Recurrence and Stationarity

The recurrent nature of the past

We could say that a good forecast for tomorrow’s weather would be today’s, plus or minus a couple of raindrops. This is to say that the weather is a system with memory: The past has an impact on the present. Or, to put it in yet a different way, the past is recurrent.

To say that today is determined by tomorrow is certainly very intuitive. In fact, it is a view that permeates statistics into fields as diverse as history, biology, or finance. Still, we must not forget that, in many cases, this is a reasonable assumption and motivated by our perspective on the subject matter. But sometimes, it won’t necessarily be true. Nassim N. Taleb talks about this in his book, The Black Swan. Imagine a turkey that is fed every day by a butcher. The turkey will think that the normal course of life is to get food for no particular reason. If it were to predict the future based on the past, the turkey would say that food would keep coming. However, we know this is just an illusion. At some point, say Thanksgiving, the turkey will become someone’s dinner.

Stationarity

As we’ve just said, in this course, we will assume that our time series will present some form of recurrence. This means that (some of) the statistical properties of our time series have to remain the same no matter when we look at them. That a time series keeps its properties over time is essential to have recurrence. Otherwise, future inference based on past performance is not possible. Put differently, if a series doesn’t retain its statistical properties over time, we can’t confidently approximate the data-generating process in the background.

Stationarity is one of the frameworks we can use to look at the stability of properties. Broadly speaking, stationarity means that the distribution of a stochastic process is independent of time steps. To put it more simply, the characteristics of a stationary series do not depend on time. This already rules it out of the stationary series club: trending or seasonal series.

Generally speaking, a stationary time series will not show any recognizable patterns in sufficiently large time windows. On the contrary, it should look more or less horizontal on a plot, and its ups and downs should always have a similar scale.

Look at the code below. It reads and transforms the average temperature in San Francisco (in ºF). It then produces two charts: One with the actual temperature over time, the other with the difference between the present and the previous day. Could any of the ...