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with solution
a= C−1 z c zs′
, γ = m′−c
C
−1
z m z , (A7)
s ˆ′ = m′+c C
− 1 ( z −m
T
z s′
z
z
)
T
z s ′
= aT
C
T
T −
za+
σ
s
2 ′
−2a
c z s ′ = σ 2
s ′
−c
z s ′ C
1
z c zs′
(A8)
φ . (A9)
Since the above prediction is already unbiased from (A6) it
follows that it is identical with the Best Quadratic Unbiased
inhomogeneous prediction (inhomBQUP = inhomBQP).
Best quadratic homogeneous prediction (homBQP):
Unconditional minimization of φ follows from
1 ⎛ ∂φ
⎞
⎜ ⎟
2 ⎝ ∂a
⎠
T
⎛ ∂β
⎞
= β ⎜ ⎟
⎝ ∂a
⎠
T
+ C
a−
z c z s′
=
( mT
a−m
′)
m + C a−c
0 , (A10)
= z z z zs′
=
with solution
a= ( C + m mT
)
1(
c ′
=
z
z
z
+ m′
m
−
zs
z )
m′−c
C
m
T −1
z z z
C
−1
′
z c z
C
−1
′ + s
s
z
1+
mT
z C
−1
z m z
=
m
z
, (A11)
mT
−
⎛
−
z C
1
z z
mT
⎞
′=
′+
−
z C
1
z z
s ˆ
m
T
c
⎜ −
⎟
′
+
−
z
C
1
z z
m
⎝ +
− z
1 m z C
1
z m s
,
T
z
1 mT
z C
1
z m z ⎠
(A12)
c C m −m′
β =
, (A13)
m
T −1
z s ′ z z
1+
1
mT
z C−
z
z
( m′−c
C m z )
2
φ = σ
2
c
z
C
−1
′
−
T
s s ′ z c zs′
+
. (A14)
1+
m C m
T
z s ′
T
z
−1
z
−1
z
Best quadratic unbiased homogeneous prediction
(homBQUP):
Minimization of φ under the condition β = 0 , requires the
formulation of the Lagrangean function L = φ −2λβ
and the
solution is provided from the equations
1 ⎛ ∂L
⎞
⎜ ⎟
2 ⎝ ∂a
⎠
T
⎛
= β ⎜
⎝
T
∂β
⎞
⎟
∂a
⎠
+ C
z
a−c
= z z zs′
=
zs′
z
⎛
−λ
⎜
⎝
T
∂β
⎞
⎟
∂a
⎠
( β −λ)
m + C a−c
0 , (A15)
1 ∂L
=−β
= m′−a T m z = 0 , (A16)
2 ∂λ
Since β = 0 (A15) gives
a= C λ (A17)
−
z
1 c zs′
+ C z
−1 m z
which can be inserted into (A16) to provide
m′−c
λ =
m
C
m
T −1
z s ′ z z
T
z C
−1
z m z
=
(A18)
with the above values of a and λ the prediction and its
mean square error become
mT
−
⎛ −
′ =
z C
1
z z
mT
′+ −
⎜
−
′ −
z C
1
z z
s ˆ
m
T
c
z
C
1
z z
m
mT
−
z C
1
z m s
z
⎝ mT
z C
1
z m z
T −1
z s ′ z z
T 1
z C
−
z m z
z
⎞
⎟
⎠
(A19)
( m′−c
C m )
2
φ = σ
2
c
z
C
−1
′
−
T
s s ′ z c zs′
+
. (A20)
m
6