Reposted from the CHy e-forum
In his “Note on Stationarity and Nonstationarity”, Lins (2012) neatly points out that the terms “stationarity” and “nonstationarity” are technical and need to be used according to their definition. He also reminds us of the rigorous definitions of these terms, which rely on probability theory and stochastic processes.
Misuse of scientific terms occurs frequently nowadays, and perhaps there are good reasons for such misuse. Take for instance the phrase “stationarity is dead” in the title of the article by Milly et al. (2008), referred to by Lins (2012). Do the authors really mean “dead”, that is, “no longer alive”, here? Well, “stationarity” is an abstract concept, not a material (let alone a living) object and, once it was devised, it could hardly be pronounced dead. To kill an abstract concept is difficult, like killing a ghost. For example one could kill the concept of stationarity, if one proved that the definition by Kendall and Stuart, which is copied in Lins (2012), is inconsistent. Of course, this is not the case in Milly et al. (2008). Certainly, the word “dead” dramatizes the message and the technical term “stationarity” makes the message more sophisticated. However, the real message is simpler, like “things are no longer static” Even in the latter phrase, “no longer” is misleading. In fact, we all know that things were never static. Thus, if we de-dramatize and correct the phrase, we will end up with “things are not static” or, more simply, “things change.”
But an article with this de-dramatized and correct title would not attract much attention. For this message has been widely known for a long time. Perhaps the earliest, and certainly the most elegant, formulation of this general natural law is due to Heraclitus: “Panta rhei”—“Everything flows”. Science is mostly about “change” and “flow.” At the same time, it is important in science to find invariant properties within change. In this respect, “steady flow” is not an oxymoron; rather it technically describes a flow whose certain observable properties, such as flow velocity and discharge (flow rate), do not change with time. The term “stationary” describes a concept more abstract than “steady”. Namely, “stationary” applies to a mathematical (stochastic) process, rather than a fluid flow, and refers to non-observable properties of the process, such as means and variances.
As long as “steady flow” describes a flow, a “stationary process” describes a process. It is a tautology to say that in a process there is change. Even a stationary process is about change. Change at which time scale? Some, who are familiar only with purely random processes, think that in a stationary process change is apparent at small time scales, but as the process is averaged out at longer and longer time scales, change disappears. Long term averages quickly tend to stagnancy. This is correct for purely random processes and also, to a lesser degree, for processes involving some short-term dependence (Markov processes). But Nature doesn’t work like that. In Nature, changes occur at all time scales—this has been now made clear thanks to abundant geological information and proxy climate data. Thus, to model natural processes we need stochastic processes different from purely random and Markov. A simple process with the desirable properties is the Hurst-Kolmogorov (HK) process and its general behaviour is summarized by Lins (2012). In a HK process, change is prominent at all scales.
In a recent Editorial, Matalas (2012), paraphrasing Mark Twain, stated that the announced death of stationarity is premature. I generally agree with this statement. The abandonment (“death”) of stationarity as a working hypothesis has nothing to do with detecting change in Nature. We could abandon stationarity and construct nonstationary descriptions when we are able to predict the future deterministically. As correctly pointed out by Lins (2012), this is meaningful and feasible in certain cases, e.g. when we model the impacts of urbanization. But to speak about nonstationarity based on climate projections is risky. For we know that such projections are not informative and valid as far as hydrological processes are concerned. Actually, we do not have practical reasons to abandon stationary descriptions, because, as pointed out by Lins (2012), flood records from undeveloped watersheds show clusters of trends going in both directions, with no consistent trend overall that could be attributed with confidence to a climate forcing, while the variations of hydroclimatic processes recorded so far can be represented quite realistically with stationary stochastic models.
Lins, H. F, Note on Stationarity and Nonstationarity, CHy e-FORUM, viewtopic.php?f=19&t=48&sid=8899e57ff4986df6fe6130365179dfcf,2012.
Matalas, N. C., Comment on the announced death of stationarity, J. Water Resour. Plann. Manage., 138, 311-312, 2012.
Milly, P.C.D., J. Betancourt, M. Falkenmark, R.M. Hirsch, Z.W. Kundzewicz, D.P. Lettenmaier, and R.J. Stouffer, Stationarity Is Dead: Whither Water Management?, Science, 319, 573-574. 2008.
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