam deformation monitoring and prediction in different projects and regions.

HIGHLIGHTS

  • The Lévy flight was improved bat algorithm to overcome the shortcoming about speed and effectiveness of bat algorithm.

  • We integrated the advantage of Gaussian and polynomial functions as the kernel function of KELM (PGKELM).

  • The LBA-PGKELM at the top of LBA-SVM and BPNN which based on gradient descent method.

INTRODUCTION

The role of a dam in the functionalities of a reservoir is undeniable, and any failure in a dam threatens humans and causing significant damages for downstream regions (Milillo et al. 2016). Therefore, proposing a dam deformation model is important for dam safety and preventing financial losses (Mohanty et al. 2020). Dam deformation monitoring and prediction is an efficient way to discover the potential risks in the construction of dams (Chongshi et al. 2011). Due to the complexity of the dam structure, the relationship between influencing factors (e.g., water pressure, aging, and temperature) and dam deformation is non-linear (Xu et al. 2012). Therefore, traditional dam deformation models (e.g., multiple linear regression and other linear models) (Yu et al. 2018; Li et al. 2019) may not reflect the direct relationship between explanatory variables and dam deformation because of linear assumption.

Machine learning methods have shown their ability in different fields, such as epidemiological studies (Chen et al. 2021), air quality forecasting (Karimian et al. 2017), and ecological security (Gong et al. 2017). In recent years, machine learning models have shown their feasibility in dam deformation prediction and other fields due to their high prediction performance and capability to handle complex and non-linear problems (Salazar et al. 2015; Al-musaylh et al. 2018; Gupta et al. 2020). Support vector machine (SVM) (Song et al. 2011), artificial neural network (ANN) (Stojanovic et al. 2016), and random forest regression (RFR) (Dai et al. 2018) are some of the widely used models. Su et al. (2018) combined an improved particle swarm optimization (PSO) algorithm with the wavelet SVM and the radial kernel function-based SVM. The authors claimed that this model has better parameter selection that reduces the iteration number, shortens the computation time, and avoids the local optimization. Bui et al. (2018) used a neural fuzzy inference system to create a regression model, whereas PSO was employed to search the best parameters for the model. It is accepted to be an effective tool for modeling the non-linear time-varying behavior of a dam. The machine learning methods applied in dam deformation can detect patterns between variables, and resist the noise interference in monitoring data, without presupposition, which is particularly suitable for the interpretation of dam behavior (Lin 2019).

The extreme learning machine (ELM) is one of the machine learning methods that, because of its excellent generalization ability and fast learning speed, has been adopted in various fields (e.g., dam displacement prediction (Cheng & Xiong 2017), water quality prediction (Lima et al. 2015), and software fault prediction (Mao et al. 2019)). However, the machine learning algorithms have their own limits. First, while trying to improve global optimization, the local best may be ignored. Second, the accuracy of an algorithm is improved by the price of diminishing processing speed (Reddy et al. 2018). Third, if there are outliers in the training samples, the hidden layer output matrix is ill-posed, which will affect the generalization ability and robustness of the model (Zhang & Zhang 2016). Finally, due to the random mapping of ELM, even with the same set of inputs, the outputs will be different (Barzegar et al. 2016). The kernel extreme learning machine can improve the stability and accuracy of the prediction, and can overcome the limitations of random mapping of ELM (Cao et al. 2020). Kang et al. (2017) proposed a prediction model based on Gaussian kernel function and extreme learning machine. They proved that the kernel extreme learning machine (KELM) has high learning efficiency and can well adapt to the complexity of dam deformation prediction. Maimaitiyiming et al. (2019) optimized an ELM model by mixing dual activation function. The results showed that the optimized KELM is good at improving model accuracy. Moreover, the generalization performance of KELM largely depends on the choice of kernel function parameters. However, the selection of kernel functions are still challenging tasks.

