KarhunenLoeveAlgorithm

class KarhunenLoeveAlgorithm(*args)

Base class for Karhunen Loeve algorithms.

Parameters:

covModel : CovarianceModel

The covariance model.

s : float, \geq0

The minimal relative amplitude of the eigenvalues to consider in the decomposition wrt the sum of the preceeding eigenvalues. The default value is 0.

Notes

The Karhunen Loeve decomposition enables to build some finite approximations of stochastic processes which are optimal with respect to the norm L^2.

We suppose that C:\cD \times \cD \rightarrow  \cS^+_d(\Rset) is a covariance function defined on \cD \in \Rset^n, continuous at \vect{0}.

The class KarhunenLoeveAlgorithm enables to determine the solutions of the second kind Fredholm equation associated to C, ie to find the (\lambda_k,  \vect{\varphi}_k)_{k \geq 1} such that:

(1)\int_{\cD} \mat{C}(\vect{s},\vect{t}) \vect{\varphi}_k(\vect{t})\,  d\vect{t} = \lambda_k  \vect{\varphi}_k(\vect{s}) \quad \forall \vect{s} \in \cD

where (\lambda_k)_{k \geq 1} is a nonincreasing sequence of nonnegative values (the eigenvalues) and (\vect{\varphi}_k)_{k \geq 1} the associated sequence of eigenfunctions, normalized by \int_{\cD}\|\vect{\varphi}_k(\vect{s})\|^2\di{\vect{s}}=1. They form an hilbertian basis of L^2(\cD, \Rset^d).

The Mercer theorem shows that the covariance function C writes:

(2)\mat{C}(\vect{s},\vect{t}) = \sum_{k=1}^{+\infty} \lambda_k \vect{\varphi}_k(\vect{s}) \Tr{\vect{\varphi}}_k(\vect{t}) \quad \forall (\vect{s}, \vect{t}) \in \cD \times \cD

OpenTURNS uses the threshold s in order to select the most significant eigenvalues, ie all the eigenvalues such that:

(3)K = \max \{k \in \Nset \, | \, \lambda_k < s \sum_{i=1}^{k-1} \}

To solve (1), we use the functional basis (\theta_p)_{1 \leq p \leq P} of L^2(\cD, \Rset) with P elements defined on \cD. We search the solutions of type:

\tilde{\vect{\varphi}}_k(\vect{t})=\sum_{p=1}^{P} \vect{\phi}_{pk}\theta_p(\vect{t})

where \vect{\phi}_{pk} \in \Rset^d. We note:

\begin{align*}
    \vect{\Phi}_k =
    \left(
      \begin{array}{l}
        \vect{\phi}_{1k} \\
        \dots \\
        \vect{\phi}_{Pk}
      \end{array}
    \right) \in \Rset^{Pd}
 \end{align*}

and \mat{\vect{\Phi}} = (\vect{\Phi}_1\, |\, \dots \, | \vect{\Phi}_K) the matrix of the K first modes of the Karhunen Loeve decomposition.

The approximated Fredholm problem writes for all k \geq 1:

\int_{\cD} \mat{C}(\vect{s},\vect{t}) \tilde{\vect{\varphi}}_k(\vect{t})\,  d\vect{t} = \lambda_k   \tilde{\vect{\varphi}}_k(\vect{s}) \quad  \forall \vect{s} \in \cD

which enables to define the residual function \vect{r}: \cD \rightarrow \Rset^d defined by

(4)\vect{r}(\vect{s}) = \int_{\cD} \mat{C}(\vect{s},\vect{t}) \tilde{\vect{\varphi}}_k(\vect{t})\,  d\vect{t} - \lambda_k  \tilde{\vect{\varphi}}_k(\vect{s})

The Fredholm problem writes:

(5)\vect{r}(\vect{s}) = \vect{0} \quad \forall \vect{s} \in \cD

which is solved either by the Galerkin approach or the collocation approach.

The integrals in (4) can be evaluated with:

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 threshold used to select the most significant eigenmodes.
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)

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

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 threshold used to select the most significant eigenmodes.

Returns:

s : float, positive

The threshold s.

Notes

OpenTURNS truncates the sequence (\lambda_k,  \vect{\varphi}_k)_{k \geq 1} at the index K defined in (3).

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, \geq 0

The threshold s defined in (3).