s - Athanasios Dermanis Home Page

First International Symposium
on the Mathematical and Physical Foundations of Theoretical Geodesy
September 7th-9th, 1993, Stuttgart, Germany
Quadratic Collocation and Robust Quadratic Collocation
Athanasios Dermanis
Department of Geodesy and Surveying
The Aristotle University of Thessaloniki
1. Introduction
Estimation and prediction in geodesy is typically limited
to seeking optimal (minimum mean square error) estimators
and/or predictors in the class of linear functions of the parameters.
This can be justified in view of the well known
fundamental result (see e.g. Rao, 1973) that when the random
parameters of the model follow a Gaussian distribution,
the linear optimal estimators and predictors cannot be improved,
i.e. they are indeed the optimal non-linear ones.
Since Gaussianity is rather a mathematical convenience that
an absolute observational fact, it seems worthwhile to seek
non-linear estimators and predictors which minimize the
mean square error, without any restrictive assumptions on
the distribution of the relevant random variables.
The idea of using nonlinear predictors goes back to Grafarend
(1972). Dermanis and Sanso (1993) have studied
some basic characteristics of non linear estimators and came
to the conclusion that estimation of deterministic model
parameters is feasible only within a Bayesian point of view.
There is no need to seek rescue in Bayesian ideas in the case
of random effects models which involve no deterministic
but only random parameters. Prediction of random parameters,
either directly present in the model or stochastically
related ones, is of much geodetic interest especially for the
determination of the gravity field of the earth conceived as a
stochastic process in the framework of the statistical approach
of collocation (Moritz, 1980, Sanso, 1986).
Among the various types of non-linear predictors the simplest
and perhaps the only practically tractable ones are the
quadratic ones. Quadratic estimation is widely used for the
determination of variance components (e.g. Schaffrin, 1983)
and rarely for combined variance component and parameter
estimation (.XEiþHN
Dermanis and Sanso (1993) have derived Bayesian quadratic
estimators which can be directly applied to random
effect models such as the models used in geodetic collocation.
Here optimal quadratic predictors will be derived directly
for the simplest collocation model and then extended
to non-linear ones. As expected the results extend the usual
(linear) collocation equations in a way which involves the
effect of third and fourth order central moments which describe
the deviation of the distribution of the random parameters
from the Gaussian assumption.
2. The simple collocation model
We start with the simplest possible collocation model of
direct observation of random signals s under additive zeromean
observational errors v
y = s+
v , E { s } = m , E { v } = 0 . (1)
We assume that s and v are stochastically independent, but
no further assumptions are made about their probability distributions.
The objective is to seek optimal predictors for any single
random variable s′ , which is stochastically related to s ,
within the class of quadratic functions of the observations y,
i.e. of the form
s ˆ ′ = γ + dT
y + y
T
Qy (inhomogeneous) (2a)
or
s ˆ ′= dT
y + y
T
Qy (homogeneous) (2b)
We shall call the predictor (2a) inhomogeneous and the
predictor (2b) homogeneous, not in respect to the property
of homogeneity itself (of course both predictors are inhomogeneous
functions of y ) but in order to establish a correspondence
with the terminology used in the linear case obtained
by letting Q= 0 . These two predictors can be combined
with the possibility of introducing the property of unbiasedness
or not, in order to produce four basic classes of
best quadratic predictors:
Best inhomogeneous Quadratic Prediction (inhomBQP)
Best homogeneous Quadratic Prediction (homBQP)
Best inhomogeneous Quadratic Unbiased Prediction
(inhomBQUP)
Best homogeneous Quadratic Unbiased Prediction
(homBQUP)
Best (optimal) prediction here refers to the minimization
of the mean square prediction error :
φ = E {( sˆ
′−s′)
2
} = E{
ε
2}
= min
(3)
where ε = s ˆ′−s′
is the prediction error. The optional prop-