With the increase of kernel number, the parameter optimization is necessary to maximize the performance of a model. Optimization is an essential component of machine learning. As one of the optimization techniques, the swarm intelligence optimization algorithm has showed strong optimization performance (Igiri et al. 2019). Swarm intelligence algorithms include cuckoo search algorithm (Wong et al. 2015), fruit fly optimization algorithm (FOA), particle swarm algorithm (PSO) and bat algorithm (BA) (Degang & Ping 2019). Based on simplicity, strong searching ability and fast convergence speed, BA is widely applied to gray image edge detection (Dhar et al. 2017), capacitive vehicle routing problems (Zhou et al. 2016), and power systems (Sathya & Mohamed Thameem Ansari 2015). The BA algorithm is initialized as a set of random solutions to enhance the local searching ability and increase the processing speed.

This paper proposed a novel framework based on KELM and Lévy flight bat algorithm (LBA-PGKEML). We proposed a hybrid kernel for KELM. Moreover, the role of intelligence optimization algorithm in improving the performance of KELM is investigated. To the best of our knowledge, this is the first attempt at the application of the hybrid kernel function, which is done by exploiting the advantages of both Gaussian and polynomial functions as the hybrid kernel function for the KELM. Considering the high generalization and robustness of KELM models, we proposed a KELM model to solve the non-linear characteristics of dam deformation process. For this purpose, the global kernel (polynomial functions) and local kernel (Gaussian functions) are combined, and from a hybrid kernel (PGKELM). Our hybrid kernel exploits the generalization ability of global kernel and learning ability of local kernel. PGKELM does not suffer from the sensitivity of single kernel KELM. Consequently, it improves the prediction accuracy of the model. Moreover, for efficient selection of parameters, we use the Lévy flight bat algorithm (LBA), which aims to reduce long searching time and solve the problem of optimal parameters’ selection. Results are also compared with that of backpropagation neural networks (BPNN) and the SVM models.

MATERIALS AND METHODS

Data process

In general, dam deformation is mainly considered in reversible deformations (caused by external variations of the hydrostatic, water pressure, and temperature) and irreversible deformations (function of time and materials lifetime) (Chrzanowski et al. 1991). Based on Chrzanowski & Szostak-Chrzanowski (1993), the prediction model of dam deformation can be expressed as:
formula
(1)
where, is the observed displacements, is the water pressure difference between the upstream and downstream, represents the change of temperature, and is the vertical displacement of dam over time.
In this paper, we mainly considered the settlement displacement of a dam, which consists of displacement due to temperature and age . Therefore, the displacement model of the dam in the vertical direction can be written as follows:
formula
(2)
The temperature effect acts on the dam structure through periodic variation, which can be calculated by Equation (3) (continuous temperature data) and Equation (4) (discontinuous temperature data). The ageing displacement can be calculated by Equation (5):
formula
(3)
formula
(4)
formula
(5)
In the above equations, (i = 0, 1, 2, 3, 4, 5, 6) are unknown coefficients that need to be estimated, , and is the cumulative number of monitoring days. , where t is the current observation date, is the beginning date of the observation. is the average temperature between day p and day q. Thus, Equation (2) can be written as:
formula
(6)
where is the coefficient of the model, is the constant term, and n is the number of influential factors. Therefore, the input samples for dam deformation prediction are:
formula
(7)
formula
(8)

Hybrid kernel extreme learning machine (KELM)

Huang et al. (2015) initially introduced the ELM, which is a feedforward neural network with a single layer or multiple layers of hidden nodes. To avoid instability in the ELM, due to the random parameters in the hidden layers, and enhance the generalization ability of ELM, Huang (2014) suggested getting the solution of optimization theory with SVM kernel function. In the KELM model, the Lagrange factor transforms the training of ELM into a dual problem (Equation (9)):
formula
(9)
where matrix T is the training sample, and C is the regularization coefficient. The advantage of this improvement is that when the feature mapping h(x) of the hidden layer is unknown, it can be solved easily by pointwise multiplication of the kernel element. The final output can be expressed as Equation (10):
formula
(10)

From Equation (10), it can be inferred that the ELM used kernel mapping instead of random mapping. Besides the aforementioned advantages of our proposed method, there is no need to set the dimension of hidden layer feature mapping. Instead, the pointwise multiplication of the kernel function is used that reduces the computational complexity.

