Regarding coding it correctly, I did it manually so I could check each step, assuming I understood it correctly. Basically, the F-P algorithm works by collapsing the data into smaller and smaller sizes, by using “blocks” of data. Each block is made by averaging adjacent data pairs in the previous dataset, to make a dataset half the size or the original, with each element in the dataset representing a block of twice the size. This incrementally reduces the autocorrelation, while not changing the standard deviation or the mean. The relevant formulas are:

This is repeated until the autocorrelation disappears or the process cannot continue, and then the variance of the mean is determined in the usual fashion. The paper I cited above gives the relevant mathematical derivation of the method.

However, experimentation with larger datasets shows that the F-P method sometimes gives smaller answers, and sometimes larger answers, than the Nychka method. In addition, because F-P reduces the size of the dataset by a factor of two each time, the autocorrelation changes by large steps in the final stages, and the zero point falls between two steps. This introduces a margin of error into the F-P algorithm.

Finally, this algorithm is designed for very large datasets (the smallest example in the paper has 62 thousand data points, and the largest, 31 million). On small datasets, the F-P algorithm seems not to converge entirely on non-detrended datasets before it stops because no more blocking operations can be done. In addition, the difference between the Nytchka and the F-P algorithm decreases as the dataset size n increases. On a dataset of 1024 data points, the difference on the detrended dataset is quite small, although it is still relatively large on the un-detrended dataset.

Because of this, I reckon that the Nytchka algorithm does a better job for the typical size of climate datasets (a few thousand points max) than the F-P algorithm.

w.

]]>I tried it with both the raw data, and the detrended data (n=128). In both cases, the Flyvberg-Peterson algorithm gave a larger standard error than the Nychka algorithm.

With the detrended data, the Nychka algorithm SEM is about 80% of the F-P algorithm (SEM 0.027 vs 0.033)

With the raw data, the Nychka algorithm SEM is about 33% of the F-P algorithm (SEM 0.034 vs 0.10)

This would seem to indicate that the Nychka algorithm underestimates the actual effect of autocorrelation.

I repeated the experiment with Jones monthly data, which has a larger autocorrelation, and with a longer time series (n=512). Both the raw and detrended data showed similar numbers. The Nychka algorithm SEM from the raw data was about 20% of the F-P algorithm. The detrended data Nychka algorithm was about 63% of the F-P algorithm.

Again, this suggests that the Nychka algorithm underestimates the true effect of the autocorrelation. The greater the autocorrelation, the greater the difference between the two methods. The Nytchka algorithm is an ad-hoc method, whereas the F-P algorithm is based on mathematical derivation …

All of this means that the effect of autocorrelation on temperature data series is worse than I had estimated using the Nychka algorithm …

w.

]]>All the best,

w.

]]>Unforunately I’ve noticed that humility is rare full stop. E.g. our capacity to inflict damage to the environment, and our fellow man, seems to be ‘increasing faster than our ability to predict its consequences.’ ]]>

With or without a theory discussions could be equally meaningless (i.e. the theory could be wrong), but, knowledge always being incomplete, that shouldn’t stop us having discussions about things that concern us, and hypothesising about cause-effect, as long as we remain humble.

You may have noticed that humility amongst climate modellers is a vanishingly rare commodity.

]]>If you do things right, then the estimate with the lowest uncertainty will give you the most accurate estimate of the answer.

That’s a huge “if.” Remember, the original statement of mine that you had such a big problem with: “narrowing the uncertainty range would make the methodology more precise, but *not necessarily* more accurate.”

The case made in #22 (that you responded to in #24) was that the Hadley Centre methodology was “fundamentally flawed,” which in my mind means “not doing things right.” You said a more accurate methodology would reduce the uncertainty below the Hadley Centre’s uncertainty. As I have shown and you have admitted, a narrower uncertainty can only imply greater accuracy over another method if things are done “right” in both methods. So if the Hadley Centre’s uncertainty in #22 is produced by incorrect methodology, why would an alternative methodology need to have a smaller uncertainty in order to be considered more accurate?

In the end, it all comes down to the fact that we don’t have a direct measurement of average global temperature to compare the method results to, so there is no way to determine which methodology produces the most accurate estimate. But it should be clear by now that narrowing the uncertainty by itself wouldn’t imply greater accuracy.

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