# Weak Multiple Fault Detection Based on Weighted
## 3 The Proposed Algorithm
### Adaptive Weight Parameter  
The value of the weights \(w_{i}\) in the OGS problem have a great effect on the accuracy of fault feature extraction. In view of this dependence on the weight selection, the weights \(w_{i}\) are constructed by analyzing the salient features of the fault signals. The purpose of modifying the values of the weights is to obtain a good reconstruction of the impulse signal. Introducing the corrected weights to penalty item \(G(\theta)\) can be written as follows:  
\[G(\theta)=\sum_{i=1}^{n}w_{i}\|\theta_{i,K}\|_{2} \tag{17}\]  
\(k\in\text{supp}(y)\) should denote the position of the significant feature, such as the impulse sequence; \(\widetilde{\Gamma}_{k}\subset\Gamma\) should be the set of groups containing position k; and finally, define a shrinkage operator on the weights \(P:w\in R_{+}\to R_{+}\) by:  
\[P(w_{i})\equiv\hat{w}_{i}=\left\{\begin{array}{ll}a_{i}w_{i}&\text{if}\\ w_{i}&\text{otherwise}\end{array}\right.\quad g(i)\in\widetilde{\Gamma}_{k} \tag{18}\]  
where \(a_{i}\in[0;1]\). For the given group \(g(i)\), if \(\hat{w}_{i}\leq w_{i}\), then the weights are correspondingly weakened in the optimization process. Therefore, preserving the coefficients of the fault-related salient features during the regularization process is advantageous.  
\(|\theta_{j,n}|\) should be the wavelet coefficient at position n and scale \(j\) under the Morlet wavelet transform. The normalized wavelet coefficients can be written as:  
\[\delta(n):=\left(\prod_{j=l_{0}}^{J}\gamma\left(|\theta_{j,n}|\right)\right)^ {1/\left(J-l_{0}+1\right)} \tag{19}\]where \(\gamma\left(\cdot\right)\) is expressed as  
\[\gamma(\theta_{j,n})=\frac{\theta_{j,n}-\min\theta_{j,}}{\max\theta_{j,}-\min \theta_{j,}}. \tag{20}\]  
In (20), \(\gamma\left(\cdot\right)\) denotes the normalization function. At the location of the impact sequence, the normalized wavelet coefficients \(\delta\) are close to 1. When the signal is in the smoothing phase, the normalized wavelet coefficients \(\delta\) are approximately zero. Threshold and the minimum interval length of \(r\) are defined to find the location of the impulse points. We define set \(K_{\tau,r}(1),K_{\tau,r}(2),K_{\tau,r}(S)\) as a non-empty, disjoint, non-adjacent interval that must satisfy the interval length greater than \(r\) and \(\delta(n)>\tau\). The position of the salient feature in the observed signal \(y\) is determined by the \(S\) set of positions \(K_{\tau,r}\) in the following equation:  
\[K_{\tau,r}:=\left\{\max\limits_{n\in K_{\tau,r}(s)}\delta(n)\right\}_{s=1}^{S} \tag{21}\]  
Algorithm 1 gives the process of estimating the set of positions \(K_{\tau,r}\) and the value of weights \(\left\{w_{i}\right\}\), which are determined by the choice of parameters \(\tau\) and \(r\).  
```
Input:\(y\); \(r,n_{0}\) Output:\(\left\{w_{i}\right\}\) Procedure: \(\theta\leftarrow\) Morlet\(CWT(y)\) \(\delta(n)\leftarrow\theta\) \(\left\{\tau_{q}\right\}\leftarrow\delta\) \(K_{\tau_{q},r}\leftarrow\left\{n:\delta(n)>\tau_{q}\right\}\) \(K_{\tau_{q},r}(i)\leftarrow\arg\max\limits_{n\in K_{q},r(i)}\delta(n)\) \(\bar{\Gamma}_{k}\gets K_{\tau_{q},r}\) \(\phi^{2}\leftarrow\) Estimate noise variance from \(\theta\) \(a\leftarrow\) Set shrinkage factor according to \(\phi\) \(\left\{w_{i}\right\}\leftarrow\left(\bar{\Gamma}_{k},a\right)\) End
```  
**Algorithm 1**Determination of the set of positions \(K_{\tau,r}\) and adaptive weights \(\left\{w_{i}\right\}\).