# Tag Archives: mathematics

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.

## L is for “linear”

A previous post addressed some issues with linear regression, “linear” meaning we’re fitting a straight line to some data. Let’s devote another post to scrutinizing the issue — so this post is all about the math, readers who aren’t that interested can rest assured we’ll get back to climate science soon.

It was mentioned in a comment that least-squares regression is BLUE. In this acronym, “B” is for “best” meaning “least-variance” — but for practical purposes it means (among other things) that if a linear trend is present, we have a better chance to detect it with fewer data points using least-squares than with any other linear unbiased estimator. “U” is for “unbiased,” meaning that the line we expect to get is the true trend line. Both of these are highly desirable qualities.

Finally, “L” is for “linear,” which in this context has nothing to do with the fact that our model trend is a straight line. It means that the best-fit line we get is a linear function of the input data. Therefore if we’re fitting data x as a linear function of time t, and it happens that the data x are the sum of two other data sets a and b, then the best-fit line to x is the sum of the best-fit line to a and the best-fit line to b. In some (perhaps even many) contexts that is a remarkably useful property.

## Gutenberg-Richter

In a comment on the last post, it was mentioned that the frequency of earthquakes of any given magnitude or greater will be given by the Gutenberg-Richter law. It states that the expected number of earthquakes in a given region over a given span of time, of a given earthquake magnitude or greater, will be

$N = 10^{a-bM}$,

where $M$ is the quake magnitude, $a$ and $b$ are constants, and $N$ is the expected number. For active regions, the constant $b$ usually has a value near 1.

## Trend and Uncertainty

A reader by the name of “Will” fails to understand why trends can be established with confidence even if the uncertainty of the individual values is unknown, even if those values are averages rather than raw data values. It’s a failure he shares with William M. Briggs, numerologist to the stars. Perhaps we can enlighten reader “Will” — I very much doubt anyone can enlighten William M. Briggs.

But first, let’s dispense with a challenge issued by “Will”:

Please make sure that you explain why the time series I presented (102, 97, 98), given the experiment Gator described, is in fact showing a negative trend.

That time series does not show a negative trend. More to the point, nobody claimed it does.

However, we can use a similar example to illustrate when we might actually need to know the uncertainty to detect change, and when we can detect change (in particular, trend) in the absence of knowing the uncertainty.

## 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.

## 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?

## Markov 2

In the last post we showed how Harold Brooks has applied a 1st-order Markov Chain model to the phenomenon of a significant tornado day (“STD”), in particular to explain the frequency of occurrence of long runs of consecutive STDs. An STD is defined as any day with at least one (possibly many more) tornados of strength F2 or greater (on the Fujita scale).

## Math Fun: the Markov Tornado

While looking around for tornado data, I found a fascinating page by Harold Brooks in which he builds a model of the likelihood of a “significant tornado day,” which I’ll call an “STD” (yeah, it’s a funny choice). This is defined as a day with at least one tornado of Fujita scale F2 or stronger.