Taylor expansion momentsΒΆ
In this page, we consider the Taylor expansion of a function.
One way to evaluate the central dispersion (expectation and variance) of the
variable
is to use the Taylor expansion of the function
at
the mean point
. Depending on the order of the Taylor expansion
(classically first or second order), we get different approximations
of the mean and variance of
.
We use the notations introduced in Taylor Expansion.
In the remainder, let be the covariance matrix of
, defined by:
where is the input covariance matrix:
for .
Notice that each diagonal element of the covariance matrix
, is
equal to the variance of an input variable (
).
Case 1:
,
ΒΆ
In this section, we analyse the special case where and
.
The second-order Taylor expansion of at the point
is:
when .
The expectation and variance of the first-order expansion are:
In the special case where the inputs are independent, then the variance expression is simplified and we get:
The expectation of the second-order expansion is:
The second-order approximation of the variance is not implemented because it requires both the
knowledge of higher
order derivatives of and the knowledge of moments of order strictly greater
than 2 of the distribution of
.
Case 2:
,
ΒΆ
In this section, we present the general case where and
.
The second-order Taylor expansion of
at the point
for each marginal function
is:
where .
The expectation and covariance matrix of the first-order expansion are:
for .
The expectation of the second-order expansion is:
for .
The second-order approximation of the variance is not implemented because it requires both the
knowledge of higher order derivatives of and the knowledge of moments of order strictly greater
than 2 of the probability density function.