Conformal Geometric Algebra in Stochastic Optimization Problems ...

172 CHAPTER 5. PARAMETER ESTIMATION
On these implicit definitions the H-constraint becomes
with H ∈ � Nh×M and zh ∈ � Nh.
Further Proceeding
H∆θ = zh,
The whole of conditions and constraints can be rendered by the matrix formalism
as � � � � � �
X Z ∆θ zg
= .
H 0 ∆y
Recall the GH-model as stated in (5.21): X ˇ β + Zǫ = z with E(ǫ ∽ ) = 0. It is now
proceeded in same way as on page 165, i.e. the vector zg is considered a vector of
new ‘pseudo’ observations. Setting again ǫz = −Zǫ, for the present case the linear
model
is obtained with E(ǫz ∽ ) = 0.
zg + ǫz = X ˇ
∆θ
Next, this reduced estimation problem is solved, subject to the H-constraint. The
solution is then used to evaluate the update ∆y.
Least Squares Minimization
According to the previous elucidations only the system
must be taken into account.
� X
H
�
∆θ =
� zg
zh
�
zh
. (5.29)
Note that a transition to pseudo observations zg is made. Setting ∆zg := ǫz the
expression ∆z T g Σ −1
zgzg ∆zg, with
∆zg := −Zǫ
(5.28)
= X∆θ − zg, (5.30)
has to be minimized, which is why the corresponding covariance matrix Σ∆zg∆zg is
required. Due to
Cov(z ∽ g, z ∽ g) = Cov(ǫ ∽ z, ǫ ∽ z) ǫz=−Zǫ
= ZCov(ǫ ∽ , ǫ ∽ )Z T = ZCov(y ∽ , y ∽ )Z T ,
it follows 16 , as on page 165,
Σzgzg = ZΣyyZ T
16 For readability, Σzgzg = Σ∆zg∆zg is being used in the following.
(5.31)