Tom Breur

6 January 2019

Forecasting is the art and science of predicting the future (spoiler alert: it’s hard!). Some business processes are more volatile than others. Invariably, business waxes and wanes in cycles, typically *multiple* cycles superimposed on top of each other. In retail business, for instance, you expect weekly recurring patterns. Saturday will be the busiest day, week after week. But there may also be patterns *within* the month, maybe with a spike in expenses after payday. Then there might be annual cycles with December often the busiest month. Other recurring cycles might happen around Black Friday, seasonal clearance, etc. Note how all of these cyclical patterns are happening within the same year, hence “superimposed” on top of each other. “Seasonality” in a time series can take multiple forms, many different shapes, each with their own frequency. They need not be related to weather seasons that many people think of when talking about seasonality.

Analysis of all these cyclical patterns is sometimes referred to as Fourier analysis (or harmonic analysis), after the eminent French mathematician Joseph Fourier (1768-1830). Fourier asserted that a function could be decomposed in a series of sines. The extension of that is what we nowadays^{1} refer to as Fourier analysis. Fourier, btw, is *also* credited with the discovery of the greenhouse effect: he calculated that the temperature on earth ought to be much colder than it is, given our distance from the sun. Fourier considered the possibility that our atmosphere might operate like an insulator, which we have now come to call the greenhouse effect, although Fourier himself never used that term.

When you build a forecast, the “forecasting horizon” is the time period for which you deem your model to be applicable. But depending on the purpose of your forecast, sometimes projections are required beyond that time frame. That’s when we enter the realm of long-range forecasting.

When we refer to long-range forecasting, a valid question then arises: how long is *long*? In his seminal book Long-Range Forecasting, J. Scott Armstrong clarifies that “long” is relative to the periodicity of cycles. A time series can be decomposed into multiple cycles with different lengths and wave forms (so-called “harmonic analysis”). Fourier analysis will surface multiple wave pattern with different shapes and frequencies. The *longest* cycles that surface from this analysis (i.e. the lowest frequencies) determine what is “long” relative to that time series. Although there is no hard and fast rule, Scott Armstrong suggests taking a multiple of the *longest* cycles present. If the lowest frequency is made up of annual cycles, then a multiple implies that anything beyond 3-4 years should be considered long-range forecasting. Another author, Arthur Charpentier , mentions a minimum of 2-3 cycles. In Smith & Sincich, “The Relationship Between the Length of the Base Period and Population Forecast Errors” (1990), the empirical minimum that they find appears longer, more towards 5-10 cycles. In summary: there clearly appears to be no hard and fast universal rule. The idea behind these rules of thumb is that the market that you are predicting has been able to reset to equilibrium from one-time shocks to the system.

Note that this definition of long-range forecasting makes it entirely relative: some economic cycles are measured in decades, swings in global temperatures are measure in thousands of years, yet other phenomena like congestion on the internet highway are measured in seconds (or even fractions thereof). It’s all relative.

1) there is some discussion in the scientific community how much should be credited to Lagrange, whom Fourier succeeded at the prestigious Ecole Polytechnique in Paris