Steig Eigenvectors and Chladni Patterns

Last year, I did a few posts connecting spatial autocorrelation to something as mundane as the Stahle/SWM tree ring network. In the process, I observed something that I found quite interesting – that principal components applied to geometric shapes with spatially autocorrelated series generated Chladni patterns, familiar from violins and sounds. The Antarctica vortex represents an interesting example of another fairly constrained geometric shape. I’ve alluded a few times to the similarity of the supposedly “physical” Steig eigenvectors to Chladni patterns and today I’ll show why I made this comparison. In the figure below, I show left – the first 3 Steig eigenvectors and right – the first 3 eigenvectors from spatially autocorrelated sites arranged on a disk.

Both the similarities and differences are worthy of notice.

The first eigenvector of an isolated disk weights the interior points more heavily than points around the circumference – a pattern observable in the Steig eigenvector as well. The first Steig eigenvector is displaced somewhat to the east – but you’ll notice that Antarctica is by no means perfectly circular and the “center of gravity” is displaced to the east as well. My guess is that the same sort of graphic done on the actual Antarctic shape will displace to the east as well. I’ll check that some time.

The 2nd and 3rd Steig eigenvectors also have points in common and points of difference with the simple circular case. Both Steig eigenvectors are “two lobed”. Because the eigenvalues of the first two circular eigenvectors are identical, these two eigenvectors (EOFs are the same thing) are not “separable” in the sense of North et al 1982. Any linear combination is as likely as any other one. So in that sense, for a disk, there is no preferred axis. In the Antarctic case, the Transantarctic Mts look like they provide a reason for orienting the second Steig eigenvector. It would be an interesting exercise to evaluate just how much extra oomph there is in this symmetry breaking. The third Steig eigenvector is not as cleanly perpendicular to the 2nd eigenvector as in the disk example, but I think that one can readily discern this structure.

   
   
   

In the circular case, it’s not as though eigenvectors of lower order are uninterpretable. I’ve posted up a pdf showing the plots for the first 16 eigenvectors, which interesting and pretty patterns (thanks to Roman for showing how to make pdfs straight from R). In the circular case, you can get radially symmetric eigenvectors – a structure that seems like it might be highly relevant if one wants to incorporate seemingly negatively correlated data from islands in the Southern Ocean. I’m not saying that this is a relevant structure, only that this eigenvector might be empirically relevant – or it might not be.

As to how to summarize or truncate – I have no prescriptions. There are lots of schemes – Preisendorfer’s Rule N has been mentioned. But I haven’t seen any head-to-head analyses of Preisendorfer’s Rule N against eigenvectors generated from spatially autocorrelated data – so I don’t know whether it’s a good rule in these circunstances or not.

Let’s return to the description of these three eigenvectors in Steig:

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 20S–90S), 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 4,8

No proof or evidence was offered for any of these assertions. You’d think that Nature would require authors to provide evidence, but seemingly not in this case. The observation about the second eigenvector seems untrue on its face: the reported eigenvector has 2 lobes, not 3, which, in my eyes, disqualifies this interpretation.

The plots here provide more evidence (though not “proof”) that the eigenvectors are simply what you’d expect from applying principal components to spatially autocorrelated data on a not quite circular shape than Steig offered up in support of his assertion that these things are “meaningfully related to important dynamical features of high-latitude Southern Hemisphere atmospheric circulation, as defined independently by extrapolar instrumental data.”

64 Comments

  1. jeez
    Posted Feb 24, 2009 at 4:35 PM | Permalink

    You didn’t do this on the back of an envelope did you?

  2. Fred Harwood
    Posted Feb 24, 2009 at 4:40 PM | Permalink

    Perhaps:

    You didn’t do this on the back of an envelope, did you.

  3. bernie
    Posted Feb 24, 2009 at 4:44 PM | Permalink

    Steve:
    Are you saying that Steig’s analysis is an artefact of the technique that he applied to data that varied with the distance between measurement points, i.e., spatially autocorrelated? Do I have this right?

    Steve: I’m not making any assertions right now. I’m just looking at the data from a different and interesting perspective. It sure looks to me like the Steig eigenvectors are connected to spatial autocorrelation to some degree. This doesn’t mean that that’s all that’s in them. There are differences as well as similarities and maybe the differences rise to being significant. Steig says that there’s a physical interpretation. Perhaps Steig has some convincing evidence to support this. I don’t know what evidence he has or doesn;t have since he didn’t provide any supporting evidence in the article. So I can’t comment on what he hasn’t presented.

    • bender
      Posted Feb 24, 2009 at 4:53 PM | Permalink

      Re: bernie (#3),
      Careful. You must realize that the SAM itself is nothing more than an EOF and therefore subject to the same kind of domain-shape distortion effects. That one (new) PC maps to one (pre-existing) EOF and a second PC maps to a second EOF is not particuarly impressive. Nor does it necessarily imply “separability”. Climatic EOFs may be orthogonal. That doesn’t make them “separable”. In my world, ocean and atmosphere are fully connected. Separability of modes (be EOF analysis) may be an illusion of limited observation time.

      • bender
        Posted Feb 24, 2009 at 4:58 PM | Permalink

        Re: bender (#7),
        The Texas sharpshooter fallacy: this PC can be interpreted as mapping to that EOF.

