Seasonality

In statistics, many time series exhibit cyclic variation known as seasonality, periodic variation, or periodic fluctuations. This variation can be either regular or semiregular.

Examples

For example, retail sales tend to peak for the Christmas season and then decline after the holidays. So time series of retail sales will typically show increasing sales from September through December and declining sales in January and February.

Seasonality is quite common in economic time series. It is also very common in geophysical and ecological time series. A notable example is the concentration of atmospheric carbon dioxide: it is at a minimum in September and October, at which point it begins to increase, reaching a peak in April/May, before declining. Another example consists of the famous Milankovitch cycles.

Detecting seasonality

In this section, techniques for detecting seasonality are discussed. The following graphical techniques can be used to detect seasonality:

• A run sequence plot will often show seasonality
• A seasonal subseries plot is a specialized technique for showing seasonality
• Multiple box plots can be used as an alternative to the seasonal subseries plot to detect seasonality
• The autocorrelation plot can help identify seasonality
• Seasonal Index measures how much the average for a particular period tends to be above (or below) the expected value

The run sequence plot is a recommended first step for analyzing any time series. Although seasonality can sometimes be indicated with this plot, seasonality is shown more clearly by the seasonal subseries plot or the box plot. The seasonal subseries plot does an excellent job of showing both the seasonal differences (between group patterns) and also the within-group patterns. The box plot shows the seasonal difference (between group patterns) quite well, but it does not show within group patterns. However, for large data sets, the box plot is usually easier to read than the seasonal subseries plot.

Both the seasonal subseries plot and the box plot assume that the seasonal periods are known. In most cases, the analyst will in fact know this. For example, for monthly data, the period is 12 since there are 12 months in a year. However, if the period is not known, the autocorrelation plot can help. If there is significant seasonality, the autocorrelation plot should show spikes at lags equal to the period. For example, for monthly data, if there is a seasonality effect, we would expect to see significant peaks at lag 12, 24, 36, and so on (although the intensity may decrease the further out we go).

Modeling seasonality

A completely regular cyclic variation in a time series might be dealt with in time series analysis by using a sinusoidal model with one or more sinusoids whose period-lengths may be known or unknown depending on the context. A less completely regular cyclic variation might be dealt with by using a special form of an ARIMA model which can be structured so as to treat cyclic variations semi-explicitly. Semiregular cyclic variations might be dealt with by spectral density estimation.

Seasonal adjustment

Main article: seasonal adjustment

See also

• Cyclostationary process
• Decomposing of time series
• Oscillation
• Periodic function
• Periodicity
• X-12-ARIMA

References

• Seasonality

 This article incorporates public domain material from the National Institute of Standards and Technology