*3.2. Fault Identification*

Once a fault is detected, the next goal is to identify which variables are the most affected and contribute most to the monitoring statistics. Assume a fault was detected between time *t*<sup>1</sup> and *t*2, let **e***<sup>i</sup>* = (*e*<sup>1</sup> *<sup>i</sup>* ,*e*<sup>2</sup> *<sup>i</sup>* , ··· ,*e<sup>d</sup> <sup>i</sup>* ) denote the residual vector for the *l* process variables, *i* = *t*1, ··· , *t*2. The normalized residuals *E<sup>l</sup> <sup>i</sup>* can be used to evaluate the impact of the fault on each variable as:

$$E\_i^l = \frac{\mathfrak{c}\_i^l - \mu^l}{\sigma^l} \tag{18}$$

where *μ<sup>l</sup>* is the mean value and *σ<sup>l</sup>* is the standard deviation of the GRU residuals of the NOC training data.

For a clearer exhibition, the deviation *E<sup>l</sup> <sup>i</sup>* of each variable is accumulated to get the total contribution rate *CR<sup>l</sup>* = ∑*<sup>N</sup> <sup>i</sup>*=*<sup>t</sup> E<sup>l</sup> i* . With the contribution rate obtained, operators can know which variables are most sensitive to the process fault. Also, operators can use the contribution plots to identify which kind of fault has occurred.

For completeness, the overall flowchart is summarized in Figure 4, including both the offline training stage (left) and the online monitoring stage (right).

**Figure 4.** Flowchart of the GRU-support vector data description (SVDD)-based fault detection and identification methodology.

The offline monitoring stage can be summarized as follows:


The online detection stage can be summarized as follows:

