VarianceStructure

A variance structure function, recognized by Kolmogorov in relation to random fields, is a pivotal concept in stochastic processes which serves as a measure of spatial continuity for stochastic processes.

The term 'variance structure function' or simply 'variance structure' or 'structure function' consistently refers to the concept, rather than any specific depiction. When reading geostatistical literature the term 'variogram' is often used to when referring to the function, and no distinction is made between the function and its representation. TODO: cite Variogram naming papers

The structure function aims to mathematically articulate the spatial interdependence of a random field or stochastic process $Z(x)$, where $x$ represents a location in space. Specifically, it denotes the variance of the differences between values at varying locations of the random field across different distances.

Formally, for a strictly stationary random field $Z(x)$, the variance structure function, denoted by $\gamma(h)$, is defined as:

$$\gamma(h) = Var[Z(x+h)-Z(x)] = E[(Z(x + h) - Z(x))^2] = \frac{\sum_{i=1}^{N(h)} (Z(x_i + h) - Z(x_i))^2}{N(h)}$$

Here:

• $E[x]$ denotes the expected value operator, reflecting an average over all locations $x$ in the field,
• $Z(x + h)$ and $Z(x)$ are the sampled values of the random field at positions $x + h$ and $x$ respectively,
• $h$ is a vector representing the direction and separation distance between two locations,
• $\gamma(h)$ symbolizes the value of the variance structure function, i.e., the variance of the difference between sampled values of the random field separated by distance $h$.

In the case where the process $Z(x)$ is intrinsically stationary, the variance structure function can also be described in terms of the covariance function $C(h)$:

$$\gamma(h) = C(0) - C(h)$$

In geostatistics, the term 'variogram' usually refers to $2\gamma(h)$, which is twice the semivariogram. For the purposes of this discussion, however, we are not incorporating this particular notion from geostatistics and will refer to $\gamma(h)$ as the (variance) structure function, or simply the variance structure.

The graphical depiction of the variance structure function, or the 'variogram', illustrates the values of the function as a function of distance $h$. This visualization aids in the analysis and interpretation of the spatial dependency structure of the stochastic process over a range of distances.

To reiterate, the terms 'variance structure function', 'variance structure', or 'structure function' always pertain to the abstract concept and never its visual representation. When referring to the graph of this function, the term 'variogram' is used.