AMO, NAO, and Correlation

Guest Post by Willis Eschenbach

There’s a new paper over at IOP called “Forcing of the wintertime atmospheric circulation by the multidecadal fluctuations of the North Atlantic ocean”, by Y Peings and G Magnusdottir, hereinafter Peings2014. I was particularly interested in a couple of things they discuss in their abstract, which says (emphasis mine):

Abstract

The North Atlantic sea surface temperature exhibits fluctuations on the multidecadal time scale, a phenomenon known as the Atlantic Multidecadal Oscillation (AMO). This letter demonstrates that the multidecadal fluctuations of the wintertime North Atlantic Oscillation (NAO) are tied to the AMO, with an opposite-signed relationship between the polarities of the AMO and the NAO. Our statistical analyses suggest that the AMO signal precedes the NAO by 10–15 years with an interesting predictability window for decadal forecasting. The AMO footprint is also detected in the multidecadal variability of the intraseasonal weather regimes of the North Atlantic sector. This observational evidence is robust over the entire 20th century and it is supported by numerical experiments with an atmospheric global climate model.

Let me start with their claim that the AMO signal precedes the NAO by 10-15 years. Here’s the cross-correlation function for the monthly data, using the full 1856-2112 NOAA AMO and the Hurrell NAO data:

cross correlation hurrell nao amoFigure 1. Cross-correlation of the full Hurrell NAO and the NOAA AMO, 1856-2012.

Hmmmm … why am I not finding the relationship between AMO and NAO they discuss? I mean, I see that the largest correlation is at zero, and there is a correlation out 15 years, but it’s all so tiny … what’s the problem?

Well, to start with, they are not using the regular AMO index, nor are they using the full year. Instead, here is their description:

A wintertime AMO index is constructed over the 1870– 2012 period using the HadISST dataset (Rayner et al 2003). The monthly SST anomalies are determined with respect to the 1981–2010 climatology, then the winter AMO index is computed by averaging the monthly SST anomalies over the North Atlantic [75W/5W; 0/70N] from December to March (DJFM). The global anomalies of SST are subtracted in order to remove the global warming trend and the tropical oceans influence, as suggested by Trenberth and Shea (2006). A Lanczos low-pass filter is applied to the time series to remove the high-frequency variability (21 total weights and a threshold of 10 years, with the end points reflected to avoid losing data).

Nor are they using the standard NAO, viz:

A decadal NAO index is computed from the 20th century reanalysis (20CR), which is available over 1871–2010 and is based on the assimilation of surface pressure observations only (Compo et al 2011). We use the station-based formulation based on the Stykkisholmur/Reykjavik and Lisbon anomalous sea-level pressure (SLP) difference (Hurrell et al 2003). The high-frequency fluctuations are removed from the NAO index using the same Lanczos filter as for the AMO index.

They are not using the standard AMO, nor the standard NAO, and most importantly, they are using a smoothed subset of the data for calculating the correlations. While using smoothed data is fine for display purposes, it is almost always a Very Bad Idea™ to do statistics and correlations using smoothed data, for reasons discussed below.

In addition, they are not using the full year. Instead, they are using a 4-month subset of the year, DJFM. While there is no inherent problem with doing this, it definitely messes with the statistics. If you want to find a significant correlation using a 4-month subset of the annual data, to achieve a significance level of 0.05 you need to find a four-month chunk with a p-value of one minus the twelfth root of 0.95, or 0.004 …

They go on to say that they have taken autocorrelation into account, viz (emphasis mine):

cross correlation their nao amo

Figure 2 of Peing2014. ORIGINAL CAPTION: Lead–lag correlations (black curve) between the DJFM AMO and the DJFM decadal NAO indices over 1901–2010. The statistical significance of the correlation is depicted by the p-value (blue dashed curve), computed using a bootstrap method that takes into account auto-correlations in the time series. The 95% confidence level is indicated by the dashed black line.

I note that they are using p=0.05 as their significance level, despite the fact that they are using partial-year correlations.

Now it’s wonderful that they have used a “bootstrap method” to allow for auto-correlations … but that’s the sum total of the information that they give us about their whizbang bootstrap method. I generally use the method of Quenouille, viz:

quenouille_n_effectiveI digitized their data to see if I could replicate their Figure 2. Figure 3 shows that result:

correlation and pvalue short djfm nao amo peingsFigure 3. My emulation of Peing2014 Figure 2. Red shows p-values less than 0.05. “DJFM” is December-January-February-March. Auto-correlation is adjusted for by the method of Quenouille detailed above. 

While the general shape is similar to Figure 2, there are a number of differences between what I find and what they find. Overall, the correlation “R” (black line) is slightly smaller. Their correlation has a max of about 0.55 and a minimum of -0.75, while mine has a max of 0.5 and a minimum of -.67. And while their results show R = +0.2 at -30, my results show R≈0. Not a lot of difference to be sure … but I’m using their data, so it should be exact.

Next, I find higher results for the p-value. Only the lags -2 to -7, and 23 to 27, are significant at the 0.05 level.

