There are three kinds of seasonal patterns that may emerge in time series. Seasonality might be deterministic or stochastic. On the stochastic side, seasonal patterns might be either stationary or not.

All these seasonality will not be mutually exclusive. A time series can have each a deterministic and stochastic seasonal component.

Let’s describe each pattern in turn.

Deterministic seasonality

Time series with a deterministic seasonality have a continuing seasonal pattern. It all the time recurs in a predictable way, each in intensity and periodicity:

• : the extent of the seasonal pattern stays the identical over the identical seasonal period;
• : the placement of the peaks and troughs doesn’t change. In other words, the time between each repetition of the seasonal pattern is constant.

Here’s an artificial monthly time series with a deterministic seasonality:

import numpy as np
period = 12
size = 120
beta1 = 0.3
beta2 = 0.6
sin1 = np.asarray([np.sin(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
cos1 = np.asarray([np.cos(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
xt = np.cumsum(np.random.normal(scale=0.1, size=size))
series_det = xt + beta1*sin1 + beta2*cos1 + np.random.normal(scale=0.1, size=size)

This time series is customized from the book in reference [3].

Constant seasonality might be well handled by seasonal dummy explanatory variables. A categorical variable that describes the seasonal period. On this case, the month that corresponds to every time step. This categorical variable is transformed right into a set of indicator (dummy) variables by one-hot encoding.

You too can use Fourier series to model seasonality. Fourier series are sine and cosine waves with various periods. You’ll be able to learn more about these in a previous article.

Stochastic stationary seasonality

beta1 = np.linspace(-.6, .3, num=size)
beta2 = np.linspace(.6, -.3, num=size)
sin1 = np.asarray([np.sin(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
cos1 = np.asarray([np.cos(2 * np.pi * i / 12) for i in np.arange(1, size + 1)])
xt = np.cumsum(np.random.normal(scale=0.1, size=size))
# synthetic series with stochastic seasonality
series_stoc = xt + beta1*sin1 + beta2*cos1 + np.random.normal(scale=0.1, size=size)

A stochastic stationary seasonality evolves over consecutive seasonal periods (e.g. 12 months over 12 months). The intensity is less predictable, however the periodicity stays roughly the identical.

With deterministic seasonality, the most effective prediction for a given month doesn’t change no matter the 12 months. For a stochastic stationary seasonality, the most effective guess relies on the worth of the identical month from the previous 12 months.

Stochastic non-stationary seasonality

Sometimes, seasonal patterns change significantly over several seasonal periods. These changes might be brought on by seasonal unit roots, which implies that seasonality is integrated.

Besides the intensity, the periodicity of one of these seasonality also tends to vary over time. Which means that the peaks and troughs vary of their location.

Examples of one of these seasonal pattern appear in several domains. These include consumption series or industrial production data.

Changes are difficult to predict when time series have an integrated seasonality. Shocks cause everlasting changes in the information, resulting in scenarios where “spring becomes summer” — quote from reference [1].