Description

Point weights are nonnegative and normalized independently to mass one. A point set paired with a function is scaled to the function's integral, and keeps mass one if that integral vanishes. Antipodal points split mass equally between both signs. See S2Fun/discrepancy for all options and normalization conventions. The metrics are

* D_2 - the standard root cap L2 discrepancy, evaluated without truncating
point measures and counting a difference in total mass
* L2 - the root bandlimited norm
* D_cap - a finite cap-search approximation; info.isExact is false
* kernel - the historical squared truncated optimization objective, which
takes two functions of equal integral only

Syntax

d = discrepancy(v,w,'metric','D_2')

Between the point sets v and w.

[d,info] = discrepancy(v,w,'metric','D_cap','numCenters',4097)

Cap discrepancy over 4097 centres.

d = discrepancy(v,w,'metric','L2','bandwidth',32)

Bandlimited norm up to degree 32.

d = discrepancy(v,w,'weights',c,'weights2',d,'metric','D_2')

Weighted point sets.

d = discrepancy(v,f,'weights',c,'metric','D_2')

Between a point set and a density f.

d = discrepancy(v,w,'squared')

The squared discrepancy.

Input

`v`	`vector3d`
`w`	`vector3d`
`f`	`S2Fun`

Options

`metric`	`'D_2' (default), 'L2', 'D_cap' or 'kernel'`
`weights`	`weights of v`
`weights2`	`weights of w`
`bandwidth`	`harmonic degree of L2 and kernel`
`numCenters`	`number of cap centres of D_cap`

Flags

squared return the squared discrepancy

Output

`d`	`double`
`info`	`struct, info.isExact says whether d is exact`

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/vector3d.discrepancy.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX

@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.

vector3d.discrepancy