## 2 Method
### The Gross Errors Detection Method of Concrete Dam Deformation Monitoring Data Based on IGG-ELM
For the ELM output model containing \(L\) hidden layer nodes:  
\[T=\big{[}t_{1},t_{2},\cdots,t_{Q}\big{]} \tag{1}\]  
\[y_{j}=\sum_{i=1}^{l}\beta_{i}g(\omega_{i}\cdot x_{j}+b_{i}),j=1,2,\cdots,Q \tag{2}\]  
where \(y_{j}\) represents the corresponding output vector for the input \(x_{j}\) in the ELM model, \(g(\omega_{i}\cdot x_{j}+b_{i})\) represents the corresponding activation function for the input \(x_{j}\) in the ELM model, \(g(\omega_{i}\cdot x_{j}+b_{i})\) represents the corresponding activation function for the input \(x_{j}\) vector, in this paper the \"Sigmoid\" type activation function is utilized, of which expression is:  
\[g(\omega_{i}\cdot x_{j}+b_{i})=\frac{1}{1+e^{-(\omega_{i}\cdot x_{j}+b_{i})}} \tag{3}\]  
where \(\omega_{i}=\big{[}\omega_{i1}\quad\omega_{i2}\quad\cdots\quad\omega_{in} \big{]}\), \(x_{j}=\big{[}x_{1j}\quad x_{2j}\quad\cdots\quad x_{nj}\big{]}^{T}\). The traditional ELM output weight \(\beta_{i}\) utilizing LS to construct loss function:  
\[Q=\sum_{i=1}^{Q}\ (t_{i}-y_{i})^{2} \tag{4}\]  
When \(Q\) obtains the minimum value, according to Equations (1) and (2) it can be approximated as: \(H\beta=T^{\prime}\), and the output weight of the hidden layer node of the neural network can be calculated:  
\[\overset{\wedge}{\beta}=H^{+}T^{\prime}=\left(H^{T}H\right)^{-1}H^{T}T^{\prime} \tag{5}\]  
In equation:  
\(H(\omega_{1},\omega_{2},\cdots,\omega_{l},b_{1},b_{2},\cdots,b_{l},x_{1},x_{2 },\cdots,x_{Q})= \begin{bmatrix}g(\omega_{1}\cdot x_{1}+b_{1})&g(\omega_{2} \cdot x_{1}+b_{2})&\cdots&g(\omega_{l}\cdot x_{1}+b_{l})\\ g(\omega_{1}\cdot x_{2}+b_{1})&g(\omega_{2}\cdot x_{2}+b_{2})&\cdots&g(\omega _{l}\cdot x_{2}+b_{l})\\ \cdots&\cdots&\cdots&\cdots\\ g(\omega_{1}\cdot x_{Q}+b_{1})&g(\omega_{2}\cdot x_{Q}+b_{2})&\cdots&g(\omega _{l}\cdot x_{Q}+b_{l})\end{bmatrix} ,H^{+}\) is the \(Moore-Penrose\) Generalized Inverses of the output matrix for the hidden layer; \(T^{\prime}\) is the transposition of the Matrix \(T\). The mathematical model expression of applying robust estimation into the ELM is:  
\[H\beta+e=T^{\prime} \tag{6}\]  
where \(e\) is a residual vector, and the error of the \(i\) sample is expressed as:  
\[e_{i}=t_{i}-\sum_{j=1}^{L}h_{ij}\beta_{j} \tag{7}\]  
The structural loss function is expressed as:  
\[Q^{\prime}=\sum_{i=1}^{Q}\rho^{2}(e_{i})=\sum_{i=1}^{Q}\rho^{2}(t_{i}-\sum_{j =1}^{L}h_{ij}\beta_{j}) \tag{8}\]where \(\rho(\cdot)\) is the influence function, \(Q\) is the total number of samples. Let \(\psi(e_{i})=\frac{\partial\rho(e_{i})}{\partial(e_{i})}\), the condition of optimal solution between the connection weight \(\beta\) of the hidden layer and the output layer is \(Q^{\prime}\) getting the minimum, then: \(\frac{\partial\rho(e_{i})}{\partial(\beta_{i})}=2\psi(e_{i})h_{i}=0\), then it is expressed as:  
\[\sum\limits_{i=1}^{Q}\psi(e_{i})h_{i}=0 \tag{9}\]  
Let \(\psi(e_{i})=b_{i}e_{i}\), where \(b_{i}\) represents weight factor, then: \(\sum\limits_{i=1}^{Q}b_{i}e_{i}h_{i}=0\), rewrite to the matrix form:  
\[H^{\mathcal{T}}B(T^{\prime}-H\beta)=0 \tag{10}\]  
Then:  
\[\overset{\wedge}{\beta}=\left(H^{\mathcal{T}}BH\right)^{-1}\!H^{\mathcal{T}} BT^{\prime} \tag{11}\]  
Due to the weights of the actual deformation of the dam are \(P_{i}\), let \(\overset{\rightarrow}{P}=\left\{\overset{\rightarrow}{P_{i}}\right\}=\left\{ \overset{\rightarrow}{P_{i}}\cdot b_{i}\right\},\) and the matrix of the normal equations is expressed as:  
\[\overset{\wedge}{\beta}=\left(H^{\mathcal{T}}\overset{\rightarrow}{P}H \right)^{-1}\!H^{\mathcal{T}}\!\overset{\rightarrow}{P}\!T^{\prime} \tag{12}\]  
The selection iteration method in the robust estimation is to determine the weight factor according to the residue size, among which the most effective robust scheme is to utilize the IGG method to gradually reduce the weight of the abnormal value to accurately identify the outliers [38]. The IGG equivalent weight function is expressed as:  
\[\overset{\rightarrow}{P}_{i}=\left\{