How to force it, ie make W(n) as near iid as possible? A reasonable requirement is that W(n) should have zero autocorrelation for lag 1, and I think, though I haven’t checked, that this is equivalent to what the existing process is trying to achieve. But you could adopt other limits on the A-C of W(n), and vary τ to achieve them.

In terms of going to higher order, as I said earlier, I don’t think that is useful except insofar as it improves the approximation of dT/dt, and I doubt if this is critical.

I have to say that hanging this whole analysis on the assumption that observed temperatures were generated by a white noise flux seems a slender reed, but I think that is what is done.

Let me ask a question here which refers to Lucia’s next post (“What is the Climate Time Constant? Refining the Estimate I”) and to her previous post on estimating the time constant. How should one force the AR(1) model on the data?

As a non-physical scientist, my instincts are to just go ahead and estimate the AR(1) model temp(detrended) = Constant + B*lag1(temp(detrended)). No need to even say AR1 in the model since there are no other explanatory variables. For the GISS data, the AR1, i.e. the coefficient B of the lagged, detrended temperature, for annual data is about 0.63. For the monthly it is about 0.70.

But I could go to the ACF of the detrended data and do a nonlinear fit of the equation ACF(lag i) = B^(lag i) plus error of course. Now because I am using the geometric form I don’t have to worry about negative ACF values and can use the whole series. If do that, the AR1 for the annual goes up to 0.78 and the monthly goes up to 0.95.

In over the lowest to the highest that is a range on tau of 2 to 20. That range has nothing to do with the theory arguing for the estimation, just the question of what is the best AR1 representation of the data. As an econometrician I am going to say the annual AR(1) model has best “information” criterion although I really think the model is either some higher order AR process, an ARMA process, or maybe a fractionally differenced (long memory) ARIMA process (ARFIMA — AutorRegressive Fractionally Integrated Moving Average) — or maybe we just have the wrong units for the time scale altogether.

I should also note that when one uses the whole ACF the effect of higher order AR coefficients in the process affect ones estimate of the AR1 value unless they are accounted for explicitly. Thus, I would again come back to the direct estimation which comes back to the exclude explanatory variable bias, e.g. the AR1 coefficient in an annual ARMA(3,3) model is around 0.35, a tau of less than a year.

Now what I don’t know is suppose we have an AR(1) process. How often will we reject that process as the true model in comparison to some higher order ARMA model, AND when we do reject the AR(1), what is the best way to estimate the AR1 value? If the answer to the first part is a large percent, say 20-30%, then the answer to the second part, which is in the realm of statistics, with give us guidance. If the answer to the first part is a trivial percent, then it is back to the physics and the physical explanation of the higher order terms in the time series.

Marty

I like your idea of forcing the AR(1) model and estimating AR1, because the error could then give an indication of how good is the de model relating sensitivity to AR1.

You may right in that I am missing the point. But I am trying. The Schwartz article is an empirical estimate at the time constant, and what I am looking at is how he is pulling a number out of the data. So let me try saying my point another way. Look at Schwartz Figures 5 and 6. I’m saying that those patterns are AR(1). We probably all are willing to accept that whatever is generating those patterns has a positive AR1 coefficient and I am not questioning the underlying model, but I am questioning the way Schwartz goes about pulling his tau — which is just -1/ln(AR1) if the model fit well — from the data. If you say OK I am going to force the AR(1) model on the detrended data, then why not do exactly that: force it, make an estimate of the AR1, which is around 0.6 for the annual data and gives a tau of about 2 years. But taking the correlogram of the series is neither consistent or inconsistent with the underlying theory, it is — and this only my opinion, albeit a somewhat informed opinion — bad statistics.