However, remember that they have used only part of the dataset, the values from December to March. Assuming that they searched all of the 4-month periods to settle finally on DJFM, that’s a dozen different samples that they have searched. And it may be more than a dozen, because I would assume that they would first look by quarters (three months). As a result, if you search that many situations, your odds of finding a result with a p-value of 0.5 purely by pure chance is quite large …

The net result is that if you look at twelve samples, you need to find a p-value of

<blockquote>1 – o.95<sup>1/12</sup> = 0.004</blockquote>

to be statistically significant at the 0.05 level … and that’s not happening anywhere in their graph.

Next, they do not find a correlation with AMO lagging the NAO, as in my results.

Next, there is an oddity, I might even say an impossibility, in their result. Look at the left hand side of Figure 2. Remember that as the lead gets longer and longer, we are using fewer and fewer datapoints in the calculation. In addition, as the lead gets longer, the correlation ( R ) is decreasing. Now, with fewer datapoints and a lower number of years, the p-value should steadily increase. You can see that in my graph—the maximum correlation and the minimum p-value are at about a two-year lead, and then as the lead heads out to 30 years, the R decreases, and the number of datapoints decreases.

But when both the correlation and the number of datapoints go down, the p-value has to increase … and while that is visible in my results, we don’t see anything like that in their results.

I am in mystery about the difference between my results and theirs. I know that the digitization is accurate to within the widths of the lines, here’s the proof of that, a screenshot of the digitization process of their Figure S3 …

digitization of PiengsFigure 4. Screenshot of the digitization process, showing that the errors are less than half a linewidth …

Finally, I have grave reservations about this general type of analysis. Basically, the AMO and the NAO represent subsections of the global temperature record. And as the name suggest, the NAO (North Atlantic Oscillation) is in itself a subset of the AMO (Atlantic Multidecadal Oscillation), representing the northern part.

As a result, I would be shocked if we did NOT find something akin to Figures 2 or 3 above. And in fact, a Monte Carlo analysis using proxy data with autocorrelation characteristics like the highly smoothed data that they are using easily generates the kind of curves shown above. That’s what happens when one dataset is a subset of another dataset, and it should not be a surprise to anyone.

In addition, such relationships are often not stable over time. For example, Figure 5 shows the cross-correlation for the AMO and NAO datasets (1901-2010), along with the identical cross-correlation calculations for the first halves (1901-1955) and for the second halves (1956-2010) of the two datasets. As you can see, the relationship is far from consistent, with cross-correlations of the two halves being different from each other, and both being different from the full dataset as well. This increases the chance that we are looking at a spurious correlation.

correlation p-value djmf nao amo early lateFigure 5. A comparison of the cross-correlation of the 30-year smoothed AMO and NAO datasets with the cross-correlations of the first halves and the second halves of the same two datasets.

As you recall, they claim in their Abstract (above) that their results are “robust over the entire 20th Century”, but their own data says otherwise.

CONCLUSIONS

In no particular order:

• Since the NAO is a subset of the AMO, we would expect cross-correlation between the two at a number of leads and lags … and that’s what we find. The authors seem to find that impressive, but their results show levels of significance and shapes of the cross-correlation that are quite commonplace when one dataset is a subset of another and the two datasets are heavily smoothed.

• They have made no attempt to adjust their significance levels to reflect the fact that they have chosen one of twelve or more possible monthly subsets of the data. This is a huge oversight, and one that puts all of their conclusions into doubt.

• I am unable to replicate the results of their cross-correlations (what they call “lead-lag” correlations above) of the smoothed 1901-2010 DJFM NAO and AMO.

• I am also unable to replicate the results of their “bootstrap” method of calculating the p-value, although that is undoubtedly related to the fact that they did not disclose their secret method …

• They neglected to include a description of one of the most important parts of their analysis, the calculation of the significance using a bootstrap method.

• The use of smoothed data in doing cross-correlation analyses is an abomination. Nature knows nothing of the 30-year average changes. Either there is significant cross-correlation between the two actual datasets or there is not. Using smoothed datasets can even generate totally spurious correlations. I give some examples here … and lest you think that I made up the idea that smoothing can lead to totally spurious correlations, it’s actually called the “Slutsky-Yule Effect”. Their use of smoothed datasets for cross-correlation alone is enough to entirely disqualify their study.

• As a result, were I a reviewer I could not agree with the publication of this study until those problems are solved.

A couple of things in closing. First, Science magazine recently decided to add a statistician to the peer-review panel for all studies … and as this paper clearly demonstrates, all journals might profitably do the same.

And second, the AMO and the PDO and the NAO are all parts of the global temperature record. As a result, using them to emulate the global temperature record as the authors have done can best be described as cheating. When someone does that, they are using part of what they are trying to predict as an explanatory variable …

And while (as the authors show) that is often a way to get impressive results, it’s like saying that you can predict the average temperature for tomorrow, as long as you already know tomorrow’s temperature from noon to 2pm. Which is not all that impressive, is it?

My best regards to all,

w.

De Maximis: If you disagree with me, and many do on any given day, please quote the exact words that you disagree with. That way, we can all understand exactly what your objection might be.

