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<html>
<head>
<title>
LCVT_DATASET - Latin Hypercubes using CVT Startup
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LCVT_DATASET <br> Latin Hypercubes using CVT Startup
</h1>
<hr>
<p>
<b>LCVT_DATASET</b>
is a C++ program which
computes a Latin Hypercube in M dimensions,
with N points, using a CVT dataset as the initial estimate.
The resulting dataset can be written to a file.
</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>
<p>
An earlier version of this program was "very" interactive,
allowing the user to enter input in any order. This turned
out to be a little too confusing. The new version of the
program asks the user for input in a strict order. If you
find this procedure too restrictive, you can try out the
old program.
</p>
<p>
Briefly the user needs to specify the following:
<ol>
<li>
The spatial dimension M of the points;
</li>
<li>
The number of points N to be generated.
</li>
<li>
The random number seed;
</li>
<li>
How the initial points are chosen. If you have no preference,
choose UNIFORM.
<ul>
<li>
GRID, use a grid of points;
</li>
<li>
HALTON, use Halton points;
</li>
<li>
RANDOM, use RAND (C++ intrinsic);
</li>
<li>
UNIFORM, use a simple uniform random number generator;
</li>
<li>
USER, call the "user" routine;
</li>
<li>
(file_name), read the initial points from a file.
</li>
</ul>
</li>
<li>
The number of CVT iterations. If you have no preference,
try 5, 10 or 20;
</li>
<li>
How the sampling is done. If you have no preference, use UNIFORM.
<ul>
<li>
GRID, use a grid of points;
</li>
<li>
HALTON, use Halton points;
</li>
<li>
RANDOM, use RAND (C++ intrinsic);
</li>
<li>
UNIFORM, use a simple uniform random number generator;
</li>
<li>
USER, call the "user" routine;
</li>
</ul>
</li>
<li>
The number of sampling points to use. Think of this as
a sampling of the unit hypercube. So to compare it to
N, the number of points, you need to take its M-th root.
In 2D, if you're using 10 generators, and 100 sample points,
to get area and sampling computations twice as good requires
4 times the sampling. It never hurts to use more sampling
points.
</li>
<li>
The "batch size". This parameter controls how many sampling
points are to be generated at one time. You can set this
value equal to the number of sampling points, but if you
are having memory problems, it can be set lower. In such
a case, a smaller value might be 1000, for instance.
</li>
<li>
The number of CVT iterations to carry out. It's not really
necessary to compute the CVT super accurately, since we're
just going to perturb it anyway. This value could be anywhere
from 10 to 500. Convergence of the CVT is typically slow,
especially if the starting positions are poor.
</li>
<li>
The number of Latin Hypercube iterations to carry out.
Actually, the iterations don't seem to improve the data
much, so a value of 1 or 2 can be reasonable.
</li>
<li>
The name of a file into which the final pointset should be
written.
</li>
</ol>
</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_DATASET</b> is available in
<a href = "../../cpp_src/lcvt_dataset/lcvt_dataset.html">a C++ version</a> and
<a href = "../../f_src/lcvt_dataset/lcvt_dataset.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lcvt_dataset/lcvt_dataset.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cvt/cvt.html">
CVT</a>,
a C++ library which
can compute a
CVT (Centroidal Voronoi Tessellation).
</p>
<p>
<a href = "../../cpp_src/cvt_dataset/cvt_dataset.html">
CVT_DATASET</a>,
a C++ program which
can create a CVT dataset (Centroidal Voronoi Tessellation).
</p>
<p>
<a href = "../../cpp_src/faure_dataset/faure_dataset.html">
FAURE_DATASET</a>,
a C++ program which
creates a Faure quasirandom dataset;
</p>
<p>
<a href = "../../cpp_src/grid_dataset/grid_dataset.html">
GRID_DATASET</a>,
a C++ program which
creates a grid sequence and writes it to a file.
</p>
<p>
<a href = "../../cpp_src/latin_center_dataset/latin_center_dataset.html">
LATIN_CENTER_DATASET</a>,
a C++ program which
creates a Latin Center Hypercube dataset;
</p>
<p>
<a href = "../../cpp_src/latin_edge_dataset/latin_edge_dataset.html">
LATIN_EDGE_DATASET</a>,
a C++ program which
creates a Latin Edge Hypercube dataset;
</p>
<p>
<a href = "../../cpp_src/latin_random_dataset/latin_random_dataset.html">
LATIN_RANDOM_DATASET</a>,
a C++ program which
creates a Latin Random Hypercube dataset;
</p>
<p>
<a href = "../../cpp_src/lcvt/lcvt.html">
LCVT</a>,
a C++ library which
is used by
<b>LCVT_DATASET</b>; a compiled copy of that library must be
available to build the program.
</p>
<p>
<a href = "../../datasets/lcvt/lcvt.html">
LCVT</a>,
a dataset directory which
contains a collection of sample
LCVT datasets created by <b>LCVT_DATASET</b>.
</p>
<p>
<a href = "../../cpp_src/niederreiter2_dataset/niederreiter2_dataset.html">
NIEDERREITER2_DATASET</a>,
a C++ program which
creates a Niederreiter quasirandom dataset with base 2;
</p>
<p>
<a href = "../../cpp_src/normal_dataset/normal_dataset.html">
NORMAL_DATASET</a>,
a C++ program which
generates a dataset of multivariate normal pseudorandom values and writes them to a file.
