## 2 Materials and Methods
### Multi-Feature Extraction of Pile Dynamic Signal
#### 2.4.3 ICEEMDAN Decomposition Sample Entropy and Information Entropy  
Feature Extraction  
To deal with nonlinear signals, signal analysis is an imposing application for the Improved Adaptive Noise Ensemble Empirical Mode Decomposition (ICEEMDAN) approach. When the foundation pile is defective, the received waveform signal is very complex, and the signal reflected from the defective interface is feeble. The ICEMAN approach analyzes the weak signals concealed in the signal and interprets the reflective waveforms several times by deconstructing the signal into its various intrinsic mode operations (IMFS). [PERSON] completed quantitative detection of internal defects in anchor solids by processing the anchor ultrasonic guided wave reflection signals by the ICEEMDAN method [42]. [PERSON] used the ICEEMDAN method to decompose the numerical simulation signals to extract damage indicators. The results showed that the extracted damage indicators were handy for identifying structures with or without damage after processing [43]. The ICEEMDAN method significantly inhibits the phenomenon of faulty components in signal decomposition and effectively improves the efficiency of signal decomposition. Here is the precise signal decomposition procedure. Step 1: Construct N signals with controllable noise based on the original signal x:  
\[x^{(i)}=x+\beta_{0}E_{1}(w^{(i)}),(i=1,2,\cdots N) \tag{4}\]  
where \(x^{(i)}\) is i construction signal; \(\beta_{0}\) denotes the standard error of the noise at the first decomposition of the signal; \(w^{(i)}\) is the \(i\)th added zero-mean unit-variance white noise; \(E_{1}\) (\(\cdot\)) is the first IMF operator to calculate the signal. Step 2:For each \(x^{(i)}\), the local mean is calculated and averaged to obtain the first residual component:  
\[r_{1}=\frac{1}{I}\sum\limits_{i=1}^{I}M(x(i)) \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Characteristic Index** & **Expression** \\ \hline mean frequency & \(F_{mean}=\sum\limits_{K=1}^{M}\frac{y_{K}}{M}\) \\ frequency center & \(F_{c}=\sum\limits_{k=1}^{M}\frac{y_{k}}{\sqrt{y_{k}}}\) \\ root mean square frequency & \(F_{rms}=\sqrt{\frac{\sum\limits_{k=1}^{M}y_{k}^{2}}{M}}\) \\ standard deviation frequency & \(F_{std}=\sqrt{\frac{\sum\limits_{k=1}^{M}(y_{k}-F_{\text{mean}})^{2}}{M}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Frequency domain characteristic index. \(M_{1}\) (\(\cdot\)) is an average local function. Step 3: Computing the first mode (k = 1), the original signal x minus the first residual r\({}_{1}\):  
\[\widetilde{d}=x-r_{1} \tag{6}\]  
Step 4: Calculate the k-mode (k \(\geq\) 2), subtract the residual r\({}_{\text{k-1}}\) from the residual r\({}_{\text{k}}\):  
\[r_{k}=\frac{1}{l}\sum_{i=1}^{l}M(r_{k-1}+\beta_{k-1}E_{k}(w^{(i)})) \tag{7}\]  
\[\widetilde{d_{1}}=r_{k-1}-r_{k} \tag{8}\]  
Step 5: Calculate the k = k + 1 mode and return to step 4 until the iteration termination condition is satisfied. The results of ICEEMDAN decomposition of complete pile dynamic signals in layered soils are shown in Figure 10.  
The likelihood of a signal eventually containing faulty components increases with signal complexity. Spurious components increase the computational effort and affect the subsequent feature extraction. The level of correlation among the decomposed signal components X and the original signal Y can be expressed using the correlation coefficient r. The connection of IMF components with the original signal is more robust and substantially impacts the foundation pile's recognition effect as r increases [44]. It is this way:  
\[r(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}} \tag{9}\]  
_Var (X)_ is the variance of \(X\), while _Var (Y)_ is the variance of \(Y\). _Cov (X, Y)_ is the covariance of \(X\) and \(Y\). Calculate the correlation coefficient between each IMF component and the original signal.