Concepts / definitions

Time series process vs. observed time series

A time series is a series of data points indexed in time order. In this class we assume that the time difference between observations is constant. A time series process is the underlying stochastical model. This can be represented as a random vector, where each entry of the vector corresponds to one time point. A time series is an observation of that process.


A time series process is stationary if the property of the values does not depend on the time. Windows of the process (i.e. a subvector using the representation above) will look the same everywhere.

An example of a stationary time series is white noise. Any windows of white noise look (sound) the same. An example of a non-stationary time series is the sound from a car passing by. We can expect the amplitude to first increase and then decrease. The amplitude is thus dependent of the time. If we measure the sounds of all cars on a road on a friday 4pm - 5pm, this again is (approximately) stationary. There might be times with more / fewer cars, but this is not dependent of the time. If we measure car noise for multiple days, there will be a periodicity. E.g. we expect more cars during the day than at night. The series will not be stationary.

Seasonal effects vs stochastic periodic effects

A time series process has a seasonal effect if the process is dependent of the time. An example of this would be restaurant bookings during the week. The exact number of bookings per day is random, but we can expect more bookings on the weekends, as people don’t have to work.

A time series process can have stochastic periodic effects, if variables are correlated. E.g. for the yearly number of sunspots, values which are ~11 years apart seem to be correlated, and values which are ~5-6 years apart are negatively correlated. The resulting time series thus has a periodicity of ~11 years. Differently to the restaurant bookings, this is not induced through the time itself, but a result of the correlation with past values. For the restaurant bookings, the values for a given saturday are high because this is a saturday, not because the values were high last saturday. Consequently, the periodicity for restaurant bookings will be exactly 7 days. If there is a weekend with fewer / no bookings, we can still expect a higher number of bookings the following weekend. If there are a high number of sunspots in 1989, 2000 and 2012, we can expect a high number of sunspots in 2023, not 2022.