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, \geq0

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

We note (\lambda_k,  \vect{\varphi}_k)_{1 \leq k \leq K} the solutions of the Fredholm problem associated to C where K is the highest index K such that \lambda_K \geq s \lambda_1.

We note \vect{\lambda} the eigenvalues sequence and \vect{\varphi} the eigenfunctions sequence.

Then we define the linear projection function \pi_{ \vect{\lambda}, \vect{\varphi}} by:

(1)\pi_{\vect{\lambda}, \vect{\varphi}}: \left|
  \begin{array}{ccl}
    L^2(\cD, \Rset^d) & \rightarrow & \cS^{\Nset} \\
    f & \mapsto &\left(\dfrac{1}{\sqrt{\lambda_k}}\int_{\cD}f(\vect{t}) \vect{\varphi}_k(\vect{t})\, d\vect{t}\right)_{k \geq 1}
  \end{array}
\right.

where \cS^{\Nset}  = \left \{ (\zeta_k)_{k \geq 1} \in  \Rset^{\Nset} \, | \, \sum_{k=1}^{\infty}\lambda_k \zeta_k^2 < +\infty \right \}.

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)\left|
  \begin{array}{ccl}
     \cM_N \times (\Rset^d)^N & \rightarrow & \Rset^K \\
     F & \mapsto & (\xi_1, \dots, \xi_K) = MF
  \end{array}
\right.

where F = (\vect{t}_i, \vect{v}_i)_{1 \leq i \leq N} is a Field and M the projection matrix.

The inverse of \pi_{\vect{\lambda}, \vect{\varphi}} is the lift function defined by:

(3)\pi_{\vect{\lambda}, \vect{\varphi}}^{-1}: \left|
  \begin{array}{ccl}
     \cS^{\Nset} & \rightarrow & L^2(\cD, \Rset^d)\\
    (\xi_k)_{k \geq 1} & \mapsto & f(.) = \sum_{k \geq 1} \sqrt{\lambda_k}\xi_k \vect{\varphi}_k(.)
  \end{array}
\right.

If the function f(.) = X(\omega_0, .) where X is the centered process which covariance function is associated to the eigenvalues and eigenfunctions (\vect{\lambda}, \vect{\varphi}), then the getEigenValues method enables to obtain the K first eigenvalues of the Karhunen Loeve decomposition of X 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()
Attributes:
thisown

The membership flag

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.
getMesh() Accessor to the mesh.
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 square-root 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.
liftAsSample(coefficients) Lift the coefficients into a sample.
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.

getEigenValues()

Accessor to the eigenvalues of the Karhunen Loeve decomposition.

Returns:
eigenVal : Point

The most significant eigenvalues.

Notes

OpenTURNS truncates the sequence (\lambda_k,  \vect{\varphi}_k)_{k \geq 1} to the most significant terms, selected by the threshold defined in KarhunenLoeveAlgorithm.

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.

getMesh()

Accessor to the mesh.

Returns:
mesh : Mesh

The mesh corresponds to the discretization points of the integral in (1).

getModes()

Get the modes as functions.

Returns:
modes : collection of Function

The truncated basis (\vect{\varphi}_k)_{1 \leq k \leq K}.

Notes

The basis is truncated to (\vect{\varphi}_k)_{1 \leq k \leq K} where K is determined by the s, defined in KarhunenLoeveAlgorithm.

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 (\vect{\varphi}_k)_{1 \leq k \leq K} are evaluated on the vertices of the mesh defining the process sample. The values of the i-th field are the values of the i-th 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 matrix M defined in (2).

getScaledModes()

Get the modes as functions scaled by the square-root of the corresponding eigenvalue.

Returns:
modes : collection of Function

The truncated basis (\sqrt{\lambda_k}\vect{\varphi}_k)_{1 \leq k \leq K}.

Notes

The basis is truncated to (\sqrt{\lambda_k}\vect{\varphi}_k)_{1 \leq k \leq K} where K is determined by the s, defined in KarhunenLoeveAlgorithm.

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 (\vect{\varphi}_k)_{1 \leq k \leq K} are evaluated on the vertices of the mesh defining the process sample. The values of the i-th field are the values of the i-th mode on these vertices.

The mesh corresponds to the discretization points used to discretize the integral
(1).
getThreshold()

Accessor to the limit ratio on eigenvalues.

Returns:
s : float, \geq 0

The threshold s used to select the most significant eigenmodes, defined in KarhunenLoeveAlgorithm.

lift(coefficients)

Lift the coefficients into a function.

Parameters:
coef : Point

The coefficients (\xi_1, \dots, \xi_K).

Returns:
modes : Function

The function f defined in (3).

Notes

The sum defining f is truncated to the first K terms, where K is determined by the s, defined in KarhunenLoeveAlgorithm.

liftAsField(coefficients)

Lift the coefficients into a field.

Parameters:
coef : Point

The coefficients (\xi_1, \dots, \xi_K).

Returns:
modes : Field

The function f defined in (3) evaluated on the mesh associated to the discretization of (1).

Notes

The sum defining f is truncated to the first K terms, where K is determined by the s, defined in KarhunenLoeveAlgorithm.

liftAsSample(coefficients)

Lift the coefficients into a sample.

Parameters:
coef : Point

The coefficients (\xi_1, \dots, \xi_K).

Returns:
modes : Sample

The function f defined in (3) evaluated on the mesh associated to the discretization of (1).

Notes

The sum defining f is truncated to the first K terms, where K is determined by the s, defined in KarhunenLoeveAlgorithm.

project(*args)

Project a function or a field on the eigenmodes basis.

Available constructors:

project(function)

project(functions)

project(values)

project(fieldSample)

Parameters:
function : Function

A function.

functions : list of Function

A list of functions.

values : Sample

Field values.

fieldSample : ProcessSample

A collection of fields.

Returns:
point : Point

The vector (\alpha_1, \dots, \alpha_K) of the components of the function or the field in the eigenmodes basis

sample : Sample

The collection of the vectors (\alpha_1, \dots, \alpha_K) of the components of the collection of functions or fields in the eigenmodes basis

Notes

The project method calculates the projection (1) on a function or a field where only the first K elements of the sequences are calculated. K is determined by the s, defined in KarhunenLoeveAlgorithm.

Lets note \cM_{KL} the mesh coming from the KarhunenLoeveResult (ie the one contained in the modesAsSample ProcessSample).

The given values are defined on the input field \cM_{KL} and the associated values are directly used for the projection.

If evaluated from a function, the project method evaluates the function on \cM_{KL} and uses (2).

setName(name)

Accessor to the object’s name.

Parameters:
name : str

The name of the object.

thisown

The membership flag