Computes the Karhunen-Loeve decomposition of a univariate stochastic process. More...

#include <kle.h>

Public Member Functions
	KLDecompUni (const Array1D< double > &tSamples)
	Constructor that takes the autocorrelation matrix "corr" ( ) of the process we are studying as well as the array "tsamples" ( ) with the points in time where snapshots of the system were taken. More...

	KLDecompUni ()

	~KLDecompUni ()
	Destructor. More...

void	Init ()

void	SetWeights (const Array1D< double > &weights)
	Set weights for computing the integral needed for Nystrom's method for solving the Fredholm integral equation. More...

void	SetWeights (const double *weights, const int npts)
	Set weights for computing the integral needed for Nystrom's method for solving the Fredholm integral equation. More...

int	decompose (const Array2D< double > &corr, const int &nKL)
	Perform KL decomposition into nKL modes and return actual number of modes that were obtained. More...

int	decompose (const double *corr, const int &nKL)
	Perform KL decomposition into nKL modes and return actual number of modes that were obtained. More...

void	KLproject (const Array2D< double > &realiz, Array2D< double > &xi)
	Project realizations $F(t,\theta_l)$ to the KL modes and store them in xi ( $\xi_k$ ) More...

const Array1D< double > &	eigenvalues () const
	Get eigenvalues in descending order. More...

void	eigenvalues (const int nEIG, double *eigs) const

const Array2D< double > &	KLmodes () const
	Get associated KL modes. More...

void	KLmodes (const int npts, const int nKL, double *klModes) const
	Get associated KL modes. More...

void	meanRealiz (const Array2D< double > &realiz, Array1D< double > &mean_realiz)
	Calculate (in meanRealiz) the mean realizations. More...

void	truncRealiz (const Array1D< double > &meanrea, const Array2D< double > &xi, const int &nKL, Array2D< double > &trunc_realiz)
	Returns the truncated KL sum. More...

Private Member Functions
	KLDecompUni (const KLDecompUni &)
	Dummy default constructor, which should not be used as it is not well defined. More...

Private Attributes
bool	decomposed_
	Flag to determine whether KL decomposition has taken place (and consequently that the interal data structures contain meaningful eigenvalues and vectors ... ) More...

Array2D< double >	whcwh_
	Matrix to hold the upper triangular part of the matrix to get eigenvalues of. More...

Array1D< double >	w_
	Array to hold weights for Nystrom's method for Fredholm integral equation solution. More...

Array1D< double >	wh_
	Array to hold square roots of weights. More...

char	jobz_
	Option to determine what to compute (eigenvalues and eigenvectors) More...

char	eigRange_
	Option to set the type of range for eigenvalues. More...

char	uplo_
	Option to indicate how matrix is stored. More...

double	vl_
	Lower bound for range of eigenvalues. More...

double	vu_
	Upper bound for range of eigenvalues. More...

int	il_
	Lower index of range of eigenvalues requested. More...

int	iu_
	Upper index of range of eigenvalues requested. More...

double	absTol_
	Absolute tolerance for convergence. More...

Array1D< double >	eig_values_
	Array to store eigenvalues. More...

Array2D< double >	KL_modes_
	Matrix to store KL modes. More...

int	eig_info_
	info on success of the eigenvector solutions More...

Array1D< int >	ifail_
	Array to store indices of eigenvectors that failed to converge. More...

Detailed Description

Computes the Karhunen-Loeve decomposition of a univariate stochastic process.

$F(t,\theta) = \left < F(t,\theta) \right >_{\theta} + \sum_{k=1}^{\infty} \sqrt{\lambda_k} f_k(t) \xi_k$

Constructor & Destructor Documentation

◆ KLDecompUni() [1/3]

KLDecompUni::KLDecompUni ( const Array1D< double > & tSamples )

Constructor that takes the autocorrelation matrix "corr" ( ) of the process we are studying as well as the array "tsamples" ( ) with the points in time where snapshots of the system were taken.

Constructs weights ( ) needed for the Nystrom method to solve the Fredholm integral equation

$\int C(s,t)f(t)dt=\lambda f(s) \rightarrow \sum w_j C(s_i,t_j) f_k(t_j) = \lambda_k f_k(s_i)$

◆ KLDecompUni() [2/3]

KLDecompUni::KLDecompUni ( )

◆ ~KLDecompUni()

KLDecompUni::~KLDecompUni ( )

inline

Destructor.

◆ KLDecompUni() [3/3]

KLDecompUni::KLDecompUni ( const KLDecompUni & )

inlineprivate

Dummy default constructor, which should not be used as it is not well defined.

Dummy copy constructor, which should not be used as it is currently not well defined

Member Function Documentation

◆ decompose() [1/2]

int KLDecompUni::decompose	(	const Array2D< double > &	corr,
		const int &	nKL
	)

Perform KL decomposition into nKL modes and return actual number of modes that were obtained.

Further manipulation of the discretized Fredholm equation leads to the eigenvalue problem

$A g=\lambda g$

where and , with being the diagonal matrix, $W_{ii}=\sqrt{w_i}$ and $K_{ij}=Cov(t_i,t_j)$ . Solutions consist of pairs of eigenvalues $\lambda_k$ and KL modes $f_k=W^{-1}g_k$ .

◆ decompose() [2/2]

int KLDecompUni::decompose	(	const double *	corr,
		const int &	nKL
	)

Perform KL decomposition into nKL modes and return actual number of modes that were obtained.

Further manipulation of the discretized Fredholm equation leads to the eigenvalue problem

$A g=\lambda g$

where and , with being the diagonal matrix, $W_{ii}=\sqrt{w_i}$ and $K_{ij}=Cov(t_i,t_j)$ . Solutions consist of pairs of eigenvalues $\lambda_k$ and KL modes $f_k=W^{-1}g_k$ .

