## 3 Method
### Model Parameter Calculation  
As discussed previously, the proposed methodology leverages a boosting algorithm with weak classifiers to facilitate fault diagnosis and prognosis, eliminating the reliance on HMMs. The boosting algorithm determines crucial model parameters, including the weighting coefficients of the weak classifiers, during its training phase. In scenarios where extensive historical data are unavailable, alternative methods for model parameter calculation are explored. Extensive-scale testing involves evaluating significant samples of faulty machines, enabling the natural progression of using methods like finite element analysis, stress analysis, and nondestructive testing, while fatigue analysis looks at how the starter motor gear and flywheel wear over time. Additionally, online estimation estimates transition probabilities during vehicle operation, leveraging similarities to previous transitions. These approaches provide valuable insights into model parameter calculation, ensuring robust fault diagnosis and prognosis methodologies in machine health monitoring applications. However, practical constraints limit the feasibility of the first two options. Online estimation, the third method, is applicable only during machine operation, where faults develop naturally. For laboratory setups, heuristic methods are employed to estimate these probabilities. The proposed methodology presents a pragmatic approach to model parameter calculation, harnessing ensemble learning techniques for precise fault diagnosis and prognosis in machine health monitoring. The equations of the boosting algorithm elucidate the mathematical formulas utilized for training and prediction making with the ensemble classifier. This is very important for correctly identifying faults and predicting their outcomes in machine health monitoring. The weighted error rate \(err_{t}\) shown in Equation (18) for the \(t\)-th weak classifier \(h_{t}\) is computed by summing the weights \(w_{i}\) of misclassified samples and dividing by the total sum of sample weights. \[\text{err}_{t}=\frac{\sum_{i=1}^{N}w_{i}\cdot\text{indicator}(h_{t}(x_{i})\
eq y _{i})}{\sum_{i=1}^{N}w_{i}} \tag{18}\]  
The weight \(\alpha_{t}\), as shown in Equation (19) assigned to the \(t\)-th weak classifier \(h_{t}\), is determined based on the computed error rate \(\text{err}_{t}\). It represents the contribution of \(h_{t}\) to the final ensemble classifier. \[\alpha_{t}=0.5\cdot\log\left(\frac{1-\text{err}_{t}}{\text{err}_{t}}\right) \tag{19}\]Equation (20) explains sample weights \(w_{i}\) are updated based on the weak classifier \(h_{t}\) performance, adjusting them to emphasize correctly classified samples and de-emphasize misclassified ones. \[w_{i}=w_{i}\cdot\exp(-\alpha_{t}\cdot y_{i}\cdot h_{t}(x_{i})),\quad i=1\text{ to }N \tag{20}\]  
The sample weights \(w_{i}\) are normalized to sum up to 1, as shown in Equation (21), maintaining their probabilistic interpretation:  
\[w_{i}=\frac{w_{i}}{\sum_{i=1}^{N}w_{i}},\quad i=1\text{ to }N \tag{21}\]  
The ensemble classifier \(H(x)\) can be calculated using Equation (22), aggregating the predictions of all weak classifiers \(h_{t}\) weighted by their respective weights \(\alpha_{t}\), and producing the final prediction. The model parameter calculation is crucial to the proposed methodology for fault diagnosis and prognosis in machine health monitoring:  
\[H(x)=\text{sign}\Bigg{(}\sum_{t=1}^{T}\alpha_{t}\cdot h_{t}(x)\Bigg{)} \tag{22}\]  
By exploring alternative methods such as large-scale testing, fatigue analysis, and online estimation, this methodology offers practical solutions for scenarios where extensive historical data may be unavailable. The detailed equations presented elucidate the training and prediction process of the boosting algorithm with weak classifiers, highlighting their significance in achieving accurate fault severity predictions.