Global Warming Statics

It is often stated that global temperature has increased over some specific time frame. Few realize there are different ways to answer this question, and the increase may not actually be significant, particularly in view of persistent correlation between temperature over long time scales (LTP).

In Statistical analysis of hydroclimatic time series: Uncertainty and insights Koutsoyiannis evaluates two publications using two different approaches to this issue: the evaluation of trends as done in Cohn, T. A., and H. F. Lins (2005), or as the simple change in temperature between two points as in Rybski et al. (2006).

Cohn and Lins [2005] used as a test statistic the slope of a linear fit to the time series to test whether or not a climate variable has changed in a statistically significant sense, over the available observation period. Rybski et al. [2006] proposed essentially the statistic D(k)(i,l) := X(k)(i) – X(k)(i–l) to test whether a or not a climate variable, defined on a time scale k, has changed in a statistically significant sense, over a period of l years (starting from year i).

I did a similar thing to Rybski in a recent post July 2008 Global Temperatures observing that the temperature increase of the lower troposphere TLT satellite measurements since 1979 was not significant. This attracted a lot of criticism at the skeptical-leaning ClimateAudit, and I said I would post more details. However, the paper by Koutsoyiannis already shows the analysis using the Rybski framework and the CRU data series, over a range of parameter values, and finds the same thing — no justification for believing that the underlying global temperature has not been been other than static.

It may be of some interest to apply this pseudo-test to the CRU data series. The application is shown graphically in Figure 2, for a double-sided test for significance level 10–2 and for the SSS case, using all possible integer lags l/k from 1 (l = 30) to 4 (l = 120). In neither case the pseudo-test resulted in rejection of the null hypothesis (no change), although it comes close to rejection for 2005 for l/k = 3.

I applied the same kind of test as Rybski and Koutsoyiannis in a limited way to the satellite RSS/MSU series TLT for the last thirty years, and found that the temperature change from July 1979 to July 2008 was also not significant. The figure below tells the same story for a different temperature series (CRU) and a wider range of parameter values, that the difference between time of publication and every past temperature is not significant. As temperatures have now decreased somewhat, the difference in temperatures between all points in the record is even less significant.

Figure 2 from: Statistical analysis of hydroclimatic time series: Uncertainty and insights.

Original Caption: Figure 2 Graphical depiction of the pseudo-test based on StD[D] with known H. The continuous solid curve represents the CRU time series averaged over climatic scale k =30. The series of points represent values of D for the indicated lags l/k. Horizontal lines represent the critical values of the pseudo-test, which are the estimates of StD[D] times a factor 2.58 corresponding to a double-sided test with significance level 1% and assuming normality (only the positive critical values are plotted).

While the two approaches, fitting trends and testing difference in temperatures answer the same question there are reasons to think the Rybski approach has advantages in a LTP framework. Mainly, D(k)(i,l) does not depend on a fitted model (as e.g. a linear fitting to the data). This means that all the assumptions inherent in using linear regression for measuring trends are avoided. In particular, in LTP the correlation between terms persists even at very long time scales, and this violates the linear regression assumptions. There is less uncertainty about the form of model, providing LTP is accepted, hence less uncertainty about the variance, and about the reliability of the test.

Another advantage of ditching trends as a concept is that there are a host of other useful concepts that can be applied to the generalized model. One of these is ”mean reversion” (eg. see http://www.puc-rio.br/marco.ind/revers.html). As usual, financial time series analysis is light-years ahead of climate science. I am sure there are other concepts that could be applied, such as conservative vs dissipative dynamical theory, that can be more easily given physical interpretations. But you need to give up on trend analysis.

I was going to repeat some of the bile directed at my analysis by bender and others at CA, but frankly I couldn’t be bothered. Essentially, bender thinks trend analysis is the only ‘correct’ analysis for the global warming question. Suffice to say that if my analysis is ‘trash’ then so is Rybski et al. 2006 and Koutsoyiannis 2007. It remains for me to perform the analysis using the RSS/MSU series with a monthly time step, using a greater range of parameters and the standard deviation equation (10) derived in Koutsoyiannis 2007. I don’t expect the result will be different from the original post — no reason to believe temperatures have increased since 1979.

Refrences:

Cohn, T. A., and H. F. Lins (2005), Nature’s style: Naturally trendy, Geophys. Res. Lett.,
32(23), L23402, doi:10.1029/2005GL024476.

Koutsoyiannis, D., and A. Montanari, Statistical analysis of hydroclimatic time series: Uncertainty and insights, Water Resources Research, 43 (5), W05429.1–9, 2007.