DATA AND CODE: The digitized 30-year smoothed datasets of the AMO and the NAO are here.  The NOAA AMO data is online here, and the Hurrell NAO data is here. I haven’t posted the computer code. It is a pig’s breakfast, and as opposed to being “user-friendly”, it is actively user-aggressive … I may clean it up if I get time, but my life is a bit crazy at the moment, the data is there, and a cross-correlation is a very simple analysis that folks can do on their own.

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63 Responses to AMO, NAO, and Correlation

  1. Steven Mosher says:

    “And second, the AMO and the PDO and the NAO are all parts of the global temperature record. As a result, using them to emulate the global temperature record as the authors have done can best be described as cheating. When someone does that, they are using part of what they are trying to predict as an explanatory variable …”

    Seconded.

    I will add another way of fooling yourself.

    You will find people arguing that El nino ‘explains’ global warming.

    Its a related logical failing.

  2. Stephen Richards says:

    And second, the AMO and the PDO and the NAO are all parts of the global temperature record. As a result, using them to emulate the global temperature record as the authors have done can best be described as cheating

    I don’t believe they intended to cheat. I think they did what they did out of ignorance. Nice clean analysis Willis. Thanks !

    Mosher statement seems a little iritating to me. I don’t like explains in comma quotes and that he calls it a related logical failing. Please explain Steven !

  3. evanmjones says:

    In addition, they are not using the full year. Instead, they are using a 4-month subset of the year, DJFM. While there is no inherent problem with doing this, it definitely messes with the statistics.

    I’ve seen that a lot. The reason given is that the bulk of NAO effect seems to be seasonal. So removing the remaining seasons is considered beneficial in isolating the effects.

  4. I’m not a scientist by trade, but I’m not scientifically ignorant either. Your explanations really go in-depth and I appreciate that.

    Where I live, the University of Alaska Fairbanks is one of the premier research facilities for geophysical science. It’s well-known here that global temperatures were increasingly slightly, lagging slightly behind the sun’s most recent solar maximum. Global temperatures are now decreasing gradually following the sun’s most recent solar minimum. As my screen name suggests, aurora activity is a hobby and we went almost five years with hardly any auroral activity — consistent with a solar minimum.

    In other words, global warming and cooling are part of a natural cycle and scientists in some fields know this, but they aren’t the ones with the microphones these days.

    It’s sad when science gets hijacked by politics.

  5. Alec Rawls says:

    Typo:

    Nor are they using the standard AMO

    Should read: “Nor are they using the standard NAO”?

  6. Joe Born says:

    Willis Eschenbach: “I generally use the method of Quenouille.”

    For those of us who don’t know any statistics, can someone supply a pointer to a derivation of that formula?

  7. justaknitter says:

    “Nor are they using the standard AMO, viz:”

    NAO? not AMO…

    Best Wishes Willis…How about a heart update.

  8. Matthew R Marler says:

    You are absolutely right on main points: (1) if the selection of months was based on prior analysis of data (who would believe an assertion to the contrary?) then the correction to the significance level has to be computed and used; (2) with autocorrelated data, bootstrapping is not straightforward, so the algorithm that they used needs to be described in detail. Even p < 0.05 is not strong evidence against the null hypothesis.

    Correlating part to whole is a little like predicting adult height from pre-adolescent height. It isn't a foregone conclusion that the correlation will be high, but the hypothesis of 0 correlation is not the correct hypothesis to test for any meaningful improvement in knowledge.

    Thanks again.

  9. Juraj V says:

    Make a cumulative of MEi index and you get exactly the North Atlantic SST record. North Atlantic SST record imprints its wave shape into the whole global record. Global record is a composite of individual ocean basins SST. There is no global record, which affects lets say North Atlantic record, it is exactly vice versa.
    I miss in the article a graph of NAO and AMO compared against each other. Or winter NAO compared to winter CET (excellent corelation), or AMO compared to summer CET (again excellent correlation).

    http://www.climate4you.com/images/CentralEnglandTempSince1659%201100pixel.gif

    Some 80 years ago, NAO peaked in 20ties (bringing even warmer decade of British winters than in 2000s) and AMO peaked two decades later, making summers warmer. Then it happened, that NAO and AMO peaked close to each other in 90s/2000s, bringing warm winters and warm summers almost at the same time. Since 1990, NAO goes down and CET winters follow. Winters in Europe, Siberia and USA are trending colder since 1990.

    http://data.giss.nasa.gov/cgi-bin/gistemp/nmaps.cgi?year_last=2014&month_last=2&sat=4&sst=0&type=trends&mean_gen=1203&year1=1990&year2=2014&base1=1951&base2=1980&radius=250&pol=rob

    There is no place for any radiative theories. Climate is a function of various local climate cycles, combining against or together. Remember that Climategate e-mail “what if all that is just a multidecadal fluctuation? They will kill us probably..”

  10. Curious George says:

    To hell with standards. In war, love, and climatology anything is allowed.

  11. Matthew R Marler says:

    I didn’t find anything easy, but this paper refers to two papers by Quenouille:http://www.ism.ac.jp/editsec/aism/pdf/048_4_0621.pdf

    It’s discussed in most statistical time series books.

