# Intrinsic spatial stationarity: doesn't it only apply for small lags?

From the definition of Intrinsic stationarity:

$E[Z(x)-Z(x-h)] = 0$

This assumption is used for example in ordinary kriging, instead of assuming a constant mean over the entire space, we assume the mean is constant locally.

If the mean is constant in a neighbourhood, we logically expect the difference between two measurements close to each other to be zero. But as the mean varies over space, we don't expect the difference of values far away from each other to be zero?

So shouldn't the assumption of intrinsic stationarity be:

$E[Z(x)-Z(x-h)] = 0$ for $h \to 0$

Yes and no.

### Yes

I recall that Andre Journel long ago emphasized the points that

• Stationarity assumptions are decisions made by the analyst concerning what kind of model to use. They are not inherent properties of the phenomenon.

• Such assumptions are robust to departures because kriging (at least as practiced 20+ years ago) was almost always a local estimator based on selection of nearby data within moving search neighborhoods.

These points support the impression that intrinsic stationarity is purely a local property by suggesting that in practice it need only hold within a typical search neighborhood, and then only approximately.

### No

However, mathematically it is indeed the case that the expected differences must all be exactly zero, regardless of the distance $|h|$. In fact, if all you assumed were that the expected differences are continuous in the lag $h$, you wouldn't be assuming much at all! That weaker assumption would be tantamount to asserting a lack of structural breaks in the expectation (which wouldn't even imply a lack of structural breaks in the realizations of the process), but otherwise it could not be exploited to construct the kriging equations nor even estimate a variogram.

To appreciate just how weak (and practically useless) the assumption of mean continuity can be, consider a process $Z$ on the real line for which

$$Z(x) = U\text{ if } x \lt 0;\ Z(x) = -U\text{ otherwise }$$

where $U$ has a standard Normal distribution. The graph of a realization will consist of a half-line at height $u$ for negative $x$ and another half-line at height $-u$ for positive $x$.

For any $x$ and $h$,

$$E(Z(x)-Z(x-h)) = E(Z(x)) - E(Z(x-h)) = E(\pm U) - E(\pm U) = 0 - 0 = 0$$

yet almost surely $U\ne -U$, showing that almost all realizations of this process are discontinuous at $0$, even though the mean of the process is continuous everywhere.

### Interpretation

Diggle and Ribeiro discuss this issue [at p. 66]. They are talking about intrinsic random functions, for which the increments $Z(x)-Z(x-h)$ are assumed stationary (not just weakly stationary):

Intrinsic random functions embrace a wider class of models than do stationary random functions. With regard to spatial prediction, the main difference between predictions obtained from intrinsic and from stationary models is that if intrinsic models are used, the prediction at a point $x$ is influenced by the local behaviour of the data; i.e., by the observed measurement at locations relatively close to $x$, whereas predictions from stationary models are also affected by global behaviour. One way to understand this is to remember that the mean of an intrinsic process is indeterminate. As a consequence, predictions derived from an assumed intrinsic model tend to fluctuate around a local average. In contrast, predictions derived from an assumed stationary model tend to revert to the global mean of the assumed model in areas where the data are sparse. Which of these two types of behaviour is the more natural depends on the scientific context in which the models are being used.

### Comment

Instead, if you want control over the local behavior of the process, you should be making assumptions about the second moment of the increments, $E([Z(x)-Z(x-h)]^{2})$. For instance, when this approaches $0$ as $h\to 0$, the process is mean-square continuous. When there exists a process $Z^\prime$ for which

$$E([Z(x)-Z(x-h) - h Z^\prime(x)]^{2}) = O(h^2)$$

for all $x$, then the process is mean-square differentiable (with derivative $Z^\prime$).

### References

Peter J. Diggle and Paulo J. Ribeiro Jr., Model-based Geostatistics. Springer (2007)

• (+1): I like this notion of stationarity as modelling assumption, as it cannot be truly assessed. – Xi'an Dec 30 '14 at 19:43
• And do I understand it right that ordinary kriging derives predictions from an intrinsic model and simple kriging predicts based on a global stationary model? – Kasper Dec 31 '14 at 8:37
• My understanding of the distinction has been a little different. You can adopt the intrinsic hypothesis for both SK and OK, but SK additionally assumes a known mean. – whuber Dec 31 '14 at 23:24