Red Noise at realclimate

realclimate today has a post How Red are My Proxies? which is so weird it’s worthy of Rasmus. (Note: see lsubsequent comment here). They discuss the autocorrelation properties of North American tree ring proxies, something about which I know a lot. They say:

Using data from the North American network of seventy sets of tree rings extending from 1400 to 1980 you obtain an actual one-year AR1 mean autocorrelation factor with a value close to 0.15 (the exact number depends on the proxy series and time period chosen but is always less than about 0.3).

They are nuts. Here’s a histogram of the AR1 coefficients of the 70-series MBH98 tree ring network which we archived in a readable table in connection with our GRL paper. I’ve included a short R script here to calculate AR1 coefficients. The mean autocorrelation was not 0.15, but was 0.4. Out of 70 AR1 coefficients, only three were less than 0.15 and the mean was 0.4. The range of values was from 0.03 to 0.79. Tellingly, the highest AR1 coefficents all belonged to bristlecones.

But it’s even worse than that. If you model the series as ARMA(1,1), the AR1 coefficients increase dramatically with high negative (and nearly always) statistically significant MA1 coefficients. Many of the AR1 coefficients now become close to 1- random walk levels, especially the bristlecones. The statistical properties of this type of series – high AR1 and negative MA1 – are trickier than people think. I’ve posted up notes on them by Ai Deng for example.

I have no idea how realclimate got their results. Their whole post looks completely goofy to me.

The other salient point – and we included this histogram in our Reply to Von Storch discussed , is that the tree ring series in this network have virtually no correlation to gridcell temperature; many of correlations to precipitation and of course the bristlecones have a correlation to CO2 levels.


  1. Posted May 24, 2006 at 5:28 PM | Permalink

    I’m only vaguely knowledgeable enough to know what you are talking about with the auto-correlation statistics, but the last graph really speaks for itself. I’ve seen it before but I guess I forgot its significance. Even Blind Freddy could see that any meta-analysis of these proxies which doesn’t have any way of telling one signal from another, would be far more likely to extract precip. than anything else given those correlations.

  2. jae
    Posted May 24, 2006 at 5:36 PM | Permalink

    And here I thought they had moved on.

  3. jae
    Posted May 24, 2006 at 5:41 PM | Permalink

    From the RC post:

    If the noise component of real proxy data were really so strongly red, not only the precision of results of Mann et al. (the target of the von Storch et al’s analysis) but indeed of all previous millennial paleo-reconstructions would be substantially degraded.

    Looks like they are finally really worried about the autocorrelation problem.

  4. jae
    Posted May 24, 2006 at 5:50 PM | Permalink

    Steve: then perhaps Von Storch should have used red noise with an AR1 coefficient of 0.4, rather than 0.7?

  5. Steve Sadlov
    Posted May 24, 2006 at 7:08 PM | Permalink

    “Devestating” (and to think …. the warmers love to use the term triumphalism). Right …

  6. TAC
    Posted May 24, 2006 at 7:08 PM | Permalink


    But it’s even worse than that. If you model the series as ARMA(1,1), the AR1 coefficients increase dramatically with high negative (and nearly always) statistically significant MA1 coefficients. Many of the AR1 coefficients now become close to 1.

    How interesting. It is also known that if you model LTP (long-term persistence, say with 0.3

  7. Steve Sadlov
    Posted May 24, 2006 at 7:12 PM | Permalink

    RE: #3. I think they are also worried about the slight correlation with precip. Bristlecones correlating with precip would, I believe, “bring back” the MWP. It would also further focus attention on the seemingly “odd” behavior which has been experienced on the US West Coast, which seems to call into question the GCMs most in vogue at present.

  8. TAC
    Posted May 24, 2006 at 7:20 PM | Permalink

    #6 got truncated by an errant “less than” symbol. The post was supposed to say:


    But it’s even worse than that. If you model the series as ARMA(1,1), the AR1 coefficients increase dramatically with high negative (and nearly always) statistically significant MA1 coefficients. Many of the AR1 coefficients now become close to 1.

