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 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.
Fig 3 – Direct download video here.
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 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.
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.
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).
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.