  4. Steve McIntyre
    Posted Feb 24, 2009 at 4:45 PM | Permalink

    #1. Pretty much. Here’s how I did the simulations.

    First I made a grid of points within a circle.

    #make a data frame of physical points
    K=33 #chosen to be 32+1
    Z=cbind(as.numeric(gl(K,K)),rep(1:K,K),rep(NA,K^2))
    Z[,1:2]=Z[,1:2]- (K+1)/2 #center
    Z=data.frame(Z);names(Z)=c(“i”,”j”,”r”)

    These can be plotted:

    plot(Z$i,Z$j,pch=19,ylab=””,xlab=””)
    temp= (Z$i^2+Z$j^2 < = ((K-1)/2)^2);sum(temp) points(Z$i[!temp],Z$j[!temp],col=2,pch=19) #show exterior points
    title("Points Used in SVD")

    Then I removed the points exterior to the circle and made a correlation matrix:

    Z=Z[temp,] #remove exterior to circle
    Z[,1:2]=Z[,1:2]+(K+1)/2 #revert to index form
    N=nrow(Z);N
    dim(Z) #797 3
    #decorrelation function
    r=0.5 #this isn’t fitted, but is in the range for observed spatial decorrelation
    f= function(x,y,i=i0,j=j0,rho=r) rho^ sqrt( (x-i)^2+(y-j)^2)
    #function to give decorrelation as function of distance

    #make a square correlation matrix NxN
    index=(1: K^2)[temp];
    R=array(NA,dim=c(N,N))
    for (k in 1:N) { #for each row in Z
    i0=Z$i[k];j0=Z$j[k]
    test= outer(1:K,1:K,f) #this gives column of correlations for each row of Z with exterior points
    R[,k]=c(test)[temp] #this trims exterior points
    }
    dim(R) #797 797

    Then I did a SVD ( principal components since centered)

    W=svd(R,nu=4*K,nv=4*K)

    Then I plotted the eigenvalues shown on another thread:

    # GDD(file=”d:/climate/images/2009/toeplitz/circle_eigenvalues.gif”,type=”gif”,w=480,h=360)
    par(mar=c(3,4,4,3))
    barplot(W$d[1:50]^2/sum(W$d^2),xlim=c(0,60),ylab=”% Variance”,xlab=””)
    title(c(“Eigenvalues for Circle with”,”Exponential Decorrelation”),cex=c(1,.8))
    mtext(side=1,font=2,line=1,”First 50 Values”)
    # dev.off()

    Then I plotted.

    #pdf(file=”d:/climate/scripts/toeplitz/circle.pdf”,width=5,height=4)
    for (k in 1:16) {
    Z[,3]=W$v[,k]
    if(sum(Z[,3])<0) Z[,3]= -Z[,3]
    gridded = interp(Z[,1],Z[,2],Z[,3])
    ylim0=range(gridded$z,na.rm=T);ylim0 # 0.007698207 0.200473573
    y0=ceiling ( max(abs(ylim0))/.05) *.05;y0
    breaks0=seq(-y0,y0,.01); N=length(breaks0)
    layout(1)
    par(mar=c(2,3,3,1))
    image.plot (gridded$x,gridded$y,gridded$z, col=tim.colors(N-1),breaks=breaks0,xlab=””,ylab=””)#,xlim=c(-127,-70),ylim=c(25,50))
    contour(gridded, add=TRUE)
    title(main=paste(“PC”,k,” Weights rho=”,r,sep=””))
    }
    #dev.off()

    Simple but it made pretty patterns. Took a little thought to make it work though.

  5. bender
    Posted Feb 24, 2009 at 4:46 PM | Permalink

    No proof or evidence was offered for any of these assertions.

    Not even in the SI? When I encounter a statement like this in the literature, I assume that that the analysis was done, but simply not reported. If the appropriate analysis was NOT in fact done, I would call this a significant deception. I would ask the authors if they did the analysis, and if so could they provide it. Giving them the benefit of the doubt, my guess is that it was done and will be the subject of a complementary paper to be published elsewhere.

  6. Ryan O
    Posted Feb 24, 2009 at 4:52 PM | Permalink

    The plots here provide more evidence (though not “proof”) that the eigenvectors are simply what you’d expect from applying principal components to spatially autocorrelated data on a not quite circular shape than Steig offered up in support of his assertion that these things are “meaningfully related to important dynamical features of high-latitude Southern Hemisphere atmospheric circulation, as defined independently by extrapolar instrumental data.”

