uppose the vector-valued random
variable
is transformed by the analytical function
into a vector-valued variable
:
The variable
belongs to a certain class of random variables
(such as the class of normal variables). We would like to evaluate statistics
of
such mean and covariance matrix in some efficient way. Consequently, we select
a variable
with the same statistics and use it in place of
.
Since such replacement is an approximate, it is acceptable to seek approximate
procedure for evaluation of statistics of
.
We restrict our attention to the class
of normal vector-valued random variables. For this reason we evaluate the mean
and covariance matrix of
.
Let
.
We treat the
as a small quantity. We propose to approximate the mean
with the
sum
where the
are some deterministic vectors in the value space of
,
are some real numbers and the range of the index
is to be determined later. We seek
,
such that
would approximate
with the second
order:
Consistently, we will be seeking approximation of the covariance matrix
of the random variable
with the
expression
for some coefficients
sought to
deliver
We now derive the
and
that deliver the second order approximations.
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