Fault dependency matrix D for describing the relationship between the fault source
and sensor output.
Consider graph G = (V, E), where set V contains two sets of nodes: F ¼
f 1 ; f 2 ; . . .; f m
f
gand T ¼ t 1 ; t 2 ; . . .; t n
f
g , and requires that F can be completely covered
by T. E ¼ fe ij g is the edge connecting node v i and v j , representing the information flow
between nodes. Graph G captures the first-order causal dependency between nodes,
which is v i affect v j or v j affect v k . On this basis, we can obtain system fault dependency
matrix D. This is a dimension binary matrix m  n with fault source as row and fault
observation sensor as column. It reflects the observation ability of sensor t j 2 T. If f i
fails and can be observed by t j , then d ij ¼ 1; otherwise, d ij ¼ 0, which means that the
fault source f i cannot be observed by t j . In order to analyze the observing ability of
sensor t j 2 T to the given fault source f j 2 F, we usually use fault propagation algorithm. This algorithm starts with any given fault source f j 2 F and propagates the fault
along e ij 2 E to determine which subsequent nodes are affected by it. When the fault
arrives at the node v i , if v i is the test node, the appropriate output is generated at the
node and its copy is transmitted to the next node along the edge connected with v i . In
order to prevent the algorithm from entering a dead cycle, when another fault arrived at
one node, other faults stop replication and propagation at that node. The termination
condition of the algorithm is that the fault copy can no longer be propagated to any
other new nodes.
3 Correlation Model
3.1 Definition of Correlation
Correlation refers to the causal relationship between an entity (physical or conceptual)
and another entity. This correlation is directional. If z can be deduced from x, the z is
related to x or z depends on x; if x and z can be deduced from each other, then they are
interrelated.
Faults and testing have this correlation in the system. If the fault f i occurs can derive
that t j fails and t j pass can derive that f i did not happen, then t j and f i are interrelated, or
t j is a symmetric test. If only the fault f i occurs can derive that t j fails, but not t j pass can
derive that f i did not happen, then t j is an asymmetric test.
First-order correlation: refers to the direct causality, such as z can be directly
derived from x, which is called the first-order correlation of x and z. If a fault node is
directly connected with a measurement point in the system, then they are called firstorder correlation.
N-order correlation: also known as higher-order correlation, which refers to the
indirect causality. For example, x can be derived from w, y can be derived from x, and
z can be derived from y, it is said that z is related to w in order 3, y is related to w and
z is related to x in order 2. In the system, if a fault node is connected to a measurement
point through some intermediate nodes, they are called N-order correlation.
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and sensor output.
Consider graph G = (V, E), where set V contains two sets of nodes: F ¼
f 1 ; f 2 ; . . .; f m
f
gand T ¼ t 1 ; t 2 ; . . .; t n
f
g , and requires that F can be completely covered
by T. E ¼ fe ij g is the edge connecting node v i and v j , representing the information flow
between nodes. Graph G captures the first-order causal dependency between nodes,
which is v i affect v j or v j affect v k . On this basis, we can obtain system fault dependency
matrix D. This is a dimension binary matrix m  n with fault source as row and fault
observation sensor as column. It reflects the observation ability of sensor t j 2 T. If f i
fails and can be observed by t j , then d ij ¼ 1; otherwise, d ij ¼ 0, which means that the
fault source f i cannot be observed by t j . In order to analyze the observing ability of
sensor t j 2 T to the given fault source f j 2 F, we usually use fault propagation algorithm. This algorithm starts with any given fault source f j 2 F and propagates the fault
along e ij 2 E to determine which subsequent nodes are affected by it. When the fault
arrives at the node v i , if v i is the test node, the appropriate output is generated at the
node and its copy is transmitted to the next node along the edge connected with v i . In
order to prevent the algorithm from entering a dead cycle, when another fault arrived at
one node, other faults stop replication and propagation at that node. The termination
condition of the algorithm is that the fault copy can no longer be propagated to any
other new nodes.
3 Correlation Model
3.1 Definition of Correlation
Correlation refers to the causal relationship between an entity (physical or conceptual)
and another entity. This correlation is directional. If z can be deduced from x, the z is
related to x or z depends on x; if x and z can be deduced from each other, then they are
interrelated.
Faults and testing have this correlation in the system. If the fault f i occurs can derive
that t j fails and t j pass can derive that f i did not happen, then t j and f i are interrelated, or
t j is a symmetric test. If only the fault f i occurs can derive that t j fails, but not t j pass can
derive that f i did not happen, then t j is an asymmetric test.
First-order correlation: refers to the direct causality, such as z can be directly
derived from x, which is called the first-order correlation of x and z. If a fault node is
directly connected with a measurement point in the system, then they are called firstorder correlation.
N-order correlation: also known as higher-order correlation, which refers to the
indirect causality. For example, x can be derived from w, y can be derived from x, and
z can be derived from y, it is said that z is related to w in order 3, y is related to w and
z is related to x in order 2. In the system, if a fault node is connected to a measurement
point through some intermediate nodes, they are called N-order correlation.
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L. Ou and F. Bai