Time Series Models

Time series analysis helps organizations understand the underlying causes of trends or systemic patterns over time. A time series is simply a set of statistics that is collected at regular intervals. Ex. the daily number of live births or death.

A Stochastic process is a (possibly) infinite sequence of variables ordered in time {Y₀, Y₁, Y₂ ...}. A time series is a single realization of a stochastic process. We want to make inference about the properties of the underlying stochastic process from a single observation.

There are two assumptions in time series analysis:

1. The data sequence is stationary. This means if all the times are shifted by the same amount, the probability distribution remains the same; meaning it depends on relative and not absolute values. In other terms:
   $$ (X_{t_1}, ... , X_{t_k}) =(X_{t_{1 + h}}, ... , X_{t_{k + h}}) $$
   for all time points t and integer h