## 5 Signal Approximation and Estimation
### Combination of ARXU and MBR for Signal Approximation  
After resampling and extracting the RMS feature from the original healthy signal, the AR technique models and extracts the state-space equation from the RMS signal under normal conditions. The state-space definition of a healthy signal based on the AR technique is represented using the following equations [39]:  
\[X_{AR}(k+1)=\ell_{X}X_{AR}(k)+\ell_{e}e_{AR}(k) \tag{2}\]  
\[Y_{AR}(k)=\ell_{Y}{}^{T}X_{AR}(k) \tag{3}\]  
\[e_{AR}(k)=Y_{n,rms}(k)-Y_{AR}(k-1) \tag{4}\]  
Here, \(X_{AR}(k),\ Y_{AR}(k)\), and \((\ell_{X},\ell_{Y},\ell_{e})\), respectively, are the state of RMS signal modeling using the AR technique, the output of RMS signal modeling using the AR technique, and the state-space tuning coefficients based on the AR algorithm. The AR algorithm for modeling the signal is valuable for the time-series signal. To improve the robustness and performance against uncertainties, an ARXU is recommended. The mathematical definition of ARXU is represented as the following equations [39,40]:  
\[X_{ARXU}(k+1)=[\ell_{X}X_{ARXU}(k)+\ell_{e}\varepsilon_{ARXU}(k)]+\ell_{e}e_{ ARXU}(k) \tag{5}\]  
\[Y_{ARXU}(k)=\ell_{Y}{}^{T}X_{ARXU}(k) \tag{6}\]  
\[e_{ARXU}(k)=Y_{n,rms}(k)-Y_{ARXU}(k-1), \tag{7}\]  
\[\varepsilon_{ARXU}(k)=Y_{ARXU}(k)-Y_{n,rms}(k) \tag{8}\]  
Here, \(X_{ARXU}(k),\ Y_{ARXU}(k),\ \varepsilon_{ARXU}(k),\ Y_{n,rms}(k)\), and \((\ell_{e})\), respectively, are the state of RMS signal modeling using ARXU, the output of the RMS signal modeling using ARXU, the uncertainties effect estimation and feedback as inputs to reduce the effect of uncertainties using ARXU, the original RMS signal for a healthy condition, and the state-space tuning coefficient based on the ARXU algorithm. The main obstacle in time-series signal modeling is estimating nonlinear behavior. Nonlinear behavior modeling of bearings is complicated. To address this issue, MBR is recommended [41,42]. MBR is a learning technique to approximate the resampled RMS signal in a normal state. Therefore, the combination of ARXU and MBR is introduced in this research. The basic concept of the MBR technique is the learning method. The following function represents the mathematical definition of MBR [42]:  
\[\check{h}_{MBR}=\sum_{i}(\hat{\partial_{i}}^{+}-\hat{\partial_{i}}^{-})( \theta_{i},\theta)+\Theta. \tag{9}\]  
Here, \(\check{h}_{MBR},(\hat{\partial_{i}}^{+},\hat{\partial_{i}}^{-}),(\theta_{i},\theta)\), and \(\Theta\) are the output-modeled RMS resampled normal signal based on MBR, the Lagrange coefficients, the kernel, and the bias, respectively. Various kernel functions have been introduced in research papers; in this work, the Gaussian function is selected and represented as follows:  
\[(\theta_{i},\theta)=e^{(-\frac{1}{2\alpha^{2}}\|\theta_{i}-\theta\|^{2})}. \tag{10}\]  
Here, \(\sigma\) is variance. Moreover, the minimum value for \((\partial_{i}{}^{+},\partial_{i}{}^{-})\) can be calculated based on the following equation:  
\[\min\sum_{i}\sum_{j}(\partial_{i}{}^{+}-\partial_{i}{}^{-})\big{(}\partial_{j} {}^{+}-\partial_{j}{}^{-}\big{)}(\theta_{i},\theta). \tag{11}\]  
If \((\theta_{i},\theta)\) is defined as \(\mathcal{M}_{ij}\) and is based on Equation (11), then  
\[Y_{AR}(k)=\ell_{Y}{}^{T}X_{AR}(k) \tag{3}\]0  
Moreover, if \(\mathcal{M}=\big{[}\mathcal{M}_{ij}\big{]}\in\mathbb{R}^{n\times n},\ \partial=\left[ \begin{array}{c}\partial^{+}\\ \partial^{-}\end{array} \right]_{2n\times 1},\ \text{and}\ \aleph=\left[ \begin{array}{cc}\mathcal{M}&-\mathcal{M}\\ -\mathcal{M}&\mathcal{M}\end{array} \right]\), then Equation (12) is rewritten as:  
\[min\frac{1}{2}\partial^{T}\aleph\partial+\mathcal{L}^{T}\partial. \tag{13}\]  
Here, \(\mathcal{L}=\left[ \begin{array}{c}-Y+\beta\\ Y+\beta\end{array} \right]_{2n\times 1},\) in which \(Y\) and \(\beta\) are the resampled RMS signal under normal conditions, and the boundary of acceptable signal modeling, respectively.