Taylor expansion¶
Let be a twice differentiable mapping.
First-order Taylor expansion¶
The first-order Taylor expansion of at the point is the function defined for each marginal function of by the equation:
for which can be written as:
where is the Jacobian matrix evaluated at the point :
for and .
Second-order Taylor expansion¶
The second-order Taylor expansion of at the point is the function defined for each marginal function of by the equation:
which can be written as:
where is the Hessian tensor of order 3 evaluated at :
for and .
The first and second order Taylor expansions are used in the following cases:
to evaluate the importance factors of the input point on the output: refer to Taylor importance factors,
to get an approximation of the mean and the variance of the output: refer to Taylor expansion moments.