  12. Joe Born says:

    Matthew R Marler: Thanks a lot for the paper, although as you no doubt assumed, I remain hopeful of finding something that’s more pitched down to my level.

    Some time ago I did snag a set of lecture notes on time series, but then I got sidetracked. Having this come up may make me refocus my efforts.

  13. Alec Rawls says:

    Nice highlights Willis:

    Our statistical analyses suggest that the AMO signal precedes the NAO by 10–15 years with an interesting predictability window for decadal forecasting…

    At which point I’m thinking, since we don’t have any idea where these ocean oscillations come from, or even if they are an actual repeating pattern, that this is pure data mining, where to find a FIFTEEN YEARS AHEAD PRECURSOR SIGNAL to be statistically significant you’d need–what?–300 years of data? Then Willis highlights part II:

    robust over the entire 20th century

    Ha ha ha. If they only have a few multi-decadal maybe-oscillations to look at their findings CAN’T be “robust.”

    And then they don’t even acknowledge that they have arbitrarily picked one of many statistical relationships to focus on? That is pretending to do statistics. W:

    if you search that many situations, your odds of finding a result with a p-value of 0.5 purely by pure chance is quite large

    And who knows how many other ways they tried looking for correlations? If anything needs to be vetted by a statistician it is pure data-mining exercises like this.

  14. Willis Eschenbach says:

    evanmjones says:
    April 2, 2014 at 12:03 pm

    In addition, they are not using the full year. Instead, they are using a 4-month subset of the year, DJFM. While there is no inherent problem with doing this, it definitely messes with the statistics.

    I’ve seen that a lot. The reason given is that the bulk of NAO effect seems to be seasonal. So removing the remaining seasons is considered beneficial in isolating the effects.

    Indeed it is done a lot, Evan, and it’s not a problem. The only issue is that you have to adjust the significance levels when you start hunting for correlations at the monthly level.

    w.

  15. Willis Eschenbach says:

    Alec Rawls says:
    April 2, 2014 at 12:19 pm

    Typo:

    Nor are they using the standard AMO

    Should read: “Nor are they using the standard NAO”?

    [Thanks, Alec, fixed. -w.]

  16. Willis Eschenbach says:

    Joe Born says:
    April 2, 2014 at 12:21 pm

    Willis Eschenbach:

    “I generally use the method of Quenouille.”

    For those of us who don’t know any statistics, can someone supply a pointer to a derivation of that formula?

    Joe, it’s good to hear from you. Regarding Quenouille, I’ve always assumed that it is an ad-hoc formula. I do find a discussion here, which likely isn’t over my head but is definitely over my time allocation.

    I’ve tested the Quenouille approach using Monte Carlo simulations, and it gives a reasonable result, sometimes a bit high, sometimes a bit low, and very occasionally way off. However, if I’m serious about wanting to know about the actual statistics for a given situation, I use Monte Carlo for the calculations and only use the Quenouille formula to check my results for reasonability. Data over theory is my watchword …

    w.

  17. John A says:

    While I am not as advanced as Willis in statistical analysis (*cough*a long way behind*cough*), I do recognize Mannian statistical method when I see it (too often still these days).

    Thus to choose a sample like that by maximizing the statistical metric one wishes to maximize is called “peeking at the result”. It’s what Mann did with his Hockey Stick and it negates any and all statistical inferences to do so.

    Well done Willis for spotting it, because what looks and feels like a spurious correlation very often is.

  18. Willis Eschenbach says:

    Matthew R Marler says:
    April 2, 2014 at 12:26 pm

    Correlating part to whole is a little like predicting adult height from pre-adolescent height. It isn’t a foregone conclusion that the correlation will be high, but the hypothesis of 0 correlation is not the correct hypothesis to test for any meaningful improvement in knowledge.

    Thanks, Matt, glad to see you. Actually, it’s worse than you say. In your analogy, it is like predicting adult height from the length of the adult’s femur … so in fact it is a foregone conclusion that the correlation will be a ways from zero.

    For example, despite the small size of the Atlantic Ocean w.r.t. the total surface area, the correlation between the detrended HadCRUT4 global temperature data and the AMO data (monthly, 1856-2012) is 0.47. For the annual data, it’s larger, 0.65.

    As a result, if you try to use the AMO to “predict” the global temperature, you can be sure that you’ll get a good result … but it won’t mean anything because you are using part of what you are predicting as an explanatory variable.

    w.

  19. Greg Goodman says:

    Many good points Willis. I also distrust this idea of picking out certain months.

    However, the old “never smooth” falacy comes up again.

    “The use of smoothed data in doing cross-correlation analyses is an abomination. Nature knows nothing of the 30-year average changes. Either there is significant cross-correlation between the two actual datasets or there is not. ”

    The danger of seeing false significance is real and something that needs to be guarded against. I see no sign that the authors are aware nor that they have adjusted thier significance tests to account for the lesser degrees of freedom in the filtered data, so the criticism if justified but the dogmatic postition about ‘abominations’ is not.

