KarhunenLoeveResult¶

class KarhunenLoeveResult(*args)¶

Result structure of a Karhunen-Loeve algorithm.

Parameters

covModelCovarianceModel: The covariance model.
sfloat, $\geq0$: The threshold used to select the most significant eigenmodes, defined in KarhunenLoeveAlgorithm.
lambdaPoint: The first eigenvalues of the Fredholm problem.
modesBasis: The first modes of the Fredholm problem.
modesAsProcessSampleProcessSample: The values of the modes on the mesh associated to the KarhunenLoeve algorithm.
projectionMatrix: 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()

Methods

`drawCumulatedEigenvaluesRemainder`()	Accessor to the cumulated eigen values normalized remainder graph.
`drawEigenvalues`()	Accessor to the eigen values graph.
`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`()	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)¶

drawCumulatedEigenvaluesRemainder()¶

Accessor to the cumulated eigen values normalized remainder graph.

Draw the values of cumulated eigen values normalized remainder:

$cevr_i = 1 - \frac{\sum_{j=1}^i \lambda_j}{\sum_{j=1}^K \lambda_j}$

Returns

graphGraph: The cumulated and normalized eigen values graph.

drawEigenvalues()¶

Accessor to the eigen values graph.

Returns

graphGraph: The eigen values graph.

getClassName()¶

Accessor to the object’s name.

Returns

class_namestr: The object class name (object.__class__.__name__).

getCovarianceModel()¶

Accessor to the covariance model.

Returns

covModelCovarianceModel: The covariance model.

getEigenvalues()¶

Accessor to the eigenvalues of the Karhunen-Loeve decomposition.

Returns

eigenValPoint: 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

idint: Internal unique identifier.

getImplementation()¶

Accessor to the underlying implementation.

Returns

implImplementation: A copy of the underlying implementation object.

getMesh()¶

Accessor to the mesh.

Returns

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

getModes()¶

Get the modes as functions.

Returns

modescollection 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

modesAsProcessSampleProcessSample: 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

namestr: The name of the object.

getProjectionMatrix()¶

Accessor to the projection matrix.

Returns

projectionMatrix: The matrix $M$ defined in (2).

getScaledModes()¶

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

Returns

modescollection 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

modesAsProcessSampleProcessSample: 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

sfloat, $\geq 0$: The threshold $s$ used to select the most significant eigenmodes, defined in KarhunenLoeveAlgorithm.

lift(coefficients)¶

Lift the coefficients into a function.

Parameters

coefPoint: The coefficients $(\xi_1, \dots, \xi_K)$ .

Returns

modesFunction: 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

coefPoint: The coefficients $(\xi_1, \dots, \xi_K)$ .

Returns

modesField: 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

coefPoint: The coefficients $(\xi_1, \dots, \xi_K)$ .

Returns

modesSample: 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

functionFunction: A function.
functionslist of Function: A list of functions.
valuesSample: Field values.
fieldSampleProcessSample: A collection of fields.

Returns

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

namestr: The name of the object.

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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