# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 5 Parameter Estimation of Degradation Model  
We estimate the RUL value by developing a degradation model of the time series health indicator. The degradation model identifies a dynamic model that describes the failure process of the system behavior. It predicts the health indicator and the time to cross a specific threshold of the health indicator as the failure indicator.  
We assume that the health indicator \(h(t_{i})\) of bearings is observed at discrete times \(t_{1},t_{2},\dots\), where \(t_{i}\geq 0\). Exponential degradation models are useful when the machine component experiences cumulative degradation [8]. Besides, the linear degradation model is another useful one if the observed system does not have cumulative degradation processes [33,44]. Hence, the considered degradation model combines the exponential model and the linear model as follows:  
\[\hat{h}_{w}(t_{i})=\phi+\beta_{w}t_{i}+\alpha_{w}\exp(t_{i}) \tag{2}\]  
where \(\phi,\beta_{w}\) and \(\alpha_{w}\) are the updated parameters dependent on the health indicator over the sliding window. Compared to the existing exponential degradation model [8], the exponential part of Equation (2) reduces the flexibility of the degradation model. The sliding window method adapts the local model parameters of the time series data to compensate for the fundamental limits of the analytical degradation model. We evaluate the RUL prediction accuracy and robustness of the proposed scheme in Section 7.  
The adaptive sliding window method relies on ridge regression to balance training accuracy and generalization accuracy. We formulate the ridge regression problem by combining the prediction error term and the regulation term with different weights on model parameters \(\mathbf{x}=[\phi,\alpha_{w},\beta_{w}]^{\top}\) to avoid the overfitting and reduce the critical effect of the exponential term of Equation (2). The parameter optimization problem of the adaptive sliding window method is  
\[\min_{\mathbf{x}}\quad\sum_{j=l-s+1}^{i}(h(t_{j})-\hat{h}_{w}(t_{j}))^{2}+\mathbf{x}^{ \top}\mathbf{W}\mathbf{x}\quad\forall t_{i}\in 1,\ \dots,N \tag{3}\]where the first term is the mean square error between measured health indicators \(h(t_{j})\) and estimated health indicators \(\hat{h}_{w}(t_{j})\)) with sliding length \(s\) and the second term is the weighted regularization term with diagonal matrix \(\mathbf{W}\in\mathbb{R}^{3\times 3}\). Each element of the diagonal matrix \(\mathbf{W}\) corresponds to the weight of the decision variable of \(\mathbf{x}\).  
Since the objective function is a regularized quadratic cost function, we rewrite it as the matrix form  
\[J(\mathbf{x})=(\mathbf{h}-\mathbf{A}\mathbf{x})^{\top}(\mathbf{h}-\mathbf{A} \mathbf{x})+\mathbf{x}^{\top}\mathbf{W}\mathbf{x}\,, \tag{4}\]  
where  
\[\mathbf{A}=\begin{bmatrix}1&\exp(t_{i-s+1})&t_{i-s+1}\\ 1&\exp(t_{i-s+2})&t_{i-s+2}\\ \vdots&\vdots&\vdots\\ 1&\exp(t_{i})&t_{i}\end{bmatrix}\in\mathbb{R}^{s\times 3}\,,\]  
and  
\[\mathbf{h}=\begin{bmatrix}h(t_{i-s+1})\\ h(t_{i-s+2})\\ \vdots\\ h(t_{i})\end{bmatrix}\in\mathbb{R}^{s}\,.\]  
The optimal solution is  
\[\mathbf{x}^{*}=(\mathbf{A}^{\top}\mathbf{A}+\mathbf{W})^{-1}\mathbf{A}^{\top }\mathbf{h}\,. \tag{5}\]  
Determining the optimal length of the sliding window is a complex task since it requires to take into account the reduction of the noise effect and the fault detection delay. The sliding window length is set to 30 s based on the estimation accuracy of the health indicator of the training sets for different sliding window lengths from 10 to 102 s.