TCO asked about physical processes that can generate time series with autocorrelation properties. This is a harder question than it seems and leads into the giant topic of stochastic processes, which rapidly gets very complicated. I’m not in a position to give a thorough answer, although it’s a topic that interests me a lot. I’ve provided here some extended quotes from V Klemeš, who has spoken to the issue (although more towards persistent processes than AR1 or AMRA(1,1) processes. He also writes acidly on climate modeling, which I also excerpt here, e.g. the following:
This I would call an “honest and humble” search for signals as opposed to the boastful claims of assorted “modellers” about all kinds of climate-change effects, motivated more by politics than by science and reflecting prejudices rather than fact.
The discussion is oriented towards persistent processes, rather than simple AR1 or ARMA(1,1) processes. I’ll come back to it with a few more examples on another day. Klemeš got into hydrology via the study of civil engineering in Czechoslovakia – he reminisces that this was one of the few university courses that a person of politically incorrect background could then study.
Any hydrologist is necessarily interested in these series with highly persistent autocorrelations. Mandelbrot modeled such time series (e.g. Nile river levels, tree rings, varves) with fractional difference “long memory” processes. While the very gradually declining autocorrelations of such geophysical series could be modeled through “long memory processes”, Klemeš argued (very plausibly in my opinion) than it was inconceivable that nature engaged in such bureaucratic depreciation accounting and sought a plausible mechanism to achieve gradually declining autocorrelations. A couple of his criticisms about people who apply inappropriate mathematical models:
Somehow the operational attitude toward mathematical modeling, the exaggerated strife for mathematical tractability and convenience ("Oh Lord, please keep our world linear and Gaussian") has blurred our sense for reality…
Our mathematical models, including time-series models, by which we try to describe geophysical records are only as good as our understanding of the processes that generated them. …a geophysical process must be analysed and understood from a physical point of view before it can be adequately mathematically described. ..
There is one important but often overlooked fact which I shall cause the mischief factor of mathematics. It sometimes causes mathematics to frustrate rather than facilitate a scientific discovery: the specific mathematical method used in data analysis may introduce into the result some features which are then wrongly attributed to the physical process being discussed. Perhaps the most common case arises in correlation analysis where the mischief factor often manifests itself as spurious correlation…
To go back to TCO’s question, Klemeš’ own suggested mechanism for generating persistence is “semi-infinite storage” models, a model which seems very plausible to me, but which looks difficult to fully understand the behavior :
An exceptionally fruitful concept for the mathematical modeling of hydrological processes is the so-called semi-infinite storage reservoir, especially the type with a fixed bottom and no fixed maximum (Klemeš, 1970, 1971, 1973]. It adequately describes the basic mechanism common to such different water reservoirs as, for instance, a lake, a single dew droplet, a glacier, a groundwater basin and a man-made reservoir operated for flood control or hydroelectric generation. Their common property is on the one hand, the possibility of running dry and the other, the fact that they have no fixed limit of storage capacity (water level in a dam can rise to any elevation above the dam crest, as is demonstrated in the history of dam failures, and a glacier can cover whole continents as is documented in geological history.)
Even a very simple model of this type can reveal very disturbing properties to be expected in hydrologic processes. For instance, a single non-linear reservoir fed with white noise will produce output that is nonstationary, a first-order Markov chain with time variant serial correlation and random component [Klemeš 1973]. ..
Most geophysical processes involve strong cumulative effects: they themselves represent processes of storage fluctuations.
I can sort of see how you can picture El Nino events from this perspective. Klemeš points out that there are often successive levels of integration in geophysical processes, adding to the modeling difficulties. For example, levels of Lake Ontario integrate precipitation; flow of the St Lawrence River is an even further integration.
The difficulty of the problem increases as we move from precipitation records to records of hydrological processes, which already by themselves reflect the effect of some hydrological storage and thus in their raw form already represent cumulative processes or include them as their components.
This is a pretty simple model to picture, but you can intuitively see how it generates autoregressive features, even if it’s hard to go from the process description to the autocorrelation properties. (There are many other processes which yield autocorrelations). He has an extended discussion of how the process of integration applied several times, quickly yields "cycles" and "trends", applying Slutsky’s 1933 Econometrica article. So when he points out that:
Hydrological series have a tendency to exhibit more pronounced and smoother cycles than precipitation series.
it is entirely consistent with a Slutsky integration phenomenon.
He observes re proxy data:
Such and similar effects may contaminate interpretations of past climatic trends, in particular when they are reconstructed from proxy data.
I really will try to post up on the Slutsky issue. Meanwhile, I’ll quote from Klemeš on climate modelers:
Nor do I see any point in constructing time-series models for “scenarios” of runoff, precipitation, temperature etc., for the “doubling” of CO2, year 2050 etc.; implying an onset of some “stationary” state”.. This I would call an “honest and humble” search for signals as opposed to the boastful claims of assorted “modellers” about all kinds of climate-change effects, motivated more by polities than by science and reflecting prejudices rather than fact. …
” their thinking is insincere because their wishes discolour the facts and determine their conclusions, instead of seeking to extend their knowledge to the utmost by impartially investigating the nature of things. [quoting from Confucius)
the ease with which scenarios involving “impacts” of any arbitrarily imposed “climate change’ can now be produced even by amateurs and dilettantes whose grasp of problems does not extend beyond an ability to insert DO-loops into the various models to which they may have gained access.
This has led to “metabluffing’ where, in contrast to ordinary bluffing described above, it is now not just the various questionable approximations of real (historic) evens that is meticulously polished and presented as rigorous sciences, but concoctions produced by arbitrary and often physically incongruent changes of model parameters, ‘process realizations’ that may be unrealizable under known physical law. ..
It has often been pointed out [Klemeš 1982b, 1991b, 1992a; Rogers, 1991, Kennedy 1991] that not much more can be said about the hydrological effects of a possible climate change beyond the fact that it introduces another source of uncertainty into water management. Specifically I summarized the achievements of a decade of “climate-change-impact’ modelling in these words: “Basically the only credible information obtained from the complex hydrological modeling exercises relating to climate variability and change is that, if the climate becomes drier, there will be less water available and the opposite for a wetter climate….