Improved online sequential extreme learning machine for identifying crack behavior in concrete dam

Abstract

Prediction models are essential in dam crack behavior identification. Prototype monitoring data arrive sequentially in dam safety monitoring. Given such characteristic, sequential learning algorithms are preferred over batch learning algorithms as they do not require retraining whenever new data are received. A new methodology using the genetic optimized online sequential extreme learning machine and bootstrap confidence intervals is proposed as a practical tool for identifying concrete dam crack behavior. First, online sequential extreme learning machine is adopted to build an online prediction model of crack behavior. The characteristic vector of crack behavior, which is taken as the online sequential extreme learning machine input, is extracted by the statistical model. A genetic algorithm is introduced to optimize the input weights and biases of online sequential extreme learning machine. Second, the BC_a method is proposed to produce confidence intervals based on the improved online sequential extreme learning machine prediction. The improved online sequential extreme learning machine for identifying crack behavior is then built. Third, the crack behavior of an actual concrete dam is taken as an example. The capability of the built model for predicting dam crack opening is evaluated. The comparative results demonstrate that the improved online sequential extreme learning machine can provide highly accurate forecasts and reasonably identify crack behavior.

Keywords

bootstrap confidence intervals crack behavior genetic algorithm identification model online sequential extreme learning machine

Introduction

Crack behavior is important in dam safety evaluation by helping identify the progressive deterioration of a dam. The crack behavior of actual concrete dams is usually diagnosed based on crack opening displacement (COD) time series (Wu, 2003), which can reflect the evolution of cracks under the influence of environmental and external loads. An appropriate mathematical model for identifying crack opening must be proposed to contribute to the creation of an early warning system for dam safety.

These models involve the use of machine learning algorithms such as auto-associative neural network, factor analysis, Mahalanobis squared distance, singular value decomposition (Figueiredo et al., 2011), support vector machine (SVM) (Worden and Manson, 2007), and deep learning (Cha et al., 2017; Chen and Jahanshahi, 2018). Researchers are now exploiting a considerably greater range of algorithms; systems like extreme learning machine (ELM) are beginning to appear more regularly. Upon its introduction in 2004, ELM has attracted increasing attention from researchers (Dai and He, 2016; Huang et al., 2006, 2011, 2015; Kang et al., 2017) because of its favorable generalization at fast learning speeds. The classical ELM optimization problem is formulated in batch mode while considering all available data at once (Scardapane et al., 2015). In dam safety monitoring applications, learning must be an ongoing process because the complete set of data is usually not available at once. When new data arrive, batch learning performs training using both new and previous data, thereby consuming much time (Lan et al., 2009).

These limitations hinder the application of real-time health monitoring in which the structure is continuously monitored and the crack behavior is assessed during its development. Liang et al. (2006) proposed an online version of ELM called the online sequential extreme learning machine (OS-ELM), which can learn the training data one by one or chunk by chunk (with fixed or varying size) and discard the training data as long as the training procedure for those data is completed.

However, OS-ELM may need a higher number of hidden nodes due to the random determination of input weights and hidden biases. Given that the genetic algorithm (GA) is widely used as a global searching method for optimization (Ganjehkaviri et al., 2017; Jenkins, 1991; Srivastava et al., 2002; Whitley, 1994), hybrids of GA and OS-ELM should be combined for real-time health monitoring.

In this article, a hybrid learning algorithm that uses GA to select the input weights and biases of OS-ELM is proposed. Statistical models are the most widely applied and accepted models by practitioners (Dai et al., 2018; Salazar et al., 2016; Wu, 2003). Thus, the components of statistical models can be extracted as OS-ELM input. The trained OS-ELM can be used to predict concrete dam crack behavior based on bootstrap confidence intervals (Efron, 1979; Efron and Tibshirani, 1986). The crack behavior of an actual concrete dam is identified and assessed using a model that is built online.

OS-ELM-based prediction model of crack behavior

Overview of ELM and OS-ELM

ELM

ELM (Huang et al., 2006) is a new batch learning algorithm for single-hidden layer feed-forward neural networks (SLFNs) that randomly chooses the input weights and hidden layer biases and then analytically determines the output weights using the Moore–Penrose (MP) generalized inverse matrix. The only parameter that needs to be tuned is the number of hidden nodes. ELM aims to achieve not only the smallest training error but also the smallest norm of output weights (Mao et al., 2016).

