# 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.

### Adjacent Observations

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).

### Dependence of Adjacent Observations

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

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**time series** - 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel:
*Time Series Analysis: Forecasting and Control*(3rd ed.) ... (next): $1$: Introduction - 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel:
*Time Series Analysis: Forecasting and Control*(3rd ed.) ... (previous) ... (next):

- 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

- $2.1$ Autocorrelation Properties of Stationary Models:

- $2$: Autocorrelation Function and Spectrum of Stationary Processes:

- Part $\text {I}$: Stochastic Models and their Forecasting:

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**time series** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**time series** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**time series**