Probabilistic Performance Analysis of Fault Diagnosis Schemes

for all t ∈ T , the autocorrelation function of x is defined as
R x (s, t) := E(x s x T t ),
for all s, t ∈ T , and the autocovariance function of x is defined as
C x (s, t) := E( (xs
− m x (s) )( x t − m x (t) ) T ) ,
for all s, t ∈ T . The random process {x t } is said to be strictly stationary if
p(x t1 , x t2 ,..., x tm ) = p(x t1 +τ, x t2 +τ,..., x tm +τ)
for all finite sets of indices t 1 , t 2 ,..., t m ∈ T , where m ∈ N, and all τ ≥ 0. The random process
{x t } is said to be wide-sense stationary (wss) if for some constant ¯m ∈ R n ,
m x (t) = ¯m,
for all t ∈ T , and for any τ ∈ T ,
R x (s + τ, s) = R x (t + τ, t),
for all s, t ∈ T . If {x t } is wss, then R x only depends on the difference between its arguments
and we may write R x (s + τ, s) = R x (τ), for all s,τ ∈ T . Given a wss process {x t }, the power
spectral density of x is defined as
∫
S x (ξ) := F(R x )(ξ) =
e −2πiξτ R x (τ) dτ,
where F is the Fourier transform operator.
2.2.6 Common Probability Distributions
1. A Gaussian random variable x : Ω → R n with mean µ ∈ R n and variance Σ ∈ R n×n , such
that Σ ≻ 0, is defined by the pdf
p x (s) :=
1
(−
(2π) n |Σ| exp 1 )
2 (s − µ)T Σ −1 (s − µ) .
This distribution is denoted x ∼ N (µ,Σ). If we define z := Σ − 1/2
(x − µ), then z ∼ N (0, I ),
which is known as the standard Gaussian distribution. If z is scalar, then the cdf of z can
be written as
P z (c) = 1 2
( ( c
1 + erf 2
)),
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