I’ve just completed Mike’s Nature trick of adding in the real temps
to each series for the last 20 years (ie from 1981 onwards) amd from
1961 for Keith’s to hide the decline.

Date: 22 Dec 2004, mike

No researchers in this field have ever, to our knowledge, “grafted the thermometer record onto” any reconstruction. It is somewhat disappointing to find this specious claim (which we usually find originating from industry-funded climate disinformation websites) appearing in this forum. Most proxy reconstructions end somewhere around 1980, for the reasons discussed above. Often, as in the comparisons we show on this site, the instrumental record (which extends to present) is shown along with the reconstructions, and clearly distinguished from them (e.g. highlighted in red as here).

Date: 6 May 2009, UC

Let’s see; I think this is made by padding with zeros, but 1981-1998 instrumental is grafted onto reconstruction:

(larger image here )

I used Mann’s lowpass.m , modified to pad with zeros instead of mean of the data,

out=lowpass0(data,1/40,0,0);

Original CA link

Backup

Date: 20 Nov 2009, UC

“I’ve just completed Mike’s Nature trick of adding in the real temps
to each series for the last 20 years (ie from 1981 onwards) amd from
1961 for Keith’s to hide the decline”

Is this about the MBH99 smooth ?

http://www.climateaudit.org/?p=1553#comment-340175

http://www.climateaudit.org/?p=1553#comment-340207

Date: 20 Nov 2009, gavin

[Response: This has nothing to do with Mann's Nature article. The 50-year smooth in figure 5b is only of the reconstruction, not the instrumental data. - gavin]

Date: 21 Nov 2009, gavin

And it remains unclear why this was described as Mann’s Nature trick since no such effect is seen in Mike’s paper in any case. – gavin]

Date: 22 Nov 2009, mike

In some earlier work though (Mann et al, 1999), the boundary condition for the smoothed curve (at 1980) was determined by padding with the mean of the subsequent data (taken from the instrumental record).

Date: 24 Nov 2009, CRU

To produce temperature series that were completely up-to-date (i.e. through to 1999) it was necessary to combine the temperature reconstructions with the instrumental record, because the temperature reconstructions from proxy data ended many years earlier whereas the instrumental record is updated every month. The use of the word “trick” was not intended to imply any deception.

Date: 25 Nov 2009, Jean S

UC has corrected me on the fact that adding the instrumental series to the proxy data prior smoothing was used already in MBH98 (Figure 5b), so, unlike I claimed in #66, “Mike’s Nature trick” is NOT a misnomer.

Date: 25 Nov 2009, UC

..and here’s instrumental (81-95)+zero padded Fig 5b smooth (red):

Original CA link

Backup

1. Mann et al 2008



2. Mannian CPS



3. Craig Loehle on the Divergence Problem

After four months, it is good to check how well the simple half-integrated-white-noise model is doing. Predictions for these 4 months were

Year        Month     -2 sigma     Predict.     +2 sigma
2008              8       0.15206     0.34864      0.54521
2008              9       0.1141       0.33388      0.55365
2008             10      0.094005    0.32581      0.55762
2008             11      0.080432    0.32024      0.56005

and observations today 15th Dec 08 on HadCRU website are the following:

2008/08 0.396
2008/09 0.374
2008/10 0.438
2008/11 0.387

Here’s how these fit to the original figure:

The model is doing quite good work. I’ll tell you when we reach the upper bound of the prediction interval. And after temperatures go permanently above that bound, AGW kicks this model to the trash can.

A wise choice of embedding dimension can be made with a priori insight or perhaps more commonly may be found by simply playing with the data.

Specially, Figure 3. of that article caught my eye:

Original Caption:Fig. 3. Nonlinear and linear trends in time series of mean sea level at Brest, France, for an embedding dimension equivalent to 30 years and an individual measurement standard error of 10 mm. The 95% confidence interval for the nonlinear fit is shaded and marked by the curved lines for the linear fit.

I found the data set from http://www.pol.ac.uk/psmsl/pubi/rlr.annual.data/190091.rlrdata , the only problem is that there are some missing values in this one. If anyone finds the full series, as in Fig 3. , pl. let me know.   However, I can replicate this figure quite closely, linear trend:

..and ssatrend with 30 year embedding dimension:

With 10 mm measurement error, ssatrend outputs approx. 3 mm error at the endpoints and 1.5 at the middle. And as you can see from the original figure, indeed it seems that these errors are lower than linear trend errors. Moore:

The confidence interval of the nonlinear trend is usually much smaller than for a least squares fit, as the data are not forced to fit any specified set of basis functions.

But there’s one problem. When I got good match with the linear trend confidence limits, I used residuals to estimate the noise variance. Residual sum of squares divided by the degrees of freedom, you know that stuff. And then I just assumed that as a good estimate of the true variance, additive iid Gaussian noise over a trend. That’s how I got the match. If I’d use the 10 mm measurement noise, the confidence limits would be much more narrow:

Residual-based limits in black, 10 mm measurement noise based limits in red.  The limits are actually more narrow than in the ‘non-linear trend’! That’s what I thought originally, if you have observation

y=s+n

and you apply a linear filter F,

F(y)=F(s) +F(n)

and you define noise as F(y)-F(s), then more smoothing the better. In Wiener filtering, for example, the aim is to find F that minimizes F(y)-s. In linear least squares fit F(s)=s. In these climate papers F(s) vs. s  seem to be not interesting. See also my CA comment . But this is something I’ll talk later, now I’d like to solve this Figure 3 issue.