    .
    If you watch the temperatures evolve over time, you will see that Steig’s eigenvectors have coupled the Peninsula to the south pole. They evolve identically. That would seem to support your statement.
    .
    Here’s the script for that . . . just plug in the ant_recon.txt time series as “sat_recon” and run.
    .

    http://www.mediafire.com/file/imkymyqz4fw/MeansPlot.R

    .
    I also did a running 48-month Wilcoxon test for every data point of every series comparing the PCA recon to the AVHRR recon. I normalized the results to the 95% CI for that point. If the plot turns purple, the AVHRR is outside the LOWER 95% CI for the PCA recon, and if it turns black, it’s outside the upper. The comparison between the two shows that their differences oscillate. I’m not sure what it means . . .
    .
    http://www.mediafire.com/?znlod8mlyn1 <-script
    .
    http://www.mediafire.com/?4tynenbdd1j <-load this file to run in the script
    .
    http://www.mediafire.com/?r489m7tgg0m <-complete Wilcoxon test data set with p-values and rank stats

  7. Trent
    Posted Feb 24, 2009 at 4:54 PM | Permalink

    Steve,

    It would be interesting to see the Steig’s AVHRR eigenvector 4 plotted. If it has 4 lobes …:-)

  8. Joe Beanski
    Posted Feb 24, 2009 at 5:29 PM | Permalink

    This reminds me of Dommenget and Latif’s 2002 note in J. Climate: “A Cautionary Note on the Interpretation of EOFs”

    http://ams.allenpress.com/perlserv/?request=res-loc&uri=urn%3Aap%3Apdf%3Adoi%3A10.1175%2F1520-0442%282002%29015%3C0216%3AACNOTI%3E2.0.CO%3B2

  9. David Wright
    Posted Feb 24, 2009 at 5:45 PM | Permalink

    I’m not surprised that a PC decomposition of antarctic temperature is approximately equivilent to a multipole decomposition, with the most principal components corresponding to the lowest multipoles; that just says that weather is mostly a long-wavelength phenomenon. I’m also not surprised that the 0-pole (uniform) component is strongly correlated with the mean antarctic barometric pressure; that’s just saying that weather and barometric pressure are linked.

    But I am surprised that the temperature trend is not mostly determined by the 0-pole component. The fact that the 0-pole component gives a trend opposite the total trend is really weird. In some sense I would expect the 0-pole component, which basically measures the average temperature, to define the trend. Does the difference just arise to the fact that the most principal component isn’t quite exactly the 0-pole component?

  10. Jeff C.
    Posted Feb 24, 2009 at 6:23 PM | Permalink

    Different application, same phenomena. These are the electric field distributions within a circular waveguide. As frequency increases with respect to the waveguide diameter, higher order modes can propagate. The first pattern is the dominant and usually desired mode. As an RF engineer, we usually design transmission lines that avoid higher-order mode propagation to prevent in-band resonances.

    • Posted Feb 24, 2009 at 8:06 PM | Permalink

      Re: Jeff C. (#12),

      I’ve seen these patterns hundreds of times using holographic interferometry on vibrating plates too.

      I wondered if this is what a Chladni pattern was.

      • Pat Frank
        Posted Feb 24, 2009 at 9:44 PM | Permalink

        Re: Jeff Id (#17), Re: Steve J (#18), These are both very good observations. Vibrating plates (drum heads) show just these sorts of patterns. Apparently, optical aberration does as well. The functional form of Zernike polynomials can then be imported into the physical theory of these phenomena to describe what is actually going on, with physical parameters entered into the function to give results with physical meaning.

        None of this happens in climate science, when principal components are assigned by fiat to temperature time series, or to temperature fields, merely on the basis of inspection. This is a tendentious and subjective assignment of physical meaning to a numerical result. So far as I know, this sort of pseudo-science happens nowhere else.

  11. rwnj
    Posted Feb 24, 2009 at 7:03 PM | Permalink

    The eigenvectors referred to here are presumably eigrnvectors of the Laplace operator with some prescribed boundary conditions (Dirichlet? Neumann? Robin?) Distorting these for a disc to the shape of Antarctica requires a number of assumptions ( choice of schlicht mapping from the disc to itself). What choice was made? What possible relation can this have to temperature distributions?

    • David Wright
      Posted Feb 24, 2009 at 7:23 PM | Permalink

      Re: rwnj (#13),

      Not really.

      First off, no physical law requires the temperature field to satisfy \del^2 T = 0, nor to have any particular behavior on the boundary of a continent.

      The eigenproblem here arises from principal component analysis, a data mining technique. The idea is to take a large number of related data sets and ask: if I had to predict all these values as linear combinations of just a few few basis vectors, which vectors would I choose? The answer turns out to involve finding the eigenvectors of the correlation matrix; the eigenvectors with the largest eigenvalues are your best bet for getting as close as possible to the real data by combining the fewest possible basis vectors. Thus the pictures of the principal eigenvalues in some sense pictures of the most common temperature patterns across antartica.

      • Pat Frank
        Posted Feb 24, 2009 at 8:04 PM | Permalink

        Re: David Wright (#15), David, the PCs are numerically orthogonal, but connected to no physical theory, as such. The assignment of PC1 to a temperature field is at best an argument by analogy.

        It’s a kind of tendentiousness in an analytical overcoat. The PC1 meets a priori expectations about what the temperature field should look like, and so that’s what it is.

    • Nick Stokes
      Posted Feb 25, 2009 at 3:25 AM | Permalink

      Re: rwnj (#13),
      The Antarctic eigenvectors are what they are. But the circle eigenvectors are those of a symmetric matrix in which the elements are 0.5^d, where d is the distance from point i to j. As such is is very like the matrix of a Laplacian with Neumann conditions. They do look like eigenvectors of that operator, as you see them on drum membranes etc.