◆ eigenvalues() [1/2]

const Array1D< double > & KLDecompUni::eigenvalues ( ) const

Get eigenvalues in descending order.

◆ eigenvalues() [2/2]

void KLDecompUni::eigenvalues	(	const int	nEIG,
		double *	eigs
	)		const

◆ Init()

void KLDecompUni::Init ( )

◆ KLmodes() [1/2]

const Array2D< double > & KLDecompUni::KLmodes ( ) const

Get associated KL modes.

◆ KLmodes() [2/2]

void KLDecompUni::KLmodes	(	const int	npts,
		const int	nKL,
		double *	klModes
	)		const

Get associated KL modes.

◆ KLproject()

void KLDecompUni::KLproject	(	const Array2D< double > &	realiz,
		Array2D< double > &	xi
	)

Project realizations $F(t,\theta_l)$ to the KL modes and store them in xi ( $\xi_k$ )

Samples of random variables $\xi_k$ are obtained by projecting realizations of the random process on the eigenmodes

$\left.\xi_k\right\vert_{\theta_l}=\left <F(t,\theta_l)-\left < F(t,\theta) \right >_{\theta}, f_k(t) \right >_t/\sqrt{\lambda_k}$

... or numerically

$\left.\xi_k\right\vert_{\theta_l}=\sum_{i=1}^{N_p} w_i\left(F(t_i,\theta_l)-\left < F(t_i,\theta) \right >_{\theta} \right) f_k(t_i)/\sqrt{\lambda_k}$

◆ meanRealiz()

void KLDecompUni::meanRealiz	(	const Array2D< double > &	realiz,
		Array1D< double > &	mean_realiz
	)

Calculate (in meanRealiz) the mean realizations.

◆ SetWeights() [1/2]

void KLDecompUni::SetWeights ( const Array1D< double > & weights )

Set weights for computing the integral needed for Nystrom's method for solving the Fredholm integral equation.

◆ SetWeights() [2/2]

void KLDecompUni::SetWeights	(	const double *	weights,
		const int	npts
	)

Set weights for computing the integral needed for Nystrom's method for solving the Fredholm integral equation.

◆ truncRealiz()

void KLDecompUni::truncRealiz	(	const Array1D< double > &	meanrea,
		const Array2D< double > &	xi,
		const int &	nKL,
		Array2D< double > &	trunc_realiz
	)

Returns the truncated KL sum.

$F(t_i,\theta_l) = \left < F(t_i,\theta) \right >_{\theta} + \sum_{k=1}^{nKL} \sqrt{\lambda_k} f_k(t_i) \left. \xi_k\right\vert_{\theta_l}$

Member Data Documentation

◆ absTol_

double KLDecompUni::absTol_

private

Absolute tolerance for convergence.

◆ decomposed_

bool KLDecompUni::decomposed_

private

Flag to determine whether KL decomposition has taken place (and consequently that the interal data structures contain meaningful eigenvalues and vectors ... )

◆ eig_info_

int KLDecompUni::eig_info_

private

info on success of the eigenvector solutions

◆ eig_values_

Array1D<double> KLDecompUni::eig_values_

private

Array to store eigenvalues.

◆ eigRange_

char KLDecompUni::eigRange_

private

Option to set the type of range for eigenvalues.

◆ ifail_

Array1D<int> KLDecompUni::ifail_

private

Array to store indices of eigenvectors that failed to converge.

◆ il_

int KLDecompUni::il_

private

Lower index of range of eigenvalues requested.

◆ iu_

int KLDecompUni::iu_

private

Upper index of range of eigenvalues requested.

◆ jobz_

char KLDecompUni::jobz_

private

Option to determine what to compute (eigenvalues and eigenvectors)

◆ KL_modes_

Array2D<double> KLDecompUni::KL_modes_

private

Matrix to store KL modes.

◆ uplo_

char KLDecompUni::uplo_

private

Option to indicate how matrix is stored.

◆ vl_

double KLDecompUni::vl_

private

Lower bound for range of eigenvalues.

◆ vu_

double KLDecompUni::vu_

private

Upper bound for range of eigenvalues.

◆ w_

Array1D<double> KLDecompUni::w_

private

Array to hold weights for Nystrom's method for Fredholm integral equation solution.

◆ wh_

Array1D<double> KLDecompUni::wh_

private

Array to hold square roots of weights.

◆ whcwh_

Array2D<double> KLDecompUni::whcwh_

private

Matrix to hold the upper triangular part of the matrix to get eigenvalues of.

The documentation for this class was generated from the following files:

Public Member Functions

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ KLDecompUni() [1/3]

◆ KLDecompUni() [2/3]

◆ ~KLDecompUni()

◆ KLDecompUni() [3/3]

Member Function Documentation

◆ decompose() [1/2]

◆ decompose() [2/2]

◆ eigenvalues() [1/2]

◆ eigenvalues() [2/2]

◆ Init()

◆ KLmodes() [1/2]

◆ KLmodes() [2/2]

◆ KLproject()

◆ meanRealiz()

◆ SetWeights() [1/2]

◆ SetWeights() [2/2]

◆ truncRealiz()

Member Data Documentation

◆ absTol_

◆ decomposed_

◆ eig_info_

◆ eig_values_

◆ eigRange_

◆ ifail_

◆ il_

◆ iu_

◆ jobz_

◆ KL_modes_

◆ uplo_

◆ vl_

◆ vu_

◆ w_

◆ wh_

◆ whcwh_