You may have missed the main point of the
Naturally trendy? article,
CoolSceptic.
It is that it is quite wrong to assume that the statistics of random walks apply to a system with physical forcing.
Nowhere in Naturally Trendy are random walks used, invoked, referenced, anywhere.
LTP is not a random walk. If you don't understand something so basic about stochastics, you have no hope whatsoever of understanding the work of Cohn or Koutsoyiannis.
Nowhere does the actual article by Cohn and Lins (which I linked to, so you have no excuse for not reading it) mention random walks, or use the statistical properties of random walks.
The fact that realclimate incorrectly believe Cohn and Lins is about random walks, merely highlights the fact that realclimate didn't read the paper properly, either.
I think
CoolSkeptic is right about this, to approximately the same degree that the RealClimate blog author cited by
RealityCheck was right to say "We have heard arguments that so-called 'random walk' [sic] can produce similar hikes in temperature" and to place similar claims about AR, ARMA, ARIMA, and FARIMA models in the historical context of claims about random walks.
Here's the main thing to understand about what's going on here: To reject random fluctuations as an explanation of apparent trends, you have to select some statistical model of the kind of random fluctuations you're taking as your null hypothesis. The more complicated the class of statistical models you select for your null hypothesis, the more difficult it becomes to reject the null hypothesis at some standard threshold of significance.
As Cohn and Lins say in
their paper:
Cohn and Lins said:
Choice of trend test, however, does matter when computing trend significance. The simplest test, Tβ,{0,0,0} (which assumes no LTP), finds strong evidence of trend, a p-value of 1.8 * 10−27. Tβ,{ϕ,0,0} (which allows for short-term persistence) yields a p-value of 5.2 * 10−11, 16 orders of magnitude larger and still highly significant. The p-value corresponding to either Tβ,{0,d,θ} or Tβ,{ϕ,d,0}, an unadjusted LRT trend test that considers both short-term and long-term persistence, is about 7%, which is not significant under the null hypothesis. In changing from one test to another, 25 orders of magnitude of significance vanished.
In other words, those who wish to deny the statistical significance of climate trends will always be able to postulate a statistical model that's capable of such complex behavior that a null hypothesis based upon that statistical model cannot be rejected at customary levels of confidence.
Cohn and Lins are troubled by this:
Cohn and Lins said:
This result is somewhat troubling given the uncertainty about the stochastic process and consequently about which test to rely on.
That doesn't bother me, because I have enough knowledge of physical sciences to realize that many physical phenomena can be simulated using sufficiently complex mathematics.
That is not to say all mathematical models of physical phenomena are equivalent. Some are more accurate than others, while some may be accurate enough but lack predictive power because their free parameters are data-dependent rather than universal.
I believe the RealClimate blogger is trying to say the more complicated classes of statistical models that are being used to deny the statistical significance of apparent climate trends acquire their requisite variety of behavior by adding data-dependent parameters, which is unsatisfactory on scientific grounds. I don't think the RealClimate blogger said it very well, but I think that's roughly what he was trying to say.
I think Cohn and Lins said it better in the last sentence of their abstract:
Cohn and Lins said:
From a practical standpoint, however, it may be preferable to acknowledge that the concept of statistical significance is meaningless when discussing poorly understood systems.
I also think Cohn and Lins would have made that point more effectively had they not devoted their discussion section to an attempt to spin that valid, practical, and (statistically) fairly obvious point into a claim that "These findings have implications for both science and public policy."