For N arbitrary distinct samples $(x_{i}, t_{i})$ where $x_{i} = (x_{i 1}, x_{i 2}, \dots, x_{in})^{T} \in R^{n}$ and $t_{i} = (t_{i 1}, t_{i 2}, \dots, t_{im})^{T} \in R^{m}$ , standard SLFNs with $\tilde{N}$ hidden neurons with activation function $g (x)$ can approximate N samples with zero error means that exist $β_{i}$ , $ω_{i}$ , and $b_{i}$ such that

\sum_{i = 1}^{\tilde{N}} β_{i} g (ω_{i} \cdot x_{j} + b_{i}) = t_{j}, j = 1, 2, \dots, N

(1)

where $ω_{i} = (ω_{i 1}, ω_{i 2}, \dots, ω_{in})$ is the weight vector connecting the $i th$ hidden neuron and the input neurons, $β_{i} = (β_{i 1}, β_{i 2}, \dots, β_{im})^{T}$ is the weight vector connecting the $i th$ hidden neuron and the output neurons, and $b_{i}$ is the bias of the $i th$ hidden neuron.

Equation (1) can be written as

H β = T

(2)

where

\begin{matrix} H (ω_{1}, \dots, ω_{\tilde{N}}, b_{1}, \dots, b_{\tilde{N}}, x_{1}, \dots, x_{N}) \\ = {[\begin{matrix} g (ω_{1} \cdot x_{1} + b_{1}) & \dots & g (ω_{\tilde{N}} \cdot x_{1} + b_{\tilde{N}}) \\ ⋮ & \dots & ⋮ \\ g (ω_{1} \cdot x_{N} + b_{1}) & \dots & g (ω_{\tilde{N}} \cdot x_{N} + b_{\tilde{N}}) \end{matrix}]}_{N \times \tilde{N}} \end{matrix}

(3)

and

β = {[\begin{matrix} {β_{1}}^{T} \\ ⋮ \\ {β_{\tilde{N}}}^{T} \end{matrix}]}_{\tilde{N} \times m} and T = {[\begin{matrix} {t_{1}}^{T} \\ ⋮ \\ {t_{\tilde{N}}}^{T} \end{matrix}]}_{\tilde{N} \times m}

(4)

$H$ denotes the hidden layer output matrix of the neural network. In batch ELM, the input weights $ω_{i}$ and hidden biases $b_{i}$ are randomly assigned, and then the output weights $β$ are estimated as

\hat{β} = H^{†} T

(5)

where $H^{†}$ is the MP generalized inverse (Serre, 2002) of the hidden layer output matrix $H$ .

OS-ELM

Developed on the basis of ELM, the OS-ELM (Huang et al., 2005; Liang et al., 2006) includes two phases, namely, the boosting phase and the sequential learning phase. New observations are mostly used in the sequential learning phase, and the OS-ELM monitoring of dam safety can be updated in real time (Figure 1).

Figure 1.

Learning targets of OS-ELM framework (Huang et al., 2015).

The OS-ELM is derived as follows:

Step 1. Boosting phase: Given a small initial training set $ℵ = {(x_{i}, t_{i}) | x_{i} \in R^{n}, t_{i} \in R^{m}, i = 1, \dots, N_{0}}$ , where $N_{0} \geq \tilde{N}$ , $\tilde{N}$ represents the number of hidden nodes, $x_{i}$ and $t_{i}$ are the same as above.

(a) Assign arbitrary input weight $ω_{i}$ and bias $b_{i}$ , $i = 1, 2, \dots, \tilde{N}$ .

(b) Calculate the initial hidden layer output matrix $H_{0} = [\begin{matrix} h_{1} & \dots & h_{N_{0}} \end{matrix}]^{T}$ , where $h_{i} = [\begin{matrix} g (ω_{1} \cdot x_{i} + b_{1}) & \dots & g (ω_{\tilde{N}} \cdot x_{i} + b_{\tilde{N}}) \end{matrix}]^{T}$ , $i = 1, 2, \dots, N_{0}$ .

(c) Calculate the initial output weight vector $β^{(0)} = M_{0} {H_{0}}^{T} T_{0}$ , where $M_{0} = ({H_{0}}^{T} H_{0})^{- 1}$ and $T_{0} = [\begin{matrix} t_{1}, & \dots & , t_{N_{0}} \end{matrix}]^{T}$ .