Rybski, D., A. Bunde, S. Havlin, and H. von Storch (2006), Long-term persistence in climate and the detection problem, Geophys. Res. Lett., 33, L06718, doi:10.1029/2005GL025591.

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0 thoughts on “Global Warming Statics

  1. David,

    This is just to make the point that one of the authors of Rybski et al (2006) is the leading climatologist and statistician Hans von Storch. On 10 July 2008 I referred to von Storch as a leading statistician in climate science on ‘The “PR Challenge”’ thread at CA ( http://www.climateaudit.org/?p=3259 , post #10), as follows:

    “In addition to their major book on statistics referred to above, Francis Zwiers and Hans von Storch co-authored the paper ‘On the Role of Statistics in Climate Research’, which was published in the International Journal of Climatology in 2004 (vol.24:665-680). One key conclusion of that paper was ‘We feel that the cooperation between the statistical and climate sciences does not function as well as that between, for example, statistical and biomedical science … [B]etter communication between statisticians and climatologists requires a better understanding by statisticians of the specifics of climate science, and a greater effort by climatologists to communicate the specifics of open problems to statisticians.”

  2. David,

    This is just to make the point that one of the authors of Rybski et al (2006) is the leading climatologist and statistician Hans von Storch. On 10 July 2008 I referred to von Storch as a leading statistician in climate science on ‘The “PR Challenge”’ thread at CA ( http://www.climateaudit.org/?p=3259 , post #10), as follows:

    “In addition to their major book on statistics referred to above, Francis Zwiers and Hans von Storch co-authored the paper ‘On the Role of Statistics in Climate Research’, which was published in the International Journal of Climatology in 2004 (vol.24:665-680). One key conclusion of that paper was ‘We feel that the cooperation between the statistical and climate sciences does not function as well as that between, for example, statistical and biomedical science … [B]etter communication between statisticians and climatologists requires a better understanding by statisticians of the specifics of climate science, and a greater effort by climatologists to communicate the specifics of open problems to statisticians.”

  3. David,

    When I first looked at your analysis I also thought it was meaningless as well because you appeared to taking two data points, showing that they fell inside some significance interval and then claiming that there is no evidence of warming despite the fact that the majority of points in the early dataset were under significance interval and the majority of points in the later dataset were over the significant interval.

    I now see that there is some theoretical basis for the test that you are doing but I have trouble intuitively understanding what this test really shows. OTOH, I don’t have any trouble intuitively undestanding Lucia’s trend analysis with all of her caveats.

    Would you be able to come up with an example that would illustrate how this test works. For example, Lucia did a good job of explaining here method here:: http://rankexploits.com/musings/2008/ipcc-projections-do-falsify-or-are-swedes-tall/

  4. David,

    When I first looked at your analysis I also thought it was meaningless as well because you appeared to taking two data points, showing that they fell inside some significance interval and then claiming that there is no evidence of warming despite the fact that the majority of points in the early dataset were under significance interval and the majority of points in the later dataset were over the significant interval.

    I now see that there is some theoretical basis for the test that you are doing but I have trouble intuitively understanding what this test really shows. OTOH, I don’t have any trouble intuitively undestanding Lucia’s trend analysis with all of her caveats.

    Would you be able to come up with an example that would illustrate how this test works. For example, Lucia did a good job of explaining here method here:: http://rankexploits.com/musings/2008/ipcc-projections-do-falsify-or-are-swedes-tall/

  5. David,

    Thanks for drawing attention to the paper by Koutsoyiannis and Montanari (2007). To avoid misapprehension, I would like to note a few things:

    1. We do not criticize in K & M the method by Cohn and Lins (2005). Even though we preferred to use an approach similar to Rybski et al (2006), our results are compatible with C & L rather than with R et al.

    2. One of the reasons we preferred an approach similar to R et al is that their test statistic has several interesting properties as we outline in the K & M.

    3. Our main differences with R et al, which led to different final results, are the following.

    3a. We replaced an approximate equation of R et al for the variance of the test statistic with equation 10, which we claim it is accurate (proof: from the definition of the test statistic, D_i,l(k) := X_i(k) – X_i-l(k) we get Var[D_i,l(k)] = Var[X_i(k)] + Var[X_i-l(k)] – 2 Cov[X_i(k), X_i-l(k)] = 2 Var[X(k)] – 2 rho_l/k(k) Var[X(k)]).

    3b. We considered the bias in the standard estimator of std (equation (8)), which we demonstrated that for highly autocorrelated processes can be very high (because the equivalent sample size n’ in (8), calculated from equation (6) or (7), may be very small as shown in Table 1). We also discussed the effect of uncertainty in the simultaneous estimation of H and std.

