## Tuesday, May 17, 2011

At Climate Audit, Steve McIntyre has been frequently writing about Chladni patterns in connected with principal components of autocorrelated data. He developed it for the Steig paper, and was disappointed when J Climate wouldn't allow its inclusion in the response. And he's recently claimed an appearance in another paper.

Chladni patterns are modes of oscillation, originally of a vibrating plate. Now people more often think of a drum membrane, which is a slightly different wave equation, but the idea, and patterns, are similar.

I must admit that I hadn't heard of Chladni before Hans Erren drew attention to them in Steve's first post. Some interesting history there. But I am familiar with the modes in question.

Steve thinks that if a Chladni pattern emerges, that somehow means that the result is showing that rather than the information about the climate pattern being sought, so the information content is reduced. I don't agree - there are reasons why the patterns arise, and they are just as informative in PCA as they are in wave studies. I'll try to show why.

Warning - mathematics (and $$\LaTeX$$) after the jump.

#### Resonance and wave equations

Resonance is familiar with acoustics. If you speak in the open air, your voice propagates away in all directions, attenuating without reflection or selective amplification. If you stand in a partly enclosed cavity, your voice is slightly louder in certain frequency bands. In a totally enclosed bare room, you hear a characteristic booming response - some frequencies are much louder.

These are the ones that excite a resonant mode, in which the air oscillates, but the normal velocity at the boundary is zero. A 3D version of the modes of a vibrating stretched string, which has zero velocity at its ends.

The essential requirement is that the wave energy must be confined but not dissipated. That is why the resonance improves in deeper cavities, for example.

The wave equation for pressure, say, is (with c=speed of sound) $$\nabla^2 p = \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2}$$ If you substitute a resonant mode $$p=P \sin(\omega t)$$, then the equation is $$\nabla^2 P = -(\frac{\omega}{c})^2 P$$ P is the resonant mode, and so is an eigenvector of the Laplacian $$\nabla^2$$. The resonant frequencies correspond to the eigenvalues.

For Chladni's plate the wave equation is more complicated, but the principle is the same.

#### Spatial autocorrelation.

Steve McI also described the simple Toeplitz autocorrelation coefficient matrix that you get in one dimension for a spatial model. With N+1 equally spaced points, the coefficients can be assumed to be powers of r - the correlation of adjacent sites. The matrix is: $$R = \left(\begin{array}{ccccc} 1 & r & r^2 & \ldots & r^N\\ r & 1 & r & r^2 & \ldots \\ r^2 & r & 1 & r & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ r^N & \ldots & r^2 & r & 1 \end{array}\right)$$ The Toeplitz property is that all terms on each diagonal are the same. This correlation matrix has a simple inverse: $$R^{-1} = \left(\begin{array}{ccccc} q & -qr & 0 & \ldots & 0\\ -qr & 2q-1 & -qr & 0 & \ldots \\ 0 & -qr & 2q-1 & -qr & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & -qr & q \end{array}\right),\quad q=\frac{1}{1-r^2}$$ Still a Toeplitz matrix, almost, but also banded - tridiagonal. The deviation from Toeplitz is at the top left and bottom right corner terms.

#### Relation to Laplacian and the wave equation

From the last equation, \begin{align} R^{-1} &= qr\left(\begin{array}{ccccc} 1/r & -1 & 0 & \ldots & 0\\ -1 & r+1/r & -1 & 0 & \ldots \\ 0 & -1 & r+1/r & -1 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & -1 & 1/r \end{array}\right) \\ &= qr\left(\begin{array}{ccccc} 1 & -1 & 0 & \ldots & 0\\ -1 & 2 & -1 & 0 & \ldots \\ 0 & -1 & 2 & -1 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & -1 & 1 \end{array}\right) + q(1-r)^2 \left(\begin{array}{ccccc} 1/(1+r^2) & 0 & 0 & \ldots & 0\\ 0 & 1 & 0 & 0 & \ldots \\ 0 & 0 & 1 & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & 0 & 1/(1+r^2) \end{array}\right) \end{align} If you remember finite differences, the first matrix is just the negative of the second difference operator, corresponding to the 1D Laplacian. And the second is almost a multiple of the identity, and is small if r is close to 1.

