# Order-Based Identification of Bearing Defects under Variable Speed Condition
## 3 Validation of the Proposed Features
### Results and Discussion
#### 3.3.3 Classification  
To perform the classification process, the acquired vibration signal described in Section 3.1 are regrouped according to their corresponding health state, regardless of the operation speed condition. 12 signals of 10 s are thus considered for each health state. To increase the number of samples, the 10 s signals are divided into 5 samples of 2 s each. Thus, a total of 60 samples are considered for each condition:  
\(\blacksquare\)  
Health bearing (H)  
Faulty bearing with an outer race defect (ORD)  
Faulty bearing with an inner race defect (IRD)  
Faulty bearing with a ball defect (BD)  
Faulty bearing with combined outer race, inner race, and ball defects (CD)  
Figure 9a,b shows the 3D space representation of the acquired pre-processed data as a function of the conventional features (SPRO, SPRI, and SPRR) and the updated features (SPRO\({}_{\text{a}}\), SPRI\({}_{\text{a}}\), and SPRR\({}_{\text{a}}\)) respectively. All features are normalized using the standard MATLAB function \"range\". This function normalizes the computed features to an interval [0,1] to ensure a clear graphical representation of the data. Figure 8: Envelop order spectrum of the faulty bearing signal. Figure 7: Envelop spectrogram of the faulty bearing signal, (**a**) frequency domain, (**b**) order domain. It can be noted that in the case of the conventional features used, there is an overlap between the data representing each health state. On the other hand, with the updated features the separation between the data corresponding to different health states is obvious. This confirms the ability of the proposed features to separate between bearing defects. An MSVM-based classification is investigated to confirm the relevance of the proposed features in the identification of bearing defects under variable speed conditions. All samples considered have been pre-processed as explained in the previous section and used in the extraction of the updated features. 50% of the extracted features corresponding to each health state have been used to build (train) the predictive MSVM classification model. Then, the constructed model is tested against the remaining 50% of the data. The classification result is given in Figure 10. To highlight the necessity of the pre-processing step, another MSVM classification model is trained and tested based on the raw signals (50% for training and 50% for testing). The result of the raw data classification is given in Figure 11.  
Figure 10: Space representation of the tested 50% of pre-processed data as predicted by the (MSVM) classifier: (**a**) using the non-updated features; (**b**) using the updated ones. Figure 9: Space representation of the acquired pre-processed data: (**a**) as a function of the conventional features; (**b**) as a function of the updated features. Confusion matrices, Tables 2 and 3, are adopted to evaluate the performance of the MSVM classifier in the case of conventional and updated features extracted from raw and pre-processed signals respectively. The present study is performed in the balanced class case which makes the accuracy of classification sufficient criterion to evaluate the classifier performance. The classification accuracy is achieved as shown in Equation (14), by counting the data labeled as true positives TP, true negatives TN, false positives FP, and false negatives FN. \[\text{Accuracy}=\frac{\text{TP}+\text{TN}}{\text{TP}+\text{TN}+\text{FP}+ \text{FN}} \tag{14}\]  
The updated features based MSVM classifier built has successfully identified the correct type of the bearing defects for over 87% of the tested pre-processed data and over 79% for the raw data, whereas with the non-updated features, the accuracy of the