# Isotropic vs. Anisotropic

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I came across **isotropic** and **anisotropic** covariance functions in kjytay’s blog, and then I found more materials, chapter 4 from the book Gaussian Processes for Machine Learning, via the reference in StackExchange: What is an isotropic (spherical) covariance matrix?.

Firstly of all, a covariance function must be symmetric and positive semi-definite.

A **stationary** covariance function is a function of $\x - \x’$. Thus it is invariant to translations in the input space.

For a stochastic process,

- weakly stationary: a process has constant mean and whose covariance function is invariant to translations
- strictly stationary: all of its finite dimensional distributions are invariant to translations.

If further the covariance function is a function only of $\vert \x -\x’\vert$ then it is called **isotropic**. These are also known as **radial basis functions (RBFs)**.

A covariance is said to be **anisotropic** if it is not isotropic, such as setting

for some positive semidefinite $M$.