The choice of kernel function is the key to KELM model performance. However, there is not a unique function that can give the best performance. The kernel functions are mainly divided into the local kernel and global kernel. The local has a stronger learning ability than the global kernel. However, it has a relatively weaker generalization performance than the global kernel (Wong et al. 2015). Therefore, to improve the learning ability and generalization performance, we combined the local and global kernel. Our proposed model, PGKELM (Equation (13)), is composed of polynomial global kernel Equation (12) and Gaussian local kernel Equation (11):
formula
(11)
formula
(12)
formula
(13)
where K(x, z), with different subscript letters, is a single local or global kernel function, and is the weight of a single kernel function. From the above formulations, we can infer that when , the hybrid kernel will become a single kernel function.

Lévy flight bat algorithm

Bat algorithm is a heuristic search algorithm based on swarm intelligence proposed by Yang (2010, 2011, 2013), and it is an effective method for searching the optimal global values.

Bat algorithm simulates the characteristics of bats that use echolocation to detect prey and avoid obstacles. In the BA, it is assumed that bats search for targets in d-dimensional space. Bats in each generation have their own positions and speeds , which are affected by the previous generation:
formula
(14)
formula
(15)
formula
(16)
where is a random vector, is the optimal position in the current population, i.e., the optimal solution, is the frequency of sound waves emitted by bats, which can be modified according to practical problems.
For local search, once a solution is selected from the current best solutions, a new local solution will be obtained by random perturbation, Equation (17):
formula
(17)
In the above, is a random number, and represents the average loudness of the same generation.
To improve the efficiency of hunting, bats dynamically adjust the loudness and frequency of sound waves based on the distance to a target. This can be formulated as:
formula
(18)
formula
(19)
where is the attenuation coefficient of the sound waves, indicates frequency enhancement coefficient, and is the initial emission frequency of ith bats.

However, from the BA algorithm, it can be inferred that the searching capability mainly depends on the interaction and influence between bat individuals. Once the individual is trapped in the local extremum, it is difficult to get rid of it, due to the lack of mutation mechanism. Under this circumstance, the Lévy flight is applied.

The Lévy flight (Equation (20)) is a stable distribution proposed by Lévy and described in Mandelbrot's 1984 work The Fractal Geometry of Nature, reviewed by Sparrow (2014). It is a random walk with a step length that follows the Lévy distribution (Roy & Sinha Chaudhuri 2013) that makes moving direction and jumping steps. In the searching process, frequent short-distance local search and a few long-distance global searches are applied to enhance the local search effect (Chawla & Duhan 2018):
formula
(20)
In the above, is the random step, and is the index parameters. Figure 1 shows the trajectory simulation diagram of Lévy flight with particles’ flight speed v= 1, the number of particles’ moving steps n= 10,000, and time parameter α= 1.5. We can see that Lévy's flight has many short-distance steps and a few large step jumps.
Figure 1

Lévy flight trajectory (n = 10,000, α = 1.5, v = 1).

Figure 1

Lévy flight trajectory (n = 10,000, α = 1.5, v = 1).

In this paper, the update method of speed and position of the original BA algorithm is replaced by the flight trajectory of Lévy, to solve the shortcomings of the BA algorithm (i.e., slow convergence speed, low convergence precision, and easy to fall into a local minimum). Therefore, Equations (14)–(16) of the BA algorithm can be replaced with Lévy flight bat algorithm (LBA):
formula
(21)
where is the location of bat i in the t-th search; is the random search vector whose jump step size obeys the Lévy distribution; is the vector operation; is the optimal position in the current population. In summary, the steps to obtain the LBA algorithm are as follows:
  • 1.

    Initializing algorithm parameters: set the number of bats , individual maximum pulse frequency , maximum pulse intensity , frequency increase coefficient , sound intensity attenuation coefficient and maximum iteration coefficient or search accuracy.

  • 2.

