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,
The threshold used to select the most significant eigenmodes, defined in
KarhunenLoeveAlgorithm
. lambda :
Point
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 .
According to the Karhunen Loeve algorithm, the integral of (1) is replaced by a specific weighted and finite sum. Thus, the linear relation (1) becomes a relation between fields which allows the following matrix representation:
(2)¶
where is a
Field
and the projection matrix.The inverse of is the lift function defined by:
(3)¶
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.GaussianProcess(covariance_X, mesh) >>> s = 0.001 >>> algo_X = ot.KarhunenLoeveP1Algorithm(mesh, covariance_X, s) >>> 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 eigenvalues 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 eigenmodes basis. setName
(name)Accessor to the object’s name. 
__init__
(*args)¶ Initialize self. See help(type(self)) for accurate signature.

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.
 covModel :

getEigenValues
()¶ Accessor to the eigenvalues of the Karhunen Loeve decomposition.
Returns:  eigenVal :
Point
The most significant eigenvalues.
Notes
OpenTURNS truncates the sequence to the most significant terms, selected by the threshold defined in
KarhunenLoeveAlgorithm
. eigenVal :

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 : collection of
Function
The truncated basis .
Notes
The basis is truncated to where is determined by the , defined in
KarhunenLoeveAlgorithm
. modes : collection of

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).
 modesAsProcessSample :

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

getProjectionMatrix
()¶ Accessor to the projection matrix.
Returns:

getScaledModes
()¶ Get the modes as functions scaled by the squareroot of the corresponding eigenvalue.
Returns:  modes : collection of
Function
The truncated basis .
Notes
The basis is truncated to where is determined by the , defined in
KarhunenLoeveAlgorithm
. modes : collection of

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 used to discretize the integral
 (1).
 modesAsProcessSample :

getThreshold
()¶ Accessor to the limit ratio on eigenvalues.
Returns:  s : float,
The threshold used to select the most significant eigenmodes, defined in
KarhunenLoeveAlgorithm
.

lift
(coefficients)¶ Lift the coefficients into a function.
Parameters:  coef :
Point
The coefficients .
Returns: Notes
The sum defining is truncated to the first terms, where is determined by the , defined in
KarhunenLoeveAlgorithm
. coef :

liftAsField
(coefficients)¶ Lift the coefficients into a field.
Parameters:  coef :
Point
The coefficients .
Returns: Notes
The sum defining is truncated to the first terms, where is determined by the , defined in
KarhunenLoeveAlgorithm
. coef :

project
(*args)¶ Project a function or a field on the eigenmodes basis.
 Available constructors:
project(function)
project(functions)
project(field)
project(fieldSample)
Parameters:  function :
Function
A function.
 functions : list of
Function
A list of functions.
 field :
Field
A field.
 fieldSample :
ProcessSample
A collection of fields.
Returns: Notes
The project method calculates the projection (1) on a function or a field where only the first elements of the sequences are calculated. is determined by the , defined in
KarhunenLoeveAlgorithm
.Lets note the mesh coming from the
KarhunenLoeveResult
(ie the one contained in the modesAsSampleProcessSample
).If the given field is defined on , the associated values are directly used for the projection. Otherwise, a
P1LagrangeEvaluation
function is built using this field and then used to interpolate the values of the field on .If evaluated from a function, the project method evaluates the function on and uses (2).

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