CovarianceStructure - crowlogic/arb4j GitHub Wiki

\title{Gaussian Processes Specified By Their Variance Structure}

\author{ Stephen Crowkey \tmaffiliation{with a little help from a robot who will remain anonymous for now} }

\maketitle

Stochastic processes with a specified variance structure, often called covariance structure, are constructed using Gaussian processes. The variogram is linked to the covariance function as follows: if $C (h)$ is the covariance function and $\gamma (h)$ is the variogram, then $\gamma (h) = C (0) - C (h)$.

A Gaussian process is defined by its mean function $m (x)$ and its covariance function $C (x, y)$. For any collection of indices ${x_1, ..., x_n }$, the random variables $Z (x_1), ..., Z (x_n)$ follow a multivariate normal distribution with mean vector $m = [m (x_1), ..., m (x_n)]$ and covariance matrix $\Sigma$ with $\Sigma_{ij} = C (x_i, x_j)$.

Given a mean of 0 and a known variogram $\gamma (h)$, we can construct a Gaussian process, assuming the variogram is valid, i.e., it satisfies certain conditions such as being non-negative and being a semi-variance.

The key idea is to construct a covariance function $C (x, y)$ from the variogram. As you've noted, the relation between the covariance function and the variogram is given by $\gamma (h) = C (0) - C (h)$, hence $C (h) = C (0) - \gamma (h)$.

In the simplest case where $C (0)$, the variance of the process, is a constant (let's say $C (0) = \sigma^2$), we can construct the covariance function as follows: \begin{equation} C (h) = \sigma^2 - \gamma (h) \end{equation} For a Gaussian process $Z (x)$ with covariance function $C (x, y)$ and mean $m (x) = 0$, you can sample from this process using any method appropriate for sampling from a multivariate normal distribution, such as the Cholesky decomposition.

It's important to mention that the construction of such a process requires that the constructed covariance function $C (h)$ be a valid covariance function, i.e., the resulting covariance matrix should be positive semi-definite. If the variogram doesn't lead to a valid covariance function, you may not be able to construct a Gaussian process based on this variogram.

Finally, the Gaussian process constructed this way will have the property that its increments have a variance structure given by the variogram, i.e., $\text{Var}[Z (x + h) - Z (x)] = \gamma (h)$.

\subsubsection{Infinitesimal Generator and Self-Adjoint Operator}\label{infinitesimal-generator}

In the context of stochastic processes, especially diffusions, the infinitesimal generator is often given by a second order differential operator. For a Gaussian process with covariance structure defined by the variogram, the infinitesimal generator can be more complex to derive and it can involve the inverse of the covariance matrix, which in this case, is derived from the variogram.

In the case of a Gaussian process, the infinitesimal generator $A$ is often defined in terms of the mean function $m (x)$ and the covariance function $C (x, y)$ as: \begin{equation} Af (x) = m' (x) f' (x) + 0.5 C'' (x, x) f (x) \end{equation} Where $f (x)$ is a test function, $m' (x)$ is the derivative of the mean function, and $C'' (x, x)$ is the second derivative of the covariance function.

The self-adjoint operator associated with a Gaussian process is typically related to its infinitesimal generator. In the context of stochastic processes and stochastic differential equations, the infinitesimal generator is given by a second-order differential operator. For a zero-mean Gaussian process with covariance function $C (x, y)$, the infinitesimal generator $A$ of this process is defined as: [ Af (x) = \frac{1}{2} C'' (x, x) f (x) ]

Here, $f (x)$ is a test function and $C'' (x, x)$ is the second derivative of the covariance function.

Please note that the self-adjoint operator or infinitesimal generator is typically associated with the dynamics of the stochastic process and may not have a direct interpretation in terms of the variogram or the covariance structure. Its definition and properties are more related to the stochastic calculus and differential equations used to describe the evolution of the process.

Therefore, while you can derive a self-adjoint operator from the covariance structure, it doesn't contain direct information about the variogram but rather about the dynamic properties of the process.