# Taylor variance decompositionΒΆ

**Case**

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.

**Approximation at the order 1 - Case**

**Approximation at the order 2 - Case**

**Case**

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:

**Approximation at the order 1 - Case**

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.

**Approximation at the order 2 - Case**

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.