If you build yourself a little model of the DGP, you will see if you use an AR(1,4,10,14) model with the fitted coefficients, you will generate ACF patterns similar to Schwartz’s on a regular basis. But if you stick in the AR(1) model coefficients, you will usually get a much lower tau and seldom get the pattern. Thus, if you are using the correlogram from the data, you are implicitly contradicting the AR(1) assumption on the DGP. This is what you tell me not to sweat. OK, but by using the correlogram from the data you will also be estimating something other that the time constant of an AR(1) process. Hence, I believe that it is necessary to explicitly go about extracting, estimating or whatever the AR1 value underlying the model.

Thanks, I halfway there. And now let me cause more problems, albeit the same idea.

The physical basis for an AR(1) seems to me to me the same as that for an AR(3) process. (Let me be clear on notation here: AR(1) means a first order autoregressive process. AR1 is the first order coefficient of any autoregressive process, i.e. an AR(1), AR(6) and an ARMA(3,3) all have an AR1. A higher order AR processes, say an AR(24), might have only a few non-zero coefficients. I do not believe there is any standard notation, but I use AR(1, 3, 12, 24) to denote a 24th order AR process with non-zero coefficients at lags 1, 3, 12, and 24.) An AR(1) process has geometric decay every period upon which the processes is defined, e.g. on monthly data. An AR(4) process has geometric decay every 4th period. This is elementary. But with regard to the physical world the time unit is arbitrary. Why couldn’t the fundamental unit of time be gamma0*PI*seconds? Or maybe, within our conventional time scales, why isn’t the decay nested, i.e. a decay of a decaying process? I haven’t done any work with such series, but I suspect that they have some distinct differences from a pure AR process.

Or let me get a bit nuttier here, suppose our data were continuous (we can always fit the N observations perfectly with an N-1st degree polynomial in time). Now we don’t have to look at the “delta t” — sorry I am not certain of how to paste a Greek delta — as an interval that can continuously vary instead of varying only for the natural numbers. Of course as the nuclear physics used to say to me “Die ganze Zahl schuf der liebe Gott, alles Ubrige ist Menschenwerk.” (But they were just quote Leopold Kronecker, the tormentor of Cantor.) But maybe the positive half of the real line is a bit approximation for time than various natural numbers, and then what is fundamental delta? Not clear. We could solve for the delta which gives the largest first order autocorrelation subject to being greater than 2*(1/sqrt(N)) and say that best fits the exponential decay model. Or maybe better is keep varying delta then create the discrete time series and test the AR(1) model versus the AR(K) or ARMA(K,P) model in a likelihood ratio test. If we get a clear cluster of AR(1) non-rejections, we have a division of time consistent with an AR(1) view of the world. But right now, whether it is monthly or annual, Schwartz is fitting an AR(1) shaped model into a non-AR(1) shaped data set. (Oh, in case, I hadn’t mentioned it: the serial correlation Lagrangian Multiplier test — the Breusch-Godfrey Test — fails for lag inclusion of 3 through 12 with annual data, and an ARMA(1,1) model can’t be rejected in favor of an AR(1) one.)

I guess if wanted to do that, I think the way to proceed is to fit a trend plus AR(K?) or ARMA(K?,P?), then take the AR(1) process out of the fit and subtract the remaining part from the data. Then you have — in theory — an AR(1) process plus the residual from the “true” model. The AR1 then gives you your tau: -1 / ln(AR1). End of story. No empirical correlograms or what. Just an AR(1) simplification and an estimate consistent with that simplification.

…or more time constants are needed.

“Surely part of the validation of the climate models is to test the validity of the prediction against the data?”

is directed at the modeling process, not the research that provides an estimate by examining actual data.

I’m a bit mystified at the lack of such an elementary notion as model validation. If the models do indeed predict a single time constant, then the empirical methods that do likewise should not be subject to examination and discussion as to whether one, two, or more time constants.

Indeed, it seems that the cart has been put before the horse. Why was the empirical data fit not done in 1993? Why no previous discussion about the number of time constants necessary to describe the system. Would it not be appropriate to do the analysis and have the discussion before building a model? And are these model predictions themselves an end-product, or are they used as inputs elsewhere?

