tms::analysis Namespace Reference

Contains functions for digital nets analyses, like figures of merit and various statistics. More...

Functions
Figures of merit
BasicInt	t (DigitalNet const &net)
Statistics
tms::Point	scatter_defect (DigitalNet const &net)

Detailed Description

Contains functions for digital nets analyses, like figures of merit and various statistics.

The precision of Quasi Monte Carlo calculations is directly related to the discrepancy of the digital net (e.g., search for "Koksma–Hlawka inequality"). Discrepancy itself is computationally hard to be calculated directly which makes researchers introduce different metrics conventionally called figures of merit that help one estimate the true discrepancy.

Statistics features that are also introduced here might be useful to verify some basic properties that any "good" digital net should obtain.

Function Documentation

t()

BasicInt tms::analysis::t

(

DigitalNet const &

net

)

Calculates the precise value of \(t\)

Finds out the least value of \(t\) for which the given digital net is a \((t, m, s)\)-net in base \(2\) (we remind that values of \(m\) and \(s\) are known at the moment of net construction).

Parameters

net	A digital net.

Returns

Precise value of parameter \(t\) for the given digital net.

Exceptions

invalid_argument

If at least one generating matrix of the given net is degenerate (determinant equals zero over \(\mathbb{F}_2\)).

Note

Exceptions will never be encountered in this function if it used with Niederreiter and Sobol nets provided in this library. Their generating matrices are guaranteed to always be non-degenerate. If you discover a case when this function fails to perform on these nets, report a bug.

References

1. Marion P., Godin M., L'Ecuyer P. (2020) An algorithm to compute the t-value of a digital net and of its projections. Journal of Computational and Applied Mathematics. 371. 112669. https://doi.org/10.1016/j.cam.2019.112669

scatter_defect()

tms::Point tms::analysis::scatter_defect

(

DigitalNet const &

net

)

Calculates the scatter defect

Performs a special version of model-based principal component analysis that was specially optimised for digital \((t, m, s)\)-nets. Here, having a singular value decomposition \(C = U \Sigma V^*\) for a covariance matrix \(C\), a scatter is defined as a sum of squares of all singular values, relative influence of the \(i\)-th principal axis is defined as a ratio of the squared \(i\)-th singular value to the scatter, and a scatter defect of the \(i\)-th principal axis is defined as a difference between its relative influence and the value of \(s^{-1}\).

If points of the given digital net saturate the \(s\)-dimensional unit cube perfectly equally in all directions, the scatter defect will be \(\overrightarrow{0} \in \mathbb{R}^s\). If points are scatterred along the \(i\)-th principal axis more than it is expected under the perfectly equal saturation, then the \(i\)-th component of defect vector will be positive, otherwise, negative.

Parameters

net	A digital net.

Returns

A tms::Point the \(i\)-th component of which equals the scatter defect along the \(i\)-th principal axis.

Note

Principal axes always form an orthonormal basis in the \(s\)-dimensional space, however, there are no guarantees that they will match with the basis that is used to express the coordinates of digital net points.

References

1. Eliseev A. (2020) Properties of (t, s)-sequences and their verification. Bachelor's graduation work (in Russian).