#include <MaternKernel.h>

Inheritance diagram for muq::Approximation::MaternKernel:

Detailed Description

This class implements a kernel of the form

\[ k(x,y) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\frac{\sqrt{2\nu}}{l}\tau\right)^\nu K_\nu\left(\frac{\sqrt{2\nu}}{l}\tau\right), \]

where \(\Gamma(\cdot)\) is the gamma function, \(K_\nu(\cdot)\) is a modified Bessel function of the second kind, \(\nu\) is a smoothness parameter, \(l\) is a lengthscale, and \(\sigma^2\) is the variance. Note that we only allow values of \(\nu\) that take the form \(i-0.5\) for some positive integer \(i\). Typical choices are \(\nu=1/2\), \(\nu=3/2\), and \(\nu=5/2\). In the limit, \(\nu\rightarrow\infty\), this Matern kernel converges to the squared exponential kernel (muq::Approximation::SquaredExpKernel) and for \(\nu=1/2\), the Matern kernel is equivalent to the exponential kernel associated with the Ornstein-Uhlenbeck process.

Note: the smoothness parameter \(\nu\) is not optimized as a hyperparameter.

Definition at line 33 of file MaternKernel.h.

Public Member Functions
	MaternKernel (unsigned dimIn, std::vector< unsigned > dimInds, double sigma2In, double lengthIn, double nuIn)

	MaternKernel (unsigned dimIn, std::vector< unsigned > dimInds, double sigma2In, double lengthIn, double nuIn, Eigen::Vector2d sigmaBounds, Eigen::Vector2d lengthBounds)

	MaternKernel (unsigned dimIn, double sigma2In, double lengthIn, double nuIn)

	MaternKernel (unsigned dimIn, double sigma2In, double lengthIn, double nuIn, Eigen::Vector2d sigmaBounds, Eigen::Vector2d lengthBounds)

virtual	~MaternKernel ()

template<typename ScalarType1 , typename ScalarType2 , typename ScalarType3 >
void	FillBlockImpl (Eigen::Ref< const Eigen::Matrix< ScalarType1, Eigen::Dynamic, 1 >> const &x1, Eigen::Ref< const Eigen::Matrix< ScalarType1, Eigen::Dynamic, 1 >> const &x2, Eigen::Ref< const Eigen::Matrix< ScalarType2, Eigen::Dynamic, 1 >> const &params, Eigen::Ref< Eigen::Matrix< ScalarType3, Eigen::Dynamic, Eigen::Dynamic >> block) const

virtual std::tuple< std::shared_ptr< muq::Modeling::LinearSDE >, std::shared_ptr< muq::Modeling::LinearOperator >, Eigen::MatrixXd >	GetStateSpace (boost::property_tree::ptree sdeOptions=boost::property_tree::ptree()) const override
	Returns a state space representation of the covariance kernel. More...

Public Member Functions inherited from muq::Approximation::KernelImpl< MaternKernel >
	KernelImpl (unsigned inputDimIn, unsigned coDimIn, unsigned numParamsIn)

	KernelImpl (unsigned inputDimIn, std::vector< unsigned > dimIndsIn, unsigned coDimIn, unsigned numParamsIn)

virtual	~KernelImpl ()

virtual std::shared_ptr< KernelBase >	Clone () const override

virtual void	FillBlock (Eigen::Ref< const Eigen::VectorXd > const &x1, Eigen::Ref< const Eigen::VectorXd > const &x2, Eigen::Ref< const Eigen::VectorXd > const &params, Eigen::Ref< Eigen::MatrixXd > block) const override

virtual void	FillPosDerivBlock (Eigen::Ref< const Eigen::VectorXd > const &x1, Eigen::Ref< const Eigen::VectorXd > const &x2, Eigen::Ref< const Eigen::VectorXd > const &params, std::vector< int > const &wrts, Eigen::Ref< Eigen::MatrixXd > block) const override

void	FillPosDerivBlockImpl (Eigen::Ref< const Eigen::VectorXd > const &x1, Eigen::Ref< const Eigen::VectorXd > const &x2, Eigen::Ref< const Eigen::VectorXd > const &params, std::vector< int > const &wrts, Eigen::Ref< Eigen::MatrixXd > block) const

Eigen::Matrix< ScalarType, Eigen::Dynamic, 1 >	GetSegment (Eigen::Ref< const Eigen::Matrix< ScalarType, Eigen::Dynamic, 1 >> const &input) const

Public Member Functions inherited from muq::Approximation::KernelBase
	KernelBase (unsigned int inputDimIn, unsigned int coDimIn, unsigned int numParamsIn)

	KernelBase (unsigned int inputDimIn, std::vector< unsigned int > dimIndsIn, unsigned int coDimIn, unsigned int numParamsIn)

virtual	~KernelBase ()

virtual std::vector< std::shared_ptr< KernelBase > >	GetSeperableComponents ()
	Overridden by ProductKernel. More...

