The cyclostationarity paradigm has been found to be a useful tool in characterizing time series data. In particular, it can help distinguish between regular and irregular cyclicity.

In general, a time series is considered to have regular cyclicity if there is a clear periodicity to the data and it repeats over time. For example, monthly sales figures might show a regular pattern of increases and decreases that repeat each month. On the other hand, a time series with irregular cyclicity does not follow any predictable pattern and does not repeat over time.

How is cyclostationarity used to identify time series?

The cyclostationarity paradigm can be used to identify whether a time series has regular or irregular cyclicity. This is done by comparing the autocorrelation function (ACF) of the time series to its power spectrum density (PSD).

If both the ACF and PSD show a periodically repeating structure, then the time series is considered to have regular cyclicity. On the other hand, if there is no pattern in either plot, then it can be concluded that the time series has irregular cyclicity.

When analyzing regular cyclicity using this paradigm, it is important to understand what values are expected for each plot. The ACF value represents how often one lag value perfectly correlates with another lag value on modulo cycles (i.e., positive integer multiples of some constant period).

Therefore, if a cycle had length five days and there were three days between every sales figure plotted on consecutive days, is the concept of a system where the constituent parts at long intervals either repeat or reverse their order. The cyclicity can be seen as regular and predictable, or as irregular and chaotic.

One must consider how the cyclicity is perceived by both an observer and those within the system in order to place it on a spectrum between regular and irregular. Let’s consider how these two perspectives might manifest themselves:

The regularity of cycling may be evident to those outside of the system as cyclic, such as with solar cycles that repeat predictably about every 11 years. The irregularity of cycling may be evident to those within the system as chaotic, such as with the erratic cycling behaviors associated with bipolar disorder.

Cyclostationarity is demonstrated by a wide variety of phenomena and behaviors including:

  • Biological patterns – whether in genetic material or neural pathways.
  • Gravity waves – alternately rising and falling while traveling through space.
  • Herky-jerky patterns of electrical voltage – oscillations between high and low voltages that are too rapid to measure.
  • Thought processes – the ebb and flow of mental processing between regularity and chaos.


A cyclostationary shift is a shift in the energy of a signal that occurs at regular intervals and has the same periodicity as the original signal. The cyclostationarity paradigm shift from regular to irregular cyclicity is an important consideration for systems that use time-domain measurements, such as seismometers and hydrophones.

The engineering implications of this paradigm shift are far reaching. If one were to calculate the power spectral density using traditional frequency domain analyses without compensating for reflected signals, they would obtain an incorrect result because they would not account for reflections.

Read More : Applications Of Cyclostationary Processes & Time Series


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