      This actually gives some justification for choosing the first three. Eigenvector 1 is a kind of zero-th moment, with no directional discrimination. 2 and 3 are like first moments, picking out the spatial trend. You wouldn’t want one without the other. Beyond that it’s higher order. I don’t know if that’s what Steig had in mind though.

      • Nick Stokes
        Posted Feb 25, 2009 at 9:01 AM | Permalink

        Re: Nick Stokes (#25),
        Steve
        Following my comment in #25, I calculated the first 15 eigenvectors of a Laplacian on a disc. These come out in Bessel functions of the form Jn(r/r0) cos(nθ) etc. I’ve plotted the first 16. The boundaries of the plot may not be quite right, and I couldn’t match your colors, but I think they are very similar. They are more symmetric than the colors make them appear. I messed up the order in putting them here, the bottom left should be bottom right.

  12. Craig Loehle
    Posted Feb 24, 2009 at 7:05 PM | Permalink

    The “explanation” of the PCs is Steig strikes me as the kind of discussion or results that one is allowed as a scientist in a paper–ancillary to the main point, linking with other phenomena, implications of results and all that. So no, Bender, I don’t think there is an analysis anywhere.

  13. Steve J
    Posted Feb 24, 2009 at 8:41 PM | Permalink

    These patterns also look like Zernike polynomials (http://en.wikipedia.org/wiki/Zernike_polynomials).

  14. Steve McIntyre
    Posted Feb 24, 2009 at 9:36 PM | Permalink

    #13. The relationship to temperature distributions is the one that I referred to in the post – the model presumes spatial autocorrelation – for which scatter plots provide support. If you take principal components on stations on a disk with spatial autocorrelation (negative exponential), you get these patterns.

    There is presumably some formal similarity in other situations which presumably produce such patterns.

  15. Jesper
    Posted Feb 24, 2009 at 9:42 PM | Permalink

    Does this exercise confirm Steig’s assertion of spatial coherence?

    How do Steig’s eigenvalues compare to those in your simulation?

  16. pjm
    Posted Feb 24, 2009 at 9:56 PM | Permalink

    Steve: …the eigenvectors are simply what you’d expect from applying principal components to spatially autocorrelated data on a not quite circular shape..

    Steig: The first principal component is significantly correlated with the SAM index … the second principal component reflects the zonal wave-3 pattern…

    There is not necessarily any contradiction here. Could it be that the SAM index and the wave-3 pattern themselves appear because the physics resembles PC? If so, does this imply that the physics is in some sense autocorrelated? Apologies for the extreme vagueness. I am not sure enough how to express all this more clearly.

  17. Geoff Sherrington
    Posted Feb 25, 2009 at 2:07 AM | Permalink

    In the particular case of the Antarctic (less the distant islands), it could be the case that the sea-level weather stations on the edge of the continent have rather different responses to those up high on the plateau. Therefore, distributions and analysis based on latitude or polar (?) coordinates might unmix into rather separate groups that are correlated better within a group than outside it. I would be interested to see a distance correlation between the stations on the rim, where the distances are measured E-W around an approximate latitude; and another analysis for the plateau stations done as already shown.

  18. Posted Feb 25, 2009 at 3:05 AM | Permalink

    Bias introduced by finite eigenvector choice is very easy to see from the Chlandi diagrams. When three are added together, there would still be major deviations from zero in parts of the disc. I wonder if this effectively ‘amplifies’ deviations in Antarctica? I wonder how many eigenvectors it takes before the deviations from a sum start to cancel effectively, and it becomes ‘unbiased’?

  19. Stuart Harmon
    Posted Feb 25, 2009 at 6:29 AM | Permalink

    Steve

    You requested information from Dr P Jones re jones et al 1990.

    I sent him a rude email 12 Feb about being a UK taxpayer blah bla blah.

    He responded as follows:-

    http://www.cru.uea.ac.uk/cru/data/jonesetal1990/

    This page was put up on April 4, 2007.

    The work was done and the paper written in 1989/90. At the time I asked
    my asked three of my co-authors (e.g. Groisman for the Russian network etc)
    for a network of rural stations in each region.

    Is this what you wanted?

    Steve: Jones initially refused. I filed a FOI request, Jones refused again. I objected to the FOI officer and eventually this data was provided (And has been discussed on a previous occasion at this site.)

  20. Mike B
    Posted Feb 25, 2009 at 10:29 AM | Permalink

    Why do so many climate scientists seem to have the idea that all principal components are guaranteed to have a physical interpretation? Maximizing the amount of explained variation in a few orthogonal components is rarely compatible with meaningful physical interpretation.

    I believe Stieg’s analysis would have been better served with Factor Analysis with oblique rotation.

    • Posted Feb 25, 2009 at 11:30 AM | Permalink

      Re: Mike B (#28), Ryan O (#6),
      To put it simply, using a few low-order PC’s simply smears the well known peninsula warming over most of Antarctica, creating the false impression of widespread warming.

      Steve, you could get more evidence to support your final paragraph by repeating your calculation with a slight breaking of symmetry of the circle, adding a ‘peninsula’ to your circle. Your eigenvalues for PC2 and PC3 will then be slightly different and one of the eigenfunctions will have an axis lined up with the ‘peninsula’ like Steig’s PC2.

