# Autocovariance Coefficients

Autocovariance is a measure of the degree to which the outcome of the function f (T + t) at coordinates (T+ t) depends upon the outcome of f(T) at coordinates t. It provides a description of the texture or a nature of the noise structure

## Background

In statistics, given a real stochastic process f(t), the auto-covariance is simply the covariance of the signal against a time-shifted version of itself. If each state of the series has a mean:

$E=[f(T)]=\mu_T$

Then the auto-covariance is given by the following equation:

$G_{xx}(T,S)=E[(f_T - \mu_T)*(f_S - \mu_S)]=(E(f_T * f_S) - (\mu_T - \mu_S))$

If f(t) is wide sense stationary function, then

$\mu_T = \mu_S = \mu$

And for all T and S

$G_{xx} (T,S)=G_{xx}(S-T)=G_{xx}(t)$

Where t=S-T is the amount of time by which the signal has been shifted.

When normalized by the variance $\sigma^2$ the auto-covariance function becomes the auto-covariance coefficient:

$\rho = G_{xx}(t)/\sigma^2$

Calculation of the auto-covariance function G(t) of a randomly varying function i(T) may be done in the time domain using the following equation:

$G(t) = \left \langle (i(T) * i(t+t) \right \rangle$

Where, the <> brackets indicate integration over time.

Correspondingly, the auto-covariance function may be calculated in the spatial domain, as

$g(\xi)=\delta_i (x) * \delta_i (x+\xi)$

$\delta_i = \frac{(i(T)- <(i(T)>}{<i(T)>}$

Where, the <> brackets indicate integration over time.

This gives us the normalized auto-covariance function or auto-covariance coefficient:

$g(t)= \left \langle \sigma_i (T) * \sigma_i (T+t)\right \rangle = [i(T) * i(T+t) - \left \langle i(T)\right \rangle ^2] / \left \langle i(T)\right \rangle ^2 = [G(t)/\left \langle i(T)\right \rangle ^2] -1$

When the data consist of a set of N discrete points, the averaging is performed as sums, then in spatial domain the 1D auto-covariance function is calculated by

If the random intensity variable, i, is a function of two independent variables, x and y, then, it is possible to define a corresponding two-dimensional auto-covariance function:

For a discrete set of data this becomes: