Category Archives: mathematics

Fun with averages and trends

Lots of time series, especially in geophysics, exhibit the phenomenon of autocorrelation. This means that not just the signal (if nontrivial signal is present), even the noise is more complicated than the simple kind in which each noise value is independent of the others. Specifically, nearby (in time) noise values tend to be correlated, hence the term “autocorrelation.”

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Hiatus

I haven’t posted much lately because I’ve been hard at work on my new book. It’s titled Understanding Statistics, and I expect to finish in a week or two. I’ll be sure to post here when I do, hoping that lots of you will buy it. Even if you don’t need one for yourself, you might know somebody who would enjoy and make good use of it. Who knows, maybe 20 of you will send a copy to Anthony Watts. Maybe he would learn something from it. Irony of the richest kind.

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Your Servant

Every mathematician develops his own preferences for notation. This is necessary because there are often (I’m tempted to say “usually”) many notations for the same concept.

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Skin a Cat

Before I begin let me make it clear that this is not about abusing cats. I love cats. We have a cat. We treat him very well. He treats us as though it’s our duty to worship him. He’s a cat.

This is about the old adage that “there’s more than one way to skin a cat.”

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Nothin’ but Noise

Pat Michaels claims (also here) that the journal Nature has lost its credibility. That’s an extraordinary claim, considering that Nature is one of the most prestigious peer-reviewed science journals in the world. There are those who believe Pat Michaels is the one lacking any credibility.

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To robust, or not to robust? … that is the question

From time to time it is suggested that ordinary least squares, a.k.a. “OLS,” is inappropriate for some particular trend analysis. Sometimes this is a “word to the wise” because OLS actually is inappropriate (or at least, inferior to other choices). Sometimes (in tamino’s humble opinion) this is because an individual has seen situations in which OLS performs poorly, and is sufficiently impressed by robust regression as a substitute, to form the faulty opinion that it’s superior to OLS generally. For the record, this comment is not one of those cases.

In reality, OLS is the workhorse of trend analysis and there are very good reasons for that. It’s founded on some very simple, and very common, assumptions about the data, and if those assumptions hold true, OLS is the best method for linear trend detection and estimation. It can be dangerous to use the word “best” in a statistical analysis, but in this case I feel justified in doing so.

Of course that raises some nontrivial questions. What are those assumptions? When might they not hold true? How could we tell? What should we do if we can establish that the OLS assumptions aren’t valid?

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Trend and Cycle Together

Many climate signals show both trend and cycle (usually an annual cycle) together. A typical example is the concentration of carbon dioxide (CO2) in the atmosphere. If you look at the data (say, from the Mauna Loa atmospheric obsevatory) both the trend and the cycle are obvious.

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The Value of Data

On a cold January morning in 1986, the space shuttle Challenger lifted off its launch pad at the Kennedy Space Center in Florida. Morale was high, especially as the Challenger flight was to inaugurate the teacher-in-space program with astronaut/high school teacher Christa MacAuliffe in its crew. Alas, 73 seconds into the flight the shuttle disintegrated, destroying the spacecraft and killing all the astronauts on board. The cause of the accident was a leak of hot gas from one of the solid rocket boosters. The leak occured because of the failure of rubber “O-rings” which were supposed to seal the joints between rocket sections, and part of the reason they failed is that the temperature was so cold at the time of the launch — the O-ring material becomes more stiff at low temperature so it’s less likely to make a proper seal.

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Why Not Weighted?

The question arose on another blog, when analyzing the Berkeley data, why not use weighted least squares with weights determined by the uncertainty levels listed in the Berkeley data file?

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Decadal Variations and AMO — Part II

In this post I’d like to examine another claim made in one of the Berkeley papers, that there is a periodic fluctuation in the AMO (Atlantic Multidecadal Oscillation) with period about 9.1 years.

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