# Study on Denoising Method of Weld Defect Signal Based on SSA-VMD-WPD
## 3 Results and Discussion
### Evaluation of Denoising Effect  
3.3.1. Evaluation Indicators  
Traditional evaluation metrics for denoising effectiveness mainly include signal-to-noise ratio (\(SNR\)) [41], root mean square error (\(RMSE\)) [42], correlation coefficient, variance contribution rate, and smoothness [43]. 1. Signal-to-Noise Ratio (\(SNR\)) \[SNR=10\times\lg\left(\frac{\sum\limits_{t=1}^{n}f^{2}(t)}{\sum\limits_{t=1}^{ n}\left(f(t)-f^{\prime}(t)\right)^{2}}\right)\] (16)  
In this equation, \(f(t)\) represents the original signal, and \(f^{\prime}(t)\) represents the denoized signal. A higher value of \(SNR\) for the denoized signal indicates better denoising effectiveness. 2. Root Mean Square Error (\(RMSE\)): \[RMSE=\sqrt{\frac{1}{n}\sum\limits_{t=1}^{n}\left(f(t)-f^{\prime}(t)\right)^{2}}\] (17)  
In this equation, \(f(t)\) represents the original signal, \(f^{\prime}(t)\) represents the denoized signal, and n is the length of the signal. \(RMSE\) reflects the differences between signals. A smaller \(RMSE\) value for the denoized signal indicates better denoising effectiveness. 3. Correlation Coefficient: \[R=\text{cov}(f(t),f^{\prime}(t))/\sigma_{1}\sigma_{2}\] (18)  
In this equation, \(f(t)\) represents the original signal, \(f^{\prime}(t)\) represents the denoized signal, \(\sigma_{1}\) is the standard deviation of \(f(t)\), and \(\sigma_{2}\) is the standard deviation of \(f^{\prime}(t)\). A higher correlation coefficient value for the denoized signal indicates better denoising effectiveness. 4. Variance Contribution Rate: \[VCR_{i}=\frac{\frac{1}{n}\sum\limits_{t=1}^{n}u_{i}(t)^{2}-\left(\frac{1}{n} \sum\limits_{t=1}^{n}u_{i}(t)\right)^{2}}{\sum\limits_{i=1}^{n}\left[\frac{1} {n}\sum\limits_{t=1}^{n}u_{i}(t)^{2}-\left(\frac{1}{n}\sum\limits_{t=1}^{n}u_ {i}(t)\right)^{2}\right]}\] (19)  
According to reference [44], when the variance contribution rate of (5) Smoothness:  
\[RVR=\frac{\sqrt{\sum\limits_{t=1}^{n-1}\left[f^{\prime}(t+1)-f^{\prime}(t)\right] ^{2}}}{\sqrt{\sum\limits_{t=1}^{n-1}\left[f(t+1)-f(t)\right]^{2}}} \tag{20}\]  
Smoothness is an important metric for evaluating denoising effectiveness, reflecting the smoothness of the signal. A smaller smoothness value for the denoized signal indicates better denoising effectiveness. (6) Composite Evaluation Metric \(T\)  
From the denoising principles of variational mode decomposition, wavelet analysis, and wavelet packet analysis, it is known that there are multiple factors affecting denoising effectiveness. Therefore, it was necessary to select appropriate metrics to evaluate denoising effectiveness to achieve optimal parameter selection and denoising method selection. Traditional signal denoising effectiveness is generally evaluated using a single metric with a monotonic relationship with denoising effectiveness, such as signal-to-noise ratio, root mean square error, smoothness, correlation coefficient, and noise mode. However, in practical denoising processes, the true clean signal is unknown, leading to certain limitations. Therefore, considering the combination of multiple indicators to construct composite evaluation metrics, both the detailed information and approximation information of the signal were comprehensively evaluated [45]. In traditional evaluation metrics, signal-to-noise ratio, root mean square error, and cross-correlation coefficient can be used to evaluate denoising effectiveness from the perspective of signal details, while smoothness can be measured from the perspective of signal approximation [46]. This paper adopted the method of composite evaluation metrics, combining root mean square error and smoothness to construct the metric. Firstly, normalization was applied to these two metrics:  
\[PRMSE=\frac{RMSE-\min(RMSE)}{\max(RMSE)-\min(RMSE)} \tag{21}\]  
\[Rr=\frac{r-\min(r)}{\max(r)-\min(r)} \tag{22}\]  
In the equation, \(PRMSE\) and \(Pr\), respectively, represent the normalized values of \(RMSE\) and \(r\), where \(\max_{()}\) denotes the maximum value operation, and \(\min_{()}\) denotes the minimum value operation.