Sampling Rate, part 2

“Suppose we have a signal which is band-limited, say it’s limited to the frequency band from 0 to 0.5 cycles per day,” says the engineering professor to the class in digital signal processing. “If we observe this signal at regular intervals with a sampling rate which is at least twice the bandwidth — in this case, at least once per day — then we can use Fourier analysis to reconstruct the signal. We can even interpolate it to fill in the gaps. This is one of the most common applications of Fourier analysis in the real world — we observe a signal, then use its Fourier transform either to reconstruct the signal or simply to identify its Fourier components (and therefore its physical nature).”

Sampling Rate

A reader recently asked:

T, from my mechanical engineering world we have strict rules on sampling rates vs. signal frequency rates. Ie you cannot reliably measure a 60hz ac sine wave with a 5hz analog sampling device. The result ends up being strange results that don’t show spikes well and also might not show averages well either. Can you help me understand how 120 year sampling proxies can resolve relatively high frequency temperature spikes?

This objection comes up so often from those who are accustomed to data which are evenly sampled in the time domain, and the misconception is so firmly imprinted on so many people, that it’s worth illustrating how uneven time sampling overcomes such limitations.

Back to School

Much of what’s wrong with the online discussion of global warming is revealed by a recent reader comment on RealClimate.

Greg Goodman thinks that he’s taking climate scientists to school — he actually “lectures” the RealClimate readership about their supposed need to “dig a bit deeper” into the data on Arctic sea ice (both extent and area). He shows a graph based on some analysis which — unbeknownst to him — actually reveals that he doesn’t know what the hell he’s doing. He thinks he has established the presence of “cyclic variations” of which the climate science community is ignorant, and concludes that climate scientists are missing “important clues” about “internal fluctuations” which, of course, those inadequate computer models just can’t handle.

One would be hard pressed to find a more clear-cut example of hubris.

Climate scientists who study sea ice have been all over the data, every piece of it, but instead of making the mistakes Goodman makes they’ve been as careful and rigorous as their expertise and experience allow. They have certainly dug a whole helluva lot deeper than Greg Goodman has, or probably is capable of. It’s Goodman who needs to go back to school.

Theil-Sen

A reader recently inquired about using the Theil-Sen slope to estimate trends in temperature data, rather than the more usual least-squares regression. The Theil-Sen estimator is a non-parametric method to estimate a slope (perhaps more properly, a “distribution-free” method) which is robust, i.e., it is resistant to the presence of outliers (extremely variant data values) which can wreak havoc with least-squares regression. It also doesn’t rely on the noise following the normal distribution, it’s truly a distribution-free method. Even when the data are normally distributed and outliers are absent, it’s still competitive with least-squares regression.

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.”

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.

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.

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.”

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.

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?