Usually in least mean squares you want to whiten the model error. I would like to suggest that it may be better to only whiten the parts of the spectrum that do not contain a common driver signal. It may be better to suppress the parts of a spectrum containing a common driver signal. This way although less information will be available at least the estimate will be less sensitive to assumptions about the model error (noise).

w(k+1)=w(k) – k //w(k+1)>0
w(k+1)=0 otherwise

The second is a moving equilibrium
w(k+1)=alpha w(x) +(1-alpha) w_typical(k)

In the second model the constant alpha is between zero an one and governs how long the tree takes to get back to normal. w_typical(k) is the typical growth rate for a tree at age k. The time constant is defined as the k such that (alpha)^(k)=alpha/e and is a good measure of the time a tree takes to get back to normal. The first model is a long memory process. Wagmann suggested that trees may be best modeled by long memory processes. In both models the inputs can be dealt with by a moving average component. Short term effects can be introduced into the first model by cancellation. For instance:

w(k+1)=w(k) – k + u(k) – u(k-1) //w(k+1)>0

where u is an input. So we notice that even though the model has infinite memory with respect to it’s state it is possible to choose the moving average part so no inputs are remembered. We can combine the two models as well to give a long term memory part and a short term memory part as follows:

w1(k+1)=w1(k) – k //w1(k+1)>0
w2(k+1)=alpha w2(x) +(1-alpha) w_typical(k)
w(k+1)=k1 w1+ k2 w2 (k+1)

As I mentioned in my previous post death can be dealt with though multiplication. If we multiply by the previous state this gives us an exponential decay in tree growth. Since we know tree growth is nearly linear a model that gives an exponential decay in growth is not a good choice. An alternative is to multiply by a factor of the form:

(w(k)/w_max)^(1/n)

where w_max is the maxium possible tree growth. This way the factor is very close to one and does not dominate the difference equation unless the previous growing season was really bad like in the case that the tree died.

Once we figure out how to best determine that with the instrumental data we will be in that much better a position to figure out how to construct past temperatures. As for what kind of model to use for reconstructing past data it would depend on the proxy. It would be necessary to consider a Varity of models to see what works and for some proxies like tree ring nonlinearities are essential to include in the model. A tree wring model may look as follows:

W(k*dt)=sum(aj * si ((k-j) dt))* W((k-1)*dt)
sj=(1-((T((k-j) dt))-Topt)/Tw)^2)*log(c) for T-Topt>0
0 otherwise

si the quality of the growing conditions
Topt is the optimal temperature
Tw the temperature difference from optimal conditions at which the tree doesn’t grow
K the time index
dt the width of the time step.
W(k*dt) is the width of the tree ring at time k*dt
C is the partial preassure of carbon dioxide

The model I am suggestion is that tree width growth is a moving average model of the quality of the annual growing conditions which I define by the function sj but I include an auto regressive part in a unusual way. The auto regressive part is multiplied by the moving average part instead of added to it. This is interesting because the model is such that if the tree doesn’t grow the previous year it won’t grow the next year. Moreover it models the decrease of growth of a tree with respect to age. If I want to include more auto regressive terms I might do it as follows:

W(k*dt)=sum(sum(a_{j,n} * si ((k-j) dt))* prod(W((k-n)*dt))^(1/n)

So what we are now doing is considering various orders of geometric averages of past growing years. This is interesting because each of the geometric averages are such that if the tree didn’t grow at all in the past year the geometric average will be zero. Thus we are considering past growth in such a way to model the fact that trees die.

The other part which I didn’t mention is the non station aspects. This can be done by treating the nonlinear parameters we identify as stats with noise as the input. For example a random walk. Since trees are probably quite non stationary the low frequency data like pollen and boar holes will probably contribute the most information to past climate well the tree rings may help fill in the missing high frequency information.

30. I wouldn’t say that we NEED a PCA expert, in the sense that we (really Steve…ok…he does all the work and we kibbitz…unless you are going to start.) I just think that it might help. Totally agree on the time series expertise, Steve has mentioned it. It is more of an issue then the PCA transforms. Both consultants would be nice! Steve is pretty good on his own, don’t get me wrong. But more smart heads will help. That’s just how things go.

There is a large body of academics which believe it is necessary to first toughly study a field before attempting to conjecture on the subject and apply ones deductive logic to reach their own concussions. For instance I have gotten the response before on a newsgroup on symbolic mathematics, if you don’t have anything constructive to say don’t say anything at all.

I however believe the deductive process used to arrive at the answer is as important as the answer. I think basic problem solving and brain storming skills are as important as the solution. If this wasn’t the case why are mathematics students require to spend so much time at proving things which are already known. Understanding goes far beyond what is typically found enumerated in the average textbook. Memorization does not equal understanding.

