Let be a function, let
be an input point and let
be the corresponding output.
First-order Taylor expansionΒΆ
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 expansionΒΆ
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.