cmp::density::KDE Class Reference

Multivariate Kernel Density Estimation (KDE) class. More...

#include <kde.h>

Public Types
enum class	BandWidthSelectionMethod { SCOTT , SILVERMAN }

Public Member Functions
	KDE ()
	Default constructor for KDE. Creates a KDE with identity bandwidth and Gaussian kernel in 1D.
	KDE (std::shared_ptr< kernel::Bandwidth > bandwidth, std::shared_ptr< kernel::Kernel > kernel)
	Constructor for KDE with specified bandwidth and kernel.
	KDE (const KDE &other)
	KDE (KDE &&other) noexcept
KDE &	operator= (const KDE &other)
KDE &	operator= (KDE &&other) noexcept
void	set (const std::shared_ptr< kernel::Bandwidth > &bandwidth, const std::shared_ptr< kernel::Kernel > &kernel)
	Set the bandwidth and kernel for the KDE.
void	condition (const Eigen::Ref< const Eigen::MatrixXd > &data, bool copyData=false)
	Condition the KDE on the provided data.
double	eval (const Eigen::VectorXd &x) const
	Evaluate the KDE at a given point.

Private Attributes
std::shared_ptr< kernel::Bandwidth >	bandwidth_ {nullptr}
	Bandwidth matrix of the kernel.
std::shared_ptr< kernel::Kernel >	kernel_ {nullptr}
	The density kernel function.
Eigen::MatrixXd	data_ {0, 0}
	Owning data storage.
std::optional< Eigen::Ref< const Eigen::MatrixXd > >	pData_
	Non-owning reference wrapper to data.
bool	isOwning_ {false}
	Flag indicating whether the KDE owns the data.
size_t	dim_ {0}
	Dimension of the dataset.
size_t	nPoints_ {0}
	Number of data points.

Detailed Description

Multivariate Kernel Density Estimation (KDE) class.

Mathematical Formulation Evaluates the multivariate probability density function estimate of a random vector \(\mathbf{x} \in \mathbb{R}^D\) based on observations \(\{\mathbf{x}_i\}_{i=1}^N\):

\[ \hat{f}(\mathbf{x}) = \frac{1}{N} \sum_{i=1}^N \frac{1}{\det(\mathbf{H})} K\left( \mathbf{H}^{-1} (\mathbf{x} - \mathbf{x}_i) \right) \]

where \(\mathbf{H}\) is the bandwidth matrix and \(K\) is the multi-dimensional kernel function.

• Scott's Rule: \(H_{dd} = N^{-1/(D+4)} \hat{\sigma}_d\)

• Silverman's Rule: \(H_{dd} = \left(\frac{4}{D+2}\right)^{1/(D+4)} N^{-1/(D+4)} \hat{\sigma}_d\) where \(\hat{\sigma}_d\) is the standard deviation of coordinate \(d\).

  Implementation Algorithm

1. condition() registers the dataset references.
2. eval() averages kernel evaluations over all conditioning observation points, scaled by the determinant of the bandwidth matrix.

Member Enumeration Documentation

BandWidthSelectionMethod

enum class cmp::density::KDE::BandWidthSelectionMethod

strong

Enumerator
SCOTT
SILVERMAN

Constructor & Destructor Documentation

KDE() [1/4]

cmp::density::KDE::KDE

(

)

Default constructor for KDE. Creates a KDE with identity bandwidth and Gaussian kernel in 1D.

KDE() [2/4]

cmp::density::KDE::KDE	(	std::shared_ptr< kernel::Bandwidth >	bandwidth,
		std::shared_ptr< kernel::Kernel >	kernel
	)

Constructor for KDE with specified bandwidth and kernel.

Parameters

bandwidth	Shared pointer to the bandwidth object.
kernel	Shared pointer to the kernel object.

KDE() [3/4]

cmp::density::KDE::KDE

(

const KDE &

other

)

KDE() [4/4]

cmp::density::KDE::KDE ( KDE &&  other )

noexcept

Member Function Documentation

condition()

void cmp::density::KDE::condition	(	const Eigen::Ref< const Eigen::MatrixXd > &	data,
		bool	copyData = false
	)

Condition the KDE on the provided data.

Parameters

data	The data matrix where each row is a data point.
copyData	If true, the data is copied into the KDE object; if false, a reference is kept.

eval()

double cmp::density::KDE::eval

(

const Eigen::VectorXd &

x

)

const

Evaluate the KDE at a given point.

Parameters

x	The point at which to evaluate the density estimate.

Returns

The estimated density at point x.

operator=() [1/2]

cmp::density::KDE & cmp::density::KDE::operator=

(

const KDE &

other

)

operator=() [2/2]

cmp::density::KDE & cmp::density::KDE::operator= ( KDE &&  other )

noexcept

set()

void cmp::density::KDE::set	(	const std::shared_ptr< kernel::Bandwidth > &	bandwidth,
		const std::shared_ptr< kernel::Kernel > &	kernel
	)

Set the bandwidth and kernel for the KDE.

Parameters

bandwidth	Shared pointer to the bandwidth object.
kernel	Shared pointer to the kernel object.

Member Data Documentation

bandwidth_

std::shared_ptr<kernel::Bandwidth> cmp::density::KDE::bandwidth_ {nullptr}

private

Bandwidth matrix of the kernel.

data_

Eigen::MatrixXd cmp::density::KDE::data_ {0, 0}

private

Owning data storage.

dim_

size_t cmp::density::KDE::dim_ {0}

private

Dimension of the dataset.

isOwning_

bool cmp::density::KDE::isOwning_ {false}

private

Flag indicating whether the KDE owns the data.

kernel_

std::shared_ptr<kernel::Kernel> cmp::density::KDE::kernel_ {nullptr}

private

The density kernel function.

nPoints_

size_t cmp::density::KDE::nPoints_ {0}

private

Number of data points.

pData_

std::optional<Eigen::Ref<const Eigen::MatrixXd> > cmp::density::KDE::pData_

private

Non-owning reference wrapper to data.

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

• include/kde.h
• src/kde.cpp