(d) Set $k = 0$ .

Step 2. Sequential learning phase: For each coming observation $ℵ = {(x_{i}, t_{i}) | x_{i} \in R^{n}, t_{i} \in R^{m}, i = N_{0} + 1, \dots, N}$ , do

(a) Calculate the hidden layer output vector $h_{k + 1} = [\begin{matrix} g (ω_{1} \cdot x_{i} + b_{1}) & \dots & g (ω_{\tilde{N}} \cdot x_{i} + b_{\tilde{N}}) \end{matrix}]^{T}$ .

(b) Calculate the latest output weight $β^{(k + 1)}$ as follows

M_{k + 1} = M_{k} - \frac{M_{k} h_{k + 1} {h_{k + 1}}^{T} M_{k}}{1 + {h_{k + 1}}^{T} M_{k} h_{k + 1}}

(6)

and

β^{(k + 1)} = β^{(k)} + M_{k + 1} h_{k + 1} ({t_{i}}^{T} - {h_{k + 1}}^{T} β^{(k)})

(7)

Input determination

Under the loads of water pressure (H), temperature (T), and other factors, the concrete dam crack will produce displacement K at any point. Based on its origin, the displacement K can be divided into hydraulic components $K (H)$ , temperature components $K (T)$ , and aging components $K (θ)$ . The parameters for representing the thermal effect are selected in two models, namely, the HTT (hydrostatic, thermal, time) models and the HST (hydrostatic, seasonal, time) models (Mata et al., 2014).

If the hydration heat of the concrete has been distributed and the quasi-steady temperature field has been reached in the dam body, then the multi-period harmonic can be selected as a factor.

Thus, the statistical model (HST) for crack opening can be expressed in the following form

\begin{matrix} K & = K (H) + K (T) + K (θ) \\ = a_{0} + \sum_{i = 1}^{m_{1}} a_{i} H^{i} + \sum_{i = 1}^{m_{2}} (b_{1 i} \sin iS + b_{2 i} \cos iS) \\ + c_{1} θ + c_{2} \ln θ \end{matrix}

(8)

where $a_{0}$ expresses a constant, $a_{i}$ , $b_{1 i}$ , $b_{2 i}$ , $c_{1}$ , and $c_{2}$ represent the coefficients, and H is the upstream water depth. The value of $m_{1}$ differs across each dam type, where 3 is generally taken for the gravity dam, while 4 or 5 is desirable for the arch dam. S is computed as $S = 2 π t / 365$ , i denotes the period ( $i = 1$ for the annual cycle and $i = 2$ for half a cycle), $t_{d}$ denotes the cumulative number of days from the initial value to the monitoring value, $m_{2}$ is usually taken as 1 or 2, and $θ = t_{d} / 100$ .

Parameter selection

OS-ELM hidden nodes and feature mappings

The number of hidden neurons $\tilde{N}$ ranges from 1 to $N_{max}$ taking 1 as the step length. Given the number of hidden nodes $\tilde{N}$ and the mapping function $g (ω, b, x)$ (the four activation functions are summarized in Table 1), we develop a multi-objective genetic OS-ELM that provides appropriate input weights and biases.

Table 1.

Commonly used mapping functions in OS-ELM.

Sigmoid function	$g (ω, b, x) = \frac{1}{1 + \exp (- (ω \cdot x + b))}$
Sine function	$\sin (ω \cdot x + b)$
Gaussian function	$g (ω, b, x) = \exp (- ‖ x - ω ‖ \cdot b)$
Hard limit function	$g (ω, b, x) = {\begin{matrix} 1, & if ω \cdot x + b \leq 0 \\ 0, & otherwise \end{matrix}$

GA

OS-ELM works by randomly choosing the input weight $ω_{i}$ and bias $b_{i}$ of the hidden nodes and then analytically obtaining the output weights and biases for an SLFN (Huang et al., 2005; Lan et al., 2009; Liang et al., 2006). Our algorithm attempts to find the SLFN that simultaneously minimizes the mean square error (MSE).

The first step in implementing GA is to generate an initial population $P_{1}$ . Each individual in the population $P_{1}$ is defined as follows

(ω, b) = (ω_{11}, \dots, ω_{n 1}, ω_{12}, \dots, ω_{n 2}, \dots, ω_{1 \tilde{N}}, \dots, ω_{n \tilde{N}}, b_{1}, \dots, b_{\tilde{N}})

(9)

where n and $\tilde{N}$ are similarly defined as above.

After creating an initial population, each individual $(ω, b)$ (sometimes referred to as a chromosome) is evaluated and assigned a fitness value. In this study, fitness is defined by the OS-ELM training MSE as follows

fitness = \frac{1}{N} \sum_{s = 1}^{N} \sum_{i = 1}^{m} {(t_{si} - f (x_{s}))}^{2}

(10)

where N and m have the same meaning as above, and $f (x_{s})$ is the value provided by the OS-ELM for the input vector $x_{input}$ .

The GA starts with the initial population, where selection is applied to create an intermediate population. Recombination and mutation are then applied to the intermediate population to create the next population. The initialization, selection, recombination, and mutation processes form one generation in the execution of a GA (Whitley, 1994).

Bootstrap confidence intervals

A confidence interval is usually interpreted as a range of values that encompass the population or “true” value—which is estimated by a certain statistic—with a given probability (Nakagawa and Cuthill, 2007). Bootstrapping (Efron, 1979) was originally introduced as a nonparametric device for estimating confidence intervals. In this work, an improved version of the percentile method called the bias-corrected and accelerated (BC_a) method was adopted. The BC_a method, as can be seen in line 4 of Table 2, is an automatic algorithm for producing highly accurate confidence intervals (DiCiccio and Efron, 1996; Efron and Tibshirani, 1986, 1994).

Table 2.

Four methods of setting approximate confidence intervals for a real valued parameter $θ$ (Efron and Tibshirani, 1986).

Method	Abbreviation	$α$ level point
1. Standard	$θ_{s} (α)$	$θ (α) + \hat{σ} z^{(α)}$
2. Basic percentile	$θ_{P} (α)$	${\hat{G}}^{- 1} (α)$
3. Bias-corrected (BC)	$θ_{BC} (α)$	${\hat{G}}^{- 1} (Φ {2 z_{0} + z^{(α)}})$
4. Bias-corrected and accelerated (BC_a)	$θ_{B C_{a}} (α)$	${\hat{G}}^{- 1} (Φ {z_{0} + \frac{(z_{0} + z^{(α)})}{1 - a (z_{0} + z^{(α)})}})$

Define $\hat{G} (s)$ as the parametric bootstrap cumulative distribution function (CDF) of ${\hat{θ}}_{*}$

\hat{G} (s) = {Prob}_{*} ({\hat{θ}}_{*} < s)

(11)

where $P r o b_{*}$ indicates the probability that is computed according to the bootstrap distribution of ${\hat{θ}}_{*}$ . Methods 2, 3, and 4 (Table 2) are defined in terms of the percentiles of $\hat{G}$ and the bootstrap distribution (11).

The BC_a interval (Efron and Tibshirani, 1986) of the intended coverage $1 - 2 α$ is given as follows

{BC}_{a} : ({\hat{θ}}_{lo}, {\hat{θ}}_{up}) = ({\hat{θ}}_{α_{1}}, {\hat{θ}}_{α_{2}})

(12)

where

\begin{matrix} α_{1} = Φ {z_{0} + \frac{(z_{0} + z^{(α)})}{1 - a (z_{0} + z^{(α)})}} \\ α_{2} = Φ {z_{0} + \frac{(z_{0} + z^{(1 - α)})}{1 - a (z_{0} + z^{(1 - α)})}} \end{matrix}

(13)

where $Φ (\cdot)$ is the standard normal CDF, and $z^{(α)}$ is the $100 α th$ percentile point of a standard normal distribution. The constant $z_{0}$ is given by $z_{0} = Φ^{- 1} {\hat{G} (\hat{θ})}$ , and the acceleration a is approximated as follows

a = \frac{{SKEW}_{θ = \hat{θ}} ({\overset{\cdot}{l}}_{θ} (t))}{6}

(14)

where ${SKEW}_{θ = \hat{θ}} ({\overset{\cdot}{l}}_{θ} (t))$ is the skewness at parameter value $θ = \hat{θ}$ of the score statistic ${\overset{\cdot}{l}}_{θ} (t) = (\partial / \partial θ) \log f_{θ} (t)$ .