      • Ryan O
        Posted Feb 25, 2009 at 11:41 AM | Permalink

        Re: PaulM (#32), Yep. You can see the coupling best in eigenvector 2 on the east side of the peninsula. While the actual ground stations all around the peninsula evolve nearly identically over time, in both the PCA and main reconstructions, the east side of the peninsula evolves exactly like mainland in the center around latitudes 80 – 65S.
        .
        Also, it’s pretty clear that none of the eigenvectors are weighted heavily in the peninsula – which, given the concentration of the stations in the peninsula – is opposite of what I would expect . . . and eigen 3 is weighted most in an area of West Antarctica with almost no actual station data.

      • Posted Feb 25, 2009 at 12:14 PM | Permalink

        Re: PaulM (#32),
        I think my latest post shows the smearing effect pretty clearly. It definitely exists but I’m not sure it is only the peninsula stations that cause the problems. When I clipped the JeffC regridded peninsula data and ran RegEM in another post there was a reduced but still positive trend of 0.05C/yr. Positive doesn’t mean it’s wrong in this case but the change from 0.12 – original to 0.07 JeffC regridded to 0.05 -no peninsula does mean some smearing is happening..

        ———-
        I’m starting to understand this better. The last two eigenvalue plots (of the main post) have a zero line through the center of the antarctic which has a non-linear shape. This basically represents the boundary of the PC which in physical cases of correlated stations over an area would have a plotable shape. The irregularity of the zero border represents the cutoff for each of the second two modes. The direction of the border in each case is determined by the physics of the situation so they could have substantially different shapes and off center locations. A constraint would be that PC2 and PC3 borders in autocorrelated data would be reasonably orthogonal at the point of intersection – which appears to be the case.

        It might be physically impossible to represent the antarctic with only 3 pc’s then. If the stations are pre-weighted by some scheme for area a PC1 might reveal the best overall trend (which would break down to a simple average) but when we rely on the PC’s to area weight the stations and the pc’s can represent only these modes, you’re basically guaranteed to have substantial area weighting issues.

        Really good stuff Steve, this post helped improve my understanding of the limitations of principal components. Thanks.

        • Jeff C.
          Posted Feb 25, 2009 at 1:19 PM | Permalink

          Re: Jeff Id (#34),

          It might be physically impossible to represent the antarctic with only 3 pc’s then.

          It might be possible if Antarctica had better spatial uniformity of temperature trends but we know this isn’t the case. If I understand this correctly, by using only 3 PCs we are limited to describing temperature trends using patterns comparable to the first three distributions shown in #27 (and we are throwing away information that would be included in the higher-order patterns). We have an area of out-of-family warming on the peninsula that is physically smaller than, and occurs off of, the main lobe peaks of any of those three patterns. By design, limiting the recon to these three patterns will spread around the localized, out-of family warming beyond its actual location (and reduce it where it actually does occur). Thus we have widespread warming over West Antarctica that is even greater than that of the peninsula despite a lack of measured data.

          However, including the higher-order patterns (PCs) where the localized out-of family warming area is in proportion to pattern lobe size and location, would have rectified the distorted reconstruction. That wasn’t done in the recontruction.

  21. Steve McIntyre
    Posted Feb 25, 2009 at 10:43 AM | Permalink

    There’s an interesting difference between the Chladni patterns and the Bessel/Zernike functions. The latter permit odd functions, while Chladni seem to be even (I think). The principal component patterns that I’ve observed are all even as well.

    There’s an irony here – the 2nd and 3rd Steig eigenvectors are pretty clearly 2-lobed. They blithely ignore the lobe count by likening it to the wave 3 pattern.

    To get a wave 3 pattern in the Bessel/Zernike functions, you have to get down to the 7th or 8th functions. So if there is some sort of extra “power” in the Antarctic wave 3 pattern relative to the 2-lobe patterns (and I have no information on this other than the reference in Steig et al), this would be an interesting phenomenon.

  22. Craig Loehle
    Posted Feb 25, 2009 at 11:01 AM | Permalink

    clearly this thread is the work of unauthorized amateurs ;)

    very nice

  23. bernie
    Posted Feb 25, 2009 at 11:08 AM | Permalink

    Steig et al say:

    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 20S–90S), 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 4,8

    Could somebody translate the physical climate processes Steig et al allude to in explaining their 3 PCs into plain English?

    • bernie
      Posted Feb 25, 2009 at 4:05 PM | Permalink

      Re: bernie (#31),

      bernie:
      February 25th, 2009 at 11:08 am
      Steig et al say:

      “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 20S–90S), 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 4,8″

      Could somebody translate the physical climate processes Steig et al allude to in explaining their 3 PCs into plain English?

      Let me try one more time, since the lack of response can mean either that I have asked a dumb question or that others are also not sure what exactly Steig et al are saying. If this has already been addressed my apologies and a link please.

  24. Alexander Harvey
    Posted Feb 25, 2009 at 2:33 PM | Permalink

    To begin, comparing the results to the harmonics of an elastic membrane is a great insight. Thanks.

    This may not be the right thread but I am struggling with what this is all about.

