Introduction to linear models for time-series

In this section, we are going to employ an artificial time-series to show some common linear models for time-series. The goal is not to provide an exhaustive explanation (which would require an entire book), but to introduce the reader to this kind of modeling method. The reader who is interested in the topic (and would like to read a complete mathematical background) can check Shumway R. H., Stoffer D. S., Time Series Analysis and Its Applications, Springer, 2017.

A time-series containing 100 observations with a frequency of 0.5 (2 observations per time instant) is generated by the following snippet:

import numpy as np
x = np.expand_dims(np.arange(0, 50, 0.5), axis=1)
y = np.sin(5.*x) + np.random.normal(0.0, 0.5, size=x.shape)
y = np.squeeze(y)

A graphical representation is shown in the following figure:

Synthetic time-series with 100 observations

This time-series has no particular characteristic except...