    If data is being “smoothed” to make it look smoother, the point totally valid, there is no reason to do stats on prettied up data. However, if the data was _filtered_ to remove, say, a strong annual cycle in order to study lesser magnitude inter-annual variation, then it is legit and it makes sense. This is why rail against using the term “smoothing” in place of frequency filtering, if that is what is meant.

    Another thing you state incorrectly is that NAO is a subset of AMO. It is not. NAO is a sea level pressure index not SST, so it cannot be a subset.

    In general you are right , this looks rather sloppy and has not been competently peer-reviewed.

  20. MichelE says:

    “And second, the AMO and the PDO and the NAO are all parts of the global temperature record. As a result, using them to emulate the global temperature record as the authors have done can best be described as cheating. When someone does that, they are using part of what they are trying to predict as an explanatory variable …”

    NAO is a pressure index, not a temperature index.

  21. Greg Goodman says:

    The idea of splitting the data can be a good test but the problem is that Hadley processing has seriously mangled the frequency content already and just comparing frequency spectra of different halves of HadSST3 gives very different results, despite ICOADS being quite similar.

    Their error is probably in using the early data at all.

    With all the anomalies, detrending and DJFM sub-setting and data adjustments going on I don’t know how meaning to give the results , whatever they are.

    NAO does have some interesting periodic content but I’ve not compared it to AMO.

  22. Joe Born says:

    Willis Eschenbach:

    Thanks for the response; since you and Mr. Marler have both directed me to the same paper, I feel honor-bound at least to make a more-creditable run at it once my taxes are done. (My first pass this afternoon didn’t quite do it.)

    Anyway, the n_eff issue comes up enough on this site (and occasionally on Ms. Liljegren’s) that it will be worth my while to walk around in some Monte Carlo runs myself if (as I suspect) I ultimately fail to grok the theory. Thanks for that suggestion, too.

  23. charles nelson says:

    ‘The North Atlantic sea surface temperature exhibits fluctuations on the multidecadal time scale’
    Does it?
    Does it really?
    The North Atlantic is a big place, so big in fact that anyone who considers that they have measured ‘its’ ‘temperature’ in any meaningful way on any timescale is deceiving either themselves or us.
    That’s not to say we shouldn’t try to make observations and indeed with satellites it may now be possible to make some vague generalised measurements but speaking as someone who makes a living by measuring and trying to control temperatures in air, water, and gases – let me assure you that anyone who claims to have identified a .5C fluctuation over 15 years in the temperature of THE NORTH ATLANTIC is an asshole!

  24. Bloke down the pub says:

    Not a lot of difference to be sure … but I’m using their data, so it should be exact,’

    As you say that you digitized their data, is there any room in that process for a discrepancy to be introduced?

  25. vukcevic says:

    About two and half years ago I emailed one of my AMO-NAO graphs to Dr. Judith Curry. The reply was short: “hi, looks good! my main suggestion is to go back prior to 1920; 1850 if possible (the data in north atlantic should be good enough)….. Judy”
    I found that the truncated AMO actually follows the northern NAO component by ~11 years, which is opposite to what the above article claims, the rest is here:
    http://hal.archives-ouvertes.fr/docs/00/64/12/35/PDF/NorthAtlanticOscillations-I.pdf

  26. Greg Goodman says:

    NAO power spectrum from daily data since 1950, all year data.
    http://climategrog.wordpress.com/?attachment_id=897

    There is a peak around 60 years. Though this will not be too accurately determined it could be a tie in with N.Alt SST. Like W. says this would not be too surprising.

    I have not looked at the phase relationship , but if this paper is suggesting 15 years, it may indicate one is related to the derivative of the other.

    Atm pressure links to cloud cover so NAO varying with d/dt(SST) may be somewhat likely.

  27. Greg Goodman says:

    Indeed. Cross-correlation of NAO and d/dt( ICAODS SST ) provides a nice clean spectrum.
    http://climategrog.wordpress.com/?attachment_id=898

    The lunar signal one again strongly present. This looks like the 9.1 +/-0.1 again.

    Despite their rather convoluted method , going straight for it with the least processed data seems to confirm a common circa 60y periodicity. I have not looked directly at the phase but as I guessed above, if one is related to the derivative of the other, a pi/2 lag would be expected. For ~60 years that would be 15y.

    The presence of a strong 9 y signal and their rather chopped about data may account for their vague 10-15 year result.

  28. 1sky1 says:

    If one is interested in exploring the relationship between a multidecadal phenomenon, such as the AMO, and another such variable, then simply computing the cross-correlation function of MONTHLY data will NOT produce indicative results. The high-variance, month-to-month random variablity of temperature will almost certainly produce negligible cross-correlation at all lags. Only by doing complete cross-spectrum analysis or by appropriate band-pass filtering can that relationship (if any) be revealed.

  29. Greg Goodman says:

    ” Only by doing complete cross-spectrum analysis or by appropriate band-pass filtering can that relationship (if any) be revealed.”

    Like I did just you mean?

