Probabilistic Performance Analysis of Fault Diagnosis Schemes

thesis is to provide a probabilistic framework, in which the probability of detection can
be efficiently computed for a large class of random fault signals.
2.6.3 Quickest Detection Problem
A related problem, which lends itself to more rigorous probabilistic analysis, is the quickest
detection problem. Suppose that we measure a sequence of independent random variables
{y k } k≥0 . Initially, the random variables are independent and identically distributed (iid)
according to some distribution P 0 . Then, at some random time t f , a change or fault occurs
which alters the distribution of the random sequence. After t f , the sequence {y k } k≥t f
is still
iid, but the distribution is P 1 . The goal is to detect that the distribution of {y k } has changed,
as quickly as possible, after the fault time t f . This problem is also known as statistical
change-point detection or simply change-point detection.
A quickest detection scheme is a procedure that processes the measurements {y k } and
produces an alarm time t a , which is an estimate of the fault time t f . Given a quickest
detection scheme, the performance is typically assessed by two performance metrics [2, 76,
84]. First, the mean time between false alarms is defined as
¯T := E(t a | t a < t f ),
Second, the mean delay is defined as
¯τ := E(t a − t f + 1 | t a ≥ t f ).
Although these metrics quantify the performance of the scheme in a meaningful way, their
application to fault diagnosis problems is limited. When the sets of measurements {y k } k