Tag Archives: Fourier analysis

Some Cycle Basics

A Fourier analysis is one of the first steps in analyzing cycles.

Take sunspots, for example,

There are extensive historic records on the annual number of sunspots, dating back to 1700. The annual data shown in the following graph dates back to 1700, and is currently maintained by the Royal Belgium Observatory.


This series is relatively stationary, although there may be a slight trend if you cut this span of data off a few years before the present.

In any case, the kind of thing you get with a Fourier analysis looks like this.


This shows the power or importance of the cycles/year numbers, and maxes out at around 0.09.

These data can be recalibrated into the following chart, which highlights the approximately 11 year major cycle in the sunspot numbers.


Now it’s possible to build a simple regression model with a lagged explanatory variable to make credible predictions. A lag of eleven years produces the following in-sample and out-of-sample fits. The regression is estimated over data to 1990, and, thus, the years 1991 through 2013 are out-of-sample.


It’s obvious this sort of forecasting approach is not quite ready for prime-time television, even though it performs OK on several of the out-of-sample years after 1990.

But this exercise does highlight a couple of things.

First, the annual number of sunspots is broadly cyclical in this sense. If you try the same trick with lagged values for the US “business cycle” the results will be radically worse. At least with the sunspot data, most of the fluctuations have timing that is correctly predicted, both in-sample (1990 and before) and out-of-sample (1991-2013).

Secondly, there are stochastic elements to this solar activity cycle. The variation in amplitude is dramatic, and, indeed, the latest numbers coming in on sunspot activity are moving to much lower levels, even though the cycle is supposedly at its peak.

I’ve reviewed several papers on predicting the sunspot cycle. There are models which are more profoundly inspired by the possible physics involved – dynamo dynamics for example. But for my money there are basic models which, on a one-year-ahead basis, do a credible job. More on this forthcoming.