# correlation

correlation, a MATLAB code which implements 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:

### 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 code which simulates Brownian motion in an M-dimensional region.

cnoise, a MATLAB code which generates samples of noise obeying a 1/f^alpha power law, by Miroslav Stoyanov.

colored_noise, a MATLAB code which generates samples of noise obeying a 1/f^alpha power law.

pink_noise, a MATLAB code which computes a "pink noise" signal obeying a 1/f power law.

random_walk_1d_simulation, a MATLAB code which simulates a random walk in a 1-dimensional region.

sde, a MATLAB code which illustrates the properties of stochastic differential equations (SDE), 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.