T HE SEVERITY OF A FAULT PATTERN AND THE CRITICALITY OF A FAULT STATE

CHAPTER 5 FAULT PATTERN EXTRACTION

5.2 T HE SEVERITY OF A FAULT PATTERN AND THE CRITICALITY OF A FAULT STATE

in the no-fault states, while at least one pattern among the set must be found in each fault state, the objective of fault detection using fault pattern is as follow (Definition 2)

Definition 2 (Fault detection using the fault pattern): to extract a set of fault patterns that are found in the fault sates but not in the no-fault states,, and at least one pattern among the set should be found in each fault state.

𝐦𝐢𝐧 𝑓𝑠_𝑛𝑢𝑚− 𝑑𝑓_𝑛𝑢𝑚 where

𝑓𝑠_𝑛𝑢𝑚: the total number of the given fault states;

𝑑𝑓_𝑛𝑢𝑚: the number of discernible fault states obtained by Algorithm 5.2.

Algorithm 5.2 Discernible fault states

Require: 𝐄(𝐗) (a set of event codes), 𝑝𝑎𝑡𝑡𝑒𝑟𝑛_𝑝^{𝑓𝑎𝑢𝑙𝑡} (the p^th fault pattern), 𝑓𝑠_𝑛𝑢𝑚 (the total number of the given fault states)

1: 𝑛𝑢𝑚𝑏𝑒𝑟_𝑜𝑓_𝑑𝑖𝑠𝑐𝑒𝑟𝑛𝑖𝑏𝑙𝑒_𝑓𝑎𝑢𝑙𝑡_𝑠𝑡𝑎𝑡𝑒𝑠 ← 0

2: for (𝒒 = 𝟏; 𝒒 < 𝑓𝑠_𝑛𝑢𝑚+ 𝟏; 𝒒 + +) do // For all fault states

3: If there exists any 𝑝𝑎𝑡𝑡𝑒𝑟𝑛_𝑝^{𝑓𝑎𝑢𝑙𝑡} ∀ for any 𝑝 such that it is found in the q^th fault state 4: 𝑛𝑢𝑚𝑏𝑒𝑟_𝑜𝑓_𝑑𝑖𝑠𝑐𝑒𝑟𝑛𝑖𝑏𝑙𝑒_𝑓𝑎𝑢𝑙𝑡_𝑠𝑡𝑎𝑡𝑒𝑠 + +

5: End if 6: End for

5.2.1 The severity degree of a fault pattern

In terms of online fault detection, the extracted fault patterns will be compared with the event codes of the current time segment of the monitored sensor signals. In this procedure, we want to know how severe the monitored pattern is. If a specific fault pattern is discovered in every fault state, it is reasonable to define it as a strongest reference for online fault detection. Unfortunately, if a certain pattern is found from an individual state, it is not reasonable to employ it as only one reference pattern for detecting every fault occurrence. However, it should not be neglected because a fault can be caused by various root causes, and the weak pattern can represent a minor root cause of a fault state.

Therefore, it is reasonable to employ a set of all extracted fault patterns for online detection, and provide a commensurate score with its frequency of the corresponding pattern’s occurrence as a weight. That is, a severity degree of a fault pattern is assessed in proportional to the found number of the fault pattern, and consequently it will be used to quantitatively estimate the effect of the fault pattern.

Algorithm 5.3 explains how to calculate the severity degree of each fault pattern. Because the severity degree is calculated based on the number of the corresponding pattern occurrences in every fault state, a higher value indicates that the fault pattern is found in many fault states and can therefore be considered as significant. If the same fault pattern is discovered multiple times in a single fault

Algorithm 5.3 A severity degree of a fault pattern

Require: 𝐄(𝐗) (a set of event codes), 𝐏𝐚𝐭𝐭𝐞𝐫𝐧^{𝐟𝐚𝐮𝐥𝐭} (fault patterns), 𝑝𝑎𝑡𝑡𝑒𝑟𝑛_𝑝^{𝑓𝑎𝑢𝑙𝑡} (the p^th fault pattern), 𝑓𝑠_𝑛𝑢𝑚 (the total number of the given fault states)

1: for (𝒑 = 𝟏; 𝒑 < |𝐏𝐚𝐭𝐭𝐞𝐫𝐧^{𝐟𝐚𝐮𝐥𝐭}| + 𝟏; 𝒑 + +) do

2: 𝑐𝑐𝑢𝑟𝑟𝑒𝑐𝑛𝑒𝑠(𝑝𝑎𝑡𝑡𝑒𝑟𝑛_𝑞^{𝑓𝑎𝑢𝑙𝑡}) ← 0 // 𝑃𝑎𝑡𝑡𝑒𝑟𝑛_𝑝^{𝑓𝑎𝑢𝑙𝑡} is the p^th element of 𝐏𝐚𝐭𝐭𝐞𝐫𝐧^{𝐟𝐚𝐮𝐥𝐭} 3: for (𝒒 = 𝟏; 𝒒 < 𝒇𝒔_𝒏𝒖𝒎+ 𝟏; 𝒒 + +) do

4: If there exists 𝑃𝑎𝑡𝑡𝑒𝑟𝑛_𝑝^{𝑓𝑎𝑢𝑙𝑡} that it is found in the q^th fault state 5: 𝑜𝑐𝑐𝑢𝑟𝑟𝑒𝑐𝑛𝑒𝑠(𝑝𝑎𝑡𝑡𝑒𝑟𝑛_𝑞^{𝑓𝑎𝑢𝑙𝑡}) + +

6: End if5 7: End for

8: 𝑠𝑒𝑣𝑒𝑟𝑖𝑡𝑦(𝑃𝑎𝑡𝑡𝑒𝑟𝑛_𝑝^{𝑓𝑎𝑢𝑙𝑡}) ← 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛(𝑐𝑐𝑢𝑟𝑟𝑒𝑐𝑛𝑒𝑠(𝑝𝑎𝑡𝑡𝑒𝑟𝑛_𝑞^{𝑓𝑎𝑢𝑙𝑡}), 𝑓𝑠_𝑛𝑢𝑚) 9: End for

state, it is only counted once because it describes only one fault state. After counting the number of the fault pattern occurrence, then the occurrence number is divided by the total number of the given fault state as a normalization step.

For example, Figure 5.2-(b) shows the computational procedure for calculating the severity degree of the extracted fault patterns. The first fault pattern ‘211’ (i.e., [𝒍_𝟏𝟐 𝒍_𝟐𝟏 𝒍_𝟑𝟏]^𝑻) has the highest severity degree (= 1.00) because it is found in every fault state. The other two patterns, ‘312’ and

‘231’, have smaller severity degrees (0.25) in the example, however, the degrees are larger than 0 because a pattern should correspond to at least one defect state according to Definition 1.

5.2.2 The criticality of a fault state

Although multiple fault states are given to analyze, each fault state can show different hazardous level to the system, such as critical, major, minor, warning, or indeterminate. This is because the current dangerous degree is an important factor for finding optimal maintenance actions, scheduling repair procedure, and further analyzing root causes of defects (D. Goyal & B. S. Pabla, 2015). Therefore, we also introduce a method to identify the criticality of a fault state quantitatively to answer how critical the current fault state is to the system.

Figure 5.3 Examples of the defect pattern’s degree of severity and the criticality of the defect states:

(a) a matrix of discretized state vectors, D(X), in which the number of sensors is 3 and the recording time is 1–33, (b) the procedure for the severity degree of defect patterns, and (c) the computation of the criticality of each defect state (the weight of the most severe pattern is 0.8).

If multiple patterns are identified in a fault state, it is reasonable to classify the target system as more dangerous than those with fewer patterns. Similarly, if a more severe pattern (i.e., a fault pattern which shows high severity) is identified in a fault state, the state can be classified as more dangerous than other states with a less severe pattern. Considering both the characteristics, the criticality of a fault state is assessed by an exponentially weighted average of the severity of extracted defect patterns, as the following equation:

𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙𝑖𝑡𝑦(𝑭𝑺_𝒒) = 𝛼 × 𝑠𝑒𝑣𝑒𝑟𝑖𝑡𝑦(𝑃𝑎𝑡𝑡𝑒𝑛_𝑞,1^{𝑓𝑎𝑢𝑙𝑡}) + ∑(1 − 𝛼)^𝑟−1× 𝑠𝑒𝑣𝑒𝑟𝑖𝑡𝑦(𝑃𝑎𝑡𝑡𝑒𝑟𝑛_𝑞,𝑟^{𝑓𝑎𝑢𝑙𝑡})

𝑘_𝑞

𝑟=2

where 𝑘_𝑞 is the total number of extracted fault patterns in the q^th fault state, 𝑃𝑎𝑡𝑡𝑒𝑛_𝑞,𝑟^{𝑓𝑎𝑢𝑙𝑡} is the r^th severe pattern in the q^th state, and 𝛼 is a user-defined weight for the most severe pattern.

Assume that the weight for the most severe pattern is set to 0.8 and the matrix of discretized state vectors is given as shown in Figure 5.3-(a). For the first fault state which is from time segment 5 to 10, ‘211’ is only found, therefore, the criticality of the first fault state is calculated as 0.8 × 1 = 0.8.

In contrast, the largest number of fault patterns, ‘211’, ‘322’, ‘312’. ‘231’, is extracted in the third fault state. The weights of the four patterns are assigned to be 0.8, 0.2, 0.04, and 0.008 respectively (as exponential decreasing), and then, the final criticality of the third state is 0.912, as shown in Figure 5.3-(c).

5.3 Empirical sensitivity analysis for multivariate discretization