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Periodic Function vs Non Periodic, Aperiodic, Quasiperiodic (In one picture)

  • An understanding of periodic functions is essential for any data scientist,
  • Periodic functions can be represented by Fourier series,
  • Some Non-Periodic functions can also be represented by Fourier series,
  • One picture to explain their similarities and dissimilarities.

Periodic functions[noterm], which repeat values at set intervals, are essential math for data science.  For example, they are a must for working with Fourier series [1], time series forecasting [2] , digital signal processing [3] and a variety of other periodic (or periodic appearing) data patterns.

The basic definition for a periodic function looks fairly straightforward: it repeats its values at set intervals, called periods. This common definition might lead you to think that non-periodic functions are simply those that don't repeat at set intervals. This is not really true though. For example, quasiperiodic functions (a subset of non-periodic functions) are made up of sets of periodic functions. This may lead you to wonder how can a set of periodic functions not be periodic? It's obvious at this point that the simple definition I gave above doesn't cut it. In order to truly understand the nature of these functions it's necessary to think of them in terms of Fourier series:

  • Periodic functions can be approximated by Fourier series expansion [4]
  • Non-periodic functions cannot be expressed by a Fourier series expansion using cosines and sines.

So an almost periodic function, [noterm] although technically not periodic, is "almost" like a periodic function because it can be represented by Fourier series. A subset of almost-periodic functions, quasiperiodic functions, are just those functions made up of sets of periodic functions that don't quite match (hence the "quasi", which means "resembling"). 

If you're not familiar with periodic functions, by this point your head might be swimming a little with all of those closely related terms. This one picture explains how they relate to each other in terms of subsets of each other, and in terms of Fourier series.

References

[1]  Fourier Series and Differential Equations with some applications in...

[2]  Periodic Smoothing of Time Series.

[3]  Stanford ‘SIRENs’ Apply Periodic Activation Functions to Implicit N...

[4]  Dondurur, D. (2018). Fundamentals of Data Processing: Periodic Function. in Acquisition and Processing of Marine Seismic Data.

Quasiperiodic function image: Jochen Burghardt, CC BY-SA 4.0 a href="https://creativecommons.org/licenses/by-sa/4.0%3E">https://creativecommons.org/licenses/by-sa/4.0>;, via Wikimedia Commons

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