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## More Hockey Mathmagic.

Posted by Jeff Id on August 27, 2009

In my last post we looked at the change in historic signal magnitude as it relates to the signal/noise ratio of proxies used in CPS.  This post takes the next step and explores what happens to the signal quality as we search for ever increasing R values. Although, I have a surface plot also, but the surface is difficult to interpret. The best method I found to show this variance is through video plot of the 2D graphs at different R values. This video assumes a fixed signal to noise ratio which is matched to the Schweingruber MXD latewood proxies.

First, recall that the signal has a true peak amplitude of 1 (figure 2) which is added to 10,000 arma simulated proxies. The signal is shown below.

Figure 1, Artificial signal added to proxies

Figure 3 is actually a video which is linked in YouTube that starts when you click on it, however the frame shown is CPS using a very low r value of 0.01. The only proxies rejected are those which have a negative correlation to the signal we’re looking for in this frame of the video. In this case, the temperature signal we’re looking for is a linear upslope from 0 to 1 in the last 100 years which exactly matches the artificial signal placed in the proxy data shown in Figure 1.

Note that the amplitude of the signal is reduced to about 0.3 from an initial value of 1. This is caused by scaling the standard deviation of the calibration range signal to match the standard deviation of the 0 – 1 temperature signal we’re looking for. Adding noise to the signal always increases the standard deviation! Think about that, no random signal can be added which decreases the standard deviation of the proxies on average.

When the proxies are rescaled to match the standard deviation of the temp signal, the average scaling over many proxies is always less than the initial signal values. Since very few of the proxies are thrown out at r =0.01 (the initial frame of the video), the average of all proxies balances in the calibration range (most recent 100 years) vs the historic range temp. reconstruction.

Figure 3 - Signal distortion video - Click to play on YouTube

As you can see, the gain across the entire signal is reasonably even in the beginning however the signal is substantially reduced for reasons explained above.  As the video plays, the calibration range (100 years on right end) becomes amplified relative to the whole signal.  This is due to increased rejection of proxies which don’t correlate with the calibration range signal we’re looking for.

Figure 4 is the data from the video in Figure 3 plotted as a surface.

Figure 4, Reconstruction at different correlations

Figure 5shows the gain factor for the historic signal at various r values.   This was calculated as a ratio of the average of the 200 points at the top of the square wave divided by the 200 points at the left end of the graph prior to the square wave .  This gain factor is only the change in historic signal amplitude as a function of r.  You can see there is very little shift in amplitude with increasing correlation values but it is non-zero.  Correlation sorting has little effect on the magnitude of the historic signal but the next figure shows it does affect the calibration range signal.

Figure 5, Demagnification of historic signal

The final plot is a graph of the ratio of the calibration range signal peak amplitude divided by the historic signal peak amplitude for various r values.

Figure 6, Amplification of calibration signal compared to historic signal

Sorting proxies by correlation is looking for what you want to find.  Figure 6 demonstrates clearly that the harder you look, the more you distort the signal.  In Mann08 the r coefficient was  0.1 so the distortion between the calibration range and the historic range is fairly minimal, however the total distortion of signal amplitude of Schweingruber proxies scaled by CPS is a multiplier slightly under 0.3 or 30% of actual signal as demonstrated in Figure 5.  If the Schweingruber proxies are typical for Mann08 there is 70% loss in amplitude of the temperature signal compared to the instrumental record.  It almost guarantees unprecedentedness!   In the previous post we saw that this was created almost entirely from the noise level and scaling according to standard deviation (Figure 7).

Figure 7, Magnification factor of historic signal with different noise levels r=0.6

One point I left out from before was that the Schweingruber series signal/noise ratio apples to the worst case demagnification from above – far right side of the graph. I believe this is typical of the response we would see in a Mann08 style reconstruction. Although we haven’t reached the level of knowing that value clearly….yet, it can be estimated.

1. ### Ryan Osaid

The video was quite cool. By the way, how do you do videos from R plots?

2. ### Jeff Idsaid

#1, There’s a lot of information in this post. I was pretty excited about it but it’s the first one I’ve done without any comments. Perhaps there’s not enough code along with it.

