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Modeling and Inverse Problems in the Presence of Uncertainty
book

Modeling and Inverse Problems in the Presence of Uncertainty

by H. T. Banks, Shuhua Hu, W. Clayton Thompson
April 2014
Intermediate to advanced content levelIntermediate to advanced
405 pages
13h
English
Chapman and Hall/CRC
Content preview from Modeling and Inverse Problems in the Presence of Uncertainty
214 Modeling and Inverse Problems in the Presence of Uncertainty
where R(t, s) = Cor{X(t), X(s)} and V(t, s) = Cov{X(t), X(s)}. In addition,
both R and V are non-negative definite in the sense that
l
X
j=1
l
X
k=1
c
j
R(t
j
, t
k
)c
k
0,
l
X
j=1
l
X
k=1
c
j
V(t
j
, t
k
)c
k
0 (7.7)
hold for any t
1
, t
2
, . . . , t
l
T and for any c = (c
1
, c
2
, . . . , c
l
)
T
R
l
. We note
that (7.7) can be easily established through the following equalities:
l
X
j=1
l
X
k=1
c
j
R(t
j
, t
k
)c
k
= c
T
E(XX
T
)c,
l
X
j=1
l
X
k=1
c
j
V(t
j
, t
k
)c
k
= c
T
E{(X E(X))(X E(X))
T
}c,
where X = (X(t
1
), X(t
2
), . . . X(t
l
))
T
.
Similarly, the m-time auto-correlation function of the stochastic process
{X(t) : t T} is defined by
Cor{X(t
1
), X(t
2
), . . . , X(t
m
)}
= E (X(t
1
)X(t
2
) ···X(t
m
)) ,
=
Z
R
m
m
Y
j=1
x
j
p
X
1
,...,X
m
(t
1
, x
1
, t
2
, x
2
, . . . , t
m
,
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Publisher Resources

ISBN: 9781482206432