That's the key to the connection between Chladni patterns and the autocorrelation matrix R. The inverse of R and the Laplacian of the wave equation differ by close to a multiple of the identity, which means they have almost the same eigenvectors. And the eigenvectors of R and $$R^{-1}$$ are the same. R is symmetric and positive definite.

#### Subtle differences

Well, an unsubtle difference is that Chladni oatterns are not 1D. But the same reasoning works - it's too messy to set out here.

The subtler difference is that the diagonal correction is not quite Toeplitz. That relates to the notion of boundary conditions for the wave equation, which is indeed critical for resonance.

At this stage I'll have to just arm-wave on that - it does in fact give the zero normal boundary condition, which is sufficient for resonance.

#### Conclusion

The same mathematics that gives Chladni for the wave equations gives similar eigenvectors for the spatially autocorrelated matrix. It isn't a spurious consequence. Consequently their appearance in, say, Steig et al 2009 doesn't mean that the PC's are "just Chladni" any more than they are just autocorrelation. Spatial autocorrelation is an essential part of the results, and the Chladni patterns reflect that.

I think that's enough heavy maths for now - I could later go more into the properties of Chladni patterns, and in particular into the implications for eigenvalue pairing (for PCs) which SM criticised in Steig et al. That would generate pretty pictures.

Meanwhile, I'll just relay these pictures of resonant modes from Wikipedia where more Chladni patterns on the disc and sphere can be found.

1. You need to read back over some of the classics of EOF/PCA literature. Sadly, however, I can't remember the names - but its back in the 1980's I think.

The assertion is fairly simple: that if the EOFs you're picking up are just the "resonant modes" of the plate, then they are likely noise {{citation needed}}.

The key point is that the modes are resonant modes of the EOF process (so to speak) but aren't actually modes of anything in the physical system itself. If you were analysing the actual motion of a metal plate, of course, the situation would be different: then, they would be physical modes of the system.

2. Belette,
My contention is that the eigenvectors of the wave equation (resonant mode) are the same as eigenvectors of the autocorrelation matrix, with basic spatial autocorrelation. On that basis the patterns aren't noise, but evidence of that spatial correlation, which is a real part of the PC/EOF pattern.

3. This paper talks about standing waves in the Antarctic. It was published many years before McIntyre thought of the concept. He talks about the Chladni patterns as if they are something novel he has thought up.

4. Anon,
Yes, circumferential modes etc have been studied for a long time. However, Steve M's angle is different. The pattern he talks about comes from PC analysis of the correlation matrix of temperature. He seems to think they are an artefact which diminishes the analysis.

My argument here is that mathematically, spatial correlation and standing waves should produce similar eigenfunctions, based on their form. Of course, it is also likely that standing waves contribute to the temperature correlation.

5. Nicke's post is right on, and of course, this why the Journal of Climate editor agreed to my objection to Steve's rant in the O'Donnell paper about Chladni patterns. This was just another ad hoc idea that they *claimed* might somehow make our analysis problematic, but didn't bother doing any work to show that it actually was problemetic. O'Donnell called this 'armwaving objections', but I was merely objecting to their armwaving. Of course, they'll say whatever the heck they want, substantiated or no.

6. Nick Stokes, much of what you say about origin and structure of Chladni patterns I would not doubt. However, I think your interpretation and that of SteveM are subtly different. You both agree that they arise from spatial autocorrelations, but I think that you merely stop by saying that since that correlation is the essence of the PC analysis its interpretation cannot be spurious.

What if those patterns were unique to the PC rendition of the AVHRR data and not part of the original AVHRR data? If one then attempted to use those patterns for explanatory purposes one could be misinterpreting. I have been doing some regressions of the various temperature anomaly data sets and reconstructions for the 1982-2006 period and was hoping to make some posts at your thread titled "Trends in Antarctica". Will you be responding to my previous posts at that thread? I have found that, for instance, the trends in the S(09) PC rendition and correlations of that S(09) data with other data series for that period for the 1982-2006 period is much more sensitive to latitude, longitude and altitude than the raw AVHRR anomaly data using the same comparisons.

I would suppose that one would want to look for the Chladni patterns in all the reconstructions from S(09) and O(10). The O(10) reconstruction, RLS, uses spatial correlations without the use of PCs while the EW reconstruction uses spatial correlations with the PCs.

