Dear Pan,

in a paper I think I will expect the analyst to identify the non-stationarity that is actually happening in a real case in a basin close to the study one. I suggest to consider the indices you listed above as well as the the variability in the distribution of the monthly values (1 distribution for each month - changes in the distribution parameters) and in the cyclo-stationary AR1 parameters.

Then, I suppose a good way is to set to 1 the actual change and to apply an increase in the discrepancies (say 2x) to check an increase in the non-stationarity. 

Sincerely

Michele

I performed a similar process to what you describe in my recently completed Ph.D. dissertation entitled "HYDROLOGIC RESPONSE TO CLIMATE CHANGE IN CALIFORNIA: Observational and Modeling Studies". You can read about what I did here .

Sorry the URL above has a character appended that makes it not work. Try this one

http://yourwaterfuture.org/BruceDaniels/Dissertation.pdf

Dear Pan

One approach to characterizing non-stationarity is to consider the ecological/geomorphic impacts of anomalous flows. Fore example the impact of high flows will be determined by the lifestages effects. For salmon autumn flow events will move fish upstream to spawn which is good. In contrast high flows in the winter when the eggs are in the gravel can be bad. I like Michele Rienzner's idea of exploring monthly distributions of flow but you might also group data according to its ecological significant as is suggested with salmon.

Good luck on you study

Dear Pan,

In addition to some of the already proposed approaches I like to add a few things.

First I like to correct you on equating non-stationarity with  trends or changes in trends. If you'd like a better grip on that I recommend reading:

Koutsoyiannis, D., 2011a. Hurst-Kolmogorov dynamics and uncertainty: Journal of the American Water Resources Association, 47 (3), 481–495.

Koutsoyiannis, D., 2011b. Hurst-Kolmogorov dynamics as a result of extremal entropy production: Physica A: Statistical Mechanics and its Applications, 390 (8), 1424– 1432

The paper by Milly (2008) in that sense was a bit confusing. Nonetheless the main message is like with the stock market, results of the past are not a guarantee for the future! Meaning past series analysis of observations of e.g. water flow might no cover the observed variations  in the future ( the realisations that we like to model). The latter can still be a stationary process that we just have not understood the underlying physical processes properly!

Climate change will have an affect on water resources and the way it needs to be represent however that impact is much smaller than for example, land use change and the building of dams (albeit these can be put in or change as a measure to mitigate or response to climate change effects). These types of impacts would lead to clear changes in the physical processes and hence lead to non-stationarity.  So my suggestion is to try to come up with future scenarios (e.g. different land management, increased non-rural areas, industry) that impact water use, water distribution and use appropriate for your area and try to put that in a model. Not easy and highly subjective! So you will need to come up with sounds rationalizations of your choices.

Increasing your variance, changing trends etc. in your model is strictly speaking not introducing non-stationarity. That said there is certainly value in doing that in exploring the range of impacts for example of increased variance of water flows.

Enjoy the exploration!

Peter

Dear Pan,

You have received several comments,  all representing interesting perspectives.

My view is that non-stationarity is a characteristic of the real data generating process. You can investigate the nature of that non-stationarity but you don't get to choose it. If you want to investigate the nature of non-stationarity, good questions to ask are:

1. What is the order of integration? Is the series I(1), I (2) or of higher order of integration. There are several unit root tests which can help you determine this. One early ( and popular) test is Dickey-Fuller test and its augmented version. But there are also several others.

2. Does the series have deterministic or stochastic trend or drift ? A good time series book will have good discussion on these issues.

I do not have enough information about your intended use, but your concern about the degree of non stationarity can be perhaps translated into order of integration. A series that is integerated of order 1 i.e. I(1) will become stationary if it is differenced once. An I(2) series will need to be differenced twice to achieve stationarity.

We may not be able to forecast a non-stationary series such as a random walk series, (one example of which is the stock market returns). The reason is that non stationary series are not mean reverting.

Climate change may involve non-stationarity of flow regimes. However, the good news is that we may find a stable ( stationary) relationship between two non-stationary series. When this happens, the series are cointegrated. There are again several ways you can test for the existence of cointegrating relationships, such as the ARDL test, and many others.

In my opinion, it may not be very useful to generate synthetic flows with a view predict future flows, especially when there is climate induced non-stationarity. But it may be more insightful to relate (in cointegration sense) the flows of interest to other hydrological variables, such as flows in nearby river systems, snow cover, or climate variables.

Good luck!

Khalid

Dear Pan,

I suggest you look up papers dealing with PARMA (periodic ARMA) and PAR as a special case.. These models have been studied fairly extensively for river flows. Check papers by Vecchia, Lund. Anderson, Meerschart, Tesfaye.