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<html>
<head>
<title>
LCVT - Latin Centroidal Voronoi Tessellations
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LCVT <br> Latin Centroidal Voronoi Tessellations
</h1>
<hr>
<p>
<b>LCVT</b>
is a MATLAB library which
creates Latin Centroidal Voronoi Tessellation (CVT) datasets.
</p>
<p>
A Latin Square dataset is typically a two dimensional dataset
of <b>N</b> points in the unit square, with the property that, if both the
<b>x</b> and <b>y</b> axes are divided up into <b>N</b> equal subintervals,
exactly one dataset point has an <b>x</b> or <b>y</b> coordinate in
each subinterval. Latin squares can easily be extended to the
case of <b>M</b> dimensions, and may be pedantically called <i>Latin
Hypersquares</i> or <i>Latin Hypercubes</i> in such a case.
Statisticians like Latin Squares, as
do experiment designers, and and people who need to approximate
scalar functions of many variables.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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).
</p>
<p>
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.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>LCVT</b> is available in
<a href = "../../cpp_src/lcvt/lcvt.html">a C++ version</a> and
<a href = "../../f_src/lcvt/lcvt.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lcvt/lcvt.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cvt/cvt.html">
CVT</a>,
a MATLAB library which
computes a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../m_src/faure/faure.html">
FAURE</a>,
a MATLAB library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../m_src/grid/grid.html">
GRID</a>,
a MATLAB library which
computes elements of a grid dataset.
</p>
<p>
<a href = "../../m_src/halton/halton.html">
HALTON</a>,
a MATLAB library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../m_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a MATLAB library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../m_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a MATLAB library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../f_src/hex_grid_angle/hex_grid_angle.html">
HEX_GRID_ANGLE</a>,
a FORTRAN90 library which
computes elements of an angled hexagonal grid dataset.
</p>
<p>
<a href = "../../m_src/ihs/ihs.html">
IHS</a>,
a MATLAB library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../m_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../m_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../m_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing
points at random.
</p>
<p>
<a href = "../../m_src/latinize/latinize.html">
LATINIZE</a>,
a MATLAB program which
can "latinize" a dataset.
</p>
<p>
<a href = "../../m_src/lattice_rule/lattice_rule.html">
LATTICE_RULE</a>,
a MATLAB library which
approximates multidimensional integrals using lattice rules.
</p>
<p>
<a href = "../../datasets/lcvt/lcvt.html">
LCVT_DATASET</a>,
a directory which
contains a collection of
sample LCVT datasets.
</p>
<p>
<a href = "../../m_src/lcvt_dataset/lcvt_dataset.html">
LCVT_DATASET</a>,
a MATLAB program which
can create an LCVT dataset.
</p>
<p>
<a href = "../../m_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a MATLAB library which
computes elements of a Niederreiter quasirandom sequence with base 2.
</p>
<p>
<a href = "../../m_src/normal/normal.html">
NORMAL</a>,
a MATLAB library which
computes elements of a normal pseudorandom sequence.
</p>
<p>
<a href = "../../m_src/sobol/sobol.html">
SOBOL</a>,
a MATLAB library which
computes elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../m_src/uniform/uniform.html">
UNIFORM</a>,
a MATLAB library which
computes elements of a uniform pseudorandom sequence.
</p>
<p>
<a href = "../../m_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a MATLAB library which
computes elements of a van der Corput quasirandom sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Franz Aurenhammer,<br>
Voronoi diagrams -
a study of a fundamental geometric data structure,<br>
ACM Computing Surveys,<br>
Volume 23, Number 3, September 1991, pages 345-405.
</li>
<li>
Franz Aurenhammer, Rolf Klein,<br>
Voronoi Diagrams,<br>
in Handbook of Computational Geometry,<br>
edited by J Sack, J Urrutia,<br>
Elsevier, 1999,<br>
LC: QA448.D38H36.
</li>
<li>
John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,<br>
User Manual and Supporting Information for Library of Codes
for Centroidal Voronoi Placement and Associated Zeroth,
First, and Second Moment Determination,<br>
Sandia National Laboratories Technical Report SAND2002-0099,<br>
February 2002,<br>
<a href = "../../publications/bgpb_2002.pdf">
../../publications/bgpb_2002.pdf </a>
</li>
<li>
Qiang Du, Vance Faber, Max Gunzburger,<br>
Centroidal Voronoi Tessellations: Applications and Algorithms,<br>
SIAM Review,<br>
Volume 41, Number 4, December 1999, pages 637-676.
</li>
<li>
Michael McKay, William Conover, Richard Beckman,<br>
A Comparison of Three Methods for Selecting Values of Input
Variables in the Analysis of Output From a Computer Code,<br>
Technometrics,<br>
Volume 21, 1979, pages 239-245.