    The Steig data are anomalies covering the period 1957 to 2006 for AVHRR data. How can this be? Given that the platform did not exist prior to 1978. I feel that their is a tremendous joke going on here that I am not sharing in. Or is the impossibility of the data the joke.

    Any assistance?

    Alex

    • NeedleFactory
      Posted Feb 25, 2009 at 5:21 PM | Permalink

      Re: Alexander Harvey (#36)
      Re: Pat Frank (#16)
      I’ve been following this thread and I’m confused.
      Alexander says “comparing the results to the harmonics of an elastic membrane is a great insight” but I think I agree with Pat, who seems (to me) to be saying that comparing temperature anomalies over a “circular” Antarctic continent to harmonics of a circular drum head
      is a false analogy.
      The second AVHRR eigenvector Steve plotted does indeed resemble a second harmonic of a drumhead, but for the analogy to obtain, we would expect the Antarctic peninsula to cool as the eastern continent warms, and vice versa, back and forth. Surely no one suggests that as a model!
      So I hazard this guess: the similarity between drum head eigenvectors and Antarctic temperature eigenvectors concern only the topology of disks, not with the physics of vibration.
      Am I missing something?

  25. Posted Feb 25, 2009 at 3:06 PM | Permalink

    Hi Steve, I am just calculating similar things with dozens of chosen European stations.

    I realized a thing that leads me to ask about your opinion: isn’t – because of purely mathematical reasons – the PC1 spatial pattern simply encoding the overall magnitude of (annual) temperature variations at a given place? Isn’t it true that the temperature fluctuations are smaller near the beaches?

    Also, in my European data, I don’t see any sharp or semi-sharp separation between any two consecutive PCs.

  26. Alexander Harvey
    Posted Feb 25, 2009 at 3:38 PM | Permalink

    Luboš,

    From Steig’s homepage his data are anomalies but I also wonder if they are artifacts left over, due to annual variations.

    Does anyone know where this AVHRR data comes from?

    I remember 1957, more particularly I remember what I was doing when Sputnik launched itself into the vocabulary. What is this data all about?

    Alex

  27. Jeff C.
    Posted Feb 25, 2009 at 3:40 PM | Permalink

    I experimented with changing regpar (number of PCs used) using the AWS data. Without the satellite data, we are limited in what we can do, but the results are interesting.

    I used Jeff Id’s Matlab routine to generate the AWS reconstruction using regpar = 3. The trends from 1980 to 2006 (the years we have AWS measurements) are shown in the first plot below.

    I then redid the recon using regpar=7 to see if I could reduce the presumed smearing of the peninsula trends onto West Antarctica – the second plot below. You can see that the northern peninsula trends became the largest on the continent (as we know is true from the surface measurements), and the Ross Ice shelf area cooling increased. (I’m using old Harry data so that is the big red dot in West Antarctica.) Regpar=8 is virtually identical to regpar=7.

    Not too surprising, but it does support some of the comments above.

    • Posted Feb 25, 2009 at 4:08 PM | Permalink

      Re: Jeff C. (#39),

      Nice. It’s probably a pain in the butt but it looks like you would get a negative trend from regpar=7 if you regrid the data?

      Steve has a nice new downloader which looks like it will use corrected (I didn’t verify it) data which I used in my last correlation post. He put the code up on..

      http://www.climateaudit.org/?p=5296

    • RomanM
      Posted Feb 25, 2009 at 4:41 PM | Permalink

      Re: Jeff C. (#39),

      I experimented with changing regpar (number of PCs used) using the AWS data. Without the satellite data, we are limited in what we can do, but the results are interesting.

      I think it is becoming quite clear that in fact, Steig et al did not use any raw satellite data in the reconstruction, but instead used a 3PC substitution for the raw data spread (pretty thinly) across the the 5509 grid points. If possible, I would like to suggest that someone run the RegEM procedure using the manned station data with only the three principle components from 1982 to 2006 adjoined to reconstruct the other half on the PCs in the pre-1982 period. All things being equal, it should lead to the Steig reconstruction using the same weights as in the post-1982 period.

      At the same time, one can try varying the regpar parameters to see the difference (although I think that substituting the PCs for the raw data leaves little room for any true variation in the results).

      • Posted Feb 25, 2009 at 6:13 PM | Permalink

        Re: RomanM (#42),

        If I understand you, this is what I did here last Friday.

        http://noconsensus.wordpress.com/2009/02/20/satellite-temperature-trend-regridded/

        I only did regpar=3 but the first graph shows the original pc’s reconstructed from the surface station data. I was able to replicate the original reconstruction pretty well from just these pc’s. The post was about the regridded version primarily though and I just used it to verify the second step using JeffC regridded data which happened to provide a much better looking PC3.

        • RomanM
          Posted Feb 25, 2009 at 9:01 PM | Permalink

          Re: Jeff Id (#45),

          Yes, I think that you have already done what I suggested. It is also my opinion that the same method was used in the paper. From the fact that the “detrended” reconstruction from 1982 to 2006 matches the original reconstruction exactly (subject to the subtraction of a regressed linear trend), it appears quite clear that the 3PC values from 1982 to 2006 were the starting point of the Steig RegEM analysis, not the result. The reconstruction consisted of stretching the series back to 1957. It also answers my question of why they would embed the measured data into the AWS results, but not the satellite. This was their initial “measured” data.

