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1 T
δ M = m 0m
0 , α
D
−1
−1
m 0 = Cs′ sC
m s + N s′
sN
q − m s′
-&
(47b)
Recall that C = C s + C v and N=
−vecCvec T C−
−1-T
where also - = - s + - v and = s + v .
5. Extension to the general model
A more general model involves observables a , which are
non-linear functions a (s)
of the signals s , for which up to
fourth order central moments are known. In this case the
general non-linear collocation model with additive noise is
y = a( s)
+ v
(48)
Furthermore, we want to predict a stochastic signal g
which is also a nonlinear function g (s)
of the same fundamental
signals s .
The solution is straightforward if the marginal and joint
central moments (up to 4th order) are known for the random
parameters a and g : we merely need to replace s and s′
and their moments, by a and g and their moments, respectively.
In this way the problem reduces to one of "moment
propagation", i.e. to the determination of m a , C a ,
- a , a , m g , c g a and - g a from the known m= m s , C s ,
- s , s . If the joint distribution of s and s′ were known,
the joint distribution of a and g , and therefore the required
moments, could be derived in principle though tedious numerical
techniques need to be used. However this requirement
departs from the spirit of a "fourth order theory" and
an approximate propagation law should be used instead.
Linearization of a (s)
and g (s)
provides such an approximation,
as in the case of linearized models and "second order
theory". A better and hopefully sufficient approximation
can be provided by propagation laws in quadratic approximations,
which are derived from quadratic approximations
to the nonlinear functions a (s)
and g (s)
. Using Taylor expansions
up to the second order about the known (Taylor
point) m= E{s}
we obtain
a= a( m)
+ Aδ s+
H vec(
s sT
a δ δ ) , (49)
g = g( m)
+ gT
δ s+
h vec(
δsδsT
) , (50)
where δ s=
s−m
and
T g
h g = vecH g ,
H
g
⎡
T
1 ∂ ⎛ ∂g
⎞ ⎤
= ⎢ ⎜ ⎟ ⎥
⎢2
∂s
s
⎣ ⎝ ∂ ⎠ ⎥⎦
m
, (53)
d= −H a vecC s , c= −h vecC s . (54)
Using these relations in the definitions of m a , C a , - a ,
a , m g , c ag
and - g a , while making use of the definitions
of m , C s , - s , s , the following propagation laws are
derived
m
C
a
a
= a( m)+
H vecC
(55)
a
= AC AT
+ A-T
H
T
+ H - AT
+
s
s
s
a
a
+ H ( −vecC
vec T C ) H
(56)
a
s
s
s
s
- = ⊗ + ⊗ + + ⊗
s
a d Ca
( A A)[
-s
AT
sH
T
a ] ( A H a ) vec
a
m
c
s
s
T
a
+ vec( +
T T
(57)
C a dd ) d
= ( A⊗A)
( A⊗A)
T
−dT
⊗(
C ⊗d)
−
−(
d⊗d)(
d⊗d
s
) T
+ -
a
s
⊗d
T
s
T g
+ -
a
s
s
T
a
T
T
a C a
⊗d−
− vec C ( d⊗d)
−(
d⊗d)
vec
(58)
= g( m)
h vec
(59)
g +
ag
vec
where
s
vec
T g
C s
= AC + +
T
sg
H a-
sg
A-s
h g +
+ H ( −vecC
vec T C h
(60)
a s s s )
s
- ga
ag
( s s g
= d⊗c
+ A⊗A)[
- g+
h ] +
s
g
+ ( A⊗H
)
s
g+
c vec(
C ddT
)
(61)
⎡vec
⎢
= ⎢
⎢
⎣vec
a vec
a +
s
11
s
n1
vec
vec
s
1
n
s
nn
⎤
⎥
⎥
⎥
⎦
+
(62)
while use of the "symmetric" Kronecker matrix product has
been made, defined for any two matrices A and B by
A⊗ s B=
A⊗B+
B⊗A
(63)
⎡∂a
⎤
A= ⎢
s
⎥ ,
⎣ ∂ ⎦
H
m
⎡1
∂ ⎛ ∂ai
⎞
= ⎢ ⎜ ⎟
⎢ ∂s
s
⎣ ⎝ ∂ ⎠
ai 2
T
⎤
⎥
⎥
⎦
m
,
g
T ⎡∂g
⎤
= ⎢
s
⎥
⎣ ∂
, (51)
⎦
H a
m
⎡vecT
H
⎢
= ⎢
⎢ T
⎣
vec H
a1
a n
⎤
⎥
⎥ , (52)
⎥
⎦
The above propagation laws can be interpreted either as
quadratic approximations of the true propagation laws for
the original non-linear model y = a( s)
+ v , g (s), or as exact
propagation laws for the "quadraticized" model
y = a( m)
+ Aδ s+
H a vec(
δsδsT
) + v , (64)
g = g( m)
+ gT
δ s+
h vec(
δsδsT
) . (65)
T g
4