    Randomly initialize the bat position and find the best position () in the population.

  • 3.

    Generate a random number . If , update the current position of the bat according to Equation (21), otherwise, update it with a disturbed position, which randomly disturbs the current position of the bat.

  • 4.

    Generate a random number . If and the current position of the bat is improved, then fly to the updated position.

  • 5.

    If the bat i is superior to the best bat in the group, after updating its position, replace the best bat individual and adjust and according to Equations (18) and (19).

  • 6.

    Evaluate the bat population and find out the spatial position of the best bat.

  • 7.

    Once the desired accuracy is met, or the maximum iteration is reached, go to the last step; otherwise, go to step 3 for the new search.

  • 8.

    Output the global extreme points and optimal individual values.

Model building

Based on our proposed algorithm, a dam deformation prediction model, LBA-PGKELM, is constructed. The main point of our model is optimizing PGKELM through the LBA algorithm. For the combined KELM, because the parameters are the superposition of multiple kernel functions in KELM, the optimization is more complicated. However, using the LBA algorithm can solve this problem.

The optimization parameters of the LBA-PGKELM algorithm include one Gaussian RBF kernel parameter, two polynomial kernel parameters, and a weighted parameter of the PGKELM model. Moreover, a regularization term is introduced to reduce the influence of noise points and abnormal points on the generalization performance of the model. Since the sample is divided into training and test datasets, the idea of least squares is used to improve the accuracy of the prediction results. The LBA-PGKELM algorithm is described by Equation (22):
formula
(22)
where the is the accuracy of the sample, is the accuracy of the training set, is the accuracy of the test set, is the Gaussian RBF kernel, is the polynomial kernel, x1, x2, and x3 are the parameters of the Gaussian kernel and the polynomial kernel, is the weighting parameter, and C is the regularization parameter.

After preprocessing of the existing data, the input samples are prepared, and the dam deformation prediction model based on the LBA-PGKELM algorithm is established (Figure 2).

Figure 2

LBA-PGKELM algorithm flow chart.

Figure 2

LBA-PGKELM algorithm flow chart.

Based on the above flow chart, the steps of our proposed algorithm are as follow:

  • 1.

    The data are divided randomly into training set (80%) and test set (20%). The mean and standard deviation of training data are calculated for normalization and dimensionality reduction of data.

  • 2.

    The polynomial kernel and Gaussian RBF kernel are combined by weighting to construct a PGKELM function with improved performance.

  • 3.

    Build the PGKELM model. The PGKELM Kernel function is used to realize the mapping transformation of eigenvalues.

  • 4.

    Several parameters are taken as the optimization parameters of LBA, and the minimum square sum of the MSE between the training set and the test set is taken as the optimization criterion.

  • 5.

    Validate the model performance based on the test dataset using mean square error and coefficient of determination.

Verifying models

To verify the performance of our proposed LBA-PGKELM algorithm, we compared its performance with the BPNN and the LBA-SVM model.

Neural network models are flexible, and allow modeling complex and highly non-linear phenomena. Backpropagation neural network (BPNN) (Zou et al. 2018) is one of the NN algorithms, and it realizes the non-linear mapping between input and output. Otherwise, the use of gradient descent algorithm (GDA) optimized the parameters to reduce the error and get better results of BPNN. SVM (Vapnik 1998) was proposed in the 1990s. According to the principle of structural risk minimization, a linear classifier with the maximum decision boundary is designed to minimize the generalization error in the worst case. In SVM, the data are mapped to high space through kernel function. This converts the non-linear problem to a linear problem, and avoids training falling into a local minimum.

Model performance analysis

To validate the performances of our proposed algorithm, the accuracy of the models is measured using the mean square error (MSE), and the correlation coefficient (R2) as follows:
formula
(23)
formula
(24)
where y(i) is forecasted values, yD(i) is the observed values, and is the mean value of observed values.

