KarhunenLoeveResult¶

class
KarhunenLoeveResult
(*args)¶ Result structure of a Karhunen Loeve algorithm.
 Available constructors:
KarhunenLoeveResult(implementation)
KarhunenLoeveResult(covModel, s, lambda, modes, modesAsProcessSample, projection)
Parameters: implementation :
KarhunenLoeveResultImplementation
A specific implementation.
covModel :
CovarianceModel
The covariance model.
s : float, positive
The minimal relative amplitude of the eigenvalues to consider in the decomposition wrt the maximum eigenvalue.
lambda :
NumericalPoint
The first eigenvalues of the Fredholm problem.
modes :
Basis
The first modes of the Fredholm problem.
modesAsProcessSample :
ProcessSample
The values of the modes on the mesh associated to the KarhunenLoeve algorithm.
projection :
Matrix
The projection matrix.
Notes
Structure generally created by the method run() of a
KarhunenLoeveAlgorithm
and obtained thanks to the method getResult().We consider a covariance function defined on , continuous at .
We note the solutions of the Fredholm problem associated to where K is the highest index such that .
We note the eigenvalues sequence and the eigenfunctions sequence.
Then we define the linear projection function by:
(1)¶
where .
The integral of (1) can be discretized according to the chosen Karhunen Loeve algorithm: on the vertices of the domain in the case of a algorithm, on the weighted experiment in the case of the quadrature method. Then function can be reduced to its values on that discretization domain. Besides, we can restrict the sequences to the terms associated to the highest eigenvalues. Thus, following these discretizations, the function has a matrical representation.
The inverse of is the lift function defined by:
(2)¶
If the function where is the centered process which covariance function is associated to the eigenvalues and eigenfunctions , then the getEigenValues method enables to obtain the first eigenvalues of the Karhunen Loeve decomposition of and the method getModes enables to get the associated modes.
Examples
>>> import openturns as ot >>> N = 256 >>> mesh = ot.IntervalMesher([N  1]).build(ot.Interval(1, 1)) >>> covariance_X = ot.AbsoluteExponential([1]) >>> process_X = ot.TemporalNormalProcess(covariance_X, mesh) >>> threshold = 0.001 >>> algo_X = ot.KarhunenLoeveP1Algorithm(mesh, covariance_X, threshold) >>> algo_X.run() >>> result_X = algo_X.getResult()
Methods
getClassName
()Accessor to the object’s name. getCovarianceModel
()Accessor to the covariance model. getEigenValues
()Accessor to the eigen values of the Karhunen Loeve decomposition. getId
()Accessor to the object’s id. getImplementation
(*args)Accessor to the underlying implementation. getModes
()Get the modes as functions. getModesAsProcessSample
()Accessor to the modes as a process sample. getName
()Accessor to the object’s name. getProjectionMatrix
()Accessor to the projection matrix. getScaledModes
()Get the modes as functions scaled by the squareroot of the corresponding eigenvalue. getScaledModesAsProcessSample
()Accessor to the scaled modes as a process sample. getThreshold
()Accessor to the limit ratio on eigenvalues. lift
(coefficients)Lift the coefficients into a function. liftAsField
(coefficients)Lift the coefficients into a field. project
(*args)Project a function or a field on the eigen modes basis. setName
(name)Accessor to the object’s name. 
__init__
(*args)¶

getClassName
()¶ Accessor to the object’s name.
Returns: class_name : str
The object class name (object.__class__.__name__).

getCovarianceModel
()¶ Accessor to the covariance model.
Returns: covModel :
CovarianceModel
The covariance model.

getEigenValues
()¶ Accessor to the eigen values of the Karhunen Loeve decomposition.
Returns: eigenVal :
NumericalPoint
The most significant eigenvalues.
Notes
OpenTURNS truncates the sequence at the highest index such that where is the threshold fixed by the User.

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

getImplementation
(*args)¶ Accessor to the underlying implementation.
Returns: impl : Implementation
The implementation class.

getModes
()¶ Get the modes as functions.
Returns: modes :
Basis
The truncated basis .
Notes
The basis is truncated to where is fixed by the User through the parameter.

getModesAsProcessSample
()¶ Accessor to the modes as a process sample.
Returns: modesAsProcessSample :
ProcessSample
The values of each mode on a mesh whose vertices were used to discretize the Fredholm equation.
Notes
The modes are evaluated on the vertices of the mesh defining the process sample. The values of the ith field are the values of the ith mode on these vertices.
The mesh corresponds to the discretization points of the integral in (1).

getName
()¶ Accessor to the object’s name.
Returns: name : str
The name of the object.

getProjectionMatrix
()¶ Accessor to the projection matrix.
Returns: projection :
Matrix
The projection matrix associated to the discretized version of (1).

getScaledModes
()¶ Get the modes as functions scaled by the squareroot of the corresponding eigenvalue.
Returns: modes :
Basis
The truncated basis .
Notes
The basis is truncated to where is fixed by the User through the parameter.

getScaledModesAsProcessSample
()¶ Accessor to the scaled modes as a process sample.
Returns: modesAsProcessSample :
ProcessSample
The values of each scaled mode on a mesh whose vertices were used to discretize the Fredholm equation.
Notes
The modes are evaluated on the vertices of the mesh defining the process sample. The values of the ith field are the values of the ith mode on these vertices.
 The mesh corresponds to the discretization points of the integral in
 (1).

getThreshold
()¶ Accessor to the limit ratio on eigenvalues.
Returns: s : float, positive
The minimal relative amplitude of the eigenvalues to consider in the decomposition wrt the maximum eigenvalue.
Notes
OpenTURNS truncates the sequence at the highest index such that .

lift
(coefficients)¶ Lift the coefficients into a function.
Parameters: coef :
NumericalPoint
The coefficients .
Returns: modes :
NumericalMathFunction
The function defined in (2).
Notes
The sum defining is truncated to the first terms, where is fixed by the User through the parameter.

liftAsField
(coefficients)¶ Lift the coefficients into a field.
Parameters: coef :
NumericalPoint
The coefficients .
Returns: modes :
Field
Notes
The sum defining is truncated to the first terms, where is fixed by the User through the parameter.

project
(*args)¶ Project a function or a field on the eigen modes basis.
 Available constructors:
project(function)
project(field)
Parameters: function :
NumericalMathFunction
A function.
field :
Field
A field.
Notes
The project method calculates the projection (1) on a field or a function where only the first elements of the sequence are calculated. is determined by the parameter fixed by the User.

setName
(name)¶ Accessor to the object’s name.
Parameters: name : str
The name of the object.