Time series

In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. Time series analysis comprises methods that attempt to understand such time series, often either to understand the underlying theory of the data points (where did they come from? what generated them?), or to make forecasts (predictions). Time series prediction is the use of a model to predict future events based on known past events: to predict future data points before they are measured. The standard example is the opening price of a share of stock based on its past performance.

Models for time series data can have many forms. Two broad classes of practical importance are the moving average (MA) models, and the autoregressive (AR) models. These two classes depend linearly on previous data points and are treated in more detail in the article on autoregressive moving average models (ARMA). Non-linear dependence on previous data points is of interest because of the possibility of producing a chaotic time series.

A number of different notations are in use for time-series analysis


 * $$X= \{X_1, X_2, \dots \}$$

is a common notation which specifies a time series X which is indexed by the natural numbers.

Tools for investigating time-series data include:


 * Consideration of the autocorrelation function and the spectral density function
 * Performing a Fourier transform to investigate the series in the frequency domain.
 * Use of a filter to remove unwanted noise.
 * Artificial neural networks
 * time-frequency analysis techniques:
 * Continuous wavelet transform
 * Short-time Fourier transform
 * Chirplet transform
 * Fractional Fourier transform
 * Chaotic analysis
 * Correlation dimension
 * Recurrence plots
 * Recurrence quantification analysis
 * Lyapunov exponents

Industry usage
Any associative array of times and numbers can be viewed as a time series. The times may not necessarily be of a regular interval length.

For example, the historical fluctuations in the price of a NYMEX Gold Contract can be said to be the time series for NYMEX Gold.

Analysts throughout the economy will use the tools outlined here to aid in the management of their corresponding businesses. Energy traders, for example, will often attempt to forecast power consumption based upon both weather normals and short term weather forecasts.