Von Storch et al. [2004] argued that the reduced variability of MBH and Jones et al [1998] was possibly due to the use of inverse regression. This hypothesis has received a fair of attention as a rival candidate for the microscopic honor of breaking the hockey stick. I obviously think that our analysis in terms of principal components, bristlecones and RE statistics is the correct one, but I’m sure that more climate scientists would endorse the VS analysis as being on the money. On the face of it, this is a quite plausible explanation. The problem is that it is weakly argued in the original article and simply doesn’t apply to the actual models.

There is nothing wrong with the VS observation about the fact that the variance of an estimate T* in an inverse regression is less than the variance of the target T, i.e. Var (T*) < Var (T). The trouble is that von Storch et al. fail to link this to the actual multivariate methods. The argument in von Storch et al [2004] is merely as follows:

A number of studies have attempted to reconstruct variations in global or Northern Hemisphere (NH) temperature within the last millennium by regressing proxy indicators and early instrumental time series on recent instrumental climate variables with high spatial resolution (1–3: MBH98, MBH99 and Jones et al 1996). Regression models are developed during the period of common instrumental and proxy data, and are then applied to longer proxy records to reconstruct past climates…

In the case of MBH98 and other reconstructions, the methodological process is more sophisticated, but the fundamental problem of the loss of variance due to noisy proxy data

mayexist also in these studies.

The difficulty is precisely that the inverse regression explanation fails precisely because of the multivariate methodologies of these other reconstructions, in which the variance of the reconstruction is "scaled" to the observed instrumental variance, not just in some other reconstructions, but in the type studies of Jones et al [1998] and MBH99.

The type high-variabilty case is Esper et al [2002]. In this case, the calibration period variance is matched to instrumental variance in the calibration period 1902-1980 (sd 0.189 K).

The type low-variability cases are Jones et al. [1998]. Empirically, if you wade through the actual methodology of Jones et al [1998], you will find that it does not contain any inverse regression step of the type posited by von Storch and, in fact, the variances are explicitly matched in the calibration period: the J98 reconstruction and CRU instrumental variance exactly match (sd – 0.217 K) in their calibration period 1901-1950. So this reconstruction does exactly what VS says they don’t.

Ironically, there is a virtually identical re-scaling in MBH. Although MBH contains an inverse regression step for calibration of individual proxies, later in their procedure, there is an (unreported) re-scaling procedure, which matches variances of reconstructed temperature PCs. (This step was added to the Ammann and Wahl emulation only in April 2005; I’ve verified it in source code). The net result is that there is nearly exact variance match between instrumental version and reconstruction in the calibration period 1902-1980 ( sd – 0.186 K vs 0.183 K). Thus, the VS inverse regression hypothesis does not apply even in this type case.

So whatever the actual explanation for the varying low-frequency ranges, the attenuation of variance by use of inverse regression in MBH and Jones et al [1998] is not it.

## 6 Comments

1.

This sentence is incomprehensible.

2. I don’t understand what you are proving and how you are doing it. And I’ve spent a hell of a lot of time, reading your stuff.

3. The Amman-Wahl swipe is gratitous to your point, Steve. Avoid the drive-bys. Take an issue and examine it. Don’t make connection to connection of a parenthetical nature that allows you to throw in other crimes and misdeeds.

This part of VS has worried me alot. I don’t understand the derivation of his claim and have been meaning to go back to his source for it. He states that reduction in variance is a function of the method then cites another method that doesn’t have the same problem (averaging I think) but its not explained at all. Anyway, in an inverse of y=mt+c+e the equation t=(y-c-e)/m where y is the proxy and t is the temp, the recovered temperatures are very sensitive to m, the estimated slope. Underestimates of m magnify long term variance and that would be where a problem comes in. But I am not sure the regression is even inverted always. Perhaps they just parameterize t=my+c directly?

RealClimate comments on Von Storch et al. 2004.

From their comments:

The amount of gall is awe-inspiring.

More comments on Von Storch from RC:

And:

Have they no shame?

Re #4: No. Why should they?

Steve,

You’re an intelligent guy. I’m sure you can figure it out.