# 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:

with .

# Case 1: , ΒΆ

The second-order Taylor expansion of at the point is:

The expectation and variance of the first-order expansion are:

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

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:

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 probability density function.