## 2 Materials and Methods
### Transfer-Entropy Algorithm  
[PERSON] [38] drew lessons from the basic theory of information entropy and extended it to form the transfer-entropy (TE) algorithm, which can characterize the relevance and information-transmission relationship among different time series with a unit of bit. This theory can quantify the information-transmission effect between related time series in the form of entropy and reflect the characteristics of information transmission. If the dynamic probabilities of a process \(x\) at time \(n+1\) is conditional only on previous \(k\) values, the process is called a \(k\)-th Markov process. The mathematic description of this transition probability is  
\[p(x(n+1)\mid x(n),x(n-1),\ldots,x(n-k+1))=p\Big{(}x(n+1)\mid x(n)^{(k)}\Big{)} =p\Big{(}x(1)\mid x^{(k)}\Big{)} \tag{10}\]Considering the influence of another process \(y(m)^{(l)}\) on these transition probabilities, the expression of coupling influence of \(x(n)^{(k)}\) and \(y(n)^{(l)}\) on \(x(n+1)\) is \(p\Big{(}x(1)\mid x^{(k)},y^{(l)}\Big{)}\). The transfer entropy \(T_{y\to x}\Big{(}x(1)\mid x^{(k)},y^{(l)}\Big{)}\) can be formulated as below. \[ \begin{array}{l}T_{y\to x}\Big{(}x(1)\mid x^{(k)},y^{(l)}\Big{)}\\ =\iiint p\Big{(}x(1),x^{(k)},y^{(l)}\Big{)}\log_{2}\Big{(}\frac{p\big{(}x(1) \mid x^{(k)},y^{(l)}\big{)}}{p\big{(}x(1)\mid x^{(k)}\big{)}}\Big{)}\\ \times dx(1)dx^{(k)}dy^{(l)}\end{array}  \tag{11}\]  
When the transfer entropy of \(y\) to \(x\) is greater than the transfer entropy of \(x\) to \(y\), \(y\) is referred to as the source of information transmission and \(T_{y\to x}\Big{(}x(1)\mid x^{(k)},y^{(l)}\Big{)}\) denotes the transfer entropy of \(y\) to \(x\). The research of [PERSON] and [PERSON] [39; 40] showed that defining the order \(k=l=1\) for both Markov processes \(y\) and \(x\) does not affect the directedness of process \(y\) to process \(x\). Together with a time delay \(\tau\) added to \(y(n)\), Equation (11) can be simplified to Equation (12). \[ \begin{array}{l}T_{y\to x}(\,x(1)\mid x,y(\tau))y(n)\\ =\iiint p(x(1),x,y(\tau))\log_{2}\Big{(}\frac{p\big{(}x(1)\mid x,y(\tau)\big{)} }{p\big{(}x(1)\mid x\big{)}}\Big{)}\\ \times dx(1)dxdy(\tau)\end{array}  \tag{12}\]  
where \(y(\tau)\equiv y(n+\tau)\). The assumption of order \(k=l=1\) quantifies the information gain from \(y(t)\) only. If we use conditional probabilities \(p(a\mid b)=p(a,b)/p(b)\), Equation (12) can be rewritten in entropy form as Equation (13). \[ \begin{array}{l}T_{y\to x}(\,x(1)\mid x,y(\tau))\\ =\iiint p(x(1),x,y(\tau))\log_{2}p(x(1),x,y(\tau))dx(1)dxdy(\tau)\\ +\int p(x)\log_{2}p(x)dx-\iint p(x,y(\tau))\log_{2}p(x,y(\tau))dxdy(\tau)\\ -\iint p(x(1),x)\log_{2}p(x(1),x)dx(1)dx\end{array}  \tag{13}\]  
In order to quantitatively describe the transmission regularity of vibration energy, the information-transmission rate (\(ITR\)) index is introduced [35; 36], which is calculated based on the transfer entropy and can effectively describe the information-transmission strength of vibration energy. For vibration signals \(y\) and \(x\), the \(ITR\) is:  
\[ITR_{y\to x}=\frac{\overline{T_{y\to x}(\tau)}-\overline{T_{x\to y}(\tau)}}{ \overline{T_{y\to x}(\tau)}} \tag{14}\]  
where \(T_{y\to x}(\tau)\) and \(T_{x\to y}(\tau)\) are the transfer entropy corresponding to the different transfer directions of signals \(x\) and \(y\); \(\overline{T_{y\to x}(\tau)}\) and \(\overline{T_{x\to y}(\tau)}\) are the average value of \(T_{y\to x}(\tau)\) and \(T_{x\to y}(\tau)\), respectively; \(ITR_{y\to x}\) is the information-transmission rate; the direction of \(ITR_{y\to x}>0\) is used as the positive direction of information transmission. In the case of \(T_{y\to x}-T_{x\to y}\geq 0\), when \(ITR_{y\to x}=0\), there is no information-transmission relationship between signal \(y\) and \(x\), and the two signals are independent; When \(ITR_{y\to x}=1\), the information of signal \(y\) is completely transmitted to the signal\(x\), \(y\) is the source of information transmission.