I am a bit disappointed that you have not delved more deeply into some of this subject matter that you introduced before walking away to other subjects

7. Kenneth,
I'm all too conscious of things that I had been hoping to do, but got distracted with other things. But on this one, I haven't been distracted yet. I will write more.

I think Steve's specific complaint about S09 was not the rendered AVHRR data, but interpretation of the PC's. "S09 utilize a criterion that only modes that appear visually similar to known physical modes are deemed significant. They determine the first 3 modes to meet this criterion. Specifically, the first spatial eigenvector is claimed to be correlated with the SAM index and the second is claimed to reflect the zonal wave-3 pattern."
And he says that autocorrelation could have produced the same result. I contend that that is not a refutation. Resonant modes in turn produce correlation. You can't separate the cause and effect.

BTW, I'm sorry that I went overseas just as your use of TempLS was getting interesting. But I'm back now, and catching up. I'll be happy to engage on that, and I'm very interested to hear how it is going.

8. Eric: Nicke's post is right on, and of course, this why the Journal of Climate editor agreed to my objection to Steve's rant in the O'Donnell paper about Chladni patterns

Not to be argumentative here, but I apparently missed that part in the exchanges. It is my reading that your criticism was more along the lines of "unoriginal and misleading" than "misinformed". That is not true? Also where specifically did the Editor agree with your criticism? (From the exchanges, it appears the authors voluntarily withdrew this section rather than having that mandated or recommended by the editor, who appears to have said nothing at all regarding Chaldni patterns in his decision letter). Also where did you state in your criticism anything of a similar vein to what Nick wrote here? I don't see too much commonality in rereading your original review.

Nick, the question with Chladni patterns arising from EOFs is distinguishing spurious correlations from real ones. Validation is the key here, and the onus should be on the person using the EOFs to demonstrate that he is applying them properly, and not simply chasing spurious correlations in his EOF decomposition.

Pointing out that they can arise spuriously doesn't demonstrate that they did, only that they could, and that this therefore is something that needs to be checked in the EOF analysis by the authors of the EOF analysis and not simply by its critics.

I don't think you need an entire section of O10 dedicated to Chaldni patterns if it isn't going to delve into the question in any more depth than the original paper did (and even agree with Eric's choice of words "misleading" in that context). The O10 authors (enough of them anyway) agreed to this, as is evidenced in their response to the Eric's comments:

"[...] we agree that we spend insufficient time developing this as it might apply to S09."

9. Carrick,
"Validation is the key here, and the onus should be on the person using the EOFs to demonstrate that he is applying them properly, and not simply chasing spurious correlations in his EOF decomposition. "
In the course of describing the PC's, S09 made this observation:

"The first three principal components are statistically separable and can be meaningfully related to important dynamical features of high-latitude Southern Hemisphere atmospheric circulation, as defined independently by extrapolar instrumental data. The first principal component is significantly correlated with the SAM index (the first principal component of sea-level-pressure or 500-hPa geopotential heights for 20u S–90u S), and the second principal component reflects the zonal wave-3 pattern, which contributes to the Antarctic dipole pattern of sea-ice anomalies in the Ross Sea and Weddell Sea sectors."

I don't believe that the use of the PC's as basis functions for interpolating ground temperatures obliges them to identify the physical causes. The argument is simply that, since they worked for satellite readings post 1980, they are likely to be a good basis for ground station interpolation (as opposed to, say, orthogonal polynomials). The fact that physical patterns can be recognised is a bonus.

10. Since we're dealing with satellite data here, there is certainly the possibility that some of the spatial structure is the AVHRR EOFS is associated with non-climate-related processes associated with the AVHRR systematics (e.g., shifts in time of measurement for example).

So yeah, it's kind of important to understand the origin of the EOFs you are observing, if you want to use the EOFs derived from AVHRR data to interpolate ground based data. It might not matter (assuming stationarity really holds) if you wanted to apply AVHRR derived EOFs to interpolate AHVRR data of course.

11. Eric,

You are incorrect on this. In fact, if you look at a similar spherical shape to the Antarctic and add a peninsular region to the North having spatial autocorrelation, you would see a rotation of the node of PC2 and 3.

