## 2 Methodology
### Signal Preprocess and Determination of State Boundaries
#### 2.3.2 State Boundary Identification  
Fourier transform is a traditional tool for analyzing fluctuating signals, but it is not suitable for non-stationary and nonlinear signals. Wavelet packet analysis effectively handles non-stationary signals and provides higher resolution in the high-frequency range, overcoming the shortcomings of traditional methods. Therefore, this study employs wavelet packet analysis to process operational parameter signals and combines it with K-means to classify the extracted features and determine the component's operating state. Sensor signals contain a large amount of information reflecting the operating state of the equipment and exhibit non-stationary and nonlinear characteristics. Wavelet packet analysis is an effective method for analyzing such signals. By decomposing signals using wavelet packets, distinct energy values can be obtained, which help distinguish signals under different operating conditions. The wavelet packet method decomposes the signal into a series of frequency bands of a certain width and then calculates the energy values within each frequency band, making it highly suitable for extracting feature vectors from the target signal. After wavelet packet decomposition, each subband of the signal is independent and non-redundant. Therefore, according to the law of energy conservation, we have  
\[E_{n}(x(t))=\sum_{m=0}^{2^{k}-1}E_{n}(x^{(k,m)}(i))^{\leftarrow} \tag{2}\]  
where \(E_{n}(x(t))\) represents the total energy of the signal, \(x(t)\) is the original signal, \(k\) is the decomposition level, \(m\) is the index of the sub-band, with \(m=0,1,\ldots,2^{k}-1\), \(E_{n}\Big{(}x^{(k,m)}(i)\Big{)}\) denotes the energy of the signal in the \(m\)-th sub-band, and \(\chi^{(k,m)}(i)\) represents the discrete signal within the \(m\)-th sub-band. If the original signal \(x(t)\) has a length of N, then the length of the discrete signal in the \(m\)-th sub-band, \(\chi^{(k,m)}(i)\), is \(2^{-k}N\), and its energy is given by:  
\[E_{n}\Big{(}x^{(k,m)}(i)\Big{)}=\frac{1}{2^{-k}N-1}\sum_{i=1}^{2^{-k}N}E_{n} \Big{(}x^{(k,m)}(i)\Big{)} \tag{3}\]  
After normalization, the relative energy of the signal in the \(m\)-th sub-band can be expressed as  
\[E_{n}(m)=\frac{E_{n}(x^{(k,m)}(i))}{E_{0}(x(t))} \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline
**IMF Component** & **Correlation Coefficient** & **IMF Component** & **Correlation Coefficient** \\ \hline
1 & 0.1920 & 4 & 0.7332 \\
2 & 0.0364 & 5 & 0.7825 \\
3 & 0.0280 & 6 & 0.9212 \\ \hline \end{tabular}
\end{table} 
Table 2: The correlation coefficient between the IMF component after signal Y decomposition and the original signal. According to the law of energy conservation, it is evident that \(\sum_{m=0}^{2k-1}E_{n}(m)=1\). The ratio of each sub-band's signal energy to the total signal energy serves as a feature vector \(e\), which effectively characterizes the operational state of the equipment. It is expressed as:  
\[e=\frac{\left[E_{n}\left(x^{(k,1)}(i)\right),E_{n}\left(x^{(k,2)}(i)\right), \ldots,E_{n}\left(x^{(k,m)}(i)\right)\right]}{E_{n}(x(t))} \tag{5}\]  
After undergoing the aforementioned processing and decomposition, the collected data from the equipment operation process can be used as feature inputs for classification algorithms to perform the next step in state classification. After performing wavelet packet decomposition on the sensor signals and extracting the feature vectors \(e\), K-means clustering is applied to classify the extracted features. First, \(K\) cluster centroids are randomly initialized, where \(K=4\), corresponding to the four operating states: the Normal Operation State (N), Initial Abnormal State (I), Degradation State (D), and Severe Failure State (C). Then, each feature vector \(e\) is assigned to the nearest cluster center based on Euclidean distance.