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<html>
<head>
<title>
CORRELATION - Examples of Correlation Functions
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
CORRELATION <br> Examples of Correlation Functions
</h1>
<hr>
<p>
<b>CORRELATION</b>
is a C++ library which
contains examples of statistical correlation functions.
</p>
<p>
The (nonstationary) correlation function c(s,t) must satisfy the
following properties:
<ol>
<li>
-1 ≤ c(s,t) ≤ +1;
</li>
<li>
c(s,t) = c(t,s);
</li>
<li>
c(s,s) = 1;
</li>
</ol>
</p>
<p>
Most of the correlation functions considered here determine the correlation of two
random values y(x1) and y(x2), depending only on distance, that is,
on the norm ||x1-x2||, which we will denote by "r". Such correlation functions
are called "stationary".
</p>
<p>
The stationary correlation function c(r) must satisfy the following properties:
<ol>
<li>
-1 ≤ c(r) ≤ +1;
</li>
<li>
c(0) = 1;
</li>
</ol>
</p>
<p>
It is often the case that a typical scale length "r0" is specified,
called the "correlation length". In that case, the correlation function
may be expressed in terms of the normalized distance r/r0.
</p>
<p>
Because correlation functions model physical situations, it is usually the case
that the correlation function will smoothly and steadily decrease to 0 with r,
or that it will oscillate between positive and negative values, with an
amplitude that is steadily decreasing. One of the most popular correlation
functions is the gaussian correlation, which has many desirable statistical
and mathematical properties.
</p>
<p>
Correlation functions available include:
<ul>
<li>
<i>besselj</i>:
<a href = "besselj_plot.png">(plot)</a>,
<a href = "besselj_plots.png">(plots)</a>,
<a href = "besselj_paths.png">(sample paths)</a>
</li>
<li>
<i>besselk</i>:
<a href = "besselk_plot.png">(plot)</a>,
<a href = "besselk_plots.png">(plots)</a>,
<a href = "besselk_paths.png">(sample paths)</a>
</li>
<li>
<i>brownian</i> (nonstationary):
<a href = "brownian_plots.png">(plots)</a>,
<a href = "brownian_paths.png">(sample paths)</a>
</li>
<li>
<i>circular</i>:
<a href = "circular_plot.png">(plot)</a>,
<a href = "circular_plots.png">(plots)</a>,
<a href = "circular_paths.png">(sample paths)</a>
</li>
<li>
<i>constant</i>:
<a href = "constant_plot.png">(plot)</a>,
<a href = "constant_plots.png">(plots)</a>,
<a href = "constant_paths.png">(sample paths)</a>
</li>
<li>
<i>cubic</i>:
<a href = "cubic_plot.png">(plot)</a>,
<a href = "cubic_plots.png">(plots)</a>,
<a href = "cubic_paths.png">(sample paths)</a>
</li>
<li>
<i>damped_cosine</i>:
<a href = "damped_cosine_plot.png">(plot)</a>,
<a href = "damped_cosine_plots.png">(plots)</a>,
<a href = "damped_cosine_paths.png">(sample paths)</a>
</li>
<li>
<i>damped_sine</i>:
<a href = "damped_sine_plot.png">(plot)</a>,
<a href = "damped_sine_plots.png">(plots)</a>,
<a href = "damped_sine_paths.png">(sample paths)</a>
</li>
<li>
<i>exponential</i>:
<a href = "exponential_plot.png">(plot)</a>,
<a href = "exponential_plots.png">(plots)</a>,
<a href = "exponential_paths.png">(sample paths)</a>
</li>
<li>
<i>gaussian</i>:
<a href = "gaussian_plot.png">(plot)</a>,
<a href = "gaussian_plots.png">(plots)</a>,
<a href = "gaussian_paths.png">(sample paths)</a>
</li>
<li>
<i>hole</i>:
<a href = "hole_plot.png">(plot)</a>,
<a href = "hole_plots.png">(plots)</a>,
<a href = "hole_paths.png">(sample paths)</a>
</li>
<li>
<i>linear</i>:
<a href = "linear_plot.png">(plot)</a>,
<a href = "linear_plots.png">(plots)</a>,
<a href = "linear_paths.png">(sample paths)</a>
</li>
<li>
<i>matern (NU=2.5)</i>:
<a href = "matern_plot.png">(plot)</a>,