That is the key to what Steve pointed out and that is a real issue to what was unfortunately successfully eliminated from our paper.

The only reason I'm even wasting the bits on this (because I know people don't get it) is that this has become a 'trend' in climatology. Patterns pulled from autocorrelation.

PCA is designed to find the primary axes of movement, having a near circular autocorrelated dataset (antarctic) with an extrusion (peninsula) guaranteed the axis of PC2.

12. circular not spherical

13. I asked Steve to elaborate and prove some points wrt modes being spurious. He could not/would not/did not. Old news...and repeated pattern. REal skeptics should eschew this silly crew and their blog historionics.

14. Jeff #11,
I think that's one of the misunderstandings about the modes. Yes, a circle does generate repeated eigenvalues, and that actually creates a 2D subspace of eigenvectors. Two basis values are shown, orthogonal, but you can rotate the pattern to any angle and it's still a mode.

However, as soon as you break the symmetry, the eigenvalues separate, and unique eigenvectors emerge. The higher eigenvalues separate more rapidly - ie big separation for small loss of symmetry. You've looked at just adding the peninsula and noted that the eigenvectors then split with this as the axis. I would expect that. If the shape is non-circular in other ways, as it is, then other influences will determine the orientation. But I can't see why this matters.

15. While contemplating Nick Stokes' algorithm for in-filling Antarctic temperature data from ground station and AVHRR satellite grids, I decided to download all the data series from O(10) including the RLS and EW reconstructions used in O(10), the S(09) reconstruction featured in S(09), and referred to hereafter as S09, and the raw (after cloud masking) AVHRR data. Also included were the ground station data from the manned and AWS stations.

I was primarily interested in the 1982-2006 period as I wanted to make comparisons with the AVHRR data that became available during that period. The downloaded data was put into monthly anomaly form based on the 1982-2006 time period. In my analysis I used 50 of the ground station data that were most complete for that period and further were within the area covered by the AVHRR grids.

The RLS method in O(10) uses only the spatial component of the AVHRR data for in-filling the missing ground station areas, while the EW method in O910) uses both the temporal and spatial relationships between ground stations and AVHRR data.

The S09 method uses RegEM and combines ground and AVHRR data. A major part of the S09 methodology involves the retention of just 3 AVHRR PCs.

In my initial analysis of these data sets I have included correlations and trends of these data using latitude, longitude and altitude as explanatory variables. I must admit that I was surprised at how influential these variables turned out to be. In think that perhaps a better regression would have used a reference point and distances from it, but in the analysis presented below I simply used latitude, longitude and altitude as taken from the downloads from O(10).

This thread has discussed the effects of Chladni patterns on a PCA that retains just a few PCs and I thought perhaps the differences I saw between the O910) and S09 methods with regards to latitude and longitude and possibly altitude might be related to the effects of Chladni patterns.

RLS does not use AVHRR PCs, while the EW method use 150 AVHRR PCs as compared to S09 which uses only 3 AVHRR PCs. In the linked first two tables below it can be seen that, when regressing trends or correlations versus latitude, longitude and altitude, overall S09 is more influenced by these variables than are the RLS and EW reconstructions. The questions that arise in my mind are if you see a change in influence of these variables from the raw data to that data reduced to a few PCs is that change one that you might observe because the PCs are eliminating sufficient noise or are the changes a matter of an influence of geometry that might be considered spurious?

While the ground station data is limited, it is interesting to look at the correlations and trends of the reconstructions and AVHRR data (nearest grid to station coordinates) versus the ground stations (linked below in the 4 tables in the second link below). We can see that the RLS reconstruction gives the best correlation at 0.96 and EW next at 0.87, the S09 lags far behind at 0.49 and the AVHRR data is in the middle at 0.71. The same order exists for the regressing trends of AVHRR versus the RLS, EW and S09.

