Time Series
A time series is a sequence of random variables indexed by time.
1. Definition
A time series is a collection of random variables $\{X_t\}_{t \in T}$ where $T$ is an index set representing time.
- Capital letters $X_1, X_2, \dots$ denote random variables (the theoretical process)
- Lowercase letters $x_1, x_2, \dots$ denote observed realizations
- In practice, the observed series $\{x_t\}$ is one realization of an underlying stochastic process
2. Formats
| Format | Notation | Description |
|---|---|---|
| Infinite & Random | $X_t$ | Theoretical stochastic process, $t \in \mathbb{Z}$ |
| Finite & Random | $X_t$ | Random variables for $t = 1, \dots, n$ |
| Finite & Observed | $x_t$ | Observed data for $t = 1, \dots, n$ |
Preferred notation: $\{X_t\}_{t \in T_N}$ or simply $\{X_t\}$.
3. Goals of Time Series Analysis
- Understand the structure (trend, seasonality, dependence)
- Fit a stochastic model to the data
- Forecast future values
For the distinction between the theoretical process and the observed data sequence, see Stochastic Process vs Time Series.