Introduction to time series smoothing

Time series data is composed of signals and noise, where signals capture intrinsic dynamics of the process; however, noise represents the unmodeled component of a signal. The intrinsic dynamics of a time series signal can be as simple as the mean of the process or it can be a complex functional form within observations, as represented here:

x_t = f(x_i) + ε_t for i=1,2,3, ... t-1

Here, x_t is observations and ε_t is white noise. The f(x_i) denotes the functional form; an example of a constant as a functional form is as follows:

x_t = μ + ε_t

Here, the constant value μ in the preceding equation acts as a drift parameter, as shown in the following figure:

Figure 3.1: Example of time series with drift parameter

As ε_t is white noise, this smoothing-based approach helps separate the intrinsic functional form from random...