Emrys::methods::SphericalSimplex< CenterWeight > Struct Template Reference

Spherical simplex method: \( L + 2 \) points over the joint input space of dimension \( L \), the mean with weight centerWeight \( W_0 \) and \( L + 1 \) points of weight \( (1 - W_0) / (L + 1) \) on a sphere of radius \( \sqrt{L / (1 - W_0)} \) about it, in the coordinates of a root of the joint covariance. More...

#include <EmrysFilter/Models/Methods.h>

Static Public Member Functions
static constexpr Eigen::Index	pointCount (Eigen::Index n)
template<typename Points, typename Weights>
static Eigen::Index	unitPoints (Eigen::Index n, Points &points, Weights &weights)
	Fills points, \( n \times (n + 2) \), with the center, then the \( n + 1 \) points of the recursion of Julier and Uhlmann (2004), whose coordinate \( j \) is \( -1 / \sqrt{j (j + 1) W_1} \) at points 1 to \( j \), \( j / \sqrt{j (j + 1) W_1} \) at point \( j + 1 \), and zero after, and weights with \( W_0 \) and \( W_1 = (1 - W_0) / (n + 1) \), over \( n > 0 \) coordinates.

Static Public Attributes
static constexpr double	centerWeight = CenterWeight
	CenterWeight

Detailed Description

template<double CenterWeight = 0.0>
struct Emrys::methods::SphericalSimplex< CenterWeight >

Spherical simplex method: \( L + 2 \) points over the joint input space of dimension \( L \), the mean with weight centerWeight \( W_0 \) and \( L + 1 \) points of weight \( (1 - W_0) / (L + 1) \) on a sphere of radius \( \sqrt{L / (1 - W_0)} \) about it, in the coordinates of a root of the joint covariance.

The fewest points that match a mean and a covariance, plus the center, for a function too costly to evaluate \( 2L + 1 \) times. The points are not symmetric about the mean, so odd moments of the input are not matched as the symmetric sets match them.

The spherical simplex set of Julier (2003), by the construction of Julier andUhlmann (2004), which places all but the center at one distance from it, \( \sqrt{L / (1 - W_0)} \). No free description of the construction has been read, and the original has not been. The output mean is a tangent-space mean about the center's output (decisions.md [sigma-point-sets]). Its parameter is a compile-time constant, as Unscented's are.

Template Parameters

CenterWeight

\( W_0 \), the center point's weight, in \( [0, 1) \)

Member Function Documentation

pointCount()

template<double CenterWeight = 0.0>

constexpr Eigen::Index Emrys::methods::SphericalSimplex< CenterWeight >::pointCount ( Eigen::Index n )

inlinestaticnodiscardconstexpr

Returns

the points over n coordinates, \( n + 2 \)

unitPoints()

template<double CenterWeight = 0.0>

template<typename Points, typename Weights>

Eigen::Index Emrys::methods::SphericalSimplex< CenterWeight >::unitPoints ( Eigen::Index n, Points & points, Weights & weights )

inlinestatic

Fills points, \( n \times (n + 2) \), with the center, then the \( n + 1 \) points of the recursion of Julier and Uhlmann (2004), whose coordinate \( j \) is \( -1 / \sqrt{j (j + 1) W_1} \) at points 1 to \( j \), \( j / \sqrt{j (j + 1) W_1} \) at point \( j + 1 \), and zero after, and weights with \( W_0 \) and \( W_1 = (1 - W_0) / (n + 1) \), over \( n > 0 \) coordinates.

Returns

the point the output mean is taken about, the center

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The documentation for this struct was generated from the following file:

• EmrysFilter/Models/Methods.h