# lcvt

lcvt, a FORTRAN90 code which creates Latin Centroidal Voronoi Tessellation (CVT) datasets.

A Latin Square dataset is typically a two dimensional dataset of N points in the unit square, with the property that, if both the x and y axes are divided up into N equal subintervals, exactly one dataset point has an x or y coordinate in each subinterval. Latin squares can easily be extended to the case of M dimensions, and may be pedantically called Latin Hypersquares or Latin Hypercubes in such a case. Statisticians like Latin Squares, as do experiment designers, and and people who need to approximate scalar functions of many variables.

The fact that the projection of a Latin Square dataset onto any coordinate axis is either exactly evenly spaced, or approximately so (depending on the algorithm), turns out to be an attractive feature for many uses.

However, a CVT dataset in a regular domain, such as the unit hypercube, has the tendency for the projections of the points to cluster together in any coordinate axis. This program is mainly an attempt to explore whether a dataset can be computed using techniques similar to those of a CVT, but with the constraint (whether imposed or expected) that the point projections do not clump up.

The approach used here is quite simple. First we compute a CVT in M dimensions, comprising N points. We assume that the bounding region is the unit hypercube. We are now going to adjust the coordinates of the points to achieve the Latin Hypercube property. For each coordinate direction, we simply sort the points by that coordinate, and then overwrite the original values by the values we'd expect to get for a centered Latin Hypercube, namely, 1/(2*N), 3/(2*N), ..., (2*N-1)/(2*N).

Now this process guarantees that we get a Latin Hypercube. Our hope is that the process of adjusting the point coordinates does not too severely damage the nice dispersion properties inherent in the CVT point placement.

### Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

### Languages:

lcvt is available in a C++ version and a FORTRAN90 version and a MATLAB version

### Related Data and Programs:

CVT, a FORTRAN90 code which can compute a Centroidal Voronoi Tessellation.

FAURE, a FORTRAN90 code which can compute elements of a Faure quasirandom sequence.

HALTON, a FORTRAN90 code which can compute elements of a Halton quasirandom sequence.

HAMMERSLEY, a FORTRAN90 code which can compute elements of a Hammersley quasirandom sequence.

LATINIZE, a FORTRAN90 code which can be used to "latinize" a dataset.

LATTICE_RULE, a FORTRAN90 code which approximates multidimensional integrals using lattice rules.

NIEDERREITER2, a FORTRAN90 code which computes elements of a Niederreiter quasirandom sequence with base 2.

NORMAL, a FORTRAN90 code which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a FORTRAN90 code which can compute elements of a Sobol quasirandom sequence.

UNIFORM, a FORTRAN90 code which can compute elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a FORTRAN90 code which can compute elements of a van der Corput quasirandom sequence.

### Reference:

1. Franz Aurenhammer,
Voronoi diagrams - a study of a fundamental geometric data structure,
ACM Computing Surveys,
Volume 23, Number 3, September 1991, pages 345-405.
2. Franz Aurenhammer, Rolf Klein,
Voronoi Diagrams,
in Handbook of Computational Geometry,
edited by J Sack, J Urrutia,
Elsevier, 1999,
LC: QA448.D38H36.
3. John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,
User Manual and Supporting Information for Library of Codes for Centroidal Voronoi Placement and Associated Zeroth, First, and Second Moment Determination,
Sandia National Laboratories Technical Report SAND2002-0099,
February 2002.
4. Qiang Du, Vance Faber, Max Gunzburger,
Centroidal Voronoi Tessellations: Applications and Algorithms,
SIAM Review,
Volume 41, Number 4, December 1999, pages 637-676.
5. Michael McKay, William Conover, Richard Beckman,
A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code,
Technometrics,
Volume 21, 1979, pages 239-245.
6. Vicente Romero, John Burkardt, Max Gunzburger, Janet Peterson,
Initial Evaluation of Pure and "Latinized" Centroidal Voronoi Tessellation for Non-Uniform Statistical Sampling,
Sensitivity Analysis of Model Output (SAMO 2004) Conference, Santa Fe, March 8-11, 2004.
7. Yuki Saka, Max Gunzburger, John Burkardt,
Latinized, improved LHS, and CVT point sets in hypercubes,
International Journal of Numerical Analysis and Modeling,
Volume 4, Number 3-4, 2007, pages 729-743.

### Source Code:

Last revised on 25 July 2020.