In the linked tables we can see that, although the data are small, the ground station and corresponding AVHRR data when regressed as trends and correlations show little or no influence from the latitude, longitude and altitude. The influence of these explanatory variables is seen to a greater extent in the O(10) and S09 reconstructions with by far the greatest influence being seen with the S09. The ground station data generally confirms what is seen in the analysis that uses all the AVHRR grid point data for all the reconstructions.

http://img695.imageshack.us/img695/9256/rlsews09avhrraltlatlon.png

http://img10.imageshack.us/img10/3707/groundstatrlsews09avhrr.png

16. Based on these relationships shown in my previous post, I would think we could consider the RLS reconstruction a good proxy for the ground station data. In that form we can debate whether one would expect to ground station data or the AVHRR data to better reflect the true temperatures in the Antarctic by comparing the RLS and AVHRR data for the 1982-2006 period. I am considering looking at breakpoints (if any exist) of the difference series derived between the AVHRR and the RLS series.

17. Unfortunately my previous post has not yet posted. I'll repost after I am assured it is not merely in que somewhere.

18. While contemplating Nick Stokes' algorithm for in-filling Antarctic temperature data from ground station and AVHRR satellite grids, I decided to download all the data series from O(10) including the RLS and EW reconstructions used in O(10), the S(09) reconstruction featured in S(09), and referred to hereafter as S09, and the raw (after cloud masking) AVHRR data. Also included were the ground station data from the manned and AWS stations.

I was primarily interested in the 1982-2006 period as I wanted to make comparisons with the AVHRR data that became available during that period. The downloaded data was put into monthly anomaly form based on the 1982-2006 time period. In my analysis I used 50 of the ground station data that were most complete for that period and further were within the area covered by the AVHRR grids.

The RLS method in O(10) uses only the spatial component of the AVHRR data for in-filling the missing ground station areas, while the EW method in O910) uses both the temporal and spatial relationships between ground stations and AVHRR data.

The S09 method uses RegEM and combines ground and AVHRR data. A major part of the S09 methodology involves the retention of just 3 AVHRR PCs.

In my initial analysis of these data sets I have included correlations and trends of these data using latitude, longitude and altitude as explanatory variables. I must admit that I was surprised at how influential these variables turned out to be. In think that perhaps a better regression would have used a reference point and distances from it, but in the analysis presented below I simply used latitude, longitude and altitude as taken from the downloads from O(10).

This thread has discussed the effects of Chladni patterns on a PCA that retains just a few PCs and I thought perhaps the differences I saw between the O910) and S09 methods with regards to latitude and longitude and possibly altitude might be related to the effects of Chladni patterns.

RLS does not use AVHRR PCs, while the EW method use 150 AVHRR PCs as compared to S09 which uses only 3 AVHRR PCs. In the linked first two tables below it can be seen that, when regressing trends or correlations versus latitude, longitude and altitude, overall S09 is more influenced by these variables than are the RLS and EW reconstructions. The questions that arise in my mind are if you see a change in influence of these variables from the raw data to that data reduced to a few PCs is that change one that you might observe because the PCs are eliminating sufficient noise or are the changes a matter of an influence of geometry that might be considered spurious?

While the ground station data is limited, it is interesting to look at the correlations and trends of the reconstructions and AVHRR data (nearest grid to station coordinates) versus the ground stations (linked below in the 4 tables in the second link below). We can see that the RLS reconstruction gives the best correlation at 0.96 and EW next at 0.87, the S09 lags far behind at 0.49 and the AVHRR data is in the middle at 0.71. The same order exists for the regressing trends of AVHRR versus the RLS, EW and S09.

In the linked tables we can see that, although the data are small, the ground station and corresponding AVHRR data when regressed as trends and correlations show little or no influence from the latitude, longitude and altitude. The influence of these explanatory variables is seen to a greater extent in the O(10) and S09 reconstructions with by far the greatest influence being seen with the S09. The ground station data generally confirms what is seen in the analysis that uses all the AVHRR grid point data for all the reconstructions.

http://img695.imageshack.us/img695/9256/rlsews09avhrraltlatlon.png

http://img10.imageshack.us/img10/3707/groundstatrlsews09avhrr.png

19. Kenneth,
Again, apologies for the spam filter. I see you have three comments there which seem to be identical. Please let me know if I should restore either of the other two.

20. Kenneth,
Apologies here too for the spam filter. I'm looking at your results, and will respond in a few hours. It's rather early morning here.

21. Nick, you left the one I wanted and where I wanted it. I think my problem may have occurred when I did not insure that the preview registered before I posted - or the post was lengthy and susceptible to the spam filter.

