Distance Covariance Functions

Why

It is common to consider random functions whose domain is time, space, or $n$-dimensional space.

Definition

Let $(X, d)$ be a metric space. A distance covariance function $k: X \times X \to \R $ is a covariance function satisfying

\[ k(x, y) > k(x, y) \iff d(x, y) < d(x, y). \]

In other words, the covariance decreases as the distance between the arguments decreases.

Example: squared exponential

Let $k: X \times X \to \R $ be defined by

\[ k(x, y) = \exp(-d(x, y)). \]

Then $k$ is a distance covariance function. It is often called the squared exponential covariance function.

Let $\alpha , \sigma \in \R $. Define $k': X \times X \to \R $ by

\[ k'(x, y) = \alpha \exp(-d(x, y)/\sigma ^2) \]

then $k'$ is still a covariance function. In this context $\sigma $ is often referred to as the characteristic length-scale of the process. The scalar $\alpha $ is sometimes called a “prefactor” that “controls” the “overall variance.”

Suppose $(X, d) = (\R ^n, \norm{\cdot })$. Then the squared exponential covariance function

\[ \alpha \exp(-\norm{x - y}/(2\sigma ^2)) \]

is sometimes called the radial basis function or gaussian covariance function.¹ Also called an exponentiated quadratic kernel.

For reasons that will be included in future editions. ↩︎