# Is there a difference between seasonality / cyclicality / periodicity

This is a question of definition, does the stats community differentiate these terms?

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I guess I win the award of the more useless question – RockScience Apr 12 '11 at 6:56

Perhaps. Though my take could easily be construed as a bit too anal retentive:

I tend to use the term seasonality as a metaphor for the 'seasons' of the year: i.e. Spring, Summer, Fall, Winter (or 'Almost Winter', Winter, 'Still Winter', and 'Construction' if you live in Pennsylvania...). In other words, I would expect a seasonal trend to have a periodicity of roughly 365 days.

I tend to use the term 'cyclicality' to refer to a response, which when decomposed in frequency space has a single dominant peak. Or, a bit more generally, much as one could stare at an engine, 'cyclicality' implies a dominant cycle -- the piston moves up, and then it moves down, and then it moves up again. Numerically, I would expect low, high, low, high, low, high, etc. So two things: (1) magnitude &/or sign switches from a low to high and (2) these switches occur with a predictable frequency. This rigor naturally evaporates when talking about business cycles -- however, I often find that a dominant frequency remains, e.g. every business quarter, or every year, things are slow for the first few weeks and high pressure the last few weeks... So there is a dominant period, but it could be very different from 'seasonality' which to me implies a year.

Lastly, I tend to use 'periodicity' when referring to the frequency of collecting measurements. Differing from cyclicality, the term 'periodicity' for me implies no expectation for the magnitude or sign of the data collected.

But this is just my \$0.02. And I'm just a stat student -- take from this what you will.

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I would also use seasonality for 1 year cycles (linked to the seasons) but wanted to check with you guys. Do you think that monthly patterns (for instance end of month effects) can be called seasonal patterns? – RockScience Apr 12 '11 at 6:22
The seasonal adjustment system X-12-ARIMA deals with trading day and moving holiday effects, so you can reasonably call these seasonal effects so long as your reader is clear that is what you are doing. – Henry Apr 12 '11 at 8:34

Yes, there is a difference.

A classic time series decomposition model is Y = T * S * C * I

Y = data T = trend S = seasonal = REGULAR patterns that occur with time, e.g. oatmeal sales higher in winter, or Starbucks coffee sales being highest at 7 a.m. These are usually very predictable. C = cyclical = longer term patterns like business cycles. These aren't as regular as seasonality, and may involve some subjectivity in estimation. I = irregular (i.e. error left over)

Periodicity refers to seasonal component. Periodicity could be monthly, biweekly, hourly, etc.

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since star usually means convolution in mathematical notation your decomposition model is far from classical :) – mpiktas Apr 12 '11 at 5:43
I suspect * mean actually here + or . depending if the seasonality is additive or multiplicative – RockScience Apr 12 '11 at 6:19
I believe that Trend is what's left over after removing seasonal and cyclical effects and smoothing out irregularity, so it doesn't quite fall into the frequency hierarchy of Cycles < Seasons< Irregularity, which makes it a bit tricky in my mind. I've also seen a combined TC (Trend-Cycle) factor used, where TC < S < I. (I.e. I is highest frequency, S is lower frequency, and TC is lowest frequency.) – Wayne Apr 12 '11 at 21:11