Probabilistic Performance Analysis of Fault Diagnosis Schemes

are simultaneously true, for each i and j . The four possible cases are typically given the
following names [61, 73]:
D 0,k ∩ H 0,k is a true negative,
D 1,k ∩ H 0,k is a false positive,
D 0,k ∩ H 1,k is a false negative,
D 1,k ∩ H 1,k is a true positive.
The corresponding probabilities of these events are denoted
P tn,k := P(D 0,k ∩ H 0,k ), (3.1)
P fp,k := P(D 1,k ∩ H 0,k ), (3.2)
P fn,k := P(D 0,k ∩ H 1,k ), (3.3)
P tp,k := P(D 1,k ∩ H 1,k ). (3.4)
In the literature (e.g., [31, 34, 73]), these event are often organized into an array
[
]
P tn,k P fn,k
, (3.5)
P fp,k P tp,k
called a confusion matrix or contingency table. Since, for each k, the collection of events
{D i ,k ∩ H j,k : i , j ∈ D} forms a partition of the sample space, the probabilities (3.1)–(3.4)
satisfy the following useful identities:
P tn,k + P fn,k = P(D 0,k ), (3.6)
P fp,k + P tp,k = P(D 1,k ), (3.7)
P tn,k + P fp,k = P(H 0,k ) = Q 0,k , (3.8)
P fn,k + P tp,k = P(H 1,k ) = Q 1,k , (3.9)
P tn,k + P fp,k + P fn,k + P tp,k = 1. (3.10)
The identity in equation (3.10) implies that there are only three independent probabilities.
In the sequel, we refer to the probabilities P tn,k , P fp,k , P fn,k , and P tp,k as the performance
metrics for the test V at time k.
Although the probabilities (3.1)–(3.4) quantify every possible state of affairs, with respect
to the hypotheses H 0,k and H 1,k , the numerical values of these probabilities may be difficult
to interpret. For example, suppose that Q 1,k ≈ 0. By equation (3.9), Q 1,k ≈ 0 implies that
P fn,k ≈ 0 and P tp,k ≈ 0. From the small numerical values of P fn,k and P tp,k , it may be difficult
to get a sense of how the fault diagnosis scheme will behave in the event that a fault actually
occurs. An alternative approach is to consider the relative magnitudes of the probabilities.
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