    How interesting! It is also known that if you model LTP (long-term persistence, say with d between 0.3 and 0.5) series as ARMA(1,1), the AR1 coefficients increase dramatically with high negative (and nearly always) statistically significant MA1 coefficients. Many of the AR1 coefficients now become close to 1.

    You can test this in R by loading the fracdiff package and entering the command:


  9. Terry
    Posted May 24, 2006 at 7:49 PM | Permalink

    The other salient point – and we included this histogram in our Reply to Von Storch discussed , is that the tree ring series in this network have virtually no correlation to gridcell temperature ….;

    How can this be?

    So what are they correlated to that gives them such a large weight in the reconstruction — to global temperature? If so, this alone would obviously disqualify the entire study, which makes me think I have missed something.

  10. Dave Dardinger
    Posted May 24, 2006 at 8:31 PM | Permalink

    re #9,

    No, it’s true that the claim seems to be that bristlecones can intuit global temperatures even though they can’t follow grid-cell temperatures. Now it is possible that there could be a linkage between say global temperatures and precipitation and that could be reflected in ring widths, but if so it’d require them to admit the the bristlecones are precipitation proxies first and foremost and the team definitely doesn’t want to admit that. So they just talk about this magic low-frequency signal which must be maintained at all costs.

  11. Terry
    Posted May 24, 2006 at 9:25 PM | Permalink

    I checked the paper the RealClimate post links to, and it estimates an AR1 coefficient of .15 based on data from 1400 – 1880.

    Does that explain the difference? Is the bristlecone “blade” in the twentieth century that makes the autocorrelation coefficient so high in your calculations?

  12. Terry
    Posted May 24, 2006 at 9:46 PM | Permalink

    Looking at the paper RealClimate linked to raised some more questions. I submitted the following post to RealClimate hoping to be educated a little bit.

    The article you link to uses a differencing technique “to remove the large variance highly correlated slow component from consideration prior to determining the AR1 autocorrelation component.”

    This doesn’t sound like a straight-forward estimate of an autocorrelation coefficient. What do you get if you use the standard method of estimating autocorrelation coefficients? Also, in the Von Storch analysis, which are they using, a standard AR1 model or a model that corresponds to this differencing technique that removes the highly correlated slow component?

    Thanks in advance for the clarification.

    Perhaps your results are different because you are using a standard autocorrelation estimate. If so, which is the appropriate estimate, i.e., what does the Von Storch model actually assume?

  13. Steve McIntyre
    Posted May 24, 2006 at 10:02 PM | Permalink

    The von Storch-Zorita approach to all of this doesn’t make a whole lot of sense either. I’ve spent quite a bit of time the last week or two pondering this dispute. Their main point is simple: if you mix a signal with high-frequency white noise and then rescale to match the amplitude of the signal, you lose amplitude in the signal. Because white noise is orthogonal to the signal, the underlying algebra is just the Pythagorean Theorem. So in that sense, it’s almost trivially true and using a climate model to provide empirical evidence of the Pythagorean Theorem seems a little longwinded. Equally it seems a little foolhardy for realclimate to argue that the Pythagorean Theorem is wrong.

    Underneath it all is a fairly intriguing issue though which I’m mulling over presenting as a publication.

  14. Terry
    Posted May 24, 2006 at 10:05 PM | Permalink

    Submitted another comment over at RealClimate:

    If you could indulge me some more, I need some more education.

    My reading of the Ritson paper you link to suggests that the differencing technique he uses removes ANY highly autocorrelated slow component before calculating the AR1 coefficient.

    My question then is, what happens if there is NO temperature signal in the data, but there IS an extraneous, highly autocorrelated signal that is not temperature related (say a CO2 fertilization effect or a precipitation signal). Does the procedure remove the confounding, signal? Presumably the procedure cannot tell the difference between the “true” signal and an extraneous signal with the same statistical properties. So is it possible the procedure is removing exactly the type of red noise that Von Storch is trying to simulate?

