Taylor variance decompositionΒΆ

The Taylor variance decomposition (also referred as quadratic cumul method or perturbation method) is a probabilistic approach designed to propagate the uncertainties of the input variables \uX through the model h towards the output variables \uY. It enables to access the central dispersion (expectation and variance) of the output variables.
This method is based on a Taylor decomposition of the output variable \uY towards the \uX random vectors around the mean point \muX. Depending on the order of the Taylor decomposition (classically first order or second order), one can obtain different formulas. For easiness of the reading, we first present the formulas with n_Y=1 before the ones obtained for n_Y > 1.
As Y=h(\uX), the Taylor decomposition around \ux = \muX at the second order yields to:
Case n_Y=1

Y = h(\muX) + < \underline{\nabla} h(\muX) , \: \uX - \muX> + \frac{1}{2}<<\underline{\underline{\nabla }}^2 h(\muX,\: \underline{\mu}_{\:X}),\: \uX - \muX>,\: \uX - \muX> + o(\Cov \uX)

where:

  • \muX = \Expect{\uX} is the vector of the input variables at the mean values of each component.

  • \Cov \uX is the covariance matrix of the random vector \uX. The elements are the followings : (\Cov \uX)_{ij} = \Expect{\left(X^i - \Expect{X^i} \right)\times\left(X^j - \Expect{X^j} \right)}

  • \underline{\nabla} h(\muX) = \: ^t \left( \frac{\partial y}{\partial x^j}\right)_{\ux\: =\: \muX} = \: ^t \left( \frac{\partial h(\ux)}{\partial x^j}\right)_{\ux\: =\: \muX} is the gradient vector taken at the value \ux = \muX and j=1,\ldots,n_X.

  • \underline{\underline{\nabla}}^2 h(\ux,\ux) is a matrix. It is composed by the second order derivative of the output variable towards the i^\textrm{th} and j^\textrm{th} components of \ux taken around \ux = \muX. It yields to: \left( \nabla^2 h(\muX,\muX) \right)_{ij} =\left( \frac{\partial^2 h(\ux,\ux)}{\partial x^i \partial x^j}\right)_{\ux\: =\: \muX}

  • <,> is a scalar product between two vectors.

Approximation at the order 1 - Case n_Y=1

\Expect{Y} = h(\muX)

\Var{Y} = \sum_{i,j=1}^{n_X} \frac{\partial h(\muX)}{\partial X^i} . \frac{\partial h(\muX)}{\partial X^j}.(\Cov \uX)_{ij}

Approximation at the order 2 - Case n_Y=1

\begin{aligned} \begin{split} \Expect{Y} = h(\muX) + \frac{1}{2}. \sum_{i,j=1}^{n_X} \frac{\partial^2 h(\muX,\muX)}{\partial x^i \partial x^j} . (\Cov \uX)_{ij} \end{split} \end{aligned}

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.
Case n_Y>1
The perturbation approach can be developed at different orders from the Taylor decomposition of the random vector \uY. As \uY=h(\uX), the Taylor decomposition around \ux = \muX at the second order yields to:

\uY = h(\muX) + <\underline{\underline{\nabla}}h(\muX) , \: \uX - \muX> + \frac{1}{2}<<\underline{\underline{\underline{\nabla }}}^2 h(\muX,\: \underline{\mu}_{\:X}),\: \uX - \muX>,\: \uX - \muX> + o(\Cov \uX)

where:

  • \muX = \Expect{\uX} is the vector of the input variables at the mean values of each component.

  • \Cov \uX is the covariance matrix of the random vector \uX. The elements are the followings : (\Cov \uX)_{ij} = \Expect{\left(X^i - \Expect{X^i} \right)^2}

  • \underline{\underline{\nabla}} h(\muX) = \: ^t \left( \frac{\partial y^i}{\partial x^j}\right)_{\ux\: =\: \muX} = \: ^t \left( \frac{\partial h^i(\ux)}{\partial x^j}\right)_{\ux\: =\: \muX} is the transposed Jacobian matrix with i=1,\ldots,n_Y and j=1,\ldots,n_X.

  • \underline{\underline{\underline{\nabla^2}}} h(\ux\:,\ux) is a tensor of order 3. It is composed by the second order derivative towards the i^\textrm{th} and j^\textrm{th} components of \ux of the k^\textrm{th} component of the output vector h(\ux). It yields to: \left( \nabla^2 h(\ux) \right)_{ijk} = \frac{\partial^2 (h^k(\ux))}{\partial x^i \partial x^j}

  • <\underline{\underline{\nabla}}h(\muX) , \: \uX - \muX> = \sum_{j=1}^{n_X} \left( \frac{\partial {\uy}}{\partial {x^j}}\right)_{\ux = \muX} . \left( X^j-\muX^j \right)

  • <<\underline{\underline{\underline{\nabla }}}^2 h(\muX,\: \underline{\mu}_{X}),\: \uX - \muX>,\: \uX - \muX> = \left( ^t (\uX^i - \muX^i). \left(\frac{\partial^2 y^k}{\partial x^i \partial x^k}\right)_{\ux = \muX}. (\uX^j - \muX^j) \right)_{ijk}

Approximation at the order 1 - Case n_Y>1

\Expect{\uY} \approx \underline{h}(\muX)

Pay attention that \Expect{\uY} is a vector. The k^\textrm{th} component of this vector is equal to the k^\textrm{th} component of the output vector computed by the model h at the mean value. \Expect{\uY} is thus the computation of the model at mean.

\Cov \uY \approx ^t\underline{\underline{\nabla}}\:\underline{h}(\muX).\Cov \uX.\underline{\underline{\nabla}}\:\underline{h}(\muX)

Approximation at the order 2 - Case n_Y>1

\Expect{\uY} \approx \underline{h}(\muX) + \frac{1}{2}.\underline{\underline{\underline{\nabla}}}^2 \underline{\underline{h}}(\muX,\muX) \odot \Cov \uX

This last formulation is the reduced writing of the following expression:

(\Expect{\uY})_k \approx (\underline{h}(\muX))_k + \left( \sum_{i=1}^{n_X}\frac{1}{2} (\Cov \uX)_{ii}.{(\nabla^2\:h(\uX))}_{iik} + \sum_{i=1}^{n_X} \sum_{j=1}^{i-1} (\Cov X)_{ij}.{(\nabla^2\:h(\uX))}_{ijk} \right)_k

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.