Multivariate Calibration (II)

In the previous post, I mentioned that Juckes et al INVR is essentially CCE. In addition, it was noted that CCE is not ML estimator and that Brown82 shows how to really compute confidence region in multivariate calibration problems. As Dr. Juckes made a good job of archiving his results, we can now compare his CCE (S=I) and ML estimator results Brown’s confidence region (with central point as point estimate) .

Enough talking, here are the results:

Esper et al. 2002 (ECS)


Red: central point, 95 % CI between green lines, average 2-sigma is 2.2 C, calibration residual based 0.44 C

Comparison with Juckes archived INVR:


Blue: central point, Black: archived INVR, r=0.67

Hegerl et al. 2006 (HCA)


Red: central point, 95 % CI between green lines, average 2-sigma is 6.1 C, calibration residual based 0.31 C


Blue: central point, Black: archived INVR, r=0.42

Jones et al. 1998


Red: central point, 95 % CI between green lines, average 2-sigma is 2.9 C, calibration residual based 0.71 C

Blue: central point, Black: archived INVR, r=0.93

Mann et al. 1999

Red: central point, 95 % CI between green lines, average 2-sigma is 1.4 C, calibration residual based 0.36 C
Blue: central point, Black: archived INVR, r=0.76


  1. ‘central point’ estimator and CCE give reasonably similar results (updated, see the previuous post)
  2. However, CIs from calibration residuals are always underestimated when compared to Brown’s CI formula results.

If you need the Matlab code, pl. email me.

Update 10 July 07

Of the above reconstructions, the most interesting is naturally MBH (MBH99 AD1000 step). MBH reconstuction looks quite good, even though those few peaks in (green) confidence intervals indicate that data does not always fit the model. Next question is, how does this estimator perform when we use the same calibration temperature but replace some proxies with noise?

First, lets try with all proxies i.i.d Gaussian (P=randn(975,14);)


Clearly the estimator handles this case well, confidence region gets really wide. But how about keeping the famous PC1, and replace all others with noise?


Reconstruction looks much better, the estimator takes that PC1 and almost completely neglects those proxies that are just noise. 95% CI limits are +- 1.6 C, calibration residuals would yield +- 0.5 C (hmmm, the same as original MBH99..) . This being the case, wouldn’t it be wise to use just PC1 alone? Let’s see:


It is better, 95 % CI now +- 0.7 C, and no more those empty confidence regions that indicated problems with the data. This is quite natural, added white noise just disturbs our estimator. But note that results are better than with the original 14-proxy reconstruction! So why this is not used alone? Because the wrong method, calibration residual based CIs, gives larger values than the previous example, +- 0.7 C ? IOW, inclusion of noise causes overfit to the calibration period, and if you use calibration residuals for estimating uncertainties, you’ll get better answer by adding plain noise. In the case of ICE this would be even more clear. See also Steve McIntyre’s comment :

My suspicions right now is that the role of the “white noise proxies�� in MBH98 works out as being equivalent to a “representation�� of the NH temperature curve more or less like Figure 2 from Phillips. The role of the “active ingredients�� is distinct and is more like a “classical�� spurious regression. I find the combination to be pretty interesting.

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2 Responses to “Multivariate Calibration (II)”

  1. John F. Pittman Says:

    All say “My / Juckes Archived” . Did you mean this?

  2. uced Says:

    My = reconstruction obtained using ML estimator written by me

    Juckes Archived = INVR reconstruction from Juckes supplement

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