</p>
<p>
<a href = "../../cpp_src/sobol_dataset/sobol_dataset.html">
SOBOL_DATASET</a>,
a C++ program which
computes a Sobol quasirandom sequence and writes it to a file.
</p>
<p>
<a href = "../../cpp_src/table_latinize/table_latinize.html">
TABLE_LATINIZE</a>,
a C++ program which
can read a <b>TABLE file</b> of points and "latinize" the points,
that is, "gently" rearranging them so that they are regularly
spaced in every coordinate direction.
</p>
<p>
<a href = "../../cpp_src/uniform_dataset/uniform_dataset.html">
UNIFORM_DATASET</a>,
a C++ program which
generates a dataset of uniform pseudorandom values and writes them to a file.
</p>
<p>
<a href = "../../cpp_src/van_der_corput_dataset/van_der_corput_dataset.html">
VAN_DER_CORPUT_DATASET</a>,
a C++ program which
creates a van der Corput quasirandom sequence and writes it to a file.
</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.
</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>
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.
</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.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lcvt_dataset.cpp">lcvt_dataset.cpp</a>,
the source code.
</li>
<li>
<a href = "lcvt_dataset.sh">lcvt_dataset.sh</a>,
commands to compile, link and load the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>Example 1</b> is a dataset of N=85 points with spatial
dimension M=2, using UNIFORM initialization and sampling,
and 10,000 sample points:
<ul>
<li>
<a href = "lcvt01_input.txt">lcvt01_input.txt</a>,
input commands.
</li>
<li>
<a href = "lcvt01_output.txt">lcvt01_output.txt</a>,
printed output.
</li>
<li>
<a href = "lcvt01.txt">lcvt01.txt</a>,
the LCVT dataset.
</li>
<li>
<a href = "lcvt01.png">lcvt01.png</a>,
a <a href = "../../png/png.html">PNG</a> image of
the LCVT dataset,
created by
<a href =
"../../g_src/plot_points/plot_points.html">
PLOT_POINTS</a>.
</li>
</ul>
</p>
<p>
<b>Example 2</b> is a dataset of N=85 points with spatial
dimension M=2, using RANDOM initialization and sampling,
and 250,000 sample points, 10 CVT iterations
and 2 Latinization iterations:
<ul>
<li>
<a href = "lcvt02_input.txt">lcvt02_input.txt</a>,
input commands.
</li>
<li>
<a href = "lcvt02_output.txt">lcvt02_output.txt</a>,
printed output.
</li>
<li>
<a href = "lcvt02.txt">lcvt02.txt</a>,
the LCVT dataset.
</li>
<li>
<a href = "lcvt02.png">lcvt02.png</a>,
a <a href = "../../png/png.html">PNG</a> image of
the LCVT dataset,
created by
<a href =
"../../g_src/plot_points/plot_points.html">
PLOT_POINTS</a>.
</li>
</ul>
</p>
<p>
<b>Example 3</b> is a dataset of N=200 points with spatial
dimension M=7, using UNIFORM initialization and sampling,
and 20,000 sample points, 5 CVT iterations
and 2 Latinization iterations:
<ul>
<li>
<a href = "lcvt03_input.txt">lcvt03_input.txt</a>,
input commands.
</li>
<li>
<a href = "lcvt03_output.txt">lcvt03_output.txt</a>,
printed output.
</li>
<li>
<a href = "lcvt03.txt">lcvt03.txt</a>,
the LCVT dataset.
</li>
<li>
<a href = "lcvt03_page1.png">lcvt03_page1.png</a>,
"page 1" of
a <a href = "../../png/png.html">PNG</a> image of
pairs of coordinates of the LCVT dataset,
created by
<a href =
"../../f_src/table_top/table_top.html">
TABLE_TOP</a>.
</li>
<li>
<a href = "lcvt03_page2.png">lcvt03_page2.png</a>,
"page 2" of
a <a href = "../../png/png.html">PNG</a> image of
pairs of coordinates of the LCVT dataset,
created by
<a href =
"../../f_src/table_top/table_top.html">
TABLE_TOP</a>.
</li>
<li>
<a href = "lcvt03_page3.png">lcvt03_page3.png</a>,
"page 3" of
a <a href = "../../png/png.html">PNG</a> image of
pairs of coordinates of the LCVT dataset,
created by
<a href =
"../../f_src/table_top/table_top.html">
TABLE_TOP</a>.
</li>
<li>
<a href = "lcvt03_page4.png">lcvt03_page4.png</a>,
"page 4" of
a <a href = "../../png/png.html">PNG</a> image of
pairs of coordinates of the LCVT dataset,
created by
<a href =
"../../f_src/table_top/table_top.html">
TABLE_TOP</a>.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for LCVT_DATASET.
</li>
<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_ITERATION</b> takes one step of the CVT iteration.
</li>
<li>
<b>FIND_CLOSEST</b> finds the Voronoi cell generator closest to a point X.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</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>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>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>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 05 March 2007.
</i>
<!-- John Burkardt -->
</body>
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</html>