# Definition:Time Series

## Definition

A time series is a sequence of observations taken at a sequence of instants of time.

### Timestamp

A timestamp of an observation $x$ of a time series is the instant with which $x$ is associated.

### Discrete Time Series

A discrete time series is such that the timestamps of the observations occur at well-defined instants, separated one from another by a time interval.

### Continuous Time Series

A continuous time series is one in which the set of timestamps of the observations forms a continuous function.

Two observations of a time series are adjacent if and only if the index of one of them is the immediate predecessor of the other (and the other is the immediate successor of the one).

A defining characteristic of a time series is that every pair of adjacent observations is dependent.

### Equispaced

A time series is equispaced if and only if the time intervals between the timestamps is equal for all pairs of adjacent observations.

### Past Value

A past value of a time series $T$ is an observation $x$ of $T$ whose timestamp is for some past instant, previous to the current value.

### Current Value

The current value of a time series $T$ is an observation $x$ of $T$ whose timestamp is the current time.

That is, it is the most recent observation.

### Future Value

A future value of a time series $T$ is an observation $x$ of $T$ whose timestamp is for some future instant.

### Origin

The origin of a time series is an arbitrary timestamp of an observation which is chosen in order that all other timestamps can be measured from that origin.

### Actual Value

An actual value of a time series $T$ is the result of a measurement of an observation at some time $t$.

The term is usually made in reference to a forecast value made after its lead time has elapsed, and its timestamp is now the current time.

### Forecast Value

A forecast value of a time series $T$ is an estimate of a future value at some lead time $t + l$.

## Sources

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.1$ Time Series and Stochastic Processes: Time series