Willis raised an interesting point about trying to invert series based on partial correlation coefficients with monthly temperatures. His post, together with UC’s response are collected here.
So, I figured I’d see if I could reconstruct what the tree ring widths looked like from the correlations to the Dulan temperature. Much to my surprise, I found out that there is no unique inversion from correlation to ring widths. Here are four synthetic ring widths, each with identical correlations to the Dulan month by month temperature, along with the Dulan temperature itself:
As you can see, the four synthetic ring width series are all quite different the Dulan temperature. In addition, the correlations with the quarterly (D-J-F, M-A-M, J-J-A, and S-O-N) and annual Dulan temperatures are very poor. None of the correlations are significant. Finally, the trends are very different.
Now, if we can’t compute the ring widths from the correlations, and we assume that the temperature is related to the ring widths because of the correlations, doesn’t that mean that we can’t compute the temperature from the ring widths? And since the trends can be quite different and still give the same correlations, doesn’t this mean that there is not necessarily a relation between temperature trends and tree ring width trends?
For my next excursion into dendro, I’m going to see if this is true of the Wilson study as well …
OK, six hours later, went to work (I’m currently building a handicapped access ramp on a commercial building), back again. Here’s Wilson et al. data, compared to the 1950-1990 instrumental record. Wilson used principal components, I’ve looked at the PC1 correlations. It has a number of very good correlations with various months, and it only has one negative correlation. The correlations also extend over a longer period, 21 months instead of 12 as in the Zhang et al. data. Here’s the correlations.
Prev. Jan, 0.09
Prev. Feb, 0.24
Prev. Mar, 0.31
Prev. Apr, 0.36
Prev. May, 0.34
Prev. Jun, 0.14
Prev. Jul, 0.02
Prev. Aug, 0.16
Prev. Sep, 0.22
Prev. Oct, 0.16
Prev. Nov, 0.05
Prev. Dec, -0.04
So I expected that the synthetic PC1s would be very close to each other. Once again, I was surprised … here’s four synthetic PC1s that all have the same correlation with the monthly data, month by month:
As you can see, not only are the PC1s different, but their trends are also different.
1) Very different tree ring width patterns can give identical correlations with a given set of monthly temperatures.
2) The fact that tree ring widths are correlated with monthly temperatures does not mean that tree ring width trends are correlated with temperature trends.
3) For any given set of monthly RW/temperature correlations, there exists a family of individual different RW curves which will give the same correlations with the monthly temperatures (within instrumental accuracy).
… am I crazy for thinking that this makes it very hard to put confidence intervals on a historical tree-ring based reconstruction, and that it makes the calculated trends very suspect?
UC added here
Ref discussion at CA. I use simulated monthly temperature record, data here . Simulated ring width response is here. Correlations with monthly temps are here. But as Willis suggested, we can find other time series that give exactly the same correlations.
See e.g. simul1. and simul2. Try with Matlab,
rw1 and rw2 are obtained using different inital guess (white noise and red noise). Quite different results,