**A.3.1 Missed alarm rate**

A missed alarm occurs when at least one process variable is erroneously not detected as faulty, *ej* <sup>≤</sup> *<sup>T</sup><sup>D</sup> j* , when a fault is actually present. In this case, at least one missed alarm occurs, and no false alarm occurs in any of the other variables. The MAR is calculated as:

$$\text{MAR} = \frac{\sum \text{missed alarms}}{\text{total number of samples in fault condition}} \times 100\% \tag{A4}$$

#### **A.3.2 Missed and false alarms rate**

It is possible that a fault will be missed in a faulty signal but detected in at least one fault-free signal. This gives both missed and false alarms: a fault is detected *ej* � � � �>*T<sup>D</sup> j* � � in at least one process signal, when no fault is actually present (false alarm), and at least one process signal has a fault that is not detected *ej* � � � �≤*T<sup>D</sup> j* � � (missed alarm). The M&FAR is calculated as:

$$\text{M\ $:FAR} = \frac{\sum \text{simulation} \text{ uses missed} \$ \text{false alarm}}{\text{total number of samples in fault condition}} \text{\*100\%} \tag{A5}$$

#### **A.3.3 True and false alarms rate**

It is possible that a fault will be detected in a faulty signal and also detected in at least one fault-free signal. This gives both true and false alarms: a fault is detected in at least one process signal, *ej* � � � �>*T<sup>D</sup> j* � �, when no fault is actually present (false alarm), and a fault is correctly detected in one process signal, *ej* � � � �>*T<sup>D</sup> j* � � (true alarm). The T&FAR is calculated as:

$$\text{T\\$}\text{\\$}\text{FAR} = \frac{\sum \text{simultaaneous true\\$}\text{\\$}\text{false}\text{\color{red}{\text{false}}}{\text{sample of samples in fault condition}} \text{\color{red}{\text{A\\$}}} \tag{A6}$$

#### **A.3.4 True alarm rate**

This represents the presence of only true alarms. A fault is detected in at least one process signal, *ej* � � � �<sup>&</sup>gt; *<sup>T</sup><sup>D</sup> j* � �, when a fault is actually present, and no false alarm exists in other fault-free signals (true alarm). The TAR is calculated as:

$$\text{TAR} = \frac{\sum \text{true alarm} | \text{no false alarms}}{\text{total number of samples in fault condition}} \text{\*100\%} \tag{A7}$$

#### **A.3.5 Fault detection rate**

The FDR is expressed as the ratio of the number of faulty data points detected as faulty to the total number of samples specific to a fault. In this case, a fault is detected in at least one process signal, *ej* � � � �<sup>&</sup>gt; *<sup>T</sup><sup>D</sup> j* � �, when a fault is actually present regardless of false alarms in other signals. The FDR is calculated as:

$$\text{FDR} = \frac{\sum \text{ correctly detected faults in the system}}{\text{total number of samples in fault condition}} \times 100\% \tag{A8}$$

FDR measures the ability of a model to detect the presence of the fault in a system when a fault is actually present. Thus, FDR is a summation of the M&FAR, T&FAR, and TAR, which implies that:

$$\text{FDR} = \text{M\&FAR} + \text{T\&FAR} + \text{TAR} \tag{A9}$$