Construction of the model for identifying concrete dam crack behavior

The improved OS-ELM algorithm is developed in this section. Assume that $(x_{i}, t_{i})$ , $i = 1, 2, \dots, N$ represents the training and testing datasets where $x_{i} = x_{input}$ and $t_{i}$ is the COD. The first $N_{1}$ sets are taken as the training samples to implement the training operation of GA optimizing OS-ELM and to obtain the crack opening prediction model. The last $N_{2}$ sets are identified and accessed by BC_a intervals. The flowchart is shown in Figure 2.

Figure 2.

Improved OS-ELM construction flowchart.

Example analysis

Based on the improved OS-ELM for identifying crack behavior, one typical crack near the 105 m elevation on the downstream face of a gravity arch dam is analyzed in detail. This crack has a crest elevation of 126.3 m and maximum height of 76.3 m. The crack is divided into 28 sections numbered from left to right. The pouring to basic completion of the dam takes 12 years and is divided into three phases (I, II, and III shown in Figure 3). Because the pouring level in the second phase concrete rapidly increases, the pouring interval is rather short. The shrinkage deformation of the second phase concrete is strongly restrained by the first phase concrete (Gu et al., 2011), thereby producing a crack of 300 m long and 5 m deep at the top of the first poured concrete. The COD records (as shown in Figure 3(b)) between 1 January 1973 and 24 December 2012 from X18-1 are analyzed. Around 90% sample points are chosen to train the improved OS-ELM, while the remaining 10% are set aside as a test set.

Figure 3.

(a) Typical cross-section of the dam (Xing and Fang, 1988) and (b) variation of X15-1 and X18-1.

Input determination

Given that $m_{1} = 4$ is desirable for the arch dam, according to equation (8), $x_{input}$ can be written as follows

x_{i n p u t} = (H^{1}, H^{2}, H^{3}, H^{4}, \sin S, \cos S, \sin 2 S, \cos 2 S, θ, \ln θ)

(15)

Obviously, $x_{input} \in R^{10}$ can be regarded as the input vector of the improved OS-ELM. COD $t_{i} \in R$ $(i = 1, 2, \dots, N)$ is taken as the OS-ELM output. In $x_{input}$ , a large numerical difference is observed between the water level component and the other feature factors, and such difference may reduce the fitting and prediction precision. All factors in $x_{input}$ are normalized as follows

x_{input} (i, j) = \frac{x_{input} (i, j) - min (x_{input} (:, j))}{max (:, j) - min (x_{input} (:, j))}

(16)

Genetic OS-ELM

We generate an initial population, a member of which is represented by $(ω, b)$ (equation (9)). fitness (equation (10)) measures the performance with respect to the input weight $ω_{i}$ and bias $b_{i}$ of OS-ELM. $\tilde{N} \times (n + 1)$ represents the length of a chromosome $(ω, b)$ . Recombination is then applied to randomly paired individuals with a probability of $P_{c} = 0.9$ . After recombination, we apply a mutation to the population with some low probability $P_{m}$ . After completing the selection, recombination, and mutation processes, a generation is implemented. The parameters of our algorithm are shown in Tables 3 and 4.

Table 3.

Parameter setting of genetic algorithm.

Parameter	Value
Number of iterations	20
Size of population	10
$P_{c}$	0.9
$P_{m}$	0.1

Table 4.

Parameter setting of OS-ELM.

Parameter	Value
$N_{0}$	200
$N_{max}$	20

OS-ELM: online sequential extreme learning machine.

OS-ELM hidden nodes

The variations in the fitting and prediction ability of the built model along with the changes of the hidden nodes $\tilde{N}$ are analyzed. The number of hidden neurons $\tilde{N}$ ranges from 1 to $N_{max}$ taking 1 as the step length. For a given hidden node $\tilde{N}$ , the genetic OS-ELM is repeated 50 times and the dataset errors are averaged (Figure 4). The following observations can be seen in Figure 4:

As the number of hidden nodes $\tilde{N}$ increases, the MSE gradually decreases.

When the given $\tilde{N}$ increases to 16, the fitting precision and prediction precision are not greatly improved.

As the fitting MSE decreases, the overall trend of the prediction MSE is decreasing. Thus, the fitness defined by the OS-ELM training MSE is reasonable.

The “over-learning” phenomenon is not observed.

Figure 4.

The fitting and prediction MSE under different hidden nodes.