<a href = "matern_plots.png">(plots)</a>,
<a href = "matern_paths.png">(sample paths)</a>
</li>
<li>
<i>pentaspherical</i>:
<a href = "pentaspherical_plot.png">(plot)</a>,
<a href = "pentaspherical_plots.png">(plots)</a>,
<a href = "pentaspherical_paths.png">(sample paths)</a>
</li>
<li>
<i>power (E=2.0)</i>:
<a href = "power_plot.png">(plot)</a>,
<a href = "power_plots.png">(plots)</a>,
<a href = "power_paths.png">(sample paths)</a>
</li>
<li>
<i>rational_quadratic</i>:
<a href = "rational_quadratic_plot.png">(plot)</a>,
<a href = "rational_quadratic_plots.png">(plots)</a>,
<a href = "rational_quadratic_paths.png">(sample paths)</a>
</li>
<li>
<i>spherical</i>:
<a href = "spherical_plot.png">(plot)</a>,
<a href = "spherical_plots.png">(plots)</a>,
<a href = "spherical_paths.png">(sample paths)</a>
</li>
<li>
<i>white_noise</i>:
<a href = "white_noise_plot.png">(plot)</a>,
<a href = "white_noise_plots.png">(plots)</a>,
<a href = "white_noise_paths.png">(sample paths)</a>
</li>
</ul>
</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>CORRELATION</b> is available in
<a href = "../../c_src/correlation/correlation.html">a C version</a> and
<a href = "../../cpp_src/correlation/correlation.html">a C++ version</a> and
<a href = "../../f77_src/correlation/correlation.html">a FORTRAN77 version</a> and
<a href = "../../f_src/correlation/correlation.html">a FORTRAN90 version</a> and
<a href = "../../m_src/correlation/correlation.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a C++ program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../cpp_src/colored_noise/colored_noise.html">
COLORED_NOISE</a>,
a C++ library which
generates samples of noise obeying a 1/f^alpha power law.
</p>
<p>
<a href = "../../examples/gnuplot/gnuplot.html">
GNUPLOT</a>,
examples which
illustrate the use of the gnuplot graphics program.
</p>
<p>
<a href = "../../cpp_src/pink_noise/pink_noise.html">
PINK_NOISE</a>,
a C++ library which
computes a pink noise signal obeying a 1/f power law.
</p>
<p>
<a href = "../../cpp_src/sde/sde.html">
SDE</a>,
a C++ library which
illustrates the properties of stochastic differential equations (SDE's), and
common algorithms for their analysis,
by Desmond Higham;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Petter Abrahamsen,<br>
A Review of Gaussian Random Fields and Correlation Functions,<br>
Norwegian Computing Center, 1997.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "correlation.cpp">correlation.cpp</a>, the source code.
</li>
<li>
<a href = "correlation.hpp">correlation.hpp</a>, the include file.
</li>
<li>
<a href = "correlation.sh">correlation.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "correlation_prb.c">correlation_prb.c</a>,
a sample calling program.
</li>
<li>
<a href = "correlation_prb.sh">correlation_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "correlation_prb_output.txt">correlation_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CORRELATION_BROWNIAN</b> computes the Brownian correlation function.
</li>
<li>
<b>CORRELATION_BESSELJ</b> evaluates the Bessel J correlation function.
</li>
<li>
<b>CORRELATION_BESSELK</b> evaluates the Bessel K correlation function.
</li>
<li>
<b>CORRELATION_BROWNIAN_DISPLAY</b> displays 4 slices of the Brownian Correlation.
</li>
<li>
<b>CORRELATION_CIRCULAR</b> evaluates the circular correlation function.
</li>
<li>
<b>CORRELATION_CONSTANT</b> evaluates the constant correlation function.
</li>
<li>
<b>CORRELATION_COSINE_DAMPED</b> evaluates the damped cosine correlation function.
</li>
<li>
<b>CORRELATION_CUBIC</b> evaluates the cubic correlation function.
