s - Athanasios Dermanis Home Page

erty of unbiased prediction is
β = E { sˆ
′− s′
} = E{
ε}
= 0 . (4)
Since s′ ˆ is quadratic in y , ε
2
will be of the 4th order in
y , and consequently φ will depend on up to 4th order
(central) moments of s , v , and cross-moments of s′ and
s . To describe these moments in matrix notation we introduce
the following basic definitions :
m= E{s} , δ s=
s−m
, δ y = y −m=
δs+
v (5)
m ′ = E{s′
} , δ s ′= s′−m′
(6)
C s
= E{ δsδsT
} , C E{ vvT
v = }, (7)
C = E{ δ yδy
T
}= C s + C v , (8)
δ ′ ′ , (9)
c
T
s s ′ = E{ sδs
} = E{
δyδs
} = c s ′ s
σ
2
= E{( s )
2}
, (10)
′
s ′
δ
⎡-1
⎤
- = E y y y
⎢ ⎥
{ vec(
δ δ
T
) δ
T
} = = - s + -
⎢
⎥
v , (11)
⎢⎣
- ⎥
n ⎦
- E{ s y y
T
} E{
s s sT
′ s = δ ′ δ δ = δ ′ δ δ } , (12)
s
= E{ vec(
δyδy
T
) vec T ( δyδy
T
)}=
⎡
=
⎢
⎢
⎢⎣
3. Solution
11
n1
1n
nn
⎤
⎥
⎥
=
⎥⎦
+
s v
(13)
For the determination of the optimal quadratic predictors it
is more convenient to express (2a) in the equivalent form
( y = m+
δy
)
sˆ
′ = γ + ( d+
2Qm)
T
y + vecT
Q vec(
δyδy
T
−mm
T
) =
C
c
z
zs′
⎡C
-T
⎤
= E {( z −m
z )( z −m
z )
T
} = ⎢
⎥ (17)
⎣-
−vecC
vecT
C⎦
⎡ css′
⎤
= E {( z −m
z )( s′−m′
)} = ⎢ ⎥
(18)
⎣vec-
s′
s ⎦
while for the homogeneous case (2b) becomes s ˆ ′ =a T z .
With the above "vectorized" formulation it is easy to calculate
the bias and the mean square error
β γ + a m −m
(19)
=
T
z
′
φ a C a 2a
c . (20)
= β 2 + T
T
z + σ
s
2
′ − zs′
The problem of prediction has now the same structure as
the linear problem with well known solution (Schaffrin,
1985a). A simple derivation for the various types of predictors
is given in appendix A, following Dermanis (1991). It
has the general form
s ˆ′ = α m′+
c C
− 1 ( z−α
m )
(21)
T
z s′
z
with (minimum) mean square error
z
φ σ
2 −c C
−1c
+ δφ
(22)
=
T
s ′ z s ′ z zs′
where α and δφ are different for different predictors:
inhomBQP = inhomBQUP:
α =1 , δφ = 0 , β = 0 . (23)
homBQP:
m C z
α =
, (24a)
m
T −1
z z
1+
mT
z C−
1
z
T
z s ′
T
z
z
( m′−c
C m
2
z )
δφ =
, (24b)
1+
m C m
T −1
z s ′ z z
1+
mT
z C−
1
z
−1
z
−1
z
z
z
c C m −m′
β =
. (24c)
m
⎡ y
= γ + [ dT
+ 2 mT
Q vecT
Q]
⎢
⎣vec(
δyδy
T
−mm
T
⎤
⎥
)
=
⎦
=γ +a T z
(14)
homBQUP:
m C z
α ,
T −1
z z
=
m T
z C
−1
z m z
( m′−c
C m
δφ =
, β = 0 . (25)
m
T −1 z z z ) 2
s ′
T
z C
−1
z m z
where a is a new unknown vector,
⎡ y
z = ⎢
T
⎣vec
δyδy
−mm
(
T
⎤
⎥ , (15)
) ⎦
⎡ m ⎤
m z = E{
z}
= ⎢
⎥ , (16)
⎣vec(
C−mm
T
) ⎦
In order to obtain expicit equations we only need to replace
z from (15), m from (16), C z from (17) and c z s′
from (18). Also the following identity needs to be used
C−
1 z
⎡C
= ⎢
⎣-
-T
−vecCvec
T
⎤
⎥
C⎦
−1
=
2