    Thanks again, and I apologize for my ignorance in this area.

  15. Steve McIntyre
    Posted May 24, 2006 at 10:47 PM | Permalink

    Terry, maybe you can figure out what Ritson actually did. If you look at the url of my R script, that has the data that he probably used. I supplied scripts to Ritson while I was working on our GRL article. I remember him getting all ballistic at one point. He asked what would happen if you added a fixed amount to the front part of the series. I did an experiment adding to the early portion of non-bristlecones (this is reported in our E&E article). The Mannian method flips all the series over and produces a colder estimate for the 15th century. You make 50 series “warmer” and the estimate gets “colder”. Ritson accused me of manually flipping the series over and comparing it to Rathergate. It’s amusing to see him teamed up with Wahl and Ammann and posting at realclimate.

  16. Ross McKitrick
    Posted May 25, 2006 at 8:31 AM | Permalink

    Re #9: Terry, you haven’t missed something. The ‘correlation’ can be opportunistic correlation to any pattern in the temperature principal components (which they call ‘Instrumental Training Patterns’), which may be a weighted average of temperatures from anywhere on the planet. As for the local temperature, here is Section 5.3 of our E&E paper from last year.
    5.3 Lack of a linear response to temperature in “key” proxies

    In McIntyre and McKitrick [2004b], in our criticism of bristlecone pines as an arbiter of world climate, we pointed out (as above) that a linear response to temperature had not been established for these sites (as seemingly required by MBH98). Mann et al.[2004b] replied that:

    MM04 demonstrate their failure to understand our methods by claiming that we required that “proxies follow a linear temperature response”. In fact we specified (MBH98) that indicators should be “linearly related to one or more of the instrumental training patterns”, not local temperatures.

    We doubt the authors really believe the idea of a temperature proxy exhibiting no relationship to local temperature makes much sense. It is instructive to compare this response to the policy articulated in Jones and Mann [2004], which states:

    A number of other temperature reconstructions used in earlier multiproxy composites or in review papers [e.g., Jones et al., 1998; Mann et al., 1998a, 1999; Mann and Jones, 2003] are not included. This is because they are either less resolved than decadal resolution [e.g., Dahl-Jensen et al., 1998] or correlations with local grid box temperatures are not significant …

    Jones and Mann [2004] do consider “climate field reconstructions” (CFRs), which appear to be similar to “instrumental training patterns” of MBH98. In this case, Jones and Mann [2004] argue that the CFRs should be shown to be similar to some aspect of local climate during some part of the year. This would seem to invite opportunistic use of either precipitation or temperature as a climate indicator, something for which they reproached Soon et al. [2003]. But perhaps most telling is the comment of MBH98 co-author Hughes in Hughes and Funkhouser [2003], who did not attribute the bristlecone pine growth to an “instrumental training pattern”, but stated that their anomalous 20th century growth rate is a “mystery”.


  17. kim
    Posted May 25, 2006 at 8:37 AM | Permalink

    Gaia taunted the bristlecones to get some cojones.

  18. TAC
    Posted May 25, 2006 at 10:36 AM | Permalink

    I’m trying to understand what Ritson actually computed. It is not the autocorrelation (ACF); it is not the partial autocorrelation (PACF). But might it be some kind of novel backwards PACF, where the contribution of variability associated with high-lag correlations to a shorter-lag correlation is effectively subtracted from the shorter-lag correlation? It is common to do this sort of adjustment to deal with deterministic sources of variability like seasonality. However, we are not dealing with a deterministic source of variability, so in any case I am not sure how to interpret the NBPACF statistic.

  19. Steve McIntyre
    Posted May 25, 2006 at 11:02 AM | Permalink

    #18. It looks like he calculated the AR1 on the first-differences of the data. Terry has asked about this at realclimate and Mann has denied it. However, if you read Ritson, it’s pretty clear that that’w what he’s done. I’ll do a quick run and see what happens with AR1 on first differences. What a bunch of goofs.