          I would think that you would get the same results whether you used what you did or by using the entire 5509 grid series (it’s the same information). I also suspect that trying to do a RegEM with almost 5600 columns (raw data or 3PC results) would be a nightmare so it would make sense to perform some sort of dimension reduction on the raw data before running the procedure. However, limiting it to three PCs likely imposes a severe limitation on any ability to properly represent the variations in trends from the entire continent. Virtually every grid point end up having a positive trend throughout the entire time period.

          I can’t hazard a guess as to how they got these PCs. They could have been calculated by Dr. Steig and crew in an unmentioned step or they may have actually received the values already in that form. Who knows?

        • Posted Feb 26, 2009 at 8:04 AM | Permalink

          Re: RomanM (#48),

          I didn’t see the reply, these threads are long and complicated. When you first found the 3 pcs I started thinking about how the covariance matrices are calculated and decided the same thing you did, it doesn’t matter whether you use 3 or 5509 because it’s the same information.

          Since the weighting of the 3 pc’s after 1982 doesn’t include surfacestation info I used the same values to reconstruct the continent with JeffC’s surface station info.

          Anyway it would be very very slow to run 5509 series through the masher so 3 is fine and I also think this is what they did.

          I can’t hazard a guess as to how they got these PCs. They could have been calculated by Dr. Steig and crew in an unmentioned step or they may have actually received the values already in that form. Who knows?

          How do you calculate PC’s with missing data from the cloud masking?? This is why I think we need the code and raw data for removing the cloud data.

  28. Riddi of England
    Posted Feb 25, 2009 at 5:04 PM | Permalink

    My apologies if your radar has already locked on to this.
    It is an english translation of scientific Japanese research paper and it struck me as germaine to meme of your esteemed blog.

    http://www.theregister.co.uk/2009/02/25/jstor_climate_report_translation/print.html

    Respectfully yours

  29. TommyS
    Posted Feb 25, 2009 at 6:27 PM | Permalink

    I wanted to have a look at the zonal wave 3 pattern and compare it with Steig AVHRR eigenvector 2, so I made a short pdf compilation.
    It is found here.

    The pdf contains plots of 6 months from 2001 to 2004. If you click the tiny links below the thumbnails, you will get to the source site showing the image in full.

  30. Alan Wilkinson
    Posted Feb 25, 2009 at 8:04 PM | Permalink

    To a long ago ex-chemist, these calculated patterns are remarkably similar to the wave mechanical Hamiltonian solutions for electron probability densities around atomic nuclei.

    Which raises the assumption of planarity. Given the earth’s curvature and the distance to the included island data points, can the third dimension be ignored?

    • Posted Feb 25, 2009 at 9:28 PM | Permalink

      Re: Alan Wilkinson (#47), Alan, I second your motion. As a long ago ex-chemist and ex-musician (not a drummer), so the drum head analogy didn’t strike as much a chord ;) with me as the electron orbital one did. The patterns also remind me of the x-ray crystallography structures I once studied. The third dimension must be considered, I’d think. How can altitude not be significant given Antarctica’s topography, even if you could assume planarity from the surface frame of reference? You certainly could not get away with ignoring the 3rd dimension when interpreting x-ray crystallography results.

  31. bender
    Posted Feb 25, 2009 at 9:23 PM | Permalink

    The patterns on the right are “separable” (but not necessarily informative) in terms of gross spatial pattern, but are not separable in terms of mechanistic causation (there is no specific cause for the breakdown in correlation over distance). Steig would probably assert that his eigenvectors are “separable” on the basis of their interpretability (patterning resulting from independent causes). Which would require a formal comparison of the eigenvectors (in the time domain) to the EOFs to which he claims they are related. Why he did not do that is beyond me. The text probably wouldn’t fit in Nature‘s ultrastrict word limit. Suggesting Nature has no business publishing analyses of that type.
    .
    Snip me. – snip

    • bernie
      Posted Feb 26, 2009 at 7:06 AM | Permalink

      Re: bender (#49), bender, kind heart that you are, what actual physical processes are Steig et al referring to and is Steig et al’s

      first principal component … significantly correlated with the SAM index(?)

      The thoroughness and elegance of the statistical deconstruction is impressive — what seems somewhat short changed in a discussion of the actual physical processes that might account for what Steig et al contend. In short, does their interpretation of the physical meaning of their first 3 PCs make sense?

      • bender
        Posted Feb 26, 2009 at 8:04 AM | Permalink

        Re: bernie (#57),
        Hopefully you are not asking me to explain why the southern annular mode behaves the way it does! What I am suggesting is that to the extent that SAM is a “separable” feature of earth’s climate, so might the Steig PC that maps to it. The SAM is derived from pressure anomalies. It would make sense if temperature were related to pressure. I mean, that’s how it works elsewhere in the world – SLPs and SSTs being closely related.

  32. Ryan O
    Posted Feb 25, 2009 at 9:30 PM | Permalink

    Has anybody yet figured out what they mean by “statistically separable”?

