Taylor variance decompositionΒΆ
where:
is the vector of the input variables at the mean values of each component.
is the covariance matrix of the random vector
. The elements are the followings :
is the gradient vector taken at the value
and
.
is a matrix. It is composed by the second order derivative of the output variable towards the
and
components of
taken around
. It yields to:
is a scalar product between two vectors.
where:
is the vector of the input variables at the mean values of each component.
is the covariance matrix of the random vector
. The elements are the followings :
is the transposed Jacobian matrix with
and
.
is a tensor of order 3. It is composed by the second order derivative towards the
and
components of
of the
component of the output vector
. It yields to:
Pay attention that
is a vector. The
component of this vector is equal to the
component of the output vector computed by the model
at the mean value.
is thus the computation of the model at mean.
This last formulation is the reduced writing of the following expression:
The decomposition of the variance at the order 2 is not implemented. It requires both the knowledge of higher order derivatives of the model and the knowledge of moments of order strictly greater than 2 of the pdf.