virtual Eigen::MatrixXd	Evaluate (Eigen::VectorXd const &x1, Eigen::VectorXd const &x2) const

virtual Eigen::MatrixXd	BuildCovariance (Eigen::MatrixXd const &x) const

virtual Eigen::MatrixXd	BuildCovariance (Eigen::MatrixXd const &x1, Eigen::MatrixXd const &x2) const

virtual void	FillCovariance (Eigen::MatrixXd const &xs, Eigen::MatrixXd const &ys, Eigen::Ref< Eigen::MatrixXd > cov) const

virtual void	FillCovariance (Eigen::MatrixXd const &xs, Eigen::Ref< Eigen::MatrixXd > cov) const

virtual void	FillDerivCovariance (Eigen::MatrixXd const &xs, Eigen::MatrixXd const &ys, std::vector< int > const &wrts, Eigen::Ref< Eigen::MatrixXd > cov) const

virtual Eigen::MatrixXd	GetPosDerivative (Eigen::VectorXd const &x1, Eigen::VectorXd const &x2, std::vector< int > const &wrts) const
	Returns derivatives of the kernel with respect to the first input, x1. More...

virtual Eigen::MatrixXd	GetParamBounds () const

virtual Eigen::VectorXd	GetParams () const

virtual void	SetParams (Eigen::VectorXd const &params)

Additional Inherited Members
Public Attributes inherited from muq::Approximation::KernelBase
const std::vector< unsigned int >	dimInds

const unsigned int	inputDim

const unsigned int	coDim

const unsigned int	numParams

Constructor & Destructor Documentation

◆ MaternKernel() [1/4]

muq::Approximation::MaternKernel::MaternKernel	(	unsigned	dimIn,
		std::vector< unsigned >	dimInds,
		double	sigma2In,
		double	lengthIn,
		double	nuIn
	)

inline

Definition at line 38 of file MaternKernel.h.

◆ MaternKernel() [2/4]

MaternKernel::MaternKernel	(	unsigned	dimIn,
		std::vector< unsigned >	dimInds,
		double	sigma2In,
		double	lengthIn,
		double	nuIn,
		Eigen::Vector2d	sigmaBounds,
		Eigen::Vector2d	lengthBounds
	)

Definition at line 23 of file MaternKernel.cpp.

References muq::Approximation::KernelBase::cachedParams, CheckNu(), and muq::Approximation::KernelBase::paramBounds.

◆ MaternKernel() [3/4]

muq::Approximation::MaternKernel::MaternKernel	(	unsigned	dimIn,
		double	sigma2In,
		double	lengthIn,
		double	nuIn
	)

inline

Definition at line 60 of file MaternKernel.h.

◆ MaternKernel() [4/4]

MaternKernel::MaternKernel	(	unsigned	dimIn,
		double	sigma2In,
		double	lengthIn,
		double	nuIn,
		Eigen::Vector2d	sigmaBounds,
		Eigen::Vector2d	lengthBounds
	)

Definition at line 48 of file MaternKernel.cpp.

References muq::Approximation::KernelBase::cachedParams, CheckNu(), and muq::Approximation::KernelBase::paramBounds.

◆ ~MaternKernel()

virtual muq::Approximation::MaternKernel::~MaternKernel ( )

inlinevirtual

Definition at line 79 of file MaternKernel.h.

Member Function Documentation

◆ BuildWeights()

Eigen::VectorXd MaternKernel::BuildWeights ( int p )

staticprivate

Definition at line 136 of file MaternKernel.cpp.

References Factorial().

◆ CheckNu()

void MaternKernel::CheckNu ( ) const

private

Definition at line 73 of file MaternKernel.cpp.

References nu.

Referenced by MaternKernel().

◆ Factorial()

static int muq::Approximation::MaternKernel::Factorial ( int n )

inlinestaticprivate

Definition at line 112 of file MaternKernel.h.

References nlohmann::detail::dtoa_impl::n.

Referenced by BuildWeights().

◆ FillBlockImpl()

template<typename ScalarType1 , typename ScalarType2 , typename ScalarType3 >

void muq::Approximation::MaternKernel::FillBlockImpl	(	Eigen::Ref< const Eigen::Matrix< ScalarType1, Eigen::Dynamic, 1 >> const &	x1,
		Eigen::Ref< const Eigen::Matrix< ScalarType1, Eigen::Dynamic, 1 >> const &	x2,
		Eigen::Ref< const Eigen::Matrix< ScalarType2, Eigen::Dynamic, 1 >> const &	params,
		Eigen::Ref< Eigen::Matrix< ScalarType3, Eigen::Dynamic, Eigen::Dynamic >>	block
	)		const

inline

Definition at line 83 of file MaternKernel.h.

References nu, scale, and weights.

◆ GetStateSpace()

std::tuple< std::shared_ptr< muq::Modeling::LinearSDE >, std::shared_ptr< muq::Modeling::LinearOperator >, Eigen::MatrixXd > MaternKernel::GetStateSpace ( boost::property_tree::ptree sdeOptions = boost::property_tree::ptree() ) const

overridevirtual

Returns a state space representation of the covariance kernel.

If this is a one dimensional kernel (i.e., inputDim=1 and coDim=1), this function returns a state space representation of the covariance kernel. In particular, it returns a linear time invariant stochastic differential equation, whose solution, when started with the returned stationary covariance, provides the same information as this Gaussian process. The first component of the vector-valued stochastic differential equation is related to the Gaussian process. See "Kalman filtering and smoothing solutions to temporal Gaussian process regression models," by Jouni Hartikainen and Simo Sarkka, for more information.

Reimplemented from muq::Approximation::KernelBase.

Definition at line 84 of file MaternKernel.cpp.

References muq::Approximation::KernelBase::cachedParams, muq::Modeling::LyapunovSolver< ScalarType, FixedRows, FixedCols >::compute(), muq::Modeling::LyapunovSolver< ScalarType, FixedRows, FixedCols >::matrixX(), and nu.