About a year and a half ago, I had a little spat with one of my very best friends."It's SO getting warmer in Massachusetts," I declared off-the-cuff, and without any qualifiers. "We only got snow once this year. This is totally global warming. Our kids will never get to play in the snow like we did when we were young." (As in this picture from the Good Ol' Days... when the walk to school was uphill both ways.)
"Nah," said my friend, "some years are just warmer than others. The climate goes through natural cycles, and everyone's way too hyper and quick to cry the global warming wolf these days. You are completely full of shit, Suz." (This friend is extremely articulate, and my paraphrasing here doesn't do justice to the way her razor-sharp rhetoric sliced me to shreds.) So, I put my tail between my legs, and didn't think about climate change in Massachusetts for a whole year.
BUT..... the saga continues. This past semester, I did a term project on some recently developed tools in signal processing. A signal is any time-varying quantity, and signal processing is a set of methods that allows us to glean information from a signal. In fact, one example of a signal is the temperature over a number of years in any given place.
The method I researched is called Empirical Mode Decomposition, or EMD if you want to sound cool and in-the-know. This is a way to decompose the signal into a number of modes which add back up to give the original signal, but each helps us understand different aspects of the signal. Let me explain in English. If you ever took high school physics, you might remember vectors. These were little arrows pointing in perpendicular directions (x and y directions, eg.) that you could pin together tip-to-tail to describe more general motion in two-dimensions. Here, let me refresh your memory with a picture:
So in this case, the vectors along the "Up" and "Right" directions decompose the "Up and Right" vector, which describes the actual motion of, say, a baseball flying through the air. This is a useful way to break up the motion of the baseball, as you might remember from your homework sets, because it's much easier to understand the motion in the horizontal and vertical directions separately. Then at the end of the problem, you can just add 'em back together to describe the entire motion.A standard generalization of this idea can be applied to functions or signal, where the different components that add up to give the function are sine waves of different frequencies. (A sine wave is just a nice, regular wave, as shown below. The red sine pictured below has a frequency three times greater than the blue sine wave.)
An amazing, beautiful mathematical fact is that any function can be represented as the sum of a bunch of these nice, simple sine waves (modulo a few technical details). This is exactly analogous to the way that any 2-D motion vector can be written as a sum of vectors along vertical and horizontal directions. The different sine wave modes are "perpendicular" in a general sense, just as vectors along x- and y- axes are. This lovely decomposition is called a Fourier Series for this signal, and is the bread and butter of any undergraduate education in physics or mathematics.Norbert Huang recently developed Empirical Mode Decomposition, which is a different way to decompose a signal. Instead of a pre-prescribed set of modes, as in a Fourier Series, EMD creates a decomposition which is specific to the signal being analyzed. The power of this method is that this special decomposition separates phenomena occuring on different time scales from the original data series. What's more, EMD leaves behind a "residue, after subtracting off the fluctuating, cyclic modes, which reveals the trend in the signal, if one exists.
The National Oceanic and Atmospheric Agency has free weather data on its website dating back to the 1920's. For my project, I took a look at some temperature data from 1988 to 2005 (roughly the length of my memory) taken in Amherst, MA (the closest data site I could find to where I grew up). Unfortunately, the image files of data and decompositions look like crud when I try to upload them to this blog. But, running EMD on this data subtracts off seasonal and other cyclic fluctuations, to give this punchline: The mean temperature in Amherst, MA has increased unambiguously by a little over a half a degree Fahrenheit over this 17-year period. Kind of scary... but the mathematics is absolutely beautiful.
