Yes, if you order the residuals and compute the 16% and 84% percentil you get a robust estimator for twice the standard deviation. When you then bin the residuals and compare the histogram with the gaussian for that std the agreement in shape is rather remarkable.

On the other hand, when I do a fft to the data I do get a nice peak in my 8th shortest baseline (using the 361 months), so there is possibly one significant periodicity in the data.

I always expected that everyone would come up with approximately the same number. My reasoning being that the unaffected period from 1978-1997 was essentially a flat line, masked by a Gaussian scatter. Any correct method of analysis involving substitution or elimination would thus come up with essentially the same answer.

The potential for problems is shown by a mathematical example, below. So I took my corrected UAH data for volcanoes and the 1998/2008 “”weather”" and computed a trend of 1.14 K/century, the lowest I got without cherry picking. The next step was to go to the midpoint and compute the average, which is 0.097668439. Next, I computed each point forward and backward from the midpoint where I inserted the average, replacing the actual data with the computed data generated by our slope of 1.14 K/century. So what happens if you correct for 1 K in one century on a monthly basis? With this assumption 1K/100 years/12 months equals 0.000833333. Then from the midpoint backwards to the start, I subtracted stepwise 0.0008333 each month to induce a “cooling of the past”; and I added 0.0008333 each month to induce a systemic rise such as unaccounted UHI effect in a stepwise manner, or GISS “correction”. Then I computed, using LINEST in Excel, the trend. The trend went from 1.144 k/Century to 2.144 K/century. Not really unexpected is it? A cooling of 1K per century from the present to the past, induces an addition of 1K per century to the trend.

Now, exactly what does GISS, Hadley, and others do? It is quite easy to prove that it matters; as would the local anthropogenic effect of introducing warming in the recent decades. So lets correct using a step change for a step change. Replace the 1.144 trend endpoints with the 2.144 trend endpoints and compute the LINEST, it is now 1.161, not 2.144. So using a trend to replace a stepfunction introduces a false trend in a computation. As has been pointed out several posters, whether you do this at the beginning or end, or both, can impart a false trend in your computations.

The question is “What does the homogenization by GISS impart to the trend, if a step function occurs and the data is changed incrementally rather than as a step change”.

Thanks for following up!

It appears that the statistical experts here came up with a slope as low or lower than my original analysis, confirming that my calculation was conservative. Given the coincident El Nino events, a climatologically corrected adjustment would likely have produced slopes less than 1.0K/century.

That’s not totally unexpected since the monthly residuals follow a nearly gaussian distribution with standard deviation 0.16 K (+-.01K). In these conditions least squares should perform rather nicely. One can even put an error bar in the slope from least squares. The error in the estimated slope is about 0.12 K/century, So we have an 8-sigma “warming” signal at 1.0 K/century.

** least squares «reacts badly» when replacing even a single point by a very large outlier (it has a rupture limit of zero). Methods which use the median of all possible pairs of points like Theil and Sen allow something like 29% of random outliers, while the repeat median from siegel can tolerate something like 50% random outliers. I checked both.

I would like to note also that this is just a way to infer if the data follow an increasing or decreasing behaviour in THIS time interval, and if it mimics the trend expected from models.

Joel, a bit more care when quoting, please. I believe it was Tom in Florida who said:

…

which you attributed to me.

Sorry about that. I did realize that I had made that mistake and issued a post correcting it (time stamp 07:58:38, 15/1/09) which ended up 3 posts down from my original post.

Thank you very much for your explanation and calculations. That is what WUWT is all about!

Using relaxation, the slope for the UAH in the woodfortrees data you listed in your post to Filipe, is 0.009067 plus/minus 0.0032% difference for removing the data indicated in your part 1 using linest in Excel and relaxation as I posted. As I also indicated, it improves your argument. The trend is lower, plus you replace the data, not with null data, but with the estimate of the trend, such that claims of incorrectly weighting are answered by the assumption. The assumption is that there is a linear trend, that volcanoes (plus or minus) are not part of. By replacing the data with the trend of the remaining data, your corrected slope is as neutral as possible.

For this period, assuming no net forcing for volcanoes, the predicted increase in temperature based on the average rate of the period used for the estimation will be 0.907 C degrees per century.

Question (for all): What of the effect of aerosols that do make it to the stratosphere? Would they not come out primarily at the poles and 30 N and S?
They would obviously have a much lower albedo than polar ice so they would also have a positive contribution to total solar insolation absorbed by the surface over the course of, potentially, 24 – 36 months!?! I wonder if the net effect 2 years after a major eruption has been thoroughly fleshed out…

The best thing to do seems to depend on what duration of trend you want to find and the relative sizes of ‘gap’ and oscillations in the data. If you are looking for the trend over the entire range of data (long term warming signal), I would concatenate the data minus the gaps, then fit a least squares line through it, find the slope of that line, and then use that slope for ‘infill’ data. This would be my base case.

I could see using that base case to do another fit and get an even more representative slope for the infill (iteration #2) but doubt if the change would be significant.

For shorter duration trends of interest you would need to use shorter spans of data for the LSF to create infill more representative of local conditions. At some limit short range you begin to create too much error by amplifying local oscillations where you have partial cycles in the data being least square fit… such as the ‘ramp’ leading into gap #2.

You can also get the UAH data here:
http://www.woodfortrees.org/data/uah/from:1978

I figured out how to add X values to linest() and tried removing the volcanic tainted months, as you suggested. This actually lowered the slope slightly, but at the reported precision is still 1.0 . The value changed to 1.02, which is the same as Les Johnson reported above.

I’m interested to hear what you come up with. thx.