Update 15 July 09: See also http://www.climateaudit.org/?p=6533 and http://www.climateaudit.org/?p=6473 , something wrong with the Monte Carlo code as well,

UPD Jan 2010: change

urlwrite(’http://www.climateaudit.org/data/mbh99/proxy.txt’,'proxy.txt’)

to

urlwrite(’http://www.climateaudit.info/data/mbh99/proxy.txt’,'proxy.txt’)

Some notes:

• Download to empty folder and rename to hockeystick.m
• Program downloads necessary data from the web (once), uses urlwrite.m (newish Matlab needed)
• It’s a script
• Shows what PC1_fixed does
• Only one file is downloaded from CA (AD1000 proxies), sorry RC, but I don’t know where to find morc014 elsewhere..
• Pl. tell me if it works or not, uc_edit at yahoo.com !

Updated to Ver 1.1, added cooling trends:

1. Re-scaling the Mann and Jones 2003 PC1



2. The Gift That Keeps On Giving



3. Mannomatic Smoothing and Pinned End-points



4. UC on CCE



5. Unthreaded 29

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

Conclusions

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.

(1)

(2)

where sizes of matrices are Y (nXq), E (nXq), B(pXq), Y’ (1Xq), E’ (1Xq), X (nXp) and X’ (pX1). is a column vector of ones (nX1). This is a bit less general than Brown’s model (only one response vector for each X’). n is length of the calibration data, q length of the response vector, and p length of the unknown X’. For example, if Y contains proxy responses to global temperature X, p is one and q the number of proxy records.

In the following, it is assumed that columns of E are zero mean, normally distributed vectors. Furthermore, rows of E are uncorrelated. (This assumption would be contradicted by red proxy noise.) The (qXq) covariance matrix of noise is denoted by G. In addition, columns of X are centered and have average sum of squares one.

Classical and Inverse Calibration Estimators

Classical estimator of X’ ( CCE (Williams 69) , indirect regression (Sundberg 99), inverse regression (Juckes 06) ) is obtained by generating ML estimator with known and and then replacing by and by where

(3a)

(3b)

and

(4)

( , i.e. centered Y ), yielding CCE estimator

(5)

where

(6)

Another way to go is ICE (inverse calibration estimator (Krutchkoff 67), direct regression (Sundberg 99) ) , directly regress X on Y,

(7)

Note that nobody yet has said that these estimators are optimal in any sense. It turns out that if we have special prior knowledge of ‘ (Xs and Ys sampled from normal population), ICE is optimal.

Important note (yet without proof here) is that sample variance of econstruction in the calibration period will be smaller than the reconstruction in the case of ICE, and larger with CCE. In the absence of noise, ICE and CCE yield (naturally) the same result. Update: see Gerd’s link and, and also note that ICE is a matrix weighted average between CCE and zero-matrix (Brown82, Eq 2.21).

Confidence Region for X’

Following Brown, we have per cent confidence region, all X’ such that

(8)

where is the upper per cent point of the standard F-distribution on q and v=(n-p-q) degrees of freedom and

(9)

The form of this confidence region is very interesting, and it is important to note that letting approach one the region degenerates to the CCE estimate . Update2: Central point of the region is NOT (AFAIK for now ) ML estimate, and the relation of central point and CCE is, as per Brown,

(10) , where

(11)

and

(12) .

Often calibration residuals are used to generate CIs for proxy reconstructions. We’ll see what will be missing in that case:
I simulated proxy vs. temperature cases with q=40, n=79 and SNR=1 and SNR=0.01. With SNR 1 we’ll get nice CIs, (which agree quite well with calibration residuals), but when SNR gets lower, the confidence region grows rapidly, being open from the upper side quite soon! Yet, in the latter case calibration residuals indicate relatively low noise. The dangerous situation is when true X’ is greater than calibration X (the very thing hockey sticks are trying to prove wrong).

Above: SNR=1, Below: SNR=0.01

Conlusions

1. Direct usage of calibration residuals for estimating confidence intervals is quite dangerous procedure.
2. Assumptions of ICE just do not work in proxy reconstructions

References

Brown 82: Multivariate Calibration, Journal of the Royal Statistical Society. Ser B. Vol. 44, No. 3, pp. 287-321

Williams 69: Regression methods in calibration problems. Bull. ISI., 43, 17-28

Krutchkoff 67: Classical and inverse regression methods of calibration. Technometrics, 9, 425-439

Sundberg 99: Multivariate Calibration – Direct and Indirect Regression Methodology
( http://www.math.su.se/~rolfs/Publications.html )

Juckes 06: Millennial temperature reconstruction intercomparison and evaluation

( http://www.cosis.net/members/journals/df/article.php?a_id=4661 )