CASE STUDY

We validated the capability and performance of the LBA-PGKELM model step by step. First, the accuracy of data was tested by the LBA algorithm similar to the BA algorithm. Second, the performance of the LBA-PGKELM model was compared with the LBA-RBFKELM and LBA-POLKELM. It is worth mentioning that the kernel functions which were used for KELM are the RBF kernel function and the polynomial kernel function. Third, the prediction accuracy of the LBA-PGKELM model was compared with the BPNN and LBA optimized support vector machine (LBA-SVM).

Study site and data

The study area is LiShan reservoir which is located in Zhejiang Province, China, and it can be seen in Figure 3. It is used mainly for irrigation, water supply, and flood control. The specifications of the reservoir are provided in Table 1. We utilized the daily-averaged data of two observation points, A2 and A3, from December 2018 to January 2021. The observations include the vertical displacements and temperature. Because the water level was constant during our study period, we did not consider it in our model. In addition, considering all monitoring stations collectively, does not provide better results than using a single station (Kang et al. 2017). Therefore, in our study, the data record at each monitoring point was modeled independently. In order to verify the ability of our proposed model, we utilized two monitoring stations’ data. The dataset cleaning process was achieved through excluding outliers. After that, the daily averaged data were calculated. According to the Pearson correlation analysis, the average correlation coefficients of two monitoring stations between time and dam settlements and between temperature and dam settlements are R = 0.72 and R = 0.99, respectively. This indicates that time and temperature are two critical factors in dam settlements. According to Equation (7), the input samples are set as . Part of the data is shown in Table 2. At the experiment, the maximum number of iterations was 500, the volume attenuation coefficient was 0.9, and the search frequency enhancement coefficient was set to 0.9.

Table 1

The description of LiShan reservoir

ProjectValue
Catchment area 1.74 km2 
Flood level 83.67 m 
Check flood level 84.47 m 
Total capacity 684,600 m3 
Normal storage level 81.8 m 
Normal capacity 512,000 m3 
Reservoir engineering level 
ProjectValue
Catchment area 1.74 km2 
Flood level 83.67 m 
Check flood level 84.47 m 
Total capacity 684,600 m3 
Normal storage level 81.8 m 
Normal capacity 512,000 m3 
Reservoir engineering level 
Table 2

The partial data of A2 station

Number (day)Temperature (°C)Settlement (mm)Number (day)Temperature (°C)Settlement (mm)
3.4 759.8 90 13.4 788.8 
6.2 760.5 95 14.0 790.6 
10 12.6 761.7 100 10.4 790.7 
15 9.2 761.3 105 15.0 792.4 
20 2.4 760.2 110 13.4 793.1 
25 5.6 761.2 115 22.2 794.5 
30 6.2 777.2 120 17.2 794.0 
35 6.3 778.1 125 19.4 794.9 
40 5.6 777.8 130 21.2 794.7 
Number (day)Temperature (°C)Settlement (mm)Number (day)Temperature (°C)Settlement (mm)
3.4 759.8 90 13.4 788.8 
6.2 760.5 95 14.0 790.6 
10 12.6 761.7 100 10.4 790.7 
15 9.2 761.3 105 15.0 792.4 
20 2.4 760.2 110 13.4 793.1 
25 5.6 761.2 115 22.2 794.5 
30 6.2 777.2 120 17.2 794.0 
35 6.3 778.1 125 19.4 794.9 
40 5.6 777.8 130 21.2 794.7 
Figure 3

Lishan reservoir and schematic layout of dam deformation monitoring points.

Figure 3

Lishan reservoir and schematic layout of dam deformation monitoring points.

To overcome the risk of over-fitting and verify the generalization performance of our proposed model, we applied the K-fold cross-validation (Ling et al. 2019; Maimaitiyiming et al. 2019). This method splits the whole dataset into K folds, where the k−1 folds are training set, and the remaining is the test set. We calculated the average MSE of the model after the K times iteration. Multiple partitioning of the dataset and random selection of data avoids the waste of data and improves the performance of the model (Wong 2017; Kaimian et al. 2019). The K-fold cross-validation is described by Equation (25). In this equation, the K is the iteration of the algorithm and it was set to 5.
formula
(25)

Results and discussions

LBA algorithm versus BA algorithm

In the first experiment, the PG kernel was used as the kernel function of KELM and the parameter optimization was carried out based on the BA algorithm and LBA algorithm, respectively.