Initially T=0, so you’re just steadily heating a body, so a=F/C

and with the assumption that the response to any perturbation has relaxation time τ, then b=-1/τ

therefore, when finally T has settled and dT/dt=0, then F/C=T/τ, or sensitivity T/F = τ/C

Note that it is essential that, for this logic, T actually satisfies Eq 1. That is what is quite wrong, I think, with Scafetta’s mixing of two AR(1) processes. You lose this essential requirement.

I think the problem is that the physical scientists want to connect each term in the fit to a physical meaning or at least explain where they come from. So, we would either need some explanation of what the other terms in the ARMA mean, physically, or an explanation of why they are required to get need to get rid of some known feature that comes up in the process of handling data.

So, under Schwartz’s model, the time constant of the climate is claimed to have a physical basis and be meaningful. I can justify white noise based on measurement imprecision.

The time constant comes from saying the climate obeys

dT/dt = -T/τ + α F where F is forcing and τ is the time constant and α is just another parameter.

That’s basically conservation of energy.

The idea that we can just pull that out of the series the precise way we are arises if we say F is white noise. This because an AR(1) process if and only if F is white noise. Otherwise, it’s a different process.

But I can’t connect higher order terms to the physics. Or I could come up with a physical basis for higher order terms, but I then need to say that I believe the “forcing” to be red noise or something that creates the other terms in the ARMA. (And you can see Arthur is arguing F is probably not white because T isn’t red.)

Now, mind you, the forcing isn’t white. But scientiests will often accept the possiblity of white noise because it’s the simplest thing you can pick. But if you start advancing other types of noise, they are going to demand an physically sound reason for that particular noise being appropriate for the Forcing, F.

And the problem with that to get scientists to accept other terms in F, we need to explain what they might be. I can explain adding white noise to the Temperatures– that measurement uncertainty.

On the other hand, if there is a good statistical explanation for why the terms “just arise” in a string of data even when the real underlying process for temperature, T, is red, then that can fly. (Sort of like, we use N-1 in the denominator of a standard deviation because of what happens when we estimate the variance using the sample mean.)

I just don’t know the reason the other terms in the ARMA yet. (Doesn’t mean there isn’t one. I just don’t understand the physics of math.)

Everybody appears to be working with the empirical AutoCorrelation Function (ACF). That is the estimate of the autocorrelations at each lag from the detrended time series. Coming from a different discipline, economics, I almost never work with the ACF. Rather I work with the estimated ARMA model. Then once I have an estimate of the ARMA (that is stationary), I calculate the theoretical ACF from the ARMA coefficients. The reason for this is that just about all the coefficients of the empirical ACFs are full of the small but pervasion random autocorrelation one gets with the ACF of a purely random, non-autocorrelated series. (Look at the probability of the Q-statistic move up and down for the ACF of white noise, and you will get an idea of the problem.)

Further, if you are fitting a pure AR(…) model, i.e. no MA terms, then don’t you get the time constant (or is it time constants when there are more than one AR terms?) directly from the AR coefficient?

Any help (with explicit equations if possible) would be appreciated.

Thanks, Marty

The physical reasons I can think of are imperfect, but here the are:
1) If you examine the TSI charts, you” see loads of high frequency noise. That might tend towards “white”.
2) Over the course of the year, we have somewhat random things happen, which are not related specifically to weather. These include different people spewing out more or less particulates, those particulates falling on snow in different places, volcanoes erupting etc.
3) Internal behavior of the climate could add more noise to the “f” term.

So, collectively a bunch of things might tend to result in the noise being “white” or approximately so. (Or, the assumption could be bad.)

BTW. I’m adjusting what I did for the bias– still assuming the white noise is ok. I’m doing it in this order not because the stuff you are suggesting is irrelevant, but because I need to have a clue how to deal with bias in my method first.