    • bender
      Posted Feb 25, 2009 at 9:37 PM | Permalink

      Re: Ryan O (#3),
      No. Ross, in the previous thread, outlined a good definition based in econometrics. I proposed a similar one based on my experience in ecosystem modeling. But what the authors meant by it no one knows. I suggested asking them if it’s critical. I don’t think it is. At some point this descends into endless parsing of semantics.
      .
      But this does harken back to my earlier question: who reviewed this paper and what did the reviews look like? As a reviewer I would have asked them to define and prove “separability” if it is a central theme in the paper.

  33. Ryan O
    Posted Feb 25, 2009 at 9:31 PM | Permalink

    I see bender and I were posting at the same time. :)

  34. rodent
    Posted Feb 25, 2009 at 9:34 PM | Permalink

    Re: Steve #29:
    Chladni patterns:
    Chladni (1756-1824) excited vibration modes in plates with violin bows. Fine sand that was sprinkled on the horizontal vibrating plate migrated to the nodal lines hence giving a visual image of their location. The antinodes were void of sand. So, any mode with nodal lines (straight or curved) was observable.

  35. AnonyMoose
    Posted Feb 25, 2009 at 10:16 PM | Permalink

    Looking at the 16 PCs, it’s obvious that if you sum the values of the preceding PCs the differences are greatest at PC3. With PC1+PC2+PC3 there is mostly reinforcement on either side of a 45 degree axis, BUT there is a bias toward positive (red) values in the center due to PC1’s pattern. From PC4 onward there are small patterns which break up the simple symmetry of the first three patterns.

  36. RomanM
    Posted Feb 26, 2009 at 9:02 AM | Permalink

    Oops! My bad. That should be Re: Jeff Id (#59), not 52. I actually started to reply to Jeff in another thread, updated the threads to see what had transpired, and move the reply to this one.

  37. Alexander Harvey
    Posted Feb 27, 2009 at 12:17 PM | Permalink

    NeedleFactory,

    The drum and Antarctica share two features:

    Individual membrane elements tend to drag their neighbours up and down with them as may temperature elements.

    The drum is constrained at the rim. Antarctica is constrained by the surrounding ocean. Temperature variability is likely to be least constrained away from the ocean.

    Alex

  38. Posted Oct 8, 2010 at 1:52 AM | Permalink

    I have noticed these same patterns with principle components analysis of economic time series. If rate of change is used (first differences) then more meaningful results are generally obtained.

  39. RomanM
    Posted Feb 26, 2009 at 8:58 AM | Permalink

    Re: Jeff Id (#52),

    With all of the concurrent threads, it is hard to figure out where to post a comment.

    The situation is pretty clear now. The satellite data was reduced (presumably from a total of 300 x 5509 = 1,652,700 points) to just three PC series totalling 900 points to describe all of the variation in the monthly temperatures for the entire Antarctic continent for a 25 year period. These 900 values were first stretched using the partially-complete surface data to 1800 using RegEM and then to 3,305,400 values to draw the picture on the cover of Nature.

    Was this successfully done? I agree with you. Until we can get to see at least the initial satellite values from which the three PCs were calculated, I don’t think that there is much more that we can do in directly evaluating the quality of the reconstruction.

    How do you calculate PC’s with missing data from the cloud masking?? This is why I think we need the code and raw data for removing the cloud data.

    There are many possible ways (not all necessarily appropriate for a given situation) to infill the values. Simple local geographic interpolation, “climate” using monthly means, and EM (without the Reg) which uses statistical relationships with all observed values to estimate the missing measurements are some possible methods. If the number of infilled values is small compared to the entire data set, then the impact of the infilling of the PCs extracted could be relatively low. However, there can be a problem in those cases where the size of the value is related to the fact that it is missing – certainly a possibility in this case because the presence of clouds and the surface temperature can be related.

  40. Ryan O
    Posted Feb 26, 2009 at 11:33 AM | Permalink

    Re: RomanM (#60),

    However, there can be a problem in those cases where the size of the value is related to the fact that it is missing – certainly a possibility in this case because the presence of clouds and the surface temperature can be related.

    .
    While without further information we wouldn’t know if the presence of clouds and surface temperature are strongly related (or, if they are, exactly how they are correlated), we can say with certainty that the cloud masking will preferentially affect certain geographical locations: namely West Antarctica and the coastlines. Cloud cover is least over the plateau. It may be entirely coincidental, but the areas in both the PCA and AVHRR recons that show the strongest warming from 1957-2006 are the areas with the most cloud cover . . . and, hence, the largest number of data points removed via masking.
    .
    I have not had any luck so far getting into GISMO at NSIDC (can’t get the Java applet to work), but while we await Steig’s processed data, I was going to start analyzing cloud cover in the same grid cell size as what Steig used over the same period and see if there is any actual correlation between cloud cover (and, by extension, missing data) and warming trend.

2 Trackbacks

  1. By Kriging on a Geoid « Climate Audit on Aug 26, 2010 at 1:24 PM

    [...] Treering Network”, 3/22/08, “Antarctic Spatial Autocorrelation #1″, 2/20/09, “Steig Eigenvectors and Chladni Patterns”, and follow-up [...]

  2. […] in nature that resemble Chladni Patterns. Everything from the patterns on the backs of Tortioses, weather patterns and patterns of forrest growth. It seems that sound has a significant role in the formation of […]

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