2.1 Background

Recall that a timeseries X(t) is an ordered data sequence, either continuous or discrete. On a computer, everything is discretized, so regardless of the original process, what we have is a discrete sequence.

In timeseries analysis, it is common to consider a timeseries as the sum of one or more oscillatory, stationary components (or signals), a trend, and noise`. The oscillatory components are often the reason we’re doing the analysis in the first place - it will tell us how big they are and what their frequency is.

Whether or not those oscillatory components are present, a trend is often present as well.

In today’s world the word “trend” is so overused that it has lost all meaning. Here, it will refer to a slowly-evolving, non-stationary component that dominates the behavior of the timeseries. For instance: a linear increase or decrease; a sinusoidal component so slow that its period is not resolved by the dataset; a nonlinear trend like the exponential increase of the Keeling curve (see below).

As for noise, it clearly involves a subjective definition. Under this name we usually subsume any variable component in which we are not interested. Some of this noise may be composed of actual measurement errors (what you would think of as noise, strictly speaking), but some of it could be what another analyst would call signal. If you study climate, daily fluctuations are noise; if you study weather, they are your signal. One commonly says that one analyst’s signal is another analyst’s noise. This noise is often modeled as a Normal random process with zero mean (aka Gaussian white noise).

To see these concepts in action, let us consider the iconic Mauna Loa Observatory CO2 measurements.