Conformal Geometric Algebra in Stochastic Optimization Problems ...

184 CHAPTER 6. PRACTICAL ASPECTS OF GEOMETRIC ALGEBRA
Fig. 6.2: Histogram over 3000000 samples from Z ∽ . The skewness of the distribution
is obvious as mean (9), median (8.7) and mode (≈ 8.2) do not coincide. 53.5% of
the samples lie below E(Z ∽ ) = 9.
Thus exploiting that hxiyj = hyjxi
E(h(x ∽ , y ∽ )) ≈ h(x,y) + 1
2
+ 1
2
+
s�
i,j=1
t�
i,j=1
s�
i=1 j=1
[Σxx]ij hxixj (x, y)
[Σyy]ij hyiyj (x,y)
t�
[Σxy]ij hxiyj
(x, y), (6.4)
Now consider a typical bilinear algebra product as given by equation (6.1): let
H ∽ = X ∽ ◦ Y ∽ such that the usage of Φ yields
k i k h = x∽ G
∽ ij yj . (6.5)
∽
Note that by the bilinearity of hk it cannot be differentiated twice with respect to
∽
the same variable, for instance x. It is thus clear from the preceding elucidations
∽
that applying equation (6.4) gives
k
E(h )
∽
=
�
k
h (x, y) + [Σxy]ij h
i,j
k xiyj (x, y)
= h k (x, y) + [Σxy] ij G k ij
It is crucial to note that this result is not an approximation since even a complete
Taylor series expansion will not provide terms higher than the first order derivatives:
error propagation for the mean of a bilinear function is exact irrespective of the
underlying distribution.