selectHalfwidth edit page

halfwidth of a de la Vallee Poussin kernel for a density estimate of orientations or directions

Description

Both methods judge a candidate halfwidth by the mean integrated squared error of the density estimate. In harmonics orthonormal for the uniform distribution it is

\[ \mathrm{MISE} = \sum_{\ell \ge 1} (1 - b_\ell)^2 A_\ell + b_\ell^2 V_\ell, \]

with \(b_\ell\) the multipliers of the kernel, \(A_\ell\) the degree energies of the density and \(V_\ell\) the variance of the empirical coefficients of degree \(\ell\). 'UCV' inserts unbiased estimates of the energies, 'conservative' lower confidence bounds that vanish above the bandwidth, which can only widen the kernel. Both are bounded below by a noise floor: the smallest halfwidth at which the sampling noise stays below a fraction tauMax of the estimate's maximum plus half the local density.

Syntax

hw = selectHalfwidth(ori,'UCV')
hw = selectHalfwidth(v,'conservative')
hw = selectHalfwidth(ori,'UCV','groups',grainId)

Input

`ori`	`orientation`
`v`	`vector3d, Miller`

Output

`hw`	`halfwidth`
`E`	`struct with the degree energies \|E.A\|, their errors \|E.se\| and the sample statistics`

Options

`weights`	`weights of the points`
`groups`	`ids of mutually dependent points, e.g. the grainId of EBSD pixels`
`bandwidth`	`harmonic degree of the energies, default 64 (UCV of orientations), 256 (UCV of directions), 32 (conservative)`
`halfwidths`	`candidate halfwidths, default 60 between 2° and 40° (orientations) or between 1° and 60° (directions)`
`folds`	`number of disjoint parts the energies are cross-fitted over (default 8)`
`z`	`multiple of the error the conservative energies are lowered by (default 2)`
`tauMax`	`noise floor as a fraction of the estimate's maximum (default 0.2)`

Flags

noNoiseFloor do not bound the halfwidth from below by the noise

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/selectHalfwidth.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.

selectHalfwidth