# Seasonal periods

Date

7 November 2014

Topics
forecasting
R
seasonality
statistics

I get questions about this almost every week. Here is an example from a recent comment on this blog:

I have two large time series data. One is separated by seconds intervals and the other by minutes. The length of each time series is 180 days. I’m using R (3.1.1) for forecasting the data. I’d like to know the value of the “frequency” argument in the ts() function in R, for each data set. Since most of the examples and cases I’ve seen so far are for months or days at the most, it is quite confusing for me when dealing with equally separated seconds or minutes. According to my understanding, the “frequency” argument is the number of observations per season. So what is the “season” in the case of seconds/minutes? My guess is that since there are 86,400 seconds and 1440 minutes a day, these should be the values for the “freq” argument. Is that correct?

Yes, the “frequency” is the number of observations per “cycle” (normally a year, but sometimes a week, a day or an hour). This is the opposite of the definition of frequency in physics, or in Fourier analysis, where “period” is the length of the cycle, and “frequency” is the inverse of period. When using the `ts()` function in R, the following choices should be used.

Data Frequency
Annual 1
Quarterly 4
Monthly 12
Weekly 52

Actually, there are not 52 weeks in a year, but 365.25/7 = 52.18 on average, allowing for a leap year every fourth year. But most functions which use `ts` objects require integer frequency.

If the frequency of observations is greater than once per week, then there is usually more than one way of handling the frequency. For example, data with daily observations might have a weekly seasonality (frequency=7) or an annual seasonality (frequency=365.25). Similarly, data that are observed every minute might have an hourly seasonality (frequency=60), a daily seasonality (frequency=24x60=1440), a weekly seasonality (frequency=24x60x7=10080) and an annual seasonality (frequency=24x60x365.25=525960). If you want to use a `ts` object, then you need to decide which of these is the most important.

An alternative is to use a `msts` object (defined in the `forecast` package) which handles multiple seasonality time series. Then you can specify all the frequencies that might be relevant. It is also flexible enough to handle non-integer frequencies.

Frequencies
Data Minute Hour Day Week Year
Daily 7 365.25
Hourly 24 168 8766
Half-hourly 48 336 17532
Minutes 60 1440 10080 525960
Seconds 60 3600 86400 604800 31557600

You won’t necessarily want to include all of these frequencies — just the ones that are likely to be present in the data. For example, any natural phenomena (e.g., sunshine hours) is unlikely to have a weekly period, and if your data are measured in one-minute intervals over a 3 month period, there is no point including an annual frequency.

For example, the `taylor` data set from the `forecast` package contains half-hourly electricity demand data from England and Wales over about 3 months in 2000. It was defined as

``taylor <- msts(x, seasonal.periods=c(48,336)``

One convenient model for multiple seasonal time series is a TBATS model:

``````taylor.fit <- tbats(taylor)
plot(forecast(taylor.fit))``````

(Warning: this takes a few minutes.)

If an `msts` object is used with a function designed for `ts` objects, the largest seasonal period is used as the “frequency” attribute.