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.