  20. Posted May 25, 2006 at 11:17 AM | Permalink

    You’re kidding. That would be the first difference of the first differences.

  21. Steve McIntyre
    Posted May 25, 2006 at 11:37 AM | Permalink

    #20. Ar1 on first differences yields many high negative values. Ritson’s calculation is a formula based on the assumption that the first differences are AR1. It’s pretty goofy. I’ll make a new post on this.

  22. TAC
    Posted May 25, 2006 at 12:17 PM | Permalink

    Re #19: First differences? That’s cool. I take it as a sign of progress. Last December there was that hullabaloo about non-zero values of d (as in ARIMA(p,d,q)), so seeing d=1 — first differencing — is good news indeed. Of course, my sense is that Mother Nature prefers d in the 0.3 to 0.5 range, but at least we now we have a pair of bookends.

  23. Steve McIntyre
    Posted May 25, 2006 at 12:28 PM | Permalink

    #22. Except that with first differences, a random walk has a Ritson-autocorrelation of 0. This is realclimate – you have to watch the pea under the thimble at all times. I’ve done another post on this.

  24. Steve Sadlov
    Posted May 25, 2006 at 1:42 PM | Permalink

    Terry’s got me in stiches. Excellent! Scientifically brilliant, while at the same time, entertaining in terms of subtle dashing to pieces of some absolute …. bull#$%s

  25. Willis Eschenbach
    Posted May 25, 2006 at 3:06 PM | Permalink

    Steve, just a note to say I’ve audited your work (using the same data, but doing it in Excel). My results agree exactly with yours, viz:

    Std. Dev._________________0.19
    Normality (Jarque-Bera)___2.86

    They claim the autocorrelation is “always less than 0.3”. In fact, 64%!! have a lag-1 autocorrelation greater than 0.3 … go figure. Have you written to ask them what’s going on?


  26. per
    Posted May 25, 2006 at 3:15 PM | Permalink

    more from RC
    #11 & #12, I guess I am confused like Terry. In Ritson’s paper, the third equation down, it appears that the proxy data is differenced Y(j)=X(j)-X(j-1) ??? Phil

    [Response: We’re checking with David Ritson for confirmation. However, the average raw lag-one autocorrelation coefficient for the full set of 112 (unprocessed) predictors used by MBH98 is rho=0.28 with a standard error of 0.03; if the 20th century is not included owing to the argument that the natural autocorrelation structure is contaminated by the anthropogenic trend, the value is lower, rho=0.245 +/-0.03. In either case, the inflation factor is minimal compared to what is assumed by Von Storch. -mike]

    [Response:(update) David Ritson confirms that the procedure in question is exactly as specified in the linked attachment and is designed to find, within specified approximations, the AR1 coefficient that describes the proxy associated random-noise. As mentioned above, we ourselves independently find an AR1 coefficient (for the combined noise+signal) between 0.25 and 0.30 for the MBH98 network, close to the Ritson value and qualitatively lower than the value used by von Storch. –mike]

  27. Terry
    Posted May 25, 2006 at 8:37 PM | Permalink

    RealClimate had been pretty good about responding to reasonable questions on this one.

    So far, there has been some partial reconciliation with Steve’s calculations. Including the 1880 to 1980 data increases their estimate from .15 to .30.

    I posted another question that may further reconcile things.

    Another chance to educate someone, i.e., me.

    It sounds like the Ritson paper calculates an average AR1 coefficient across all proxy series, and the average is a simple (unweighted) average. Is this the right way to do it? Intuitively, it would seem that a weighted average is more appropriate where the weights are assigned according to the relative importance of the proxy in the reconstruction. Otherwise, the inclusion of a lot of unimportant proxy series would tend to bias the results. Do you know what the autocorrelations in the most important series are?

  28. Steve McIntyre
    Posted May 25, 2006 at 8:46 PM | Permalink

    Terry, they don’t know what weights they assign and have said that it cannot be calculated. You can and I have.

