Comments on: THe Ritson Coefficient

By: Take a Ritalin, Dave « Climate Audit

Take a Ritalin, Dave « Climate Audit — Sat, 04 Dec 2010 18:19:54 +0000

[...] to Ritson’s recent postings at realclimate about autocorrelation which I discussed here and here. Ritson is now promoting the idea that autocorrelation in proxy series is really low and that we [...]

By: fFreddy

fFreddy — Thu, 22 Jun 2006 22:25:51 +0000

The LIA+-50 example was straight off the top of my head and was only intended to illustrate a point: don’t take it too seriously.

Or that there is a better model than an AR1.

Unless AR1 describes global temperature perfectly, then certainly there must be a better model. Haven’t a clue what it is, though.

By: Phil B.

Phil B. — Thu, 22 Jun 2006 20:24:29 +0000

Re #22 fFreddy, given the equation x(n+1)= a*x(n) + e(n) were e(n)~ N(0,1) i.e. e(n) is normally distributed with zero mean and a variance of 1. Then the pdf for x(n+1) is N(a*x(n),1). Which is what you and Mark T., I believe are pointing out. For the Little ice age example you have described, you are now assuming that e(n) has a time varying mean and variance i.e. e(n)~ N(m(t),sigma(t)) or nonstationary statistics. Or that there is a better model than an AR1.

By: Mark T.

Mark T. — Thu, 22 Jun 2006 18:23:47 +0000

With a Markov process, you don’t care about the path by which you got to x3: x4 will be the same no matter what the different possible x2, x1, etc.

That’s a good description. I spent a lot of time with Markov this past semester in a Stochastic Modeling class and our teacher was nice enough to make sure we understood this very point.

Mark

By: fFreddy

fFreddy — Thu, 22 Jun 2006 18:14:32 +0000

I wonder if you could make an argument that the more data you could include in your “environmental snapshot” at any one time, the less you would need multiple ARMA terms ? So the multiple terms are (sort of) acting as estimators for all the relevant real world data that you can’t measure, and that the process could only be Markovian if your data vector contained absolutely all relevant data ?
Hmm. Bit spacey …

By: fFreddy

fFreddy — Thu, 22 Jun 2006 18:08:52 +0000

Re #20, Phil B.
What you say is true, but there are lots of different combinations of x0, e0, e1 and e2 which could get you to any given x3. With a Markov process, you don’t care about the path by which you got to x3: x4 will be the same no matter what the different possible x2, x1, etc.

Demetris’ point is basically that a climate system must have some “momentum”. You could imagine a point 50 years before the depths of the Little Ice Age, and a point 50 years after, where a snapshot of the global mean temperature would be the same at both points. But it would not be reasonable to say that the environment one year after each of those points has the same probability distribution.

By: Phil B.

Phil B. — Thu, 22 Jun 2006 17:31:44 +0000

Re #3&4, It is straight forward to obtain the z transform tranfer function for a moving average. Multiply this transfer function by the AR1 transfer function. By downsampling the output to the annual rate you now create a transfer function that contains this transfer function plus aliases. My guess is that your ARMA(1,1) modeling is a good approximation to this downsampled transfer function. A good reference for this downsampling is chapter 3 in Strang/Nguyen “Wavelets and Filter Banks” book.

By: Phil B.

Phil B. — Thu, 22 Jun 2006 17:07:52 +0000

Re #4 Determis, you wrote “Recall from the theory of stochastic processes that a Markovian process is by definition “a stochastic process whose past has no influence on the future if its present is specified” (Papoulis, 1991, p. 635). ” But isn’t the present just a weighted sum of the past. ie for a first order discrete markov process x(n+1)= a*x(n) + e(n) => x(3) = e(2) + a*e(1) + a^2*e(0) + a^3*x(0). Or did I miss your point?

By: UC

UC — Thu, 22 Jun 2006 15:31:57 +0000

IMO Ritson Coeff works well if the assumptions, very low freq signal and AR1 additive noise, are valid. Works for random walk as well. But if the signal has high-frequency components, the coefficient will be underestimated. And I don’t get it, why the signal (local annual temperature, right?) should have only slow components? It would imply that we could predict near future local annual temperatures with infinitesimal error. Makes no sense / I’m completely lost.

BTW, #14 Do you mean the Cooley&Tukey Tukey?

By: TCO

TCO — Wed, 21 Jun 2006 17:39:15 +0000

How would Ritson do on the Akiake Criteria? What about you? Should there be more factors? Physicality arguments (the summing to annual is very unsatisfying given that the vast majority of the series are annual resolution already).