Comment on the Announced Death of Stationarity

Milly et al. (2008) have announced the death of stationarity and state that stationarity can never be revived. The cause of death is given as anthropogenically induced changes in climate, the effects of which can not be undone, even with concerted effort directed to that effect. To contend with a nonstationary hydrology, Milly et al. state that an effort as focused as the Harvard Water Resources Program (1955–1960) may be necessary. These claims are questioned, whether stationarity is indeed dead or now has only secondary status.

The aim of time-series analysis is to develop an analytical structure that provides a good approximation of the “mechanism” that generated a time-ordered sequence of observations. The view that the structure of a time series is a composition of deterministic and stochastic components dates back at least to Wold (1938). This view has been widened to four components—trend, fluctuations about the trend, seasonal movement, and irregular or random movement [see, e.g., Kendall (1976)]. If a time series reflects trend, it does not mean that irregular movement is precluded from being a part of the structure of the time series nor does it preclude irregular movement being best described as a stationary random process. To state so emphatically that hydrologic sequences are nonstationary—stationarity is dead—is to suggest that hydrologic time series are devoid of stationary random components or that stationarity has only a secondary role to play. That claim is open to philosophic and scientific argument, but it is doubtful that observed hydrologic time series support the claim in operational terms of water management. Time series are not either stationary or nonstationary—stationarity and, consequently, nonstationarity may assume various forms. A time series stationary in, for example, the second moment, is stationary in the first moment but not necessarily in the third or higher ordered moment.

The assertion that the past was stationary, that the present is not stationary, and that the future will never be stationary needs to be qualified. It is doubtful whether nonstationarity in the form of a trend, positive or negative, can be sustained well into the future without catastrophic consequences. One can never be sure whether a trend is indeed a trend, however long the “trend” has persisted, unless the sequence itself terminates [see, e.g., Kendall (1976)]. Perhaps the stationarity of the past will never recur, but that does not rule out a future state of stationarity relative to an average different from that before. Fortunately, water-resource systems have inherent degrees of robustness, redundancy, and resilience that allow them to be operated quite reliably under extreme stresses, at least over relatively short periods of time (Matalas and Fiering 1977). Thus, water-resource systems are to an extent adaptable to contending with trending hydrologic circumstances, though the systems may have been designed to deal with stationary inputs.

Several techniques for contending with nonstationary processes are to be found in the literature. For some, stationarity is an explicit part of the structure of time series, as in the case of additive components; and for others, stationarity that is implicit may be revealed, e.g., by differencing, as in the case of autoregressive integrated moving average (ARIMA) processes [see, e.g., Box et al. (1994)]. The stationary component is basic to probabilistic analyses of a time series. That the development of additional means of dealing with nonstationary processes calls for as focused an effort as the Harvard Water Resources Program is questionable. An account of the program is given by Maass et al. (1962).

A principal aim of the program was to formally link the disciplines of economics and engineering. In part, the linkage was based on the notion of a production function that translates the hydrologic inputs to a water-resource system into the economic outputs of the system. The production function defines the set of technologically feasible designs, designs for which specific amounts of water at specific locations and at specific times can be met by the available streamflow input. To account for the fact that the magnitudes of the observed flows and their sequential order are unlikely to be repeated in the future, streamflow sequences were assumed to be realizations of persistent stationary processes. The assumption facilitated the development of synthetic hydrology, the generation of realizable “future” sequences that in turn facilitated the handling of stochastic inputs to a production function. If nonstationarity is viewed as a deterministic component of a time series, then no conceptual difficulty is introduced in dealing with nonstationary inputs to a production function.

Detecting elements of nonstationarity, giving them analytical description, and then removing them from an observed time series is important. It is not a trivial task. It is a task that calls for research that can be funded through proposals submitted to appropriate organizations and government agencies. This is the route taken by an earlier generation of hydrologists and water managers in coming to grip with persistence, short-and long-term memory, in resolving an old issue referred to as the Hurst phenomenon, an issue as perplexing then as nonstationarity is now.

For more than four decades, hydrologists have been aware of the propensity of stationary long memory processes to generate time series of finite length exhibiting features of nonstationarity. Yet those processes have not been brought fully into the discussions of whether nonstationarities are apparent or perceived in “short” sequences of hydrologic fluxes.

Before passing judgment on whether stationarity is indeed dead or whether it has secondary status, the degree to which real or perceived nonstationarities affect the uncertainties underlying the processes of making water planning and management decisions needs to be assessed. The assumption of stationarity has not yet been pushed to the limit of its operational usefulness in the face of a changing climate.

The extensive body of literature on the theory of stationary stochastic processes continues to expand. The theory has served hydrology well over the years, and it promises to continue doing so. If only for attaining first-order approximations, stationarity greatly facilitates dealing with multivariate problems, e.g., regionalizing hydrologic information, generating synthetic traces at a given set of sites conditioned on specific properties of persistence and on specific marginal distributions, and providing a baseline for assessing the distributions of record events.

In summary, stationarity is almost always a part of the composition of a time series, explicit or otherwise. Stationarity may be overlooked, but it remains alive and well. To paraphrase Mark Twain, the announced death of stationarity is premature.