To compare the convergence effect of the improved bat algorithm (LBA) and the bat algorithm (BA), the relationship between the fitness value and the iterations is shown in Figure 4, and the maximum number of iterations is 200, the population size was 100, over the same test dataset. The optimization results in the LBA and BA algorithm were estimated by the fitness function (Ragalo & Pillay 2018a, 2018b). The higher fitness value indicates the higher optimization ability of a model. As can be seen in Figure 4, the LBA-best (best performance) and LBA-worst (worst performance) have smaller difference than those between BA-best and BA-worst curve (Figure 4(b)). The best fitness value of the LBA algorithm is 0.6, which is better than the optimal result of the BA algorithm, 0.8 (Figure 4(a) and 4(b)). During the two tests, the LBA algorithm only used 100 iterations to obtain relatively ideal results. Moreover, the convergence speed of the BA algorithm is significantly slower. This is because the searching capability of the BA algorithm mainly depends on the interaction and influence between bat individuals. Moreover, due to the lack of mutation mechanism, once the bat individual is trapped in the local extremum, it is difficult to get rid of it. Whereas in the searching process of the LBA algorithm, utilization of frequently short distance local search and infrequently long distance global search can enhance the local search effect and improve the optimization ability. From our results, it can be inferred that the LBA algorithm performs better in both convergence result and convergence speed than the BA algorithm.

Figure 4

Iterative descent curve: (a) the performance of BA-best and BA-worst; (b) the performance of LBA-best and LBA-worst; (c) the performance of LBA-best and BA-best; (d) the performance of LBA-worst and BA-worst.

Figure 4

Iterative descent curve: (a) the performance of BA-best and BA-worst; (b) the performance of LBA-best and LBA-worst; (c) the performance of LBA-best and BA-best; (d) the performance of LBA-worst and BA-worst.

Hybrid kernel algorithm versus single kernel algorithm

Unlike most studies (Cao et al. 2020), we applied the combined two functions as the kernel function of KELM. We performed the optimization comparison between PG kernel function, polynomial kernel, and RBF kernel function (Tables 3 and 4; Figures 5 and 6). As can be seen in both stations, the PGKELM as a mixed kernel function showed better performance, with overall MSE = 0.5 mm2, than other kernel functions (MSE = 0.73 mm2 and 1.35 mm2 for RBKELM and POLKELM, respectively). Although the difference in R2 among different models is not big, it is not the only indicator for evaluating model performance. The extreme learning machine is one of the machine learning methods. On the other hand, the Gaussian kernel has strong local learning ability but weak generalization ability. By combining these two kernel functions, as the hybrid PGKELM kernel, better generalization performance and learning ability than a single kernel function are obtained.

Table 3

The A2 station values of hybrid kernel and single kernel

TypesPGKELMRBFKELMPOLKELM
Test set R2 0.9989 0.9923 0.9923 
MSE (mm20.5259 2.0800 2.0390 
Training set R2 0.9982 0.9957 0.9944 
MSE (mm20.4947 1.1883 1.4224 
Overall dataset R2 0.9981 0.9982 0.9973 
MSE (mm20.5403 0.4947 0.7225 
TypesPGKELMRBFKELMPOLKELM
Test set R2 0.9989 0.9923 0.9923 
MSE (mm20.5259 2.0800 2.0390 
Training set R2 0.9982 0.9957 0.9944 
MSE (mm20.4947 1.1883 1.4224 
Overall dataset R2 0.9981 0.9982 0.9973 
MSE (mm20.5403 0.4947 0.7225 

The values in bold indicate the best results.