  30. scf says:

    I recall an earlier article where you were showing the perils of running correlations on smoothed data, it was about some other paper making the same mistake. It seems a lot of papers involve all sorts of random data fluctuations until they find something that appears to look interesting.

  31. 1sky1 says:

    Greg:

    You did NOT do a complete cross-spectrum analysis. It produces a complex-valued, frequency-dependent result C – iQ, where C is the co-spectrum C and Q is the quadrature spectrum Q; those spectra are obtained, respectively, by the cosine- and sine-transforms of the sample cross-covariance function between two variables. whose power densities are P1 and P2. What’s truly indicative in those results is the squared coherence (C^2 + Q^2)/(P1*P2) and the relative phase arctan(-Q/C) between the variables in each frequency band. What you present is far removed from that–and is plotted inappropriately as a function of period.

    P.S. Of course, the sample cross-covariance function needs to be extended to lags long enough to encompass periods of interest (which Willis fails to do) and be properly windowed to avoid spurious Gibbs effects in the results.

  32. bushbunny says:

    Bit O/T. But my ex was a vulcan pilot with the nuclear strike force. He went on several missions to Norway, to do cold weather survival with experienced Norwegian instructors. Even ate whale meat. But one of the things they learned was if they survived a crash in the North Atlantic, they had little time to get into their survival dinghy. They would die in a very short time from hypothermia. So has the water heated up enough now to save people. No. Like the Titanic passengers.

  33. Richard M says:

    Using subsets of a data set to predict future positions of an entire data set can have its uses. Consider that IBM is part of the DJI. If the IBM movement for one week predicted the DJI trend for the following week I’m pretty sure some people would find that useful.

  34. gymnosperm says:

    “There is a peak around 60 years.”

    We really need to get to the bottom of this sixty year thing.

  35. charles nelson says:

    Calculate this please someone….
    http://en.wikipedia.org/wiki/File:Golfstrom.jpg

  36. Edim says:

    I agree with Juraj V. It is all just a natural multidecadal fluctuation. NAO is not a temperature index. It’s the difference of atmospheric pressure at sea level between the Icelandic low and the Azores high. Positive and negative PDO phases are correlated with increasing and decreasing temperature indices (-PDO cooling, +PDO warming).
    http://www.woodfortrees.org/plot/jisao-pdo/normalise/plot/hadcrut4gl/mean:120/derivative/normalise

  37. Dr. Strangelove says:

    Willis
    The significance of PDO, AMO and NAO being able to emulate global temperature record is they are natural cycles. It means climate change is driven by natural cycles. No small feat considering all the “science is settled” of AGW.

  38. Willis Eschenbach says:

    Greg Goodman says:
    April 2, 2014 at 1:32 pm

    Many good points Willis. I also distrust this idea of picking out certain months.

    However, the old “never smooth” falacy comes up again.

    “The use of smoothed data in doing cross-correlation analyses is an abomination. Nature knows nothing of the 30-year average changes. Either there is significant cross-correlation between the two actual datasets or there is not. ”

    The danger of seeing false significance is real and something that needs to be guarded against. I see no sign that the authors are aware nor that they have adjusted thier significance tests to account for the lesser degrees of freedom in the filtered data, so the criticism if justified but the dogmatic postition about ‘abominations’ is not.

    If data is being “smoothed” to make it look smoother, the point totally valid, there is no reason to do stats on prettied up data. However, if the data was _filtered_ to remove, say, a strong annual cycle in order to study lesser magnitude inter-annual variation, then it is legit and it makes sense. This is why rail against using the term “smoothing” in place of frequency filtering, if that is what is meant.

    Thanks, Greg. I was not talking about removing seasonal variations, sorry for the confusion. I was talking about the kind of egregious smoothing that they are doing … and sadly, that’s the common kind.

    Another thing you state incorrectly is that NAO is a subset of AMO. It is not. NAO is a sea level pressure index not SST, so it cannot be a subset.

    Since on average the SLP is a function of SST, measuring pressure is just another way of measuring the temperature. You can see this by comparing e.g. the SOI and ENSO3.4. Despite the fact that one is measuring pressure over a large area and the other is measuring temperature over a smaller area, their annual correlation is a staggering 0.8 …

    Or you can just compare the NAO and the AMO directly. Despite one measuring pressure and one temperature, on an annual basis, they have a correlation of -0.43 …

    Regards,

    w.

  39. Willis Eschenbach says:

    Bloke down the pub says:
    April 2, 2014 at 3:40 pm

    ‘ Not a lot of difference to be sure … but I’m using their data, so it should be exact,’

    As you say that you digitized their data, is there any room in that process for a discrepancy to be introduced?

    See Figure 4, Bloke.

    w.

  40. Greg Goodman says:

    1sky1: “What’s truly indicative in those results is the squared coherence (C^2 + Q^2)/(P1*P2) and the relative phase arctan(-Q/C) between the variables in each frequency band. What you present is far removed from that–and is plotted inappropriately as a function of period. ”

    Thanks , it was not clear what you’d meant by “complete” cross-spectrum analysis.

    Now isn’t squared coherence (C^2 + Q^2)/(P1*P2) just what I did ? I agree there would be additional information to be obtained from looking at the phase relationship.