    AS to the effect of one HS series in a system of white noise, take a look at down the page which illustrates that one series is enough to bias their methods.
    I’ll try to post up in the next day or two a graphic illustrating what’s wrong with the entire premise here.

  29. Terry
    Posted May 25, 2006 at 10:12 PM | Permalink

    Sounds like there might be a minor research opportunity here.

    It could go something like this:

    This paper provides a method for estimating the AR1 coefficient of the noise in the Von Storch analysis and applies that model to the data used in MBH98 to obtain an estimated AR1 coefficient of xx%. Inputting this estimate into the Von Storch model results in an estimated [blah blah] of yy%.

    Estimate the average AR1 over the entire time period (footnote: I do not truncate the data to avoid concerns about data mining) using weights proportional to the importance of the series in the MBH98 reconstruction.

    Explain the weighting scheme, include an intuitive justification for the weighting along the lines of my comment above, provide a rigorous mathematical derivation of the weighting scheme.

    Footnote thanking RealClimate for suggesting this approach to the problem.

  30. Steve Sadlov
    Posted May 26, 2006 at 10:56 AM | Permalink

    RE: #29. You’ve got your abstract. Looks like a really nice opportunity.

  31. Dano
    Posted May 26, 2006 at 12:15 PM | Permalink


    It’s a research proposal, not an abstract.

    Until you do the work, you don’t get to write an abstract.


  32. Steve Sadlov
    Posted May 26, 2006 at 12:24 PM | Permalink

    I always write my abstract first. It helps to scope the paper.

  33. Dave Dardinger
    Posted May 26, 2006 at 12:52 PM | Permalink

    re: #30

    Until you do the work, you don’t get to write an abstract.

    I think you need to tell the Hockey Team and the other warmers that fact.

  34. Dano
    Posted May 26, 2006 at 1:43 PM | Permalink


    So you know your results before you collect data? Impressive study design!

    But, to break with this site’s implicit rules, I’ll admit to being overly nitpicky on the importance of semantics here. I’m sure you mean you write a ‘research proposal’ to scope the paper.



  35. Steve Sadlov
    Posted May 26, 2006 at 2:46 PM | Permalink

    RE: #34. Of course the results are not known. So, at this juncture, the statement is of the approach and the overall scope. The one liner (and that is all it should be) is put in at the end. A few other mods are possible. Why are you nit picking? Ad hominem again?

  36. TCO
    Posted May 26, 2006 at 10:28 PM | Permalink

    Don’t argue with DanO. He is not up on complicated subjects. He is a good liberal and strong biker and is our tree coring bitch data-gatherer, for when we get funding.

  37. Crust
    Posted Jun 12, 2006 at 9:15 AM | Permalink

    You write:

    Out of 70 AR1 coefficients, only three were less than 0.15 and the mean was 0.4. The range of values was from 0.3 to 0.79

    This is trivially inconsistent (if the bottom of the range is 0.3, there can’t be any values below 0.15). What did you intend to write?

  38. Posted Jun 12, 2006 at 10:19 AM | Permalink

    Good catch. My guess is he meant the range of values was from 0.03 to 0.79. The graph shows one value between 0.00 and 0.10, and two between 0.10 and 0.20, consistent with his stateent that three were less than 0.15 (presumably two were between 0.10 and 0.15 and the other was 0.03).

  39. Posted Jun 12, 2006 at 10:22 AM | Permalink

    OK, now that I look at that, I realize my last comment was wrong, I was looking at the wrong graph. The second graph, the ARMA(1,1) Model, shows a range starting at 0.3, but his comment about 0.3 to 0.79 was before the ARMA(1,0) Model graph. Now I’m thinking the sentence should have been in the following paragraph. But, I’ll shut up now and let Mr. McIntyre sort it out…

  40. Steve McIntyre
    Posted Jun 12, 2006 at 11:56 AM | Permalink

    #40. If you look at the script here,, I try to include cited values at the script line where they are calculated. The 0.3 – now changed – is a typo for 0.03 (as seen in the script.)

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