Internal structures of time series

In this section, we will conceptually explain the following special characteristics of time series data that requires its special mathematical treatment:

• General trend
• Seasonality
• Cyclical movements
• Unexpected variations

Most time series has of one or more of the aforementioned internal structures. Based on this notion, a time series can be expressed as x_t = f_t + s_t + c_t + e_t, which is a sum of the trend, seasonal, cyclical, and irregular components in that order. Here, t is the time index at which observations about the series have been taken at t = 1,2,3 ...N successive and equally spaced points in time.

The objective of time series analysis is to decompose a time series into its constituent characteristics and develop mathematical models for each. These models are then used to understand what causes the observed behavior of the time series and to predict the series for future points in time.