I’m up to 15 years or so. I need to proof read though.

Is there a theoretical basis for the “white noise” assumption? I don’t see the term “white noise” in Schwartz 2007 at all…

I.e. don’t you somehow need to separate out the driving causes of temperature change from the temperature response?

Yep!!! In the full problem, you need to know the spectral properties of both the random component and the mean component.

Schwartz gets around the need to know the forcing using the classic method: Making an assumption!

Schwartz assumes the random component of the external forcing for the temperature anomaly is white. This is certainly not exact, and may or may not be a good approximation. In principle, the definition of “anomaly” takes care of the annual periodicity (and this can be shown.) However, if the random component of the external component is not spectrally white (or nearly so), this will introduce errors.

In his Schwartz’s first paper, and in his reply to comments, Schwartz assumed the mean component is a linear ramp function that’s been going on “forever”. That’s the assumption that permits linear detrending.

So, Schwartz made two assumptions to deal with the exact issue you are pondering. As with all assumptions, either one or the other could be sufficiently wrong to result in inaccurate answers. (This is why showing the autocorrelation looks “right” is an important fiduciary step. If the data looked self-inconsistent with the assumptions in any way, that would cast serious doubt on the method.)

Scafetta tried to account for the non-linear nature of the mean component by by detrending using GISS model mean data but also did a linear detrend. This made very little difference to the answer. (This likely means the amount of ‘false noise’ introduce by detrending is relatively small compared to the amount of real, honest to-goodness random “noise” in the extrenal forcing to the system. )

Isn’t the whole point of the relaxation time (tau) to be a measure of the response of temperature to forcing? I.e. don’t you somehow need to separate out the driving causes of temperature change from the temperature response? So how can we possibly hope to extract it from the temperature time series alone? I feel I’m missing some key insight here!

However, what I am saying is:
a) there is a bias.
b) the adding the white noise to the synthetic process may make a difference in the magnitude of the bias. (I don’t know how much though, but it’s not zero.)
d) adding white noise to the synthetic process may make a difference to the standard deviation in the population of time constant determined experimentally. (I don’t know if it makes it larger or smaller, and in particular, I don’t know if this will matter to my method or Schwartz’s.)
e) Tamino (FASM) didn’t determine the magnitude of bias or standard deviation with white noise added to the synthetic process process.

So, while there is a bias and there is a large standard devaition, I don’t currently know how large the bias or the standard deviation is/are!

I want to run my method with a bias correction. Also, for my method, I need to know the bias over a large range of ρ because I apply a linear fit to the ρ vs lag time data, so the information in FASM isn’t enough for me to go on. (And given what Schwartz said in his response, it appears there is no standard method. So, as long as I’m doing the synthetic data thing, I’m just going to estimate the biase with the synthetic data!)

I’m going to deal with detrending issues after looking at the problem with no detrending. I’ll look at the issue Arthur addresssed then too. The reason for this is that the bias/scatter issue exists both with and without detrending, and with/without the “spikes” in the spectrum. So, to deal with all of them, I need to do this first.

I’ll probably be posting today.

However, the part of Tamino’s work that I was noting was where he generated AR(1) data with a 5 year lag, put it through the analysis, and came up with a shorter relaxation time. I can’t see that that would be affected by the noise of experimental measurement. It’s just a statistical processing issue. Incidentally, I note that UC seems to have done the same thing, but with sometimes longer relaxation times as a result. Neither T nor UC where very explicit about what they did at this stage (though UC gave R code for the GISS treatment).

It’s actually a bit unclear to me what you are suggesting. Is it that data should be synthesized with an AR(1) process, then have white noise added? Is it the difference from the pink noise of the AR(1) that matters?

Anyway, I’ll keep trying to get through your Dec post, UC’s posts and the CA discussion. Incidentally I noticed that Francois O and UC both brought up the detrending issue, and I haven’t seen a real answer to it.