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 noise by canceling it. The smoothing forecasting methods can be considered as filters that take inputs and separate the trend and noise components, as...