Table 4

The A3 station values of hybrid kernel and single kernel

TypesPGKELMRBFKELMPOLKELM
Test set R2 0.9702 0.8997 0.9128 
MSE (mm20.2050 0.8121 0.7644 
Training set R2 0.9867 0.9347 0.9614 
MSE (mm20.9623 1.4137 2.5159 
Overall dataset R2 0.9797 0.9148 0.9355 
MSE (mm20.4969 0.9678 1.9684 
TypesPGKELMRBFKELMPOLKELM
Test set R2 0.9702 0.8997 0.9128 
MSE (mm20.2050 0.8121 0.7644 
Training set R2 0.9867 0.9347 0.9614 
MSE (mm20.9623 1.4137 2.5159 
Overall dataset R2 0.9797 0.9148 0.9355 
MSE (mm20.4969 0.9678 1.9684 

The values in bold indicate the best results.

Figure 5

Comparison between the performance of hybrid kernel and single kernel at the A2 station: (a) correlation coefficient and (b) mean square error.

Figure 5

Comparison between the performance of hybrid kernel and single kernel at the A2 station: (a) correlation coefficient and (b) mean square error.

Figure 6

Comparison between the performance of hybrid kernel and single kernel at the A3 station: (a) correlation coefficient and (b) mean square error.

Figure 6

Comparison between the performance of hybrid kernel and single kernel at the A3 station: (a) correlation coefficient and (b) mean square error.

LBA-PGKELM versus other algorithms

To verify the performance of our proposed LBA-PGKELM algorithm, we compared its performance with the BPNN and the LBA-SVM model. The BPNN comprised a hidden layer with 80 neurons, and the learning rate was 0.01. The maximum number of iteration cycles was set to 1,000, and the sigmoid function was selected as activation function. It is worth mentioning that the optimization parameters of LBA-SVM including SVM type, kernel function type, loss function and gamma function, were chosen similar to those of the KELM model.

As can be seen in Tables 5 and 6, and Figures 7 and 8, the LBA-PGKELM shows the best performance among all three models, with MSE = 0.7225 mm2. Considering the significant difference between MSE values of different models, we can infer that the LBA algorithm improves the capability of searching optimal parameters. The BPNN showed the worst performance. This may have been caused by using gradient descent algorithm to generate a local minimum that may lead to local optimum. Moreover, the BPNN has no optimization method to improve the generalization and global optimization capability. Thanks to the superior optimization ability of the LBA algorithm, the LBA-PGKELM model and the LBA-SVM model showed better prediction accuracy. On the one hand, the LBA algorithm has a better generalization. On the other hand, the LBA-PGKELM and LBA-SVM have a clear advantage in solving small-sample and non-linear problems due to their solid theoretical basis. Moreover, under the same swarm intelligence optimized KELM algorithm, the PGKELM algorithm performs better than non-hybrid kernel algorithm. This suggests that the mixed kernel function gives full play to the advantages of the two kernel functions and improves the prediction accuracy of the model. Our proposed model focuses on improving the prediction accuracy. Compared with LSTM (Yang et al. 2020), the prediction accuracy of non-linear small sample data is higher than that of time series method, the MSE of test is 0.2050 mm2 and 1.06 mm2, respectively. The comparison yields a conclusion consistent with the concrete dam case; the performance of the proposed LBA-PGKELM model is superior, with BPNN, LBA-SVM, and LSTM. Therefore, the effectiveness and universality of the proposed methodology are verified.

Table 5

The A2 station values of LBA-PGKELM, LBA-SVM, and BPNN models

TypesLBA-PGKELMLBA-SVMBPNN
Test set R2 0.9973 0.9941 0.9940 
MSE (mm20.7225 1.5356 6.4120 
Training set R2 0.9982 0.9972 0.9949 
MSE (mm20.4947 0.7804 5.7209 
Overall set R2 0.9981 0.9967 0.9945 
MSE (mm20.5403 0.9314 5.8639 
TypesLBA-PGKELMLBA-SVMBPNN
Test set R2 0.9973 0.9941 0.9940 
MSE (mm20.7225 1.5356 6.4120 
Training set R2 0.9982 0.9972 0.9949 
MSE (mm20.4947 0.7804 5.7209 
Overall set R2 0.9981 0.9967 0.9945 
MSE (mm20.5403 0.9314 5.8639 

The values in bold indicate the best results.