</li>
<li>
<b>CORRELATION_DAMPED_COSINE</b> evaluates the damped cosine correlation function.
</li>
<li>
<b>CORRELATION_DAMPED_SINE</b> evaluates the damped sine correlation function.
</li>
<li>
<b>CORRELATION_EXPONENTIAL</b> evaluates the exponential correlation function.
</li>
<li>
<b>CORRELATION_GAUSSIAN</b> evaluates the Gaussian correlation function.
</li>
<li>
<b>CORRELATION_HOLE</b> evaluates the hole correlation function.
</li>
<li>
<b>CORRELATION_LINEAR</b> evaluates the linear correlation function.
</li>
<li>
<b>CORRELATION_MATERN</b> evaluates the Matern correlation function.
</li>
<li>
<b>CORRELATION_PENTASPHERICAL</b> evaluates the pentaspherical correlation function.
</li>
<li>
<b>CORRELATION_POWER</b> evaluates the power correlation function.
</li>
<li>
<b>CORRELATION_RATIONAL_QUADRATIC:</b> rational quadratic correlation function.
</li>
<li>
<b>CORRELATION_SINE_DAMPED</b> evaluates the damped sine correlation function.
</li>
<li>
<b>CORRELATION_SPHERICAL</b> evaluates the spherical correlation function.
</li>
<li>
<b>CORRELATION_TO_COVARIANCE:</b> covariance matrix from a correlation matrix.
</li>
<li>
<b>CORRELATION_WHITE_NOISE</b> evaluates the white noise correlation function.
</li>
<li>
<b>COVARIANCE_TO_CORRELATION:</b> correlation matrix from a covariance matrix.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>MINIJ</b> returns the MINIJ matrix.
</li>
<li>
<b>R8_B0MP</b> evaluates the modulus and phase for the Bessel J0 and Y0 functions.
</li>
<li>
<b>R8_BESI1</b> evaluates the Bessel function I of order 1 of an R8 argument.
</li>
<li>
<b>R8_BESI1E</b> evaluates the exponentially scaled Bessel function I1(X).
</li>
<li>
<b>R8_BESJ0</b> evaluates the Bessel function J of order 0 of an R8 argument.
</li>
<li>
<b>R8_BESK</b> evaluates the Bessel function K of order NU of an R8 argument.
</li>
<li>
<b>R8_BESK1</b> evaluates the Bessel function K of order 1 of an R8 argument.
</li>
<li>
<b>R8_BESK1E</b> evaluates the exponentially scaled Bessel function K1(X).
</li>
<li>
<b>R8_BESKES:</b> a sequence of exponentially scaled K Bessel functions at X.
</li>
<li>
<b>R8_BESKS</b> evaluates a sequence of K Bessel functions at X.
</li>
<li>
<b>R8_CSEVL</b> evaluates a Chebyshev series.
</li>
<li>
<b>R8_GAML</b> evaluates bounds for an R8 argument of the gamma function.
</li>
<li>
<b>R8_GAMMA</b> evaluates the gamma function of an R8 argument.
</li>
<li>
<b>R8_INITS</b> initializes a Chebyshev series.
</li>
<li>
<b>R8_KNUS</b> computes a sequence of K Bessel functions.
</li>
<li>
<b>R8_LGMC</b> evaluates the log gamma correction factor for an R8 argument.
</li>
<li>
<b>R8_MACH</b> returns real ( kind = 8 ) real machine-dependent constants.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8MAT_CHOLESKY_FACTOR</b> computes the Cholesky factor of a symmetric matrix.
</li>
<li>
<b>R8MAT_IS_SYMMETRIC</b> checks an R8MAT for symmetry.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_NORMAL_01</b> returns a unit pseudonormal R8VEC.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>SAMPLE_PATH_CHOLESKY</b> computes a sample path for a correlation function.
</li>
<li>
<b>SAMPLE_PATH_EIGEN</b> computes a sample path for a correlation function.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TQL2</b> computes all eigenvalues/vectors, real symmetric tridiagonal matrix.
</li>
<li>
<b>TRED2</b> transforms a real symmetric matrix to symmetric tridiagonal form.
</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 12 November 2012.
</i>
<!-- John Burkardt -->
</body>
</html>