### 4.1 Parameters setting strategy

The false negative rate is calculated by

In the algorithm flowchart of the proposed method, the delay time and the decreasing function are the most important parameters. The value of the delay time directly determines the response time to fault. And the decreasing function determines the distance to decrease. It not only has an impact on the false negative rate, but also on the response time. Therefore, it is necessary to choose those two parameters appropriately. Because the relationship between the false negative rate and the response time is similar to that between the precision and the recall in machine learning, referring to the F1-score, a score is set up by us:

where β represents the different weight of the false negative rate and the response time. It could be set to different value according to the need. Because in the experiments we consider the false negative rate and the response time equally, β is set to 1. So the formula becomes the following form:

Different from the precision and the recall in machine learning, we would like the false negative rate and the response time both to be small. So, the smaller the value of F1 score, the better.

### 4.2 A case study on gear about fault re-decision

00The experiments were performed on the rolling bearing bench shown in Figure 4. Artificial fault was imposed both on the gearings and the outer race of the test bearings.

In this experiment, the shaft rotational speed and sampling frequency were set to be 1800 rpm and 25600 Hz, respectively. The magnetic brake load is set as 2A. The data acquisition system is consisted of four channels, with three acceleration signals and one torque signal. The sensors placed in the different positions on the machine. The original signals are shown in the Figure 5. The red line represents the signals collected under normal conditions, while the blue one represents those collected under the conditions that there is a broken tooth in the 00000000gear.

The SVM model is used as the IFDM. The normal signal and the abnormal signal each take the first 12000 data points for training. And 10000 data points from the normal signal is used as test data. After training, the SVM model could achieve 64.83% accuracy without the proposed method. It is not a satisfying result because its false negative rate is 35.17% according to the formula (2), which means the monitored machine system might often be shut down when it should be run normally.

When one sample is input, the SVM model firstly gives an initial result. Then, the initial result will be handled by the proposed method instead of being used directly to control the machine. Through the re-decisioning process, which not only considers the current results but comprehensively considers the previous results, the proposed method would give a final result about the status of machine.

0000With our proposed method, the false negative rate decreases from 35.17% to 13.46%, which is shown in Figure 6. The best false alarm rate is 1.83%, using “f(x) = log(x+1)" as the decreasing function and 10 as the delay time. When the delay time is less than or equal to 6, the false negative rates of four decreasing functions all go down. But if the delay time is bigger than 6, the false negative rates remain unchanged at 3.48% with the decreasing function of “f(x) = 2x" and at 2.88% when it is “f(x) = x2" (see the blue bars and the red bars in the Figure 6). It indicates that our proposed method works.

It could be seen in Figure 6 that for all the decreasing function, their F1-scores are smallest when the response time is 1. However, when the response time is bigger than 5 and as the response time increases, the F1-scores for the decreasing functions of “f(x) = x” and “f(x) = log(x+1)" are both decreasing while the other two keep rising (see the yellow line and the purple line in Figure 6). Therefore, the proposed method was carried on again with a larger delay time, whose false negative rates and F1-socure are shown in Fig 7. As expected, their F1-scores keep go down (see the yellow line and the purple line in the Figure 7). However, even when the delay time is 20, their F1-scores are still larger than that when the delay time is 1.

0000Summary, the proposed method could achieve a significant reduction in the false negative rate. For a low false negative rate and a relatively low F1 score, we recommend using the f(x) = x as the decreasing function and 10 as the delay time, which could reduce the false negative rate from 35.17% to 1.89% in current experiment.