To achieve a lower number of hidden nodes and a lower MSE, 16 is chosen as the number of hidden nodes $\tilde{N}$ .

OS-ELM mapping functions

The OS-ELM mapping function is taken in turn as a sigmoid function, sine function, Gaussian function, and hard limit function. Let the number of hidden nodes be $\tilde{N} = 16$ . We apply the improved OS-ELM algorithm 50 times, and the fitting and prediction MSE under different mapping functions is shown in Figure 5. This chart shows the mean, median, upper and lower quartiles, and minimum and maximum COD values for the residuals. The following observations can be seen in Figure 5:

MSE varies when the mapping function differs. Among the aforementioned mapping functions, the sigmoid function MSE is the smallest, thereby indicating that this function has the highest precision.

The fitting and prediction MSE under the sigmoid function slightly vary each time.

Figure 5.

(a) Boxplot of the fitting MSE under different mapping functions and (b) boxplot of the prediction MSE under different mapping functions.

According to the above analysis, the sigmoid function must be chosen as the mapping function of the COD records in the study.

Real-time learning

The data arrive sequentially in dam safety monitoring. In this context, using a sequential algorithm is preferable over collecting data before processing. The OS-ELM algorithm is a sequential implementation of the ELM algorithm with two outstanding characteristics, namely, (1) only the newly arrived data are learned and (2) a block of data is discarded as soon as these data are learned by the algorithm. These two characteristics contribute to real-time health monitoring, in which the structure is continuously monitored and the crack behavior is assessed online. If block equals number of data for each month, then the data for each month can be learned online. The time-varying characteristic of the sample series will be learned sufficiently in the sequential learning phase.

Performance verification

The performance of four model types, namely, the statistical model, ELM, OS-ELM, and improvedOS-ELM, are tested in this example analysis. The following quantitative evaluation indexes ( $K_{MAPE}$ , $K_{M S E}$ , $K_{NMSE}$ , and $R^{2}$ ) are used to evaluate the performance of the model

K_{MAPE} = \frac{100}{T} \sum_{i = 1}^{T} | \frac{t_{i} - {\hat{t}}_{i}}{t_{i}} |

(17)

K_{MSE} = \frac{1}{T} \sum_{i = 1}^{T} {(t_{i} - {\hat{t}}_{i})}^{2}

(18)

K_{NMSE} = \frac{1}{T σ^{2}} \sum_{i = 1}^{T} {(t_{i} - {\hat{t}}_{i})}^{2}

(19)

and

R^{2} = \sum_{i = 1}^{T} {({\hat{t}}_{i} - \frac{1}{T} \sum_{i = 1}^{T} t_{i})}^{2} / \sum_{i = 1}^{T} {[t_{i} - \frac{1}{T} \sum_{i = 1}^{T} t_{i}]}^{2}

(20)

where $t_{i}$ is the measured value, ${\hat{t}}_{i}$ represents the prediction, T is the length of the test data, and $σ^{2}$ denotes the data variance for the test set $N_{2}$ .

The determined parameters and train set $N_{1}$ are used to train the improved OS-ELM. The prediction of the test set $N_{2}$ can be obtained. The predictive results of the statistical model, ELM, OS-ELM, and improved OS-ELM are shown in Figure 6. Table 5 lists the performance evaluation indexes for these three prediction methods.

Figure 6.

Observations, predictions of (a) statistical model, (b) ELM, (c) OS-ELM, and (d) the improved OS-ELM.

Table 5.

Comparisons of three prediction methods.

	Prediction ability
	$K_{MAPE}$	$K_{M S E}$	$K_{NMSE}$	$R^{2}$
Statistical model	7.06%	0.0641	0.2972	0.8990
ELM	8.69%	0.1023	0.4748	0.9003
OS-ELM	6.84%	0.0607	0.2817	0.8230
Improved OS-ELM	4.47%	0.0267	0.1239	0.9311

ELM: extreme learning machine; OS-ELM: online sequential extreme learning machine.

The best results are highlighted in bold.

The improved OS-ELM is much more effective than the other models for predicting crack behavior. Standard OS-ELM outperforms ELM except in terms of $R^{2}$ . Given the OS-ELM real-time learning, the time-varying characteristic of the sample points can be learned sufficiently. The improved OS-ELM outperforms OS-ELM in terms of regression accuracy because our algorithm optimizes the input weight $ω_{i}$ and bias $b_{i}$ .

