Functions
template<typename DerivedA, typename DerivedQ, typename DerivedL>
double	calcChi2_bootstrap (const Eigen::MatrixBase< DerivedA > &a, const Eigen::MatrixBase< DerivedQ > &Q, const Eigen::MatrixBase< DerivedL > &L_LTDL_Q, const Eigen::Index &ncand=2)
	Calculates $\chi^2$ , the size of the ellipsoidal region, via bootrapping.
template<typename DerivedD>
double	calcChi2_volume (const Eigen::MatrixBase< DerivedD > &D, const Eigen::Index &ncand=2, double factor=1.5)
	Calculates $\chi^2$ , the size of the ellipsoidal region, via volume of the ellipsoidal region.
template<typename DerivedL, typename DerivedZ, typename DerivedA>
void	gauss (Eigen::MatrixBase< DerivedL > &L, Eigen::Index i, Eigen::Index j, Eigen::MatrixBase< DerivedA > &a, Eigen::MatrixBase< DerivedZ > &Z, Eigen::Index n)
	[11] Chang 2005, Integer Gauss Transformations algorithm
template<typename DerivedL, typename DerivedD, typename DerivedA, typename DerivedZ>
void	permute (Eigen::MatrixBase< DerivedL > &L, Eigen::MatrixBase< DerivedD > &D, Eigen::Index k, double delta, Eigen::MatrixBase< DerivedA > &a, Eigen::MatrixBase< DerivedZ > &Z, Eigen::Index n)
	[11] Chang 2005, Permutations algorithm

Function Documentation

◆ calcChi2_bootstrap()

template<typename DerivedA, typename DerivedQ, typename DerivedL>

double NAV::Ambiguity::internal::calcChi2_bootstrap	(	const Eigen::MatrixBase< DerivedA > &	a,
		const Eigen::MatrixBase< DerivedQ > &	Q,
		const Eigen::MatrixBase< DerivedL > &	L_LTDL_Q,
		const Eigen::Index &	ncand = 2 )

Calculates $\chi^2$ , the size of the ellipsoidal region, via bootrapping.

$\boldsymbol{\check{z}}_B$ is a good candidate for setting the size of the search space ([48] Springer Handbook GNSS, ch. 23.4.2, eq. 23.58)

$\begin{equation} \label{eq:eq-GNSS-chi2-bootstrap} \chi^2 = || \boldsymbol{\hat{z}} - \boldsymbol{\check{z}}_B ||_{\mathbf{Q}_{z}}^2 = (\boldsymbol{\hat{z}} - \boldsymbol{\check{z}}_B)^T \mathbf{Q}_{z}^{-1} (\boldsymbol{\hat{z}} - \boldsymbol{\check{z}}_B) \end{equation}$

Parameters

[in]	a	Float ambiguity vector [cycles]
[in]	Q	Variance/covariance matrix of the ambiguities
[in]	L_LTDL_Q	Lower-triangular matrix from the L^T * D * L decomposition of Q_z
[in]	ncand	Requested number of candidates (default = 2)

Returns: Size of the ellipsoidal region $\chi^2$

Note: See [13] de Jonge 1996, ch. 4.11

Definition at line 80 of file EllipsoidalRegion.hpp.

◆ calcChi2_volume()

template<typename DerivedD>

double NAV::Ambiguity::internal::calcChi2_volume	(	const Eigen::MatrixBase< DerivedD > &	D,
		const Eigen::Index &	ncand = 2,
		double	factor = 1.5 )

Calculates $\chi^2$ , the size of the ellipsoidal region, via volume of the ellipsoidal region.

The volume, expressed in $[\text{cycles}^n]$ , of the ellipsoidal region is ([13] de Jonge 1996, ch. 4.9, eq. 4.19)

$\begin{equation} \label{eq:eq-GNSS-chi2-volume} E_n = \chi^n \sqrt{|Q_{\hat{a}}|} V_n \end{equation}$

The volume function is ([13] de Jonge 1996, ch. 4.9, eq. 4.20)

$\begin{equation} \label{eq:eq-GNSS-chi2-volume-function} V_n = \frac{2}{n} \frac{\pi^{\frac{n}{2}}}{\Gamma \left( \frac{n}{2} \right)} \end{equation}$

The determinant of the vairance covariance matrix is ([13] de Jonge 1996, ch. 4.9, eq. 4.24)

$\begin{equation} \label{eq:eq-GNSS-chi2-volume-detQ} |Q_{\hat{a}}| = \prod_{i=1}^{n} \sigma^2_{\hat{a}_{i|i+1, \dots, n}} \end{equation}$

The volume is good indicator for the number of candidates ([13] de Jonge 1996, ch. 4.10)

$\begin{equation} \label{eq:eq-GNSS-chi2-volume-cand} n_{\text{cand}} = \text{(int)}E_n \end{equation}$

Parameters

[in]	D	Vector containing all the variances of the ambiguities (L^T * D * L decomposition)
[in]	ncand	Requested number of candidates (default = 2)
[in]	factor	Multiplication factor for the volume of the resulting search ellipsoid (default = 1.5)

Returns: Size of the ellipsoidal region $\chi^2$

Definition at line 53 of file EllipsoidalRegion.hpp.

◆ gauss()

template<typename DerivedL, typename DerivedZ, typename DerivedA>

void NAV::Ambiguity::internal::gauss	(	Eigen::MatrixBase< DerivedL > &	L,
		Eigen::Index	i,
		Eigen::Index	j,
		Eigen::MatrixBase< DerivedA > &	a,
		Eigen::MatrixBase< DerivedZ > &	Z,
		Eigen::Index	n )

[11] Chang 2005, Integer Gauss Transformations algorithm

Parameters

[in,out]	L	(L^T * D * L) decomposition of Q_z
[in]	i	Row index
[in]	j	Col index
[in,out]	a	Float ambiguity vector [cycles]
[in,out]	Z	Decorrelation transformation matrix
[in]	n	Dimension

Definition at line 40 of file Decorrelation.hpp.

◆ permute()

template<typename DerivedL, typename DerivedD, typename DerivedA, typename DerivedZ>

void NAV::Ambiguity::internal::permute	(	Eigen::MatrixBase< DerivedL > &	L,
		Eigen::MatrixBase< DerivedD > &	D,
		Eigen::Index	k,
		double	delta,
		Eigen::MatrixBase< DerivedA > &	a,
		Eigen::MatrixBase< DerivedZ > &	Z,
		Eigen::Index	n )

[11] Chang 2005, Permutations algorithm

Parameters

[in,out]	L	(L^T * D * L) decomposition of Q_z
[in,out]	D	(L^T * D * L) decomposition of Q_z
[in]	k	Index
[in]	delta	Delta parameter
[in,out]	a	Float ambiguity vector [cycles]
[in,out]	Z	Z trafo
[in]	n	Dimension

Definition at line 62 of file Decorrelation.hpp.

Functions

Function Documentation

◆ calcChi2_bootstrap()

◆ calcChi2_volume()

◆ gauss()

◆ permute()