“Gaussian random vector with a covariance matrix C,” which implies it is a vector of white noise.

It is white noise only if C is diagonal matrix. (If the vector contains time-indexed data)

]]>This para is from the 2nd round review:

The current inconsistency of the wording masks the layers of underlying maintained hypotheses, as spelled out in Allen and Tett (1999). These should be spelled out, by adding a quote directly from Allen and Tett, immediately after line 34, as follows: “It is important to recognise that the demonstration of internal consistency is all that can ever be expected from a formal attribution study. Proof that the model is “correct”, meaning that every alternative has been taken into account and rejected, is a logical impossibility.”

These are all from the 1st round review, and were merrily ignored:

This paragraph raises a key point about estimates of internal variability. By dismissing observation-based estimates of residual variability on the grounds that the time series are “short relative to the timescales of interest” you are implying that the variance changes so much over time that even 150 years of data does not suffice to estimate it. If this is really true, then why is the same data sufficient for detection, as well as everything else in the report? Also, you are effectively saying the data are nonstationary. But the entire rest of your statistical model, including the Appendix 1 material, relies on the assumption of stationarity.

Turning to the methodology, line 16 says “u is a realization of internal climate variability”, which, based on the material on page 8, I assume means it is an output of a climate model. But then line 17 says it is a “Gaussian random vector with a covariance matrix C,” which implies it is a vector of white noise. These statements are contradictory to each other, and are further contradicted by the GLS formula in line 18, which is derived assuming u is a regression residual. In these 3 lines you have given 3 conflicting definitions of the vector u. Is it a model-generated internal variability vector, or a vector of white noise, or a regression residual? I consider each possibility in turn in the next 3 cells.

If u is a model-generated estimate of internal variability, then in order to solve for the coefficient vector a there must either be a further residual vector in the regression equation or the equation is a perfect fit. But if it’s a perfect fit then there would not be any error bars around the signal coefficients. So that’s not it. Hence there must be an unstated residual vector. But the formula given for the coefficient vector a is derived by minimizing the sum of squared u terms (u’u), not the sum of squared residuals. This makes no sense as presented. The other possibility is that u is in the regression equation and its coefficient is assumed to be 1, so the regression equation is y=u+Xa+e, where e is a residual. But then the formula for a should be the usual restricted least squares formula, a = inv(X’X)X’y + inv(X’X)R'[R.inv(X’X)R’](r-Ra) where R is a matrix with 1 in the top left corner and 0 everywhere else and r is a vector with first element 1 and 0 everywhere else (see any econometrics text under the heading “linear restrictions”, e.g. Johnson 1984 pp 204-05).

]]>*The instrumental record is not long enough to provide a reliable estimate and may also be contaminated by the effects of external forcing.*

It is understood that CGCMs may not simulate natural internal climate variability accurately,..

*In the standard approach, detection of a postulated climate change signal occurs when its amplitude in observations is shown to be significantly different from zero. This is handled by testing the null hypothesis H : a = 0 where 0 is a vector of zeros.*

..but they do not mention that the global temperature record passes that test, with p=0.93 (*). I don’t want to read further. Well, OK, the divergence in Figure 1 looks nice.

(*)Mann argues that p=0.93 leads to unphysically long noise persistence. And then he applies median filter to the spectrum so that such peaks (caused by p=0.93) are removed. Problem solved, robust p=0.38. Higher than Ritson p=0.33. This is art.

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