    3c. We tried to demonstrate that, while proxies contain useful statistical information and, in particular, they advocate the presence of LTP with a very high H, they involve several uncertainties and are statistically incompatible to each other, so that it is not wise to use their statistical estimates in statistical testing. Thus, in our testing we used only statistics from the instrumental CRU series (similar to C & L).

    3d. We avoided the use of the detrended fluctuation analysis (DFA) for the estimation of H, because we think it has several problems (as a result of which R et al. had estimated some H values greater than 1) and, in particular, it hides the statistical uncertainty involved in estimation.

    Despite these differences, I think that Rybski et al. (2006) is a very nice paper and I share the opinion of Ian about its authors.

    4. The focus of our study (K & M, 2007) was on understanding rather than on proposing an exact test and so we preferred analytical formulations rather than numerical (e.g. Monte Carlo) methods. The cost for this preference was the fact that the test we formed is a “pseudotest” rather than a formal one. Specifically, the test was based on the assumption that the true value of H or rho is known. This is not usually the case, so the rejection rate of the pseudotest is lower than it should (i.e. if the test does not reject your hypothesis, you are on the safe side).

    Demetris

    PS. As I have noted elsewhere, K & M (2007) has an interesting prehistory (downloadable from http://www.itia.ntua.gr/en/docinfo/781) .

  6. David,

    Thanks for drawing attention to the paper by Koutsoyiannis and Montanari (2007). To avoid misapprehension, I would like to note a few things:

    1. We do not criticize in K & M the method by Cohn and Lins (2005). Even though we preferred to use an approach similar to Rybski et al (2006), our results are compatible with C & L rather than with R et al.

    2. One of the reasons we preferred an approach similar to R et al is that their test statistic has several interesting properties as we outline in the K & M.

    3. Our main differences with R et al, which led to different final results, are the following.

    3a. We replaced an approximate equation of R et al for the variance of the test statistic with equation 10, which we claim it is accurate (proof: from the definition of the test statistic, D_i,l(k) := X_i(k) – X_i-l(k) we get Var[D_i,l(k)] = Var[X_i(k)] + Var[X_i-l(k)] – 2 Cov[X_i(k), X_i-l(k)] = 2 Var[X(k)] – 2 rho_l/k(k) Var[X(k)]).

    3b. We considered the bias in the standard estimator of std (equation (8)), which we demonstrated that for highly autocorrelated processes can be very high (because the equivalent sample size n’ in (8), calculated from equation (6) or (7), may be very small as shown in Table 1). We also discussed the effect of uncertainty in the simultaneous estimation of H and std.

    3c. We tried to demonstrate that, while proxies contain useful statistical information and, in particular, they advocate the presence of LTP with a very high H, they involve several uncertainties and are statistically incompatible to each other, so that it is not wise to use their statistical estimates in statistical testing. Thus, in our testing we used only statistics from the instrumental CRU series (similar to C & L).

    3d. We avoided the use of the detrended fluctuation analysis (DFA) for the estimation of H, because we think it has several problems (as a result of which R et al. had estimated some H values greater than 1) and, in particular, it hides the statistical uncertainty involved in estimation.

    Despite these differences, I think that Rybski et al. (2006) is a very nice paper and I share the opinion of Ian about its authors.

    4. The focus of our study (K & M, 2007) was on understanding rather than on proposing an exact test and so we preferred analytical formulations rather than numerical (e.g. Monte Carlo) methods. The cost for this preference was the fact that the test we formed is a “pseudotest” rather than a formal one. Specifically, the test was based on the assumption that the true value of H or rho is known. This is not usually the case, so the rejection rate of the pseudotest is lower than it should (i.e. if the test does not reject your hypothesis, you are on the safe side).

    Demetris

    PS. As I have noted elsewhere, K & M (2007) has an interesting prehistory (downloadable from http://www.itia.ntua.gr/en/docinfo/781) .

  7. Raven and Demetris. Thanks for the comments. I can see it would be worthwhile to develop a parallel analysis to Lucia’s only using the Rybski approach. I don’t really understand why some people are having a problem with it, but perhaps its because I come from a maths background where the training is to constantly generalize and abstract. Its part of the charm (or addiction) of blogs — this feedback.

  8. Raven and Demetris. Thanks for the comments. I can see it would be worthwhile to develop a parallel analysis to Lucia’s only using the Rybski approach. I don’t really understand why some people are having a problem with it, but perhaps its because I come from a maths background where the training is to constantly generalize and abstract. Its part of the charm (or addiction) of blogs — this feedback.

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