TIME SERIES

In statistics, signal processing, and econometrics, a 'time series' is a sequence of data points, measured typically at successive times, spaced at (often 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.
As shown by Box and Jenkins in their book, models for time series data can have many forms and represent different stochastic processes. When modeling the mean of a process, three broad classes of practical importance are the ''Autoregressive'' (AR) models, the ''Integrated'' (I) models, and the ''Moving Average'' (MA) models (the MA process is related but not to be confused with the concept of Moving average ). These three classes depend linearly on previous data points and are treated in more detail in the articles Autoregressive Moving Average Models (ARMA) and Autoregressive Integrated Moving Average (ARIMA). The Autoregressive Fractionally Integrated Moving Average (ARFIMA) model generalizes the former three. Non-linear dependence on previous data points is of interest because of the possibility of producing a chaotic time series.
Among non-linear time series, there are models to represent the changes of variance along time (heteroskedasticity). These models are called Autoregressive Conditional Heteroskedasticity (ARCH) and the collection comprises a wide variaty of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc).

Contents

Notation

Assumptions

Related tools

Industry usage

See also

External links

Notation

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. We also are accustomed to
:$Y=\; \{Y\_t\; :\; t\; in\; T\; \}$

Assumptions

There are only two assumptions from which the theory is built:

★ Stationary process

★ Ergodicity
The general representation of an autoregressive model well-known as AR(p) is:

where the term $epsilon\_t$ is the source of randomness and is called white noise. It is assumed to have the following characteristics:
1. $E[epsilon\_t]=0$
2. $E[epsilon^2\_t]=sigma^2$
3.
ot=s
If it also has a Normal distribution, it is called Normal White-Noise:
$\{epsilon\_t\}\_\{(t\; in\; T)\}\; :\; Normal-WN$

Related tools

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.

★ Principal components analysis (or empirical orthogonal function analysis)

★ 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.

See also

★ Analysis of rhythmic variance

★ Anomaly time series

★ Autocorrelation

★ Partial autocorrelation

★ Linear prediction

★ Longitudinal study

★ Model (macroeconomics)

★ Moving average (finance)

★ Prediction interval

★ Predictive analytics

★ Seasonal adjustment

★ System identification

★ Time series database

★ Trend estimation

External links

★ A First Course on Time Series Analysis - an open source book on time series analysis with SAS

★ Software for Time Series Analysis

★ Free online chaos theory based analysis

★ Online Tutorial 'Recurrence Plot' (Flash animation); lots of examples

★ A Professional Environment for Time Series and Signal Analysis

★ Adaptive Harmonic Components Detection and Forecasting in Wave Non-Periodic Time Series Using Neural Networks