Straightening the Hockey Stick Blade

This is a follow up to my post on Dr. Craig Loehles paper on divergence, The 800lb Gorrilla in the Hockey Stick’s Locker Room. Dr. Loehle demonstrated what will happen to historic temperature trends in tree ring data if we have a non-linear system, citing several papers which refer to an inverse quadratic relationship of tree growth to temperature. Inverse quadratic means that as temp warms, trees first grow more to some limit where additional temperature has the reverse effect, slowing growth. This would explain why tree ring data has so little up or down movement in history and also why extreme tree ring widths are not found in nature (the hockey stick handle).

What we need to understand first is that any noisy signal sorted by correlation is distorted into what you wish to find. In this post I use R to generate ARMA matched tree ring proxies, I inserted an inverse quadratic signal in the proxies and then went looking for it.

The top pane is hypothetical actual temperature the bottom pane is an inverse quadratic response to temperature as found by a number of scientific papers. The shape is arbitrary and normalized to 1.

This is a sample proxy created by an Autoregressive Moving Average Process ARMA with ar=0.85 and ma=-0.4 which are similar in magnitude to tree ring proxies. — They do vary though. I made 10,000 proxies like this with the inverse quadratic response signal added in.

Next I normalized all the proxies using Mannian CPS normalization and correlation sorting. I was looking for the original “linear response” temperature signal.

First I looked at a low correlation factor of r=0.3.

The top pane is the actual signal in the data due to the tree response to the bottom pane. the middle pane is the correlation sorted graph, the bottom is the temperature signal we are looking for. In this case there is minor distortion of the curve but it doesn’t look like our expected linear temp curve so I looked harder.

With an r value of 0.5, the middle pane has taken on the linear shape we would expect from a linear response. Climatology incorrectly refers to these proxies as “more temperature sensitive”.

Let’s go a step further.

Ok, at an r of 0.7 we finally see the most “temperature senstive” proxies (as stated by climatologists) in the group and they are highly linear on the average. The inverse quadratic relationship of growth to temperature is lost completely. But the key question is “What happens to historic temperature signals in this case?”

First, to remind my readers of my past work. We must remember that Mannian CPS distorts the true temperature scale. To demonstrate that effect again I simply added a signal from 1100-1200ad (in the historic data) with an equal magnitude to the temperature data in the calibration range 1900-2000.

The signals are of equal magnitude so what happens when we use Mannian CPS and correlation to go find our linear temp signal (very first graph of this post in the first pane of the graph).

Temperature scale is clearly reduced in history as compared to the 1900ad Calibration period. I say that because the historic 1150 temp was exactly the same magnitude at the 2000 temp. For those who don’t know CPS is mathematically simple.

1 – Take mean of all proxies in calibration range.

2 – Scale each proxy standard deviation to match temperature standard deviation in calibration range.

3 – Sort all proxies according to their correlation to temp – Keep the higher correlation.

4 – Average remaining proxies

In this case the process takes high upslope proxies and shrinks their magnitude while low upslope proxies are magnified. The net in practice however is almost always a reduction in historic trend because the “signal has noise (in the form of frost, moisture, co2, bugs, etc.) added on to it increasing the standard deviation in the calibration period.

So my next step was to add an inverse quadratic response to a signal with a high historic value.

The top pane is the actual temperature which rises to 1.5 degrees which is a half degree higher than recent temps 2000 in this example. The hypothetical response is shown in the bottom pane with a maximum of 1 degree in both historic and recent temps.

What happens to the signal after Mannian CPS?

This strong inverse quadratic response is an extreme but not unreasonable example. Imagine a less extreme version where a non-linear response stopped increasing growth after a certain temperature, we would get similar results. Certainly, linearity needs to be demonstrated in some fashion before proxies can be used as temperature. Also, it would be nice for climatology in general to begin rejecting CPS and correlation sorting. Besides the obvious distortion of the temperature scale, matching individual proxy scales to expected temperature inappropriately weights weaker proxies with a stronger value and demagnifies stronger proxies.

Amazingly, the one consistent thing in Mann 08 is that the net of almost every single one of the math effects and several other oddities is a demagnification of historic temperatures and the extraction of an unprecedented temperature rise in recent times.