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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 C++ 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 = "../../cpp_src/box_behnken/box_behnken.html">
BOX_BEHNKEN</a>,
a C++ library which
computes a Box-Behnken design,
that is, a set of arguments to sample the behavior
of a function of multiple parameters;
</p>
<p>
<a href = "../../cpp_src/cvt/cvt.html">
CVT</a>,
a C++ library which
computes a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../cpp_src/faure/faure.html">
FAURE</a>,
a C++ library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/grid/grid.html">
GRID</a>,
a C++ library which
computes elements of a grid dataset.
</p>
<p>
<a href = "../../cpp_src/halton/halton.html">
HALTON</a>,
a C++ library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a C++ library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a C++ 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 = "../../cpp_src/ihs/ihs.html">
IHS</a>,
a C++ library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../cpp_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../cpp_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../cpp_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing
points at random.
</p>
<p>
<a href = "../../cpp_src/latinize/latinize.html">
LATINIZE</a>,
a C++ program which
"latinizes" a dataset.
</p>
<p>
<a href = "../../datasets/lcvt/lcvt.html">
LCVT</a>,
a dataset directory which
contains a collection of sample LCVT datasets.
</p>
<p>
<a href = "../../cpp_src/lcvt_dataset/lcvt_dataset.html">
LCVT_DATASET</a>,
a C++ program which
creates an LCVT dataset.
</p>
<p>
<a href = "../../cpp_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a C++ library which
computes elements of a Niederreiter quasirandom sequence with base 2.
</p>
<p>
<a href = "../../cpp_src/normal/normal.html">
NORMAL</a>,
a C++ library which
computes elements of a normal pseudorandom sequence.
</p>
<p>
<a href = "../../cpp_src/sobol/sobol.html">
SOBOL</a>,
a C++ library which
computes elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
computes elements of a uniform pseudorandom sequence.
</p>
<p>
<a href = "../../cpp_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a C++ 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 = "lcvt.cpp">lcvt.cpp</a>, the source code;
</li>
<li>
<a href = "lcvt.hpp">lcvt.hpp</a>, the include file;
</li>
<li>
<a href = "lcvt.csh">lcvt.csh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lcvt_prb.cpp">lcvt_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "lcvt_prb.csh">lcvt_prb.csh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "lcvt_prb_output.txt">lcvt_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>CLUSTER_ENERGY</b> returns the energy of a dataset.
</li>
<li>
<b>CVT</b> computes a Centroidal Voronoi Tessellation.
</li>
<li>
<b>CVT_ITERATION</b> takes one step of the CVT iteration.
</li>
<li>
<b>CVT_WRITE</b> writes a CVT dataset to a file.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>FIND_CLOSEST</b> finds the Voronoi cell generator closest to a point X.
</li>
<li>
<b>GET_SEED</b> returns a random seed for the random number generator.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two integers.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two integers.
</li>
<li>
<b>I4_TO_HALTON</b> computes an element of a Halton sequence.
</li>
<li>
<b>LCVT_WRITE</b> writes a Latinized CVT dataset to a file.
</li>
<li>
<b>PARAM_PRINT</b> prints the program parameters.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>R8_EPSILON</b> returns the roundoff unit for R8 arithmetic.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8MAT_LATINIZE</b> "Latinizes" an R8MAT.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8TABLE_DATA_READ</b> reads the data from an R8TABLE file.
</li>
<li>
<b>R8VEC_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R8VEC.
</li>
<li>
<b>REGION_SAMPLER</b> returns a sample point in the physical region.
</li>
<li>
<b>S_EQI</b> reports whether two strings are equal, ignoring case.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>TEST_REGION</b> determines if a point is within the physical region.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> returns the current YMDHMS date as a string.
</li>
<li>
<b>TUPLE_NEXT_FAST</b> computes the next element of a tuple space, "fast".
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 22 September 2006.
</i>
<!-- John Burkardt -->
</body>
</html>