## 3 Method
### Model Parameter Calculation
By leveraging insights from robust techniques for parameter calculation, the subsequent pseudocode defined in Algorithm 1 delineates the implementation of AdaBoost with weak classifiers for fault detection in DC motor gears. This algorithm is very good at finding patterns that point to gear problems by improving the classification model over and over again using weighted training data. Consequently, it facilitates the development of dependable predictive maintenance strategies for DC motor systems. ```
1:Input: Training dataset \(D=\{(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\}\) where \(x_{i}\) are feature vectors and \(y_{i}\in\{-1,+1\}\) are labels, number of weak classifiers \(T\)
2:Output: Final strong classifier \(H(x)\)
3:procedureAdBoost(\(D\), \(T\))
4: Initialize weights \(w_{i}=\frac{1}{n}\) for \(i=1,2,...,n\)
5: Initialize an empty list \(H\) to store weak classifiers
6:for\(t=1\) to \(T\)do
7: Train a weak classifier \(h_{t}(x)\) using weighted dataset \(D_{t}=\{(x_{i},y_{i},w_{i})\}_{i=1}^{n}\)
8: Calculate error \(\varepsilon_{t}=\sum_{i=1}^{n}w_{i}\cdot\kappa^{\prime}(h_{t}(x_{i})\
eq y_{i})\)
9: Ensure \(\varepsilon_{t}<0.5\) to ensure classifier performance above random guessing
10: Compute weak classifier weight \(\alpha_{t}=\frac{1}{2}\ln\Big{(}\frac{1-\varepsilon_{t}}{\varepsilon_{t}}\Big{)}\)
11: Update weights: \(w_{i}\gets w_{i}\cdot\exp(-\alpha_{t}y_{i}h_{t}(x_{i}))\) for \(i=1,2,...,n\)
12: Normalize weights: \(w_{i}\leftarrow\frac{w_{i}}{\sum_{i=1}^{n}w_{i}}\) for \(i=1,2,...,n\)
13: Append weak classifier \(h_{t}(x)\) to list \(H\)
14:endfor
15:return Final strong classifier \(H(x)=\text{sign}\Big{(}\sum_{t=1}^{T}\alpha_{t}h_{t}(x)\Big{)}\)
16:endprocedure
```  
**Algorithm 1** AdaBoost with weak classifiers  
The AdaBoost algorithm with weak classifiers defined in Algorithm 1 operates by iteratively refining a solid classifier from a collection of weak learners. Initially, the algorithm assigns equal weights to all training samples, ensuring each observation contributes equally to the learning process. Through a series of iterations, known as rounds, weak classifiers are trained on weighted subsets of the training data. These weak classifiers are typically simple models, such as decision stumps, which focus on capturing specific patterns in the data. After training each weak classifier, its performance is evaluated by calculating the error rate, which measures the discrepancy between the predicted and actual labels. To maintain the classifier's performance above random guessing, the algorithm requires each weak classifier's error rate to remain below 0.5. The weight of each weak classifier is then determined based on its error rate, with more accurate classifiers assigned higher weights. Additionally, the weights of the training samples are adjusted to prioritize misclassified instances, effectively emphasizing challenging data points in subsequent iterations. After normalization, the weak classifiers are combined into a final robust classifier using a weighted voting scheme, where each weak classifier's contribution is weighted based on its performance. This ensemble approach allows AdaBoost to effectively leverage the collective knowledge of the weak classifiers, resulting in a robust classifier capable of accurately predicting DC motor gear faults. Through its iterative learning process and adaptive weight adjustment, AdaBoost with weak classifiers enhances the reliability of predictive maintenance strategies for DC motor systems, facilitating early detection and mitigation of gear faults to ensure uninterrupted operation and prolong the lifespan of the equipment. This holistic approach underscores the importance of robust model parameter calculation techniques in developing effective machine health monitoring systems. The explanation effectively delineates the methodology's approach to determining vital model parameters for fault diagnosis and prognosis in machine health monitoring.