CORRELATION
Examples of Correlation Functions
CORRELATION
is a MATLAB library which
contains examples of statistical correlation functions.
The (nonstationary) correlation function c(s,t) must satisfy the
following properties:

1 ≤ c(s,t) ≤ +1;

c(s,t) = c(t,s);

c(s,s) = 1;
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 x1x2, which we will denote by "r". Such correlation functions
are called "stationary".
The stationary correlation function c(r) must satisfy the following properties:

1 ≤ c(r) ≤ +1;

c(0) = 1;
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.
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.
Correlation functions available include:

besselj:
(plot),
(plots),
(sample paths)

besselk:
(plot),
(plots),
(sample paths)

brownian (nonstationary):
(plots),
(sample paths)

circular:
(plot),
(plots),
(sample paths)

constant:
(plot),
(plots),
(sample paths)

cubic:
(plot),
(plots),
(sample paths)

damped_cosine:
(plot),
(plots),
(sample paths)

damped_sine:
(plot),
(plots),
(sample paths)

exponential:
(plot),
(plots),
(sample paths)

gaussian:
(plot),
(plots),
(sample paths)

hole:
(plot),
(plots),
(sample paths)

linear:
(plot),
(plots),
(sample paths)

matern (NU=2.5):
(plot),
(plots),
(sample paths)

pentaspherical:
(plot),
(plots),
(sample paths)

power (E=2.0):
(plot),
(plots),
(sample paths)

rational_quadratic:
(plot),
(plots),
(sample paths)

spherical:
(plot),
(plots),
(sample paths)

white_noise:
(plot),
(plots),
(sample paths)
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
CORRELATION is available in
a C version and
a C++ version and
a FORTRAN77 version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
BROWNIAN_MOTION_SIMULATION,
a MATLAB program which
simulates Brownian motion in an Mdimensional region.
CNOISE,
a MATLAB library which
generates samples of noise obeying a 1/f^alpha power law,
by Miroslav Stoyanov.
COLORED_NOISE,
a MATLAB library which
generates samples of noise obeying a 1/f^alpha power law.
CORRELATION_CHEBFUN,
a MATLAB library which
uses the chebfun library to compute truncated KarhunenLoeve expansions
of stochastic processes with a given correlation function.
PINK_NOISE,
a MATLAB library which
computes a "pink noise" signal obeying a 1/f power law.
RANDOM_WALK_1D_SIMULATION,
a MATLAB program which
simulates a random walk in a 1dimensional region.
SDE,
a MATLAB library which
illustrates the properties of stochastic differential equations (SDE's), and
common algorithms for their analysis,
by Desmond Higham;
Reference:

Petter Abrahamsen,
A Review of Gaussian Random Fields and Correlation Functions,
Norwegian Computing Center, 1997.

Claude Dietrich, Garry Newsam,
Fast and exact simulation of stationary Gaussian processes through
the circulant embedding of the covariance matrix,
SIAM Journal on Scientific Computing,
Volume 18, Number 4, pages 10881107, July 1997.
Source Code:

correlation_besselj.m,
evaluates the Bessel J correlation function.

correlation_besselk.m,
evaluates the Bessel K correlation function.

correlation_brownian.m,
evaluates the Brownian correlation function K(s,t).

correlation_brownian_display.m,
displays four "slices" of the Brownian correlation function K(s,t).
Included for comparison only.

correlation_circular.m,
evaluates the circular correlation function.

correlation_constant.m,
evaluates the constant correlation function.

correlation_cubic.m,
evaluates the cubic correlation function.

correlation_damped_cosine.m,
evaluates the exponentially damped cosine correlation function.

correlation_damped_sine.m,
evaluates the damped sine correlation function.

correlation_exponential.m,
evaluates the exponential correlation function.

correlation_gaussian.m,
evaluates the gaussian correlation function.

correlation_hole.m,
evaluates the hole correlation function.

correlation_linear.m,
evaluates the linear correlation function.

correlation_matern.m,
evaluates the Matern correlation function.

correlation_pentaspherical.m,
evaluates the pentaspherical correlation function.

correlation_power.m,
evaluates the power correlation function.

correlation_rational_quadratic.m,
evaluates the rational quadratic correlation function.

correlation_spherical.m,
evaluates the spherical correlation function.

correlation_to_covariance.m,
converts a correlation matrix to a covariance matrix.

correlation_white_noise.m,
evaluates the white noise function.

covariance_to_correlation.m,
converts a covariance matrix to a correlation matrix.

minij.m,
computes the matrix whose entries are A(i,j) = min(i,j).

r8mat_is_symmetric.m,
checks that an R8MAT is symmetric.

r8mat_print.m,
prints an R8MAT.

r8mat_print_some.m,
prints some of an R8MAT.

r8vec_print.m,
prints an R8VEC.

sample_paths_cholesky.m,
computes sample paths of data with a given stationary correlation function,
using the Cholesky factorization of the symmetric correlation matrix.

sample_paths_eigen.m,
computes sample paths of data with a given stationary correlation function,
using the eigen decomposition of the symmetric correlation matrix.

sample_paths_fft.m,
computes sample paths of data with a given stationary correlation function,
embedding the symmetric correlation matrix (which is actually Toeplitz)
into a circulant matrix and then using fft techniques for fast factorization and
multiplication.

sample_paths2_cholesky.m,
computes sample paths of data with a given nonstationary correlation function,
using the Cholesky factorization of the symmetric correlation matrix.

sample_paths2_eigen.m,
computes sample paths of data with a given nonstationary correlation function,
using the eigen decomposition of the symmetric correlation matrix.

timestamp.m,
prints the YMDHMS date as a timestamp.
Examples and Tests:
You can go up one level to
the MATLAB source codes.
Last modified on 12 November 2012.