    A period plot is not “inappropriate” it’s a damned sight easier to read, since most people are discussing cycle periods and don’t think in “per year” units of frequency. If you want to start integrating the area under the curve to assess the power of a peak, you will want to be working in frequency. The periodogram is “appropriate” for the discussion of what periods are present and relative height of peaks.

  41. Willis Eschenbach says:

    Richard M says:
    April 2, 2014 at 8:13 pm

    Using subsets of a data set to predict future positions of an entire data set can have its uses. Consider that IBM is part of the DJI. If the IBM movement for one week predicted the DJI trend for the following week I’m pretty sure some people would find that useful.

    That’s true, Richard, but that’s not what these folks are doing. They are predicting the DJI for next week by using the IBM price for next week … how you gonna do that?

    w.

  42. Willis Eschenbach says:

    Dr. Strangelove says:
    April 3, 2014 at 1:02 am

    Willis
    The significance of PDO, AMO and NAO being able to emulate global temperature record is they are natural cycles. It means climate change is driven by natural cycles. No small feat considering all the “science is settled” of AGW.

    Thanks, Doc. You cannot assume causation simply because they are correlated. In fact, the AMO and the global temperature are correlated because the AMO is a part of the global temperature record. If

    A = B + C

    then A and B will be correlated. But that doesn’t mean that B is “driving” A as you say. Instead, it is simply a consequence of the fact that B is a part of A

    w.

  43. lgl says:

    Since the NAO is a subset of the AMO

    No it’s not. AMO is the integral of NAO (long term)
    http://virakkraft.com/NAO-AMO
    High NAO gives more sunshine hours to northern Europe so it makes sense.
    http://virakkraft.com/Sunshine-duration.pptx

  44. Edim says:

    Willis, AMO is not a part of the SH temperature indices and they still correlate.
    http://www.woodfortrees.org/plot/esrl-amo/plot/hadcrut4sh/detrend:0.8/plot/hadcrut4nh/trend/detrend:0.8

    The mode of variability known as AMO is actually global and the North Atlantic SST is part of that. The secular trend in this global oscillation is another longer quasi-oscillation.

  45. Greg Goodman says:

    For those who have a built-in reciprocal button here is the NAO vs N.Atl SST cross-spectrum plotted against frequency.
    http://climategrog.wordpress.com/?attachment_id=899

  46. Greg Goodman says:

    lgl says:
    No it’s not. AMO is the integral of NAO (long term)
    http://virakkraft.com/NAO-AMO

    Thanks, that is the guess I was working on doing the cross-correlation using d/dt(SST).

    This is like the d/dt(CO2) vs SST thing all over again. Unless you are looking at the correct form of the variables you will not get the right relationship. When the paper found a lag of 15 and a cycle of 60, I saw the obvious indication of a pi/2 lag of two quantities in quadrature.

    Integrating NAO or diffing SST amounts to the same thing (except the latter is better for FFT etc).

  47. Greg Goodman says:

    Since lgl would like a complete analysis, here’s the phase plot too:
    http://climategrog.wordpress.com/?attachment_id=900

    The circa 60y and 9y peaks are close to anti-phase, when comparing NAO to d/dt(SST), the strongest peak (on the inter-annual scale) at 2.68y is close to being in-phase.

    spectra were calculated using d/dt(N_Atl_SST) lagging NAO as positive correlation.

  48. Greg Goodman says:

    Just a note on visualising the phase plot: since phase cycles in 2*pi radians, +3.142 = -3.142 , so it’s not jumping, it’s cycling evenly around in phase. The upper limit and lower limit represent the same phase lag: the two variables being in anti-phase.

  49. lgl says:

    Greg
    Integrating NAO or diffing SST amounts to the same thing

    It’s perhaps a bit confucing but diffing the AMO gives something similar to ENSO, not the NAO, so I guess ENSO is the dominant driver on short term and NAO on longer term.

  50. Craig says:

    “Abstract: Our statistical analyses suggest that the AMO signal precedes the NAO by 10–15 years with an interesting predictability window for decadal forecasting.”

    The AMO signal appears to proceed the ENSO by at least ~74 years, and with a lot less manipulation of the data.
    http://wattsupwiththat.com/2014/02/13/a-relationship-between-sea-ice-anomalies-ssts-and-the-enso/

  51. Kristian says:

    Edim says, April 3, 2014 at 2:22 am:

    “Willis, AMO is not a part of the SH temperature indices and they still correlate.
    http://www.woodfortrees.org/plot/esrl-amo/plot/hadcrut4sh/detrend:0.8/plot/hadcrut4nh/trend/detrend:0.8

    The mode of variability known as AMO is actually global and the North Atlantic SST is part of that. The secular trend in this global oscillation is another longer quasi-oscillation.”

    Good observation, Edim.

    This is because that ‘global’ variability is not the AMO, it is the PDV (Pacific Decadal Variability). The Pacific is what runs the global climate, propagated from the tropics to both hemispheres by the ENSO process.

