Previously “A New Temperature Reconstruction” used random data with long term persistence (LTP) to illustrate the circular reasoning behind the ‘hockey stick’ reconstruction of past temperatures. This one shows the potential for false positives due to the statistics used in the ‘hockey stick’. The dynamic simulation below shows future temperatures predicted using a random fractional differencing algorithm that generates realistic LTP behavior. Future temperatures and validation statistics are calculated each time the page is reloaded. One unusual statistic used in MBH98 suggests the future can be predicted using random numbers.

*Note: This is a first version of the application and may contain errors and be improved considerably. The code is freely available under the GPL to order to promote open science. See The Reference Frame for more information.*

The code is written in php and available for download here.

Open application by itself in a window here.

To embed this simulation in a web page, copy and paste the following html:

But you are probably thinking — how can random numbers be a prediction? The series is generated by fractional differencing, a way of integrating random fluctuations across multiple scales. Let me explain…

The validation is based on the 11 points at the end of the temperature record not used in generating the simulated points. Two statistics were calculated and can be seen on the figure:

- The R2 correlation is ubiquitously used for quantifying the strength of association of two variables. A critical value of 0.1 would indicate a possible mild correlation, but values closer to one indicate significance.
- The RE reduction of error statistic is used in dendroclimatology and in the ‘hockey stick’ reconstruction of MBH98, where critical values greater than zero are claimed to indicate significance of the model. RE is claimed to be superior to the R2 statistic in WA06.

Hit reload a few times to get a feel for the average of the statistics. The R2 statistic is usually close to zero indicating the prediction has no statistical skill over the validation period. The RE statistic, however, is always greater than zero, and often greater than 0.5.

MBH98 uses an RE benchmark of zero to indicate significance. The random numbers here give RE statistics greater than the critical value of zero. Therefore, using the RE statistic with a critical value of zero would attribute statistical skill to random numbers. That is, under criteria used in MBH98, random numbers could be regarded as skillful predictors of future temperatures.

This example illustrates (if the code is correct) a situation, similar to MBH98, where the R2 statistic correctly indicates no statistical skill in the predictions, but the RE statistic erroneously indicates statistical skill.

Conclusions hinge on the choice of statistic and where you set the benchmark. MM05 obtain a critical value for RE of greater than 0.5 using random red-noise data in a replication of the procedure used in MBH98. Non-existent statistical skill of the models is one of the main arguments in MM05 against the reconstruction method in MBH98.

### References

WA06 — Wahl, Eugene R. and Caspar M. Ammann, 2006 (under review). Robustness of the Mann, Bradley, Hughes Reconstruction of Surface Temperatures: Examination of Criticisms Based on the Nature and Processing of Proxy Climate Evidence.

MBH98 — Mann, M.E., Bradley, R.S. and Hughes, M.K., 1998. Global-Scale Temperature Patterns and Climate Forcing Over the Past Six Centuries, Nature, 392, 779-787.

MM05 — McIntyre, S., and R. McKitrick, 2005. Hockey sticks, principal components, and spurious significance, Geophys. Res. Lett., 32, L03710, doi:10.1029/2004GL021750