KarhunenLoeveAlgorithm¶

class
KarhunenLoeveAlgorithm
(*args)¶ Base class for Karhunen Loeve algorithms.
 Available constructors:
 KarhunenLoeveAlgorithm(covModel, s)
Parameters: covModel :
CovarianceModel
The covariance model.
s : float, positive
The minimal relative amplitude of the eigenvalues to consider in the decomposition wrt the maximum eigenvalue.
See also
Notes
The Karhunen Loeve decomposition enables to build some finite approximations of stochastic processes which are optimal with respect to the norm .
We suppose that is a covariance function defined on , continuous at .
The class
KarhunenLoeveAlgorithm
enables to determine the solutions of the second kind Fredholm equation associated to , ie to find the such that:(1)¶
where is a nonincreasing sequence of nonnegative values (the eigenvalues) and the associated sequence of eigenfunctions, normalized by . They form an hilbertian basis of .
The Mercer theorem shows that the covariance function writes:
(2)¶
OpenTURNS uses the threshold in order to select the most significant eigenvalues, ie all the eigenvalues such that .
To solve (1), we use the functional basis of with elements defined on . We search the solutions of type:
where . We note:
and the matrix of the first modes of the Karhunen Loeve decomposition.
The approximated Fredholm problem writes for all :
which enables to define the residual function defined by
(3)¶
The Fredholm problem writes:
(4)¶
which is solved either by the Galerkin approach or the collocation approach.
The integrals in (3) can be evaluated with:
 a approach: see
KarhunenLoeveP1Algorithm
,  a quadrature approach: see
KarhunenLoeveQuadratureAlgorithm
,  a singular values decomposition approach: see
KarhunenLoeveSVDAlgorithm
.
Methods
getClassName
()Accessor to the object’s name. getCovarianceModel
()Accessor to the covariance model. getId
()Accessor to the object’s id. getImplementation
(*args)Accessor to the underlying implementation. getName
()Accessor to the object’s name. getResult
()Get the result structure. getThreshold
()Accessor to the limit ratio on eigenvalues. run
()Launch the algorithm. setCovarianceModel
(covariance)Accessor to the covariance model. setName
(name)Accessor to the object’s name. setThreshold
(threshold)Accessor to the limit ratio on eigenvalues. 
__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.

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.

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

getResult
()¶ Get the result structure.
Returns: resKL :
KarhunenLoeveResult
The structure containing all the results of the Fredholm problem.
Notes
The structure contains all the results of the Fredholm problem.

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 .

run
()¶ Launch the algorithm.
Notes
It launches the algorithm and creates a
KarhunenLoeveResult
, structure containing all the results.

setCovarianceModel
(covariance)¶ Accessor to the covariance model.
Parameters: covModel :
CovarianceModel
The covariance model.

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

setThreshold
(threshold)¶ Accessor to the limit ratio on eigenvalues.
Parameters: 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 .