TaylorExpansionMoments¶

class
TaylorExpansionMoments
(*args)¶ First and second order Taylor expansion formulas.
Refer to Taylor decomposition importance factors.
 Parameters
 limitStateVariable
RandomVector
It must be of type Composite, which means it must have been defined with the class
CompositeRandomVector
.
 limitStateVariable
Notes
In a probabilistic approach the Taylor expansion can be used propagate the uncertainties of the input variables through the model towards the output variables . 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 towards the random vectors around the mean point . Depending on the order of the Taylor decomposition (classically first order or second order), one can obtain different formulas introduced hereafter.
As , the Taylor decomposition around at the second order yields to:
where:
is the vector of the input variables at the mean values of each component.
is the covariance matrix of the random vector uX. 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:
Expectation:
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.
Variance:
Approximation at the order 2:
Expectation:
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 distributionbased 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
(self)Draw the importance factors.
getClassName
(self)Accessor to the object’s name.
getCovariance
(self)Get the approximation at the first order of the covariance matrix.
getGradientAtMean
(self)Get the gradient of the function.
getHessianAtMean
(self)Get the hessian of the function.
getId
(self)Accessor to the object’s id.
getImportanceFactors
(self)Get the importance factors.
getLimitStateVariable
(self)Get the limit state variable.
getMeanFirstOrder
(self)Get the approximation at the first order of the mean.
getMeanSecondOrder
(self)Get the approximation at the second order of the mean.
getName
(self)Accessor to the object’s name.
getShadowedId
(self)Accessor to the object’s shadowed id.
getValueAtMean
(self)Get the value of the function.
getVisibility
(self)Accessor to the object’s visibility state.
hasName
(self)Test if the object is named.
hasVisibleName
(self)Test if the object has a distinguishable name.
setName
(self, name)Accessor to the object’s name.
setShadowedId
(self, id)Accessor to the object’s shadowed id.
setVisibility
(self, visible)Accessor to the object’s visibility state.

__init__
(self, \*args)¶ Initialize self. See help(type(self)) for accurate signature.

drawImportanceFactors
(self)¶ Draw the importance factors.
 Returns
 graph
Graph
Graph containing the pie corresponding to the importance factors of the probabilistic variables.
 graph

getClassName
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getCovariance
(self)¶ 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.
 covariance

getGradientAtMean
(self)¶ 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.
 gradient

getHessianAtMean
(self)¶ 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.
 hessian

getId
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getImportanceFactors
(self)¶ Get the importance factors.
 Returns
 factors
Point
Importance factors of the inputs : only when randVect is of dimension 1.
 factors

getLimitStateVariable
(self)¶ Get the limit state variable.
 Returns
 limitStateVariable
RandomVector
Limit state variable.
 limitStateVariable

getMeanFirstOrder
(self)¶ 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.
 mean

getMeanSecondOrder
(self)¶ 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).
 mean

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getShadowedId
(self)¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getValueAtMean
(self)¶ 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.
 value

getVisibility
(self)¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

hasName
(self)¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
(self)¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

setName
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.