  52. Greg says:

    lgl says:
    April 3, 2014 at 6:19 am

    Greg
    Integrating NAO or diffing SST amounts to the same thing

    It’s perhaps a bit confucing but diffing the AMO gives something similar to ENSO, not the NAO, so I guess ENSO is the dominant driver on short term and NAO on longer term.
    ====

    OK, I’m confucked ;)

    I was meaning it was the same with respect to phase lags.
    Do you have that AMO/ENSO idea in a plot?

  53. Greg says:

    @lgl :
    int x(t).dt = k . y(t) + c
    x(t) = k . dy(t) / dt

    It’s identical except for the constant of integration.

  54. Greg says:

    Craig says: “The AMO signal appears to proceed the ENSO by at least ~74 years, and with a lot less manipulation of the data.
    http://wattsupwiththat.com/2014/02/13/a-relationship-between-sea-ice-anomalies-ssts-and-the-enso/

    That analysis you posted was very questionable, you just dumped it in a post and did not did not reply to the problems raised. Please don’t wave it around like it’s an established relationship.

    The whole idea you just suggested , that AMO goes to hide and then pops out 74 years later to create ENSO, is frankly laughable.

  55. Greg says:

    Greg says:
    April 3, 2014 at 8:25 am

    @lgl :
    int x(t).dt = k . y(t) + c
    x(t) = k . dy(t) / dt

    It’s identical except for the constant of integration.
    ….
    except that it’s less good for FFT. Having integrated out all the short periods, all that is left is long period that peaks at 58.75 years. There’s very little lower down so it’s fairly pure harmonic form.

    Looking at the correlation fn, it seems to peak with SST lagging int(NAO) by about 29 months.

  56. Matthew R Marler says:

    Willis Eschenbach: In your analogy, it is like predicting adult height from the length of the adult’s femur

    8-)

  57. Greg says:

    thanks, that’s interesting. derivative of Atlantic SST correlating with Nino3.4 SST.

    At least at first glance it does seem to correlate.

  58. Craig says:

    Greg says:
    April 3, 2014 at 8:30 am

    “That analysis you posted was very questionable, you just dumped it in a post and did not did not reply to the problems raised. Please don’t wave it around like it’s an established relationship.

    The whole idea you just suggested , that AMO goes to hide and then pops out 74 years later to create ENSO, is frankly laughable.”

    Last time I checked, “appears” does not mean “established.” I have never claimed it to be anything more than an observation, and never I suggested anything even remotely similar to what you wrote. If you want to read between the lines, you might try natural cycles.

    I don’t see where you raised any problems when I originally posted it? I’m curious what you see or don’t see that makes you so quick to write off a possible relationship. Perhaps you have something more substantive to offer than “laughable?”

  59. Not using the standard AMO or NAO, not using the whole year, just 4 months of it, and not using raw data for correlations, but using smoothed data. As pointed out above, this is a lesson in how well-meaning people, but too clever by half, wind up fooling themselves with misused statistics.

    It is also an example of how an intellectual community in a young field has not yet matured to be sufficiently self-critical, or to have internalized the skills needed to impose true rigor on their colleagues. Other sciences have gone through this phase, including medicine. The rules of statistics are independent of specialty. Why can’t climate scientists learn from the mistakes made in older disciplines, instead of repeating them?

  60. vukcevic says:

    Hi lgl
    Mechanical, electric and other types of natural oscillations are based on exchange between two types of energy, which in practice always means there is a degree of dissipation, hence for sustained oscillation external source of energy input is required.
    In the above case it is an exchange between the SST and the relevant atmospheric pressure, while sun replenishes dissipated amount.
    May I remind you of our short exchange from 2012 and the graph in this comment:
    http://wattsupwiththat.com/2012/07/06/the-nao-seafood-oscillation/#comment-1027468

  61. 1sky1 says:

    Greg Goodman says:
    April 3, 2014 at 1:33 am

    “Now isn’t squared coherence (C^2 + Q^2)/(P1*P2) just what I did ? I agree there would be additional information to be obtained from looking at the phase relationship.”

    It’s up to you–not me–to specify exactly what you did. In any event, none of your results appear to be credible cross-spectral estimates with decent degrees of freedom. Their very high frequency-resolution suggests that you simply FFT’d the sample cross-covariance function computed out to lags considerably longer than a quarter of the record length–without any windowing. Did you then compute the co- and quad-spectra separately and normalize their sum of squares by the product of the individual power densities? If so, then the “cross power spectrum” is exceptionally low and the phase spectrum would be expected to show randomly scattered values, instead of a persistent linear trend (modulo 2pi) characteristic of highly coherent variables with a fixed time-offset.

    BTW, no professionals in signal analysis ever display their spectra as a function of period; nor is such a display called a “periodogram.” Also, the standard unit of frequency is CYCLES per unit time, as in Herz = cycles per second, not just time inverse.

  62. Dr. Strangelove says:

    Willis
    A = B + C
    This is silly. A is a function of B. Change B and A changes too, if C is constant. That’s cause and effect, not merely correlation. Anyway climate science is not simple arithmetic. Global temperature is the effect. There are various causes. Natural cycles are probably one of them, if not the dominant one.

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