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

where
⎡C
= ⎢
⎣
−1
+ C
−1-T
N
−N
−1
-&
−1
−1
-&
-&
−1
−C
−1-T
N
−1
N
−1
⎤
⎥
⎦
(26)
N= −
C C−
−1
T
. (27)
vec vec T -
The general solution has the form
ˆ T T 1
s s ′
s s ′
s ′ = α m′+
c C
−1(
s−α
m)
+ n N
−
[ q(
y)
−α
m ] (28)
with
n
-&
=
1
, (29)
−
ss′ css′
−vec - s′
s
q = q(
y)
=
m q
-&
-&
−1 T T
y − vec[(
y − m)(
y − m)
q
− mm
] , (30)
=
− 1 m−vec(
C−mm
T ) = E{
q(
y )} . (31)
The mean square error of the prediction is
φ σ
2
−c C−1c
−n
N
−1n
+ δφ . (32)
=
T T
s ′ ss
′ ss
′ ss
′ s s′
For the particular prediction types we have
inhomBQP = inhomBQUP:
α =1 , δφ = 0 , β = 0 . (33)
homBQP:
mT
C
−1y
+ mT
q N
−1q
α =
, (34a)
+ + m
1 mT
C
−1m
mT
q N
−1
( m′−cT s ′
C
δφ = s
1+
mT
C−
c
β =
C
−1
m−n
m+
m
1
m−n
N
T
s s ′
T
q
N
N
T −1
T −1
s s ′
s s ′
1+
mT
C−
1m+
mT
q N
−1
homBQUP:
T
q
−1
q
−1
m
q)
m
q−m′
q
q
q
2
(34b)
(34c)
mT
C
−1y
+ mT
q N
−1q
α =
, (35a)
mT
C−
1m+
m N
−1m
( m′−
−
δφ =
β = 0 . (35b)
m
cT s
C−1 m nT s
N
−1q
2
s ′
s ′
)
T
C
−1m
+ mT
q N
−1m
q
The values from equations (33) are a quadratic extension
of what is usually described as "best linear prediction" and
is commonly called "collocation" in the geodetic literature.
The other solutions offer some alternatives which are "robust"
with respect to incorrect (prior) information about the
signal mean m . The homBQUP solution in particular given
by equations (35) is a quadratic extension of what has been
called "robust collocation" by Schaffrin (1985b). Note that
the quadratic predictors differ from the linear ones in (a) the
additional third tern in equation (28) and (b) in the additional
terms appearing in the definitions of α , β and δφ
(terms where the matrix N
−1
appears).
4. Vector generalization
For the prediction of a vector of new signals s′ we can
predict each component s′
i independently and combine the
solutions ŝ′
i in a single matrix equation for s′ ˆ . We also
need to introduce
m s ′ = E {s′}
, (36)
C E{( s m )( s m )
T
′ = ′− ′ − } , (37)
s s
s
s
= E{( s′−m
)( s′
m )
T
}
(38)
C s′ s′
− s′
= E{ vec[(
s−m
)( s−m
)
T
]( s′
m )
T
}, (39)
- s′ s
s s − s′
-&
=
1
(40)
N
−
s′ s Cs′
s −-s′
s
(using the notation m s = E{s}
instead of m for the sake of
distinction), as well as the bias vector and the mean
square error matrix M
= E { sˆ
′−s′}
= E{
0}
, ( 0 =ˆ s′−s′
) (41)
M = E{( sˆ
′−s′)(ˆ
s′−s′
)
T
} = E{
00T
}. (42)
With the above notation the general solution becomes
−1
−1
sˆ ′
s s s
s s s
= α m ′ + C ′ C ( s −α
m ) + N ′ N [ q(
y)
−α
m ] (43)
with α and q (y)
as before and mean square error matrix
M
C
−CT C
−1C
−NT
N
−1N
+ δM
. (44)
= s′
s′
s s′
s s′
s s′
s
For the particular prediction types we have
inhomBQP = inhomBQUP:
α =1 , δ M = 0 , = 0 . (45)
homBQP:
T
s
T
s
−1
m C y + m
α =
−1
1+
m C m + m
s
T −1
qN
q
T −1
qN
m
q
α
≡
α
1 −1
−1
= ( Cs′ sC
m s + N s′
s N q − m s′
α D
N
D
)
q
(46a)
(46b)
δ M = α T
D
(46c)
homBQUP:
T
s
T
s
−1
s
T
q
T
q
m C y + m N
α =
−1
m C m + m N
−1
−1
q
m
q
α
≡
α
N
D
, = 0 . (47a)
3

