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<html>
<head>
<title>
DISCRETE_PDF_SAMPLE_2D - Create and Sample a PDF Based on Sample Data
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
DISCRETE_PDF_SAMPLE_2D <br> Create and Sample a PDF Based on Sample Data
</h1>
<hr>
<p>
<b>DISCRETE_PDF_SAMPLE_2D</b>
is a C++ program which
demonstrates how to construct a Probability Density Function (PDF)
from a table of sample data over a 2D region, and then to use that PDF to create new samples.
</p>
<p>
The program presented here is hard-wired to handle a specific problem.
However, the ideas used in the program are easily extended to other regions
and other dimensions.
</p>
<p>
For the problem given here, we assume we have sample values of a function
<b>F(X,Y)</b> for each subregion of a region. These values might actually
represent population counts, a density, the integral of some function over
the subregion, or simply an abstract function. We implicitly assumed that
all the values are positive.
</p>
<p>
The particular region studied here is the unit square, which has been
broken down into a 20x20 array of equal subsquares.
</p>
<p>
If we normalize by the sum of the data values, the result is a PDF
associated with each subregion.
By assigning an arbitrary order to the subregions, we can add the PDF
values up to the given subregion to get a CDF (cumulative density function)
for that subregion. Now given an arbitrary random value U, we can locate
the subregion whose CDF value just exceeds U. Choosing a random point
within this subregion gives us the sample point. If we choose many such
sample points, the statistics for this sample will tend to the discrete
PDF that we defined from the data we were given.
</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>DISCRETE_PDF_SAMPLE_2D</b> is available in
<a href = "../../c_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">a C version</a> and
<a href = "../../cpp_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">a C++ version</a> and
<a href = "../../f77_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fem1d_sample/fem1d_sample.html">
FEM1D_SAMPLE</a>,
a C++ program which
samples a scalar or vector finite element function of one variable,
defined by <b>FEM files</b>,
returning interpolated values at the sample points.
</p>
<p>
<a href = "../../cpp_src/fem2d_sample/fem2d_sample.html">
FEM2D_SAMPLE</a>,
a C++ program which
evaluates a finite element function defined on an order 3 or order 6 triangulation.
</p>
<p>
<a href = "../../cpp_src/prob/prob.html">
PROB</a>,
a C++ library which
evaluates and inverts a number of probabilistic distributions.
</p>
<p>
<a href = "../../cpp_src/random_data/random_data.html">
RANDOM_DATA</a>,
a C++ library which
generates sample points for
various probability distributions, spatial dimensions, and geometries;
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "discrete_pdf_sample_2d.cpp">discrete_pdf_sample_2d.cpp</a>, the source code.
</li>
<li>
<a href = "discrete_pdf_sample_2d.sh">discrete_pdf_sample_2d.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>TEST01</b> computes 1000 samples, based on a 20x20 PDF table
that is heavily biased toward the northwest corner.
<ul>
<li>
<a href = "test01.txt">test01.txt</a>,
1000 sample points generated by the program.
</li>
<li>
<a href = "test01.png">test01.png</a>,
a PNG image of the data.
</li>
</ul>
</p>
<p>
<b>TEST02</b> computes 1000 samples, based on a 12x8 PDF table
that is loosely based on the population of counties in Iowa.
<ul>
<li>
<a href = "test02.txt">test02.txt</a>,
1000 sample points generated by the program.
</li>
<li>
<a href = "test02.png">test02.png</a>,
a PNG image of the data.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for DISCRETE_PDF_SAMPLE_2D.
</li>
<li>
<b>DISCRETE_CDF_TO_XY</b> finds XY points corresponding to discrete CDF values.
</li>
<li>
<b>GET_DISCRETE_PDF</b> returns the value of the discrete PDF function in each cell.
</li>
<li>
<b>R8MAT_COPY_NEW</b> copies one R8MAT to a "new" R8MAT.
</li>
<li>
<b>R8MAT_SCALE</b> multiplies an R8MAT by a scalar.
</li>
<li>
<b>R8MAT_SUM</b> returns the sum of an R8MAT.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_UNIFORM_01_NEW</b> returns a new unit pseudorandom R8VEC.
</li>
<li>
<b>SET_DISCRETE_PDF</b> sets a CDF from a discrete PDF.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 modified on 03 January 2012.
</i>
<!-- John Burkardt -->
</body>
</html>