TaylorExpansionMoments

class TaylorExpansionMoments(*args)

First and second order Taylor expansion formulas.

Available constructors:
TaylorExpansionMoments(limitStateVariable)
Parameters:

limitStateVariable : RandomVector

This RandomVector must be of type Composite, which means it must have been defined with the fourth usage of declaration of a RandomVector (from a Function and an antecedent Distribution) or with the class CompositeRandomVector.

Notes

In a probabilistic approach the Taylor expansion can be used 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, 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 introduced hereafter.

As \uY=h(\uX), the Taylor decomposition around \ux = \muX at the second order yields to:

\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)

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}

  • \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}

  • <\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)

  • <<\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}

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 in the standard version of OpenTURNS. 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.RandomVector(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)

x.__init__(…) initializes x; see help(type(x)) for signature

drawImportanceFactors()

Draw the importance factors.

Returns:

graph : Graph

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

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getCovariance()

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

Returns:

covariance : CovarianceMatrix

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

getGradientAtMean()

Get the gradient of the function.

Returns:

gradient : Matrix

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:

hessian : SymmetricTensor

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:

id : int

Internal unique identifier.

getImportanceFactors()

Get the importance factors.

Returns:

factors : Point

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

getLimitStateVariable()

Get the limit state variable.

Returns:

limitStateVariable : RandomVector

Limit state variable.

getMeanFirstOrder()

Get the approximation at the first order of the mean.

Returns:

mean : Point

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

getMeanSecondOrder()

Get the approximation at the second order of the mean.

Returns:

mean : Point

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:

name : str

The name of the object.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getValueAtMean()

Get the value of the function.

Returns:

value : Point

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:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

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

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.