# Application of Improved MFDFA and D-S Evidence Theory in Fault Diagnosis
## 2 Theoretical Basis
### Improved MFDFA Method  
The MFDFA method processes the fault signal by assuming the local trend of the signal as a polynomial for removal, which affects its applicability. Thus, a multifractal analysis method of EMD-LS-MFDFA is proposed to address this problem. The proposed approach integrates the automatic determination of EMD trend term method [17; 18] and the least squares (LS) method [19]. This removes limitations associated with the complete decomposition of EMD, while simultaneously overcoming the defects of modal confusion and the endpoint effect. In addition, this approach also successfully avoids the influence of the original signal on the LS fitting accuracy. The EMD-LS-MFDFA method can therefore be assured to accurately obtain the trend terms of the faulty signal at different scales, allowing extraction of the multi fractal spectrum parameters.  
For the original signal \(x(i)(i=1,2,\cdots,N)\), the fitting process of trend terms by EMD-LS can be expressed as:  
\[x(i)=\sum\limits_{j=1}^{n}c_{j}(i)+r_{n}(i) \tag{1}\]  
In Equation (1), \(n\) is the number of intrinsic mode functions (IMFs) decomposed out of \(x(i)\) by EMD according to its own scale characteristics, \(c_{j}(i)\) is the \(j_{th}\) IMF component \((j=1,2,\cdots,n)\) and \(r_{n}(i)\) is the residual component of the signal.  
Ideally, \(r_{n}(i)\) is used as the final trend term of the original signal [17]. However, in practice, the trend term also contains some low-frequency IMF components, and if \(r_{n}(i)\) alone is used as the final trend term, there will be a large error. Therefore, if the residual component is set as the last IMF component, considering that the trend term of the original signal may be the sum from the \(T_{th}\) IMF component to the last IMF component, it can be expressed by the following formula:  
\[M(i)=\sum\limits_{j=T}^{n}c_{j}(i) \tag{2}\]  
For determining the value of \(T\), an automatic criterion for EMD trend terms is introduced, which can be expressed by the following formula:  
\[Z=\frac{\sum\limits_{j=T}^{n+1}\sum\limits_{i=0}^{N}c_{j}(i)}{\sum\limits_{i= 0}^{N}x(i)} \tag{3}\]  
Ideally, \(Z=1\), but in practice, real conditions are too harsh, so after extensive simulation analysis, the value \(Z=0.95\) was determined.  
After automatic determination, let the trend term \(M(i)\) be a series \((i=1,2\cdots,N)\) with sampling frequency \(f\), then the trend term can be expressed as below:  
\[w(i)=\sum\limits_{k=1}^{K}b_{k}a^{k} \tag{4}\]  
where \(a=i/f\) and \(b_{k}\) is the coefficient of the trend term. The latter can be solved using the least squares method.  
Based on this, the final signal \(x^{\prime}(i)\) can be extracted using EMD-LS:  
\[x^{\prime}(i)=x(i)-(M(i)-w(i)) \tag{5}\]  
In turn, for the signal \(x^{\prime}(i)\), the MEDFA method is applied to carry out the feature extraction [18], i.e., the signal can be expressed as a \(q\) fluctuation function of order.  
\[F_{q}(s)=as^{h(q)} \tag{6}\]  
where \(s\) is the length of the equal subinterval divided by the cumulative deviation of \(x^{\prime}(i)\), \(a\) is a constant, and \(h(q)\) is the generalized Hurst exponent. Its value is the gradient of the slope obtained by fitting the least squares method \(F_{q}(s)\) to the scale \(s\). When the multiplicative fractal process is carried out when the value \(h(q)\) varies with \(q\) it may be described as a multifractal process.  
According to the Legendre transform, the generalized Hurst index \(h(q)\) is related to the mass index \(\tau(q)\), singularity index \(\alpha\), and multifractal spectrum \(f(\alpha)\) in the multifractal as follows:  
\[\tau(q)=qh(q)-1 \tag{7}\]  
\[\alpha=h(q)+q\frac{dh(q)}{dq} \tag{8}\]  
\[f(\alpha)=q[\alpha-h(q)]+1 \tag{9}\]