</li>
<li>
Vicente Romero, John Burkardt, Max Gunzburger, Janet Peterson, <br>
Initial Evaluation of Pure and "Latinized" Centroidal Voronoi
Tessellation for Non-Uniform Statistical Sampling,<br>
Sensitivity Analysis of Model Output (SAMO 2004) Conference,
Santa Fe, March 8-11, 2004,<br>
<a href = "../../publications/rbgp_2004.pdf">rbgp_2004.pdf</a>.
</li>
<li>
Yuki Saka, Max Gunzburger, John Burkardt, <br>
Latinized, improved LHS, and CVT point sets in hypercubes, <br>
submitted to IEEE Transactions on Information Theory,<br>
<a href = "../../publications/sgb_submitted.pdf">sgb_submitted.pdf</a>.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ch_cap.m">ch_cap.m</a>
capitalizes a single character.
</li>
<li>
<a href = "ch_eqi.m">ch_eqi.m</a>
is TRUE if two characters are equal, ignoring case.
</li>
<li>
<a href = "ch_to_digit.m">ch_to_digit.m</a>
returns the integer value of a base 10 digit.
</li>
<li>
<a href = "cluster_energy.m">cluster_energy.m</a>
returns the energy of a dataset.
</li>
<li>
<a href = "cvt.m">cvt.m</a>
computes a Centroidal Voronoi Tessellation.
</li>
<li>
<a href = "cvt_iteration.m">cvt_iteration.m</a>
takes one step of the CVT iteration.
</li>
<li>
<a href = "cvt_write.m">cvt_write.m</a>
writes a CVT dataset to a file.
</li>
<li>
<a href = "file_column_count.m">file_column_count.m</a>
counts the number of columns in the first line of a file.
</li>
<li>
<a href = "file_row_count.m">file_rows_count.m</a>
counts the number of rows in a file.
</li>
<li>
<a href = "find_closest.m">find_closest.m</a>
finds the Voronoi cell generator closest to a point X.
</li>
<li>
<a href = "get_seed.m">get_seed.m</a>
returns a seed for the random number generator.
</li>
<li>
<a href = "i4_to_halton.m">i4_to_halton.m</a>
computes an element of a vector Halton sequence.
</li>
<li>
<a href = "lcvt_write.m">lcvt_write.m</a>
writes a LCVT dataset to a file.
</li>
<li>
<a href = "param_print.m">param_print.m</a>
prints the program parameters.
</li>
<li>
<a href = "prime.m">prime.m</a>
returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<a href = "r8_uniform_01.m">r8_uniform_01.m</a>
is a portable pseudorandom number generator.
</li>
<li>
<a href = "r8mat_latinize.m">r8mat_latinize.m</a>
"Latinizes" an R8MAT.
</li>
<li>
<a href = "r8mat_transpose_print.m">r8mat_transpose_print.m</a>
prints an R8MAT, transposed, with an optional title.
</li>
<li>
<a href = "r8at_transpose_print_some.m">r8mat_transpose_print_some.m</a>
prints some of an R8MAT, transpose, with an optional title.
</li>
<li>
<a href = "r8vec_sort_heap_index_a.m">r8vec_sort_heap_index_a.m</a>
does an indexed heap ascending sort of a real vector.
</li>
<li>
<a href = "region_sampler.m">region_sampler.m</a>
returns a sample point in the physical region.
</li>
<li>
<a href = "s_cap.m">s_cap.m</a>
replaces any lowercase letters by uppercase ones in a string.
</li>
<li>
<a href = "s_eqi.m">s_eqi.m</a>
is TRUE if two strings are equal, ignoring case.
</li>
<li>
<a href = "s_len_trim.m">s_len_trim.m</a>
returns the length of a string to the last nonblank.
</li>
<li>
<a href = "s_to_i4.m">s_to_i4.m</a>
reads an I4 from a string.
</li>
<li>
<a href = "s_to_r8.m">s_to_r8.m</a>
reads an R8 from a string.
</li>
<li>
<a href = "s_to_r8vec.m">s_to_r8vec.m</a>
reads an R8VEC from a string.
</li>
<li>
<a href = "s_word_count.m">s_word_count.m</a>
returns the number of words in a string.
</li>
<li>
<a href = "test_region.m">test_region.m</a>
determines if a point is within the physical region.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a timestamp.
</li>
<li>
<a href = "tuple_next_fast.m">tuple_next_fast.m</a>
computes the next element of a tuple space, "fast".
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lcvt_test.m">lcvt_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "lcvt_test01.m">lcvt_test01.m</a>,
repeats a simple calculation using the different CVT sampling options.
</li>
<li>
<a href = "lcvt_test02.m">lcvt_test02.m</a>,
carries out a simple calculation in which the generators are
initialized to gridpoints.
</li>
<li>
<a href = "lcvt_test_output.txt">lcvt_test_output.txt</a>,
the output from a run of the sample program.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 30 May 2007.
</i>
<!-- John Burkardt -->
</body>
</html>