📂Statistical Analysis

Isotropy of Variogram

Definition ¹

The semivariogram $\gamma \left( \mathbf{h} \right)$ of a spatial process can be expressed as depending solely on the magnitude $d := \left\| \mathbf{h} \right\|$ rather than on the direction vector $\mathbf{h} \in \mathbb{R}^{r}$, in which case the variogram $2 \gamma$ is said to be isotropic. If it is not isotropic, it is referred to as anisotropic. $$ \gamma \left( \left\| \mathbf{h} \right\| \right) = \gamma (d) $$ Especially if it is isotropic and also intrinsically stationary, it is also called homogeneous.

Explanation

That point reference data possesses isotropy means, for instance, the relationship between any two given points is explained solely by their distance rather than by the direction such as east or west.

Of course, isotropic data is much easier to analyze, and considering that spatial statistical analysis itself is not easy in statistics, dealing with anisotropic data is in practice handled by specialists… In other words, ’those who specialized in spatial statistics among statistics majors’. In reality, compared to other fields (including R), the ecosystem of libraries is still not mature, and it is true that individual coding skills are still heavily relied upon.

Anisotropic Data

Examples of data lacking isotropy can easily include environmental observation data around rivers or mountains.

• Pollution around chemical plants: If a plant discharges waste into a nearby river, the quantity of a certain substance of interest would flow from upstream to downstream, hence exhibiting a certain directionality. Logically, data from a point downstream and across the river should be very similar if the river is not wide, yet, the directionality following the river flow makes it difficult to assume that any point shares the same directionality.
• Weather observation: Fundamentally, everything in the air diffuses displaying Brownian motion, but as the wind blows according to the terrain, they create flows with causality stronger than correlation. If these have statistical significance beyond coincidence at a local scale, separate handling is required for them.

Isotropy Check

The phrases checking for isotropy and detecting anisotropy essentially mean the same, but two known visual methods for checking this include the directional semivariogram (left, A) and the rose diagram (right, B). ² Both figures present the semivariograms of the data calculated separately by dividing them every 45º, and at a glance, an anomaly of $\gamma \left( \mathbf{h} \right)$ in the 135º direction can be seen.

Alternatively, the Empirical Semivariogram Contour (ESC) plot can also be referred.

Semivariogram Models

In the case of $\gamma \left( \left\| \mathbf{h} \right\| \right) = \gamma (d)$, $\gamma$ can be expressed not as a complex matrix form, but as a one-dimensional scalar function, that is, as ▷eq8◀.

This topic continues in the post Models of Semivariograms.

Radial Functions

An isotropic variogram is a kind of radial function.