 ### The importance of being stationary ?

All is flux: nothing is stationary - Democritus

The various different versions of geostatistics are based on fundamental assumptions which are required by regionalised variable theory and its modifications and extensions. One of the key assumptions is that of stationarity. Unfortunately there is much misunderstanding about this among both geostatistical practitioners and sceptics. Donald Myers wrote a paper in Mathematical Geology (1989) which attempted to clear up the misunderstandings, but unfortunately created a few new ones in the process. He emphasised that stationarity is a property of the random function (i.e. the regionalized variable), not of the data. We shall return to this question later.

The common view that stationarity simply means that ‘things are the same everywhere' within the zone being modelled is a gross over-simplification. In fact, there are several types of stationarity which commonly have been required by different geostatistical methods.

Strong stationarity requires that the distribution of the regionalised variable Z(x) is independent of location x. This implies that (if they exist) the mean, variance, and all other distribution parameters are everywhere the same. This type of stationarity is required for such nonlinear kriging methods as disjunctive kriging and multigaussian kriging, and also (as Myers himself accepts) for indicator and probability kriging.

Second-order stationarity, also known misleadingly as the ‘weak-stationarity' assumption, requires that the expected value (i.e. the mean) is the same everywhere, and that the spatial covariance function (hence also the semivariogram for each lag h) is also the same everywhere.

The intrinsic hypothesis, which is an even weaker form of stationarity, does not impose a constant expected value, but does require that the expected value of [Z(x) - Z(x+h)] be zero for all vectors h separating any two points in the region of interest. The variance of this increment VAR[Z(x)-Z(x+h)] = 2 (h), and defines the variogram (hence (h) is the semivariogram) which is independent of location x. This is the form of stationarity required by ordinary linear kriging.

In 'universal kriging', an even weaker form of stationarity is allowed, in which the expected values of both Z(x) and [Z(x)-Z(x+h)] may vary regularly with location. In universal kriging, this regular variation is modelled by a polynomial function. There are also still weaker stationarity assumptions which can be used, and have been identified in the theory of generalised random intrinsic functions. The penalty paid for such relaxation of the stationarity assumption in geostatistical models which have been developed to use the weakest stationarity assumptions is that they are all much more difficult to use and more difficult to understand, and also - like universal kriging - have methodological problems such as the difficulty of obtaining a valid semivariogram model. Without a semivariogram - which requires the intrinsic form of stationarity (or a simple transformation to it, such as universal kriging purports to offer) - there is no geostatistics.

One key aspect of stationarity is the concept of homoscedasticity: that the variance does not change with location. In many geological situations of interest to the mining industry this is quite obviously untrue. For example the variance of mineral grade can be very different depending on location in the middle, on the margins, or outside an orebody. Geostatisticians have sometimes tried to overcome this by asserting that there might be a ‘proportional effect' (local variance is proportional to local mean value), but this would hold true only for lognormal distributions, which are usually inappropriate (indeed this is why lognormal kriging is relatively little used). The proportional effect, if it exists, is usually true only in a very qualitative sense. Another way in which geostatistical practitioners have tried to avoid this problem is by definition of ever smaller ‘homogeneous' zones. This, of course, often leads them into the trap of identifying zones which contain too little data from which useful statistics of any kind may be derived.

Donald Myers asserts (p.348) that we cannot test any data set for stationarity because a data set is only one realization of the random function. This seems to me an odd way to view reality. We have only one universe, and in an observational science such as geology any spatial data set, however large, is of course drawn from an unique population: we cannot carry out multiple replicate experiments to obtain different versions of an orebody. There is just one realization. The key question that properly should be asked is whether or not this is modelled well by a particular type of random function.

It is of course possible to play mathematical games, as in conditional simulation, and develop multiple computed realizations of a random function - but these are conditioned on properties of the data set that we have. These multiple possible realizations may help us to understand the range of potential variation in the regions between the data points, but they are only as good as the assumptions implicit in the random model and the particular computational method used. If the true underlying behaviour of the sampled region does not match the supposed random function, then it is likely that any realizations based on that random function will fail to approximate reality. It is also just as unlikely that any kriged estimates will give satisfactory approximations.

If any statistical methods can validly be applied in the observational sciences - and quite clearly they can - then it must be possible to develop tests of hypotheses based upon the data, even if those data constitute a single available realization. If it is not possible to test hypotheses against data, there is no scientific method, and the application of statistical (including geostatistical) methods would yield results which are of completely unknown significance. Therefore, if Donald Myers's statement is true and it is impossible even in principle to test a data set for stationarity, then the entire body of geostatistical methods built upon any stationarity assumptions must be rejected as unscientific. Alternatively, this statement itself could be wrong in principle. That there is no general test for stationarity of spatial data is perhaps simply because such a test has not yet been devised, not because it cannot be devised. In developing such a test, it will be most important to frame the right question - and there could indeed be multiple tests for different forms of stationarity. In fact, in the one-dimensional case, such tests have already been developed. Some of these are described in Gibbons (1971):

1. Versions of the runs test for randomness can be used as tests for trend at any particular scale - documented by Mood (1940) and Mosteller (1941).
2. Both Kendall's tau statistic (rank correlation coefficient) and Spearman's rank correlation coefficient can be used as tests against trend: what is computed is autocorrelation over all lag intervals to be considered, and this can be tested for significance in just the same way as the correlation coefficient can itself be tested in the general case.

Another approach to the problem, however, would be to view stationarity (of any form) as an unlikely property of a model suitable for fitting data which quite clearly vary in both expected value and variance from one place to another, and to seek an alternative model which does not require any such assumptions.

References:
Gibbons, J.D., 1971: Nonparametric statistical inference, McGraw Hill Kogakusha, Tokyo, 306pp
Mood, A.M., 1940: The distribution theory of runs, Ann.Math.Statistics, v.11, p.367-392
Mosteller, F., 1941: Note on an application of runs to quality control charts, Ann.Math.Statistics, v.12, p.228-232
Myers, D., 1989: To be or not to be... stationary ? That is the question, Mathematical Geology, v.21 no.3, p. 347-362

Stephen Henley
Matlock, England
steve@silicondale.co.uk
www.SiliconDale.com