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.