Also my post at the original thread was my mistake.

22. McI has been babbling Chladni for years, but has never buckled down and made a clear assertion of an artifact. I challenged him on this years ago. Told him to make a clear assertion of differences and the like. He just refused and went back to his yuck yuck blog games. Pushing the Chladni into the Steig crit paper was just stupid kitchen sinkism. The guy is so used to running his own blog that he has lost the ability to make clear arguments. Could you imagine any business person paying this guy for a report? He's just all over the map disorganized, adhom, and illogical.

23. I am responsible for some of the best acoustic guitars on the planet. We used to mess with Chladni patterns on guitar tops. It's mildly interesting, but they suck. They're for the marketing department: a sucker is born every whatever stuff...

Give me a man with an ear and a sharp tool every single time.

I wonder if he used them to find minerals?

24. Nick. I think your interpretation

"Spatial autocorrelation is an essential part of the results, and the Chladni patterns reflect that."

is very sound. If you analyse a given dataset with important spatial autocorrelations, the leading EOF will (obviously) tend to have a Buell (Chladni) pattern. Obviously, if you change the domain (including soma Antarctic Peninsula), other variability appears, but this is self-evident since you have changed the dataset. Additionally, I don't understand why spatial autocorrelation should automatically mean there-is-no-signal-here. It happens that the signal shows spatial correlation, no that there is no signal.

Quoting from:
Monahan, Adam H., John C. Fyfe, Maarten H. P. Ambaum, David B. Stephenson, Gerald R. North, 2009: Empirical Orthogonal Functions: The Medium is the Message. J. Climate, 22, 6501–6514.
doi: 10.1175/2009JCLI3062.1

"When a field is characterized by spatially homogeneous statistics (i.e., invariant from place to place), the EOFs will be strongly influenced by the size and shape of the domain (e.g., Buell 1975, 1979; Richman 1986; Dommenget 2007)."

The starting point is that "When a field is characterized by spatially homogeneous statistics" ... then it will appear a "Buell pattern". This does not mean that "In any event that the EOF pattern is Buell-like .... then, the field is just spatially homogeneous noise"

I think that there are some more severe criticisms that can be done to EOFs. Still, they are useful, particularly for data compression.

jon

25. Isn't this basic Sturm-Liouville theory? The wave equation is S-L, so will produce mutually orthogonal eigenfunctions, just like EOFs.

26. Anon - yes, I agree with all that, including the last.

Martin, Yes, my 1-D wave equation example is just Sturm-Liouville. The 2-D and 3-D patterns of alternating high-low also are governed by S-L concepts, though the theory doesn't apply directly. The orthogonality can be derived similarly (symmetric, positive definite).

My main point is that the autocorrelation matrices have S-L like operators as their inverses.

27. Chladni patterns can look very much like the atomic orbital shapes from the Schrodinger wave equations. I have no reason to doubt that Chladni patterns have meaning for violin acoustics.

The leap in reasoning here appears to be that since Chladni patterns are a function of spatial correlations and that they have meaning in some applications that that somehow means that it has meaning for all applications and lends some physical meaning to the patterns beyond the spatial correlation. The point that you all appear to be ignoring is that SteveM related the S09 Chladni patterns to a geometry of a circle. Would that not be that much different than a wave equation of atomic orbitals telling us something about the geometry of the orbitals. In the case of the Antarctica and what S09 was searching for, the application would appear to be very different.

I pointed out that regressions of S09 against latitude, longitude and altitude are very different than the cloud masked raw AVHRR data and are more dependent on those variables. I did not see any discussion of that point.

28. Here is a link to what O'Donnell found with PCA and spurious teleconnections of the earth's temperatures. You might want to discuss the points he made - that PCA can be a very useful tool in some applications, but that in itself does not preclude it being abused and misrepresented.

http://noconsensus.wordpress.com/2010/03/30/pca-sampling-error-and-teleconnections/

29. Kenneth,
One thing that bothered me is that Steve related to Chladni patterns to the disc, and then deduced than the eigenvalues should be paired. But that pairing fails as soon as the circular symmetry is lost.