Figure 7 shows the residuals for statistical model, ELM, OS-ELM, and improved OS-ELM as well as highlights the excellent performance of the improved OS-ELM. An analysis of Table 5 reveals that the four quantitative evaluation indexes ( $K_{MAPE}$ , $K_{M S E}$ , $K_{NMSE}$ , and $R^{2}$ ) show their best performance in the improved OS-ELM, thereby implying narrower confidence intervals for the same level of significance. The improved OS-ELM prediction can also be used to estimate the bootstrap confidence intervals.

Figure 7.

Residuals for (a) statistical model, (b) ELM, (c) OS-ELM, and (d) the improved OS-ELM.

BC_a intervals

The improved OS-ELM is applied 50 times to obtain bootstrap confidence intervals samples. On the basis of these samples, we produce the 95% BC_a confidence intervals for the maximum and minimum COD. The extremum of these confidence intervals at each time are extracted to draw an envelope graph for crack behavior. The 95% BC_a envelope graph for crack behavior is presented in Figure 8.

Figure 8.

95% BC_a envelope graph for the crack behavior.

The BC_a intervals can reflect the crack behavior under a given environment. Larger cracks are formed during winter because of the low water level and temperature. By contrast, smaller cracks are formed during summer. The identification process is computed using an online and dynamic pattern, which will contribute to real-time health monitoring and to the formation of an early warning system for dam safety.

As shown in the graph, most observations can be enveloped between the minimum and maximum. Although very few small crack values exceed the confidence intervals, the large values of dam crack behavior are the focus of our study. The dam crack behavior in the test set does not show any abnormality according to the results.

Conclusion

This study proposes a new methodology for identifying dam crack behavior called OS-ELM, which combines genetic OS-ELM with bootstrap confidence intervals. The input vector represents the impact factors and components of concrete dam crack behavior. The improved OS-ELM not only realizes real-time prediction but also dynamically identifies whether the crack behavior is abnormal.

The OS-ELM is proposed to extend the basic ELM for online sequential data. The main problem is that the input weight $ω_{i}$ and bias $b_{i}$ of the standard OS-ELM are randomly generated. We propose the improved OS-ELM optimized by GA, which sufficiently utilizes the time-varying characteristic of crack behavior. The improved OS-ELM with typical input and optimal parameters is proposed to establish a model for predicting concrete dam crack behavior.

The ranges of values that encompass the COD values are estimated using the BC_a method, an automatic algorithm for producing highly accurate confidence intervals and narrow confidence intervals for COD values. We can judge whether crack behavior is abnormal based on the 95% BC_a envelope graph.

A typical crack of a gravity arch dam is analyzed using ELM, OS-ELM, and improved OS-ELM. Compared with other methods, the improved OS-ELM shows a higher prediction precision and better quantitative evaluation indexes ( $K_{MAPE}$ , $K_{M S E}$ , $K_{NMSE}$ , and $R^{2}$ ). This method can also efficiently identify concrete dam crack behavior. Using this method in real time can also improve its early warning function in dam safety monitoring.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The writers are grateful for the financial support from National Key R&D Program of China (Grant No. 2016YFC0401601), the Fundamental Research Funds for the Central Universities (Grant No. 2017B619X14), Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant Nos KYCX17_0428, KYZZ15_0140), National Natural Science Foundation of China (Grant Nos 51739003, 51479054, 51779086, 51579086, 51379068, 51579083, 51579085, 51609074), Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (Grant No. YS11001), Jiangsu Natural Science Foundation (Grant No. BK20160872), Special Project Funded of National Key Laboratory (Grant Nos 20145027612, 20165042112), Key R&D Program of Guangxi (Grant No. AB17195074), and Central University Basic Research Project (Grant No. 2017B11114).

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Improved online sequential extreme learning machine for identifying crack behavior in concrete dam

Abstract

Keywords

Introduction

OS-ELM-based prediction model of crack behavior

Overview of ELM and OS-ELM

ELM

OS-ELM

Input determination

Parameter selection

OS-ELM hidden nodes and feature mappings

GA

Bootstrap confidence intervals

Construction of the model for identifying concrete dam crack behavior

Example analysis

Input determination

Genetic OS-ELM

OS-ELM hidden nodes

OS-ELM mapping functions

Real-time learning

Performance verification

BC a intervals

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

BC_a intervals