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. -1 ≤ c(s,t) ≤ +1;
2. c(s,t) = c(t,s);
3. 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 ||x1-x2||, which we will denote by "r". Such correlation functions are called "stationary".

The stationary correlation function c(r) must satisfy the following properties:

1. -1 ≤ c(r) ≤ +1;
2. 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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BROWNIAN_MOTION_SIMULATION, a MATLAB program which simulates Brownian motion in an M-dimensional 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 Karhunen-Loeve 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 1-dimensional 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:

1. Petter Abrahamsen,
   A Review of Gaussian Random Fields and Correlation Functions,
   Norwegian Computing Center, 1997.
2. 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 1088-1107, 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:

• correlation_test.m, calls all the tests;
• correlation_test01.m, plots each correlation function with unit correlation length;
• correlation_test02.m, plots sample paths for each correlation function using SAMPLE_PATHS_CHOLESKY and SAMPLE_PATHS_EIGEN.
• correlation_test03.m, plots each correlation function with several correlation lengths;
• correlation_test04.m, converts between correlation and covariance matrices.
• correlation_test05.m, plots the Brownian correlation function.
• correlation_test06.m, plots sample paths of the Brownian correlation function.
• correlation_test.txt, the output file.

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