Table 6

The A3 station values of LBA-PGKELM, LBA-SVM, and BPNN models

TypesLBA-PGKELMLBA-SVMBPNN
Test set R2 0.9702 0.9147 0.8547 
MSE (mm20.3205 0.8606 1.0450 
Training set R2 0.9867 0.9544 0.9036 
MSE (mm20.8642 1.2437 1.9484 
Overall set R2 0.9797 0.9366 0.8751 
MSE (mm20.6594 0.9865 1.2452 
TypesLBA-PGKELMLBA-SVMBPNN
Test set R2 0.9702 0.9147 0.8547 
MSE (mm20.3205 0.8606 1.0450 
Training set R2 0.9867 0.9544 0.9036 
MSE (mm20.8642 1.2437 1.9484 
Overall set R2 0.9797 0.9366 0.8751 
MSE (mm20.6594 0.9865 1.2452 

The values in bold indicate the best results.

Figure 7

Comparison between the performance of PGKELM, LBA-SVM, and BPNN in dam deformation prediction at the A2 station: (a) correlation coefficients and (b) mean square error.

Figure 7

Comparison between the performance of PGKELM, LBA-SVM, and BPNN in dam deformation prediction at the A2 station: (a) correlation coefficients and (b) mean square error.

Figure 8

Comparison between the performance of PGKELM, LBA-SVM, and BPNN in dam deformation prediction at the A2 station: (a) correlation coefficients and (b) mean square error.

Figure 8

Comparison between the performance of PGKELM, LBA-SVM, and BPNN in dam deformation prediction at the A2 station: (a) correlation coefficients and (b) mean square error.

CONCLUSIONS

The LBA-PGKELM algorithm is proved to be an effective and simple method for establishing the dam deformation prediction. The hybrid kernel function for KELM and the Lévy flight optimized bat algorithm were combined to improve the dam deformation accuracy, based on machine learning method. The effectiveness and superiority of the proposed methodology are demonstrated by application to a real concrete dam, two observation points and compared with BPNN and SVM algorithms. The main conclusion are as follows.

  1. The present study investigated the two kernel KELM model, which is formed by exploiting the advantages of both Gaussian and polynomial functions as the kernel for the KELM. The proposed hybrid kernel not only avoids the instability of traditional ELM, but also improves the generalization and learning ability of the model.

  2. The modification on the BA algorithm, the LBA algorithm, can solve the disadvantages of the BA algorithm, such as slow convergence speed, low convergence precision, and easy to fall into a local minimum.

  3. Finally, compared with the conventional single-core KELM model, the PGKELM has strong learning ability and generalization ability. The high performance of the proposed method indicates that the selection, processing, and coding of the input variables have been carried out successfully.

According to statistics, there are 98,822 reservoirs in China. The high accuracy of our proposed model demonstrates the feasibility of our model in dam deformation prediction, and it can be applied to other reservoirs. In a further study we will connect all the monitoring stations of the whole dam and consider the spatiotemporal diversity in deformation behavior, aiming to construct a more competitive model for dam deformation prediction.

AUTHOR CONTRIBUTIONS

Conceptualization, Youliang Chen and Gang Xiao; Methodology, Youliang Chen; Software, Gang Xiao; Validation, Youliang Chen; Formal analysis, Gang Xiao; Data curation, Xiangjun Zhang; Writing – original draft preparation, Xiangjun Zhang; Writing – review and editing, Hamed Karimian; Visualization, Xiangjun Zhang; Supervision, Youliang Chen and Hamed Karimian; Project administration and funding acquisition, Jinsong Huang.

ACKNOWLEDGEMENTS

This research is supported by the National Dam Center Open Fund Project of China (CX2019B07) and the Science and Technology Project of Jiangxi Provincial Department of Education (GJJ170522) and the Ganzhou Key R & D Project.

CONFLICTS OF INTEREST

The authors declare no conflict of interest.

DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

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