e continue calculation of the previous
section.
We use analyticity of
and write the Taylor expansion with respect to
:
where the
is the vector component index. We take the
expectation,
where we used the direct consequence of definition of
:
The above expression is approximated with the
where we use the
notation
and
is the pth component of the vector
.
Hence, the second order of approximation would be delivered
if
Therefore, it is enough to satisfy the following conditions


(Unscented conditions for mean)

Let
be the covariance matrix
and
is any matrix with the
property
We seek the
of the
form
where the
are columns of the matrix
and
are some scaling factors. Such choice satisfies the second requirement of
(
Unscented conditions for mean
)
if
The third requirement of (
Unscented
conditions for mean
) transforms
into
Hence, we require
that
It is enough to leave a free scaling factor
,
.
Then the
recipes
would satisfy the second and third requirements of the
(
Unscented conditions for mean
).
The remaining factor
is determined by the first
requirement:
