TaylorExpansionMoments¶

class TaylorExpansionMoments(*args)¶

First and second order Taylor expansion formulas.

Refer to Taylor decomposition importance factors.

Parameters:

limitStateVariableRandomVector: It must be of type Composite, which means it must have been defined with the class CompositeRandomVector.

Notes

Assuming that $\uX$ has its two first moments finite and that $h$ is sufficiently regular, the Taylor expansion method is used to approximate the mean and variance of a random variable $\uY = h(\uX)$ with respect to the mean and variance of the input random vector $\uX$ .

We note:

$\mu_X = \Expect{\uX}$ the mean of the random vector $\uX$
$\Cov \uX$ the covariance matrix of the random vector $\uX$ . The elements are the followings : $(\Cov \uX)_{ij} = \Expect{\left(X^i - \Expect{X^i} \right) \left(X^j - \Expect{X^j} \right) }$
$\vect{\vect{\nabla}} h(\muX) = \: \Tr{\left( \frac{\partial y^i}{\partial x^j}\right)}_{\ux\: =\: \muX} = \: \Tr{\left( \frac{\partial h^i(\ux)}{\partial x^j}\right)}_{\ux\: =\: \muX}$ is the transposed Jacobian matrix with $i=1,\ldots,n_Y$ and $j=1,\ldots,n_X$ .
$\vect{\vect{\vect{\nabla^2}}} h(\ux\:,\ux)$ 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: $\left( \nabla^2 h(\ux) \right)_{ijk} = \frac{\partial^2 (h^k(\ux))}{\partial x^i \partial x^j}$

Then we have:

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

and

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

The Taylor expansion of order 2 of the model $h$ is:

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

Depending on the order of the Taylor expansion (classically first order or second order), one can obtain different formulas introduced hereafter.

Approximation at the order 1:

Expectation:

$\Expect{\uY} \approx \vect{h}(\muX)$

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.

Variance:

$\Cov \uY \approx \Tr{\vect{\vect{\nabla}}}\:\vect{h}(\muX).\Cov \uX.\vect{\vect{\nabla}}\:\vect{h}(\muX)$

Approximation at the order 2:

Expectation:

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

Variance:

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.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> myFunc = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'],
...     ['(x1*x1+x2^3*x1)/(2*x3*x3+x4^4+1)', 'cos(x2*x2+x4)/(x1*x1+1+x3^4)'])
>>> R = ot.CorrelationMatrix(4)
>>> for i in range(4):
...     R[i, i - 1] = 0.25
>>> distribution = ot.Normal([0.2]*4, [0.1, 0.2, 0.3, 0.4], R)
>>> # We create a distribution-based RandomVector
>>> X = ot.RandomVector(distribution)
>>> # We create a composite RandomVector Y from X and myFunc
>>> Y = ot.CompositeRandomVector(myFunc, X)
>>> # We create a Taylor expansion method to approximate moments
>>> myTaylorExpansionMoments = ot.TaylorExpansionMoments(Y)
>>> print(myTaylorExpansionMoments.getMeanFirstOrder())
[0.0384615,0.932544]

Methods

`drawImportanceFactors`()	Draw the importance factors.
`getClassName`()	Accessor to the object's name.
`getCovariance`()	Get the approximation at the first order of the covariance matrix.
`getGradientAtMean`()	Get the gradient of the function.
`getHessianAtMean`()	Get the hessian of the function.
`getId`()	Accessor to the object's id.
`getImportanceFactors`()	Get the importance factors.
`getLimitStateVariable`()	Get the limit state variable.
`getMeanFirstOrder`()	Get the approximation at the first order of the mean.
`getMeanSecondOrder`()	Get the approximation at the second order of the mean.
`getName`()	Accessor to the object's name.
`getShadowedId`()	Accessor to the object's shadowed id.
`getValueAtMean`()	Get the value of the function.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`setName`(name)	Accessor to the object's name.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

drawImportanceFactors()¶

Draw the importance factors.

Returns:

graphGraph: Graph containing the pie corresponding to the importance factors of the probabilistic variables.

getClassName()¶

Accessor to the object’s name.

Returns:

class_namestr: The object class name (object.__class__.__name__).

getCovariance()¶

Get the approximation at the first order of the covariance matrix.

Returns:

covarianceCovarianceMatrix: Approximation at the first order of the covariance matrix of the random vector.

getGradientAtMean()¶

Get the gradient of the function.

Returns:

gradientMatrix: Gradient of the Function which defines the random vector at the mean point of the input random vector.

getHessianAtMean()¶

Get the hessian of the function.

Returns:

hessianSymmetricTensor: Hessian of the Function which defines the random vector at the mean point of the input random vector.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getImportanceFactors()¶

Get the importance factors.

Returns:

factorsPoint: Importance factors of the inputs : only when randVect is of dimension 1.

getLimitStateVariable()¶

Get the limit state variable.

Returns:

limitStateVariableRandomVector: Limit state variable.

getMeanFirstOrder()¶

Get the approximation at the first order of the mean.

Returns:

meanPoint: Approximation at the first order of the mean of the random vector.

getMeanSecondOrder()¶

Get the approximation at the second order of the mean.

Returns:

meanPoint: Approximation at the second order of the mean of the random vector (it requires that the hessian of the Function has been defined).

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

getValueAtMean()¶

Get the value of the function.

Returns:

valuePoint: Value of the Function which defines the random vector at the mean point of the input random vector.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

hasVisibleNamebool: True if the name is not empty and not the default one.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

Examples using the class¶

Analyse the central tendency of a cantilever beam

Estimate moments from Taylor expansions

Example of sensitivity analyses on the wing weight model

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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TaylorExpansionMoments¶

Examples using the class¶