Let’s see. For the AVI style video save every R file as a JPG, other formats work also.
Get ImagetoAVI freeware. It does a nice job making AVI’s but you have to choose the right format for encoding. Use microsoft 263 or video 1 with no compression.

The AVI compression sucks so antoher freewere from AVI to MP4 is required.

Get Zune converter freeware. It converts AVI to MP4. Set your image size to be the same size as the pics (even numbers only). Set bit rate to at least 2000 and convert.

See it’s easy.

I spent about a day figuring it out the first time. :D

3. ### Ryan Osaid

Ah. Sounds pretty good.

I’m thinking we should do a video of Antarctic monthly anomalies for the supplemental info. ;)

It’s looking like I’m going to have to start reading all the papers discussing the amplitude issue. I’m not sure I see how it’s possible to claim that the CPS method does not significantly affect the amplitude.

Another thing that would be interesting to see is if the signal were not a “clean” signal. If the actual temperature you used was the above signal plus weather “noise”, and your proxies were made up of that result plus an additional noise factor, then it would seem at first glance that r would be an even worse way of sorting proxies. By the way, what were the ARMA parameters you used?

It would also be interesting to see if rms noise or average explained variance were better sorting statistics. I imagine they’ll still suck because they’ll mine for the hockey stick, but they might not be as biased as r.

4. ### Jeff Idsaid

I’m also curious about how searching a dirty signal will change the result. Mann08 used GISS which contains some odd smoothing yet is still noisy. It has to make a difference to add some random decorrelation. As far as sorting statistics, I’ve spent my time thinking about methods which will balance the effects of noise. i.e. Physical angle of the difference between least squares slopes or some odd methods of inverting the noise.

I’ve done several anomaly video’s here but they aren’t good quality. It would be possible to do a better one now.

http://noconsensus.wordpress.com/2009/03/29/satellite-anomaly-video-know-your-data-pt-3/

Also, I’ve got an idea for fixing the offset problem in the area weighted reconstruction which I will finish this morning. I won’t have time yet for reading other things, perhaps this afternoon.

5. ### Ryan Osaid

Cool.

By the way, when you have a chance to look at it, the calibration section of what I sent you may have applicability to the way Mann handles the divergence problem. It should be possible to perform the same analysis on Mann08.

The other thought I had is for the calibration period dendro samples, are they not taken in regions where the factor you are looking for is a stressor? By that I mean if you want to establish a relationship between temperature and ring width, you would take the sample near a temperature-induced treeline.

For the historic samples, this is not possible. Your samples are limited to whatever you have, which may or may not be from an area where the factor you are looking for is a stressor. This leads to an additional calibration problem, where the relationship in the calibration period is different than the relationship in the reconstruction period. I haven’t ever seen anything that discusses how such a problem would be handled – and, at any rate, it isn’t handled by Mann.

6. ### Davidsaid

Rewriting the historical temperature record
(or “Totally inventing the hockey stick chart”)

First look at:

http://www.theregister.co.uk/2009/11/30/crugate_analysis/print.html

Note the chart of the original version of the temperature record. And note what happened to it when Michael Mann got busy with his Siberian “treemometers” and his crazy statistics.

On her blog, Lucia explains how to make a hockey stick chart out of random data:

http://rankexploits.com/musings/2009/tricking-yourself-into-cherry-picking/

She notes:
“Also notice that when I do this, the “blue proxie reconstruction” prior to 1960 is quite smooth. In fact, because the proxies are not sensitive, the past history prior to the “calibration” period looks unchanging. If the current period has an uptick, applying this method to red noise will make the current uptick look “unprecedented”. (The same would happen if the current period had a down turn, except we’d have unprecedented cooling. )”

For more context:

http://www.theregister.co.uk/2009/09/29/yamal_scandal/print.html

http://bishophill.squarespace.com/blog/2008/8/11/caspar-and-the-jesus-paper.html

[The last link talks about the supposed statistical justification for the hockey stick (RE) and pretty much makes a mockery out of it.]

7. ### Tom Moriartysaid

Jeff,

Please take a moment to look at my comments concerning the Luterbacher “proxies” used by Mann in his 2008 version of the hockey stick.