*4.3. Model Performance Test*

This section verifies the effect of the proposed monitoring method, and constructs a first-order dynamic process monitoring method: DPLVM [18] and static process monitoring method: PPLSR [33], which were used to compare with the proposed method. The latent variable dimensions of the model were adjusted to 3.

The first 1000 normal samples were used to train the parameters of the model, and the trained model was monitored for three types of different fault samples. In order to distinguish between normal and abnormal samples, the first 200 samples of each type of failure test set were normal samples, and the last 200 samples were their respective failure samples. Table 5 shows the FAR and FDR of different monitoring methods under different failure test sets, and the last line calculates the average value of different indicators.

**Table 5.** FAR and FDR of the three methods under different fault cases.


It can be seen from Table 5 that the monitoring performance of the proposed method was better than that of the static model PPLSR and the first-order dynamic model DPLVM. Therefore, the detection performance was greatly improved after the autoregressive equation was added to the model to extract the dynamic and time lag information. Compared with the basic first-order dynamic DPLVM fault detection method, DALM considered the time lag characteristics, so the model performance was further improved. The detailed monitoring results of the three methods for the three types of faults are shown in Figures 6–8.

**Figure 6.** Monitoring results of fault 1. (**a**) T<sup>2</sup> of PPLSR in fault 1; (**b**) T<sup>2</sup> of DPLVM in fault 1; (**c**) T<sup>2</sup> of DALM in fault 1.

**Figure 7.** Monitoring results of fault 2. (**a**) T<sup>2</sup> of PPLSR in fault 2; (**b**) T<sup>2</sup> of DPLVM in fault 2; (**c**) T<sup>2</sup> of DALM in fault 2.

For each type of fault test set, the first 200 samples were in a normal state, and the last 200 samples were fault samples. It can be seen from Figure 8 that the static model PPLSR easily mistakenly classified normal samples into faulty samples, and it also easily classified faulty samples into normal samples. The error rate of the first-order dynamic model DPLVM was reduced a lot. Furthermore, the FAR based on the DALM fault detection method proposed in this paper was close to the significance level and the FDR was close to 1, verifying that its monitoring performance was greatly improved.

**Figure 8.** Monitoring results of fault 3. (**a**) T<sup>2</sup> of PPLSR in fault 3; (**b**) T<sup>2</sup> of DPLVM in fault 3; (**c**) T<sup>2</sup> of DALM in fault 3.
