The goal of time series decomposition is to increase our understanding of the data by breaking down the series into multiple components. It provides insight in terms of modeling complexity and which approaches to follow in order to accurately capture each of the components.

These components can be divided into two types: systematic and non-systematic. The systematic ones are characterized by consistency and the fact that they can be described and modeled. By contrast, the non-systematic ones cannot be modeled directly.

The following are the systematic components:

• level: The mean value in the series.
• trend: An estimate of the trend, that is, the change in value between successive time points at any given moment. It can be associated with the slope (increasing/decreasing) of the series.
• seasonality: Deviations from the mean caused by repeating short-term cycles.

The following is the non-systematic component:

• noise: The random variation in the series

There are two types of models that are used for decomposing time series: additive and multiplicative.

The following are the characteristics of the additive model:

• Model's form: y(t) = level + trend + seasonality + noise
• Linear model: changes over time are consistent in size
• The trend is linear (straight line)
• Linear seasonality with the same frequency (width) and amplitude (height) of cycles over time

The following are the characteristics of the multiplicative model:

• Model's form: y(t) = level * trend * seasonality * noise
• Non-linear model: changes over time are not consistent in size, for example, exponential
• A curved, non-linear trend
• Non-linear seasonality with increasing/decreasing frequency and amplitude of cycles over time

It can be the case that we do not want to work with the multiplicative model. One possible solution is to apply certain transformations to make the trend/seasonality linear. One example of a transformation could be taking the log of a series in which we observe exponential growth.

In this recipe, we present how to carry out time-series decomposition of monthly gold prices downloaded from Quandl.