That makes no sense at all, does it?

Intuitively, it seems that confidence intervals on temperature slopes (when we want to compare them with a long term trend) should depend more on the working time range than on the number of data points, or on how well those data points fit a linear regression. We should have more confidence on a 20-year trend than a 10-year trend, almost regardless of whether we use monthly data as opposed to annual data. Certainly, the standard slope confidence interval calculation is not going to do it. We need to come up with a different method to compare short-term trends with long-term ones.

I will suggest one such method in this post. First, we need to come up with a long projected trend we can test the method on. We could use a 100-year IPCC trend line, if there is such a thing. For simplicity, I will use a third-order polynomial trend line as my "projected trend." Readers can repeat the exercise with any arbitrary trend line if they so wish. I should note that the third-order polynomial trend line projects a temperature change rate of 2.2C / century from 1998 to 2008.

The following is a graph of GISS global annual mean temperatures, along with the "projected trend." For the year 2008 I'm using 0.44C as the mean temperature. You can use other temperature data sets and monthly data too. I don't think that will make a big difference.

We have 118 years of 11-year slopes we can analyze. There are different ways to do this. To make it easy to follow, I will detrend the temperature series according to our projected trend. This way we can compare apples with apples as far as slopes go. The detrended series is shown in the following graph.

The long term slope of detrended temperatures is, of course, zero. All 11-year slopes in the detrended series will distribute around zero. We know that the 1998-2008 slope is -1.53C / century. The question we want an answer for is whether the 1998-2008 slope is unusual compared to 11-year slopes observed historically, which would indicate there's likely a point of change away from the projected trend.

We can start by visualizing the distribution of 11-year slopes throughout the detrended series. The following is a graph of the number of years in slope ranges of width 0.2C / century. For example, the number of years that have slopes between 0.1 and 0.3 is 10.

This is roughly a normal distribution of years according to their slopes. In it, approximately 95% of years have slopes in the -2.7 to 2.7 range. That is, 4 years have slopes of -2.7 or lower, and 3 years have slopes of 2.7 or higher. I put forth that

**the real confidence interval for 11-year temperature slopes relative to long-term 3rd-order polynomial trend lines is approximately ± 2.7 C / century**.

The 11-year slope for 1998 is only -1.53C / century, well within the estimated confidence interval. Therefore, it's a little premature to say that the 1998-2008 trend falsifies 2C / century. Of course, if 2009 is a cold year, that might change this evaluation.

## 12 comments:

Very nice post. Could you please explain again the point you mention at the beginning, about why having more points per se does not mean tighter estimates (rather than having a longer time period)? Is it because if we take monthly data instead of annual over the same time period, the correlation in the data and noise become bigger? Or some other reason?

The way I would explain it is that getting a tighter confidence interval in the slope of a time series is fine in regards to that time series. You become better informed about the slope of that time range. The error is in assuming that you become that much better informed about the slope of a longer term series.

For example, suppose we calculate the temperature slope of the last 10 hours. We can use data on the last 600 minutes instead, and become much better informed about the temperature trend of the last 10 hours. We could even say we are somewhat informed about the 20 hour trend. But this tells us nothing about the 1-year or 10-year trends. Getting a tighter confidence interval by including more granular data can be deceiving in this sense.

What I argued in the post is that the 11-year trend can be informative about the 100-year trend, but it is informative ± 2.7 C / century.

http://www.nature.com/nature/journal/v375/n6533/abs/375666a0.html seems relevant

Thanks Hank. The URL got cut off there. The article is here.

I would assume that this slowdown in CO2 emissions would show up in the Hawaii data, and would subsequently also show up in hindcasts. I think the projected temperature change rate at the moment is about 2C/century regardless.

When you state "We know that the 1998-2008 slope is -1.53C / century", I assume this is the slope

afterdetrending?I had already come to the conclusion that the last 11 years didn't mean a lot. I get a slope of 0.09 degrees per decade (using GISS annual data with a guestimate of 0.40 for 2008), but I also get a standard deviation of 0.97.

I like what you've done, though, in showing that it isn't abnormal.

That's right. It's after detrending. Before detrending, the slope is still positive, as you note. It's clearly a misconception that the planet has cooled or warming has stopped since 1998. What's debated is whether 2C/century is falsified. It's not, though.

(I already posted this comment over at Greenfyre's, but since it's being drowned by a ton of inactivist vomit, I figured I'll post it here again...)

* * *

Interesting method, but my worry is whether there’s a meaningful probabilistic interpretation to the test results, i.e. whether the results provide us with a useful model saying “the global temperatures can be said to be generated from such and such a type of stochastic process with such and such parameters”.

(This was in fact one of my beefs against Pielke Jr. — he’d apply statistical tests in ad hoc ways without regard to what they mean.)

Of course we can't conclude anything about what is causing the temperature increase, from this analysis alone. In fact, strictly speaking, we can't even say there's a statistically significant upward temperature trend between 1998 and 2008. What we can say is that the 2C/century hypothesis cannot and should not be rejected.

I have a different analysis on causation. That's basically a detrended cross-correlation analysis (it controls for possibly coincidental trends) although I didn't know that when I first wrote it.

Joseph:

It doesn't have to model causation, just a statistical generative model that we can use for purposes of analysis.

"Of course, if 2009 is a cold year, that might change this evaluation."

Interesting caveat. Thanks for the post. :)

Talking about the time periods. These periods can be predictable, but yet can also be unpredictable. Time can only tell.

Once we a get closer look about the years that has passed, we need to examine the changes that occurred during those years. The temperature is increasing in the next 10 months by 4 degrees Celsius according to a study made by Nasa

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