The deductive process has many dead ends. Edison tried 100 times before inventing the light bulb. Not all dead ends are useless. For instance, one estimation procedure I recently attempted is to assume the signal is all noise then try to reduce the error in the prediction. That is minimize norm(E[(y-AX) (y-AX)']) with the assumption y is all noise. I thus assumed that the autocorrelation of the error y-AX was equal to the autocorrelation in y. I then applied weighted least mean squared to get an estimate of the singular values in the singular value based pseudo inverse and use the MATLAB routine fmin search to try to find the singular values which minimize: (E[(y-AX) (y-AX)']) or more precisely minimize:

norm(P-P Phi’ A’- A Phi P + A Phi P phi’ A’)

Where P=E[(y-AX) (y-AX)']

And phi is the singular value base pseudo inverse. I ran the routine several times and my estimates appeared to converge. It seemed to fit the data quite well but not as well as the assumption that the error was independent and identically distributed. I then checked to see how norm(E[(y-AX) (y-AX)']) varied with each iteration and to my surprise it didn’t change. The optimization routine was only finding local minimums due to round off error.

So what seems like a failed estimation routine reviles something seemingly obvious and surprisingly interesting. That is the noise in the residual E[(y-AX) (y-AX)'] cannot be reduced by weighted least mean squares. All that happens is the uncertainty in the residual (Are Aproi information) is spread between the uncertainly in the measurement (P) and the uncertainty in the fit (A Phi P phi’ A’). Thus for weighted least mean squares the error bars that you get for the fit are equal to the error bars that you assume in the first place. Moreover the error bars in the parameters you are trying to estimate are based on the error bars assumed aprori for the fit. This is something I have not read in the discussion of weighted least mean squares but it clearly highlights the weakness in the method when the error is not known aprori’

The paper is actually quite interesting if you are seriously interested in coping responsibly with the uncertainty problem. The reason it seems opaque to some and boring to others is that it is fairly complete and not unnecessarily brief. i.e. It is everything that Nature tries not to be. This is what climate science papers are going to look like in the future, by the way, if the auditing process has its intended effect. So don’t complain about these kinds of papers!

We don’t know why the paper was rejected from GRL, but the authors would probably happily tell you why. I suspect it is because as #1 John A suggests: uncertainty is soooooooo boring. The associate editors and reviewers themselves would probably not dare to say it is too boring a subject to publish. But the Editor might suggest it was too boring for the GRL audience.

Audits like this are generally going to have a hard time getting published because they are narrowly focused on targeting one paper or class of papers. Get used to it. The gatekeepers favor papers that have broad relevance. (A look at the criteria for publication in any major journal will show that there is a strong bias toward “interesting” and “broadly relevant” papers.) You can already hear the criticism from the AGW anti-MM camp: too focused on one paper (MBH98), one group (HT), one pattern (HS), etc. We all know that’s not true – that CA is focused broadly on the METHODS of climate science – but that message is not resonating with the believers.

The more you focus on BUILDING something positive, as opposed to DISCREDITING something negative, the more acceptable your work will be in the mainstream literature, because the more broadly useful it will be. (It is a mistake to assume the literature is a forum for “good science”, conjecture & refutation. It is a place to market tools & ideas that others are willing to pay for (in untraceable utils of goodwill). That steams science purists like Popper & Feynman. But that’s the way it is.)

The Wegman outcome is favorable in that now a fair number (if small percentage) of good statisticians may sudenly emerge to rise to the defense of science, if they are encouraged to do so.

There is a seed – a natural alliance between science purists and CC skeptics – that will grow beyond your expectations … if you let it. Fence-sitting skeptics need to know that skepticism does not make you a planet-hater.

The house of cards will fall when the reality of uncertainty starts sinking in, pushing the fragile institution beyond its tipping point. Believers do not understand just how unstable that structure is. They do not understand the uncertainty problem. It takes too much hard slogging through the dry statistical literature, like Burger, to ‘get it’. Suggestion for believers on both sides: if it bores you, or hurts your brain, you had probably better read and understand it.

As for the paper itself …

Not a particularly helpful observation. Except to point out the obvious: your hypothesis may well be correct (uncertainty is the lynchpin in the AGW house of cards), but you are going to need some heavy weight to get you there. TCO says you need a PCA expert. I think you need a time series simulation expert. You need someone who works in computational statistics (which uses computer simulation and numerical solutions) as well as classical mathematical statistics. It’s not often you can get both in one package. (I believe Wegman is of the latter variety.)