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

6. Discusion
The question that remains open is whether the newly derived
predictors are of any practical significance. The answer
depends of course on the distribution of the signals
involved i.e. in whether this distribution departs from the
Gaussian one. If the distribution is Gaussian linear predictors
cannot be improved and the additional terms in the
quadratic predictors will have to vanish. Within the framework
of the "fourth order" approach followed here, the distributions
themselves play no role and the question of departure
from the Gaussian distribution can be confined in
two specific points: (a) the existence of non-vanishing third
order central moments and (b) the departure of fourth order
central moments from the corresponding values in the Gaussian
case which are known functions of the lower order
moments.
A practical way to evaluate the actual effect of such departures,
when they exist, is to evaluate in each case the mean
square errors φ L and φ Q from the linear predictors, respectively,
and to look whether the relative improvement
φ L
−φ Q
is significant or not.
φL
Turning to problems related to the gravity field, the question
is directly related to the stochastic characteristics of the
underlying stochastic process, namely the disturbing potential
of the earth T . Sampling techniques can be used to derive
the "third order central moment function" Φ T ( P,
Q,
R)
,
which is a three-point function, and the "fourth order central
moment function" Ψ T ( P,
Q,
R,
S)
, which is a four-point
function, in a way similar to the one used for the estimation
of the covariance (second order central moment) function.
Under the usual assumptions of stationarity and isotropy the
above functions become functions of spherical distances ψ
between the points in question.
The function Φ T ( P,
Q,
R)
=ΦT
( ψ PQ , ψ PR , ψ QR ) , e.g., is
approximated by fitting a function Φ T ( ψ 1,
ψ 2 , ψ 3 ) to the 3-
dimensional histogram (3-parameter step function) derived
by taking averages of all the products
[ T ( P)
−mT
][ T ( Q)
−mT
][ T ( R)
−mT
]
where m T = E{T } , corresponding to triads of points P , Q ,
R , such that the values ψ PQ , ψ PR , ψ QR fall within a prescribed
cube.
The results of such a sampling investigation have a local
character, i.e. they cannot be taken to apply to different areas
of the earth surface.
In conclusion laborious numerical investigations in various
parts of the earth need to be undertaken before quadratic
collocation can be evaluated as a worthwhile alternative to
the usual (linear) one.
References
Dermanis, A. (1991): A unified approach to linear estimation
and prediction. IUGG General Assembly, Vienna
Aug. 1991.
Dermanis, A. and F. Sanso (1993): A Study of Nonlinear
Estimation. General Meeting of the International
Association of Geodesy, Beijing, China, August 8-
14, 1993.
5
Grafarend, E. (1972): Nichtlineare Prädiktion. ZfV, 97, 6,
245-255.
.XEiþHN / /RFDOO\ EHVW TXDGUDWLF HVWLPDWRUV
Math. Slovaca, vol. 35, no. 4, 393-408.
Moritz, H. (1980): Advanced Physical Geodesy. H. Wichmann
Verlag, Karlsruhe.
Rao, C.R. (1973): Linear statistical inference and its applications.
2nd edition. Wiley, New York.
Sanso, F. (1986): Statistical Methods in Physical Geodesy.
In: H. Sünkel (ed.) "Mathematical and Numerical
Techniques in Physical Geodesy", Lecture Notes in
Earth Sciences, Vol. 7, 49-155. Springer-Verlag,
Berlin.
Schaffrin, B. (1983): Varianz-kovarianz-komponenten-Schätzung
bei der Ausgleichung heterogener Wiederholungsmessungen.
DGK Reihe C, Heft Nr. 282.
Schaffrin, B. (1985a): Das geodätische Datum mit stochastischer
Vorinformation. DGK Reihe C, Heft
Nr. 313.
Schaffrin, B. (1985b): On robust collocation. In: F. Sanso
(ed). Proceedings First Hotine-Marussi Symposium
on Mathematical Geodesy (Roma, 1985), Milano,
1986, pp. 343-361.
Appendix A
The various types of quadratic predictors can more easily
be derived in the "vectorized" form starting with equations
(14), (19) and (20)
s ˆ ′ = γ +a T z (inhom) ,
s ˆ ′ =a T z
(hom) (A1)
β
= γ + aT m z −m′
(inhom),
β a m −m
(hom) (A2)
=
T
z
′
φ a C a 2a
c . (A3)
= β 2 + T
T
z + σ
s
2
′ − zs′
We note that
∂β
∂a =mT z
∂β
, = 1 . (A4)
∂γ
Best quadratic inhomogeneous prediction (inhomBQP):
Unconditional minimization of φ follows from the set of
equations
1 ⎛ ∂φ
⎞
⎜ ⎟
2 ⎝ ∂a
⎠
1
2
T
T
⎛ ∂β
⎞
= β ⎜ ⎟
⎝ ∂a
⎠
∂φ
∂β
= β = β = γ + a
∂γ
∂γ
+ Cz a−c
z = β m z + C za−c
z = 0 ,
T
m z
s′ s′
(A5)
−m′=
0 , (A6)

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