Another way of seeing the relation between autocorrelation and the wave equation is that the point Greens functions for the Helmholtz equation ( ie eigenfunctions of the Laplacian or Fourier Transform of the W.E) is an exponential-like function (Bessel Kn in the plane), which is pretty much what you'd write down for autocorrelation.

The reason that atomic orbitals look similar is that they are the solution, not of a Helmholtz equation $\nabla^2 \phi = \lambda \phi$
but of $\nabla^2 \phi = (\lambda + V(r)) \phi$
where V is a potential function. But since the orbitals are usually visualised in the region where $(\lambda + V(r))$ has not varied by a lot, the patterns are similar.

30. I'll try that Latex again:
The reason that atomic orbitals look similar is that they are the solution, not of a Helmholtz equation $$\nabla^2 \phi = \lambda \phi$$
but of $$\nabla^2 \phi = (\lambda + V(r)) \phi$$
where V is a potential function. But since the orbitals are usually visualised in the region where $$(\lambda + V(r))$$ has not varied by a lot, the patterns are similar.

31. Kenneth,
I've been re-reading that tAV link. I'm still trying to relate the sphere example to the wave equation interpretation. But I think this is the key para that I dispute:

The issue of teleconnections and physical meaningfulness has come up here before – most notably with the work on the Steig Antarctic temperature reconstruction. The primary point of contention there was that Steig limited his analysis to the first 3 eigenvector pairs by using the “physical meaning” argument and a sampling error argument from North (1982). Steve McIntyre showed that very similar (and physically non-meaningful) patterns result from EOF analysis on an object shaped like Antarctica with exponential correlation functions and precisely zero physical dynamics.

I think the assertion that they are physically non-meaningful is wrong. I would go back to my demonstration that the inverse of the discretized wave operator is very like an autocorrelation matrix. Now we're used to a wave equation having modes and attaching physical significance to them, but we think of autocorrelation as being just a local relation. But when you decompose a correlation matrix (derived from points in space) into eigenvectors, you are actually doing something that is not local but involved all the points. It does indeed teleconnect. That 1-D matrix that I wrote has trig function eigenvectors.

32. Nick, I see that the point that O'Donnell made with reference to teleconnections seems to be missed in your replies. That you can show that the EOF patterns has a mathematical basis and is tied to spatial correlations is not the point at all. Nobody is disagreeing with that observation and analysis. The point is that scientist can interpret these patterns as something physical with cause and effect that they are not. The scientist does not go through the analysis you did and point the patterns as having a mathematical basis but rather they point or sometimes hint that the patterns have a given and specific physical meaning. You seem not able to admit that that interpretation could be wrong and keep going back to a mathematical analysis but never saying what that means in terms of a specific physical interpretations.

O'Donnell showed a case where teleconnections did not exist in the original data but could be seen in the EOF patterns.

33. Kenneth, I don't think I'm missing that at all. My point is basically that space correlation and wave-like pde's are intertwined. So I don't think it is at all unreasonable for a scientist to seek to interpret EOF's from a correlation matrix as having physical significance in terms of wave effects. Which means teleconnections. It's saying, in effect, that a wave equation has been observed.

There's a much higher bar to claiming that the correlation proves the teleconnections. Then you have to determine how robust the patterns are to changes in the correlation matrix.

But if you have other physically based reasons for expecting resonant modes, which was Steig's situation, then I think it is appropriate to look at the correlation eigenmodes to try to identify them with standing wave patterns.

34. Actually in Steig's case I believe he attempted to place a physical interpretation on the 3 EOFs he retained in order to rationalize why he limited his selection to 3. It all gets very arbitrary at some point - but I suppose they can always fall back on the Nick Stokes defense.

You have never answered or commented on my observation that the original AVHRR data has much less dependence on latitude,longitude and altitude in regressions than the S09 reconstruction. Does that mean anything with regards to Chladni patterns? Is that occurrence an indication of a problem and/or artifact of the S09 methodology?

35. Kenneth, re:6/6/11 11:21am
When in doubt, go with the measured data. Reconstructing a data set in order to get something that is more manageable mathematically cannot really make the results more robust.

Unfortunately climate data tends to be so sparse that any in-depth analysis quickly runs in to the fact that there simply isn't enough data to extract a signal from the noise.