Time series
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In statistics, signal processing, and many other fields, 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 context of the data points (Where did they come from? What generated them?), or to make forecasts (predictions). Time series forecasting is the use of a model to forecast future events based on known past events: to forecast future data points before they are measured. A standard example in econometrics is the opening price of a share of stock based on its past performance.
The term time series analysis is used to distinguish a problem, firstly from more ordinary data analysis problems (where there is no natural ordering of the context of individual observations), and secondly from spatial data analysis where there is a context that observations (often) relate to geographical locations. There are additional possibilities in the form of spacetime models (often called spatialtemporal analysis). A time series model will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural oneway ordering of time so that values in a series for a given time will be expressed as deriving in some way from past values, rather than from future values (see time reversibility.)
Methods for time series analyses are often divided into two classes: frequencydomain methods and timedomain methods. The former centre around spectral analysis and recently wavelet analysis, and can be regarded as modelfree analyses wellsuited to exploratory investigations. Timedomain methods have a modelfree subset consisting of the examination of autocorrelation and crosscorrelation analysis, but it is here that partly and fullyspecified time series models make their appearance.
Contents 
[edit] Time series analyses
There are several types of data analysis available for time series which are appropriate for different purposes.
[edit] General exploration
 Graphical examination of data series
 Autocorrelation analysis to examine serial dependence
 Spectral analysis to examine cyclic behaviour which need not be related to seasonality
[edit] Description
 Separation into components representing trend, seasonality, slow and fast variation, cyclical irregular: see Decomposition of time series
 Simple properties of marginal distributions
[edit] Prediction and forecasting
 Fullyformed statistical models for stochastic simulation purposes, so as to generate alternative versions of the time series, representing what might happen over nonspecific timeperiods in the future (prediction).
 Simple or fullyformed statistical models to describe the likely outcome of the time series in the immediate future, given knowledge of the most recent outcomes (forecasting).
[edit] Time series models
As shown by Box and Jenkins^{[1]}, models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. These three classes depend linearly^{[2]} on previous data points. Combinations of these ideas produce autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vectorvalued data are available under the heading of multivariate timeseries models and sometimes the preceding acronyms are extended by including an initial "V" for "vector". An additional set of extensions of these models is available for use where the observed timeseries is driven by some "forcing" timeseries (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous".
Nonlinear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a chaotic time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from nonlinear models, over those from linear models.
Among other types of nonlinear time series models, 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 variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locallyvarying variability, where the variability might be modelled as being driven by a separate timevarying process, as in a doubly stochastic model.
In recent work on modelfree analyses, wavelet transform based methods (for example locally stationary wavelets and wavelet decomposed neural networks) have gained favor. Multiscale (often referred to as multiresolution) techniques decompose a given time series, attempting to illustrate time dependence at multiple scales.
[edit] Notation
A number of different notations are in use for timeseries analysis:
 X = {X_{1}, X_{2}, ...}
is a common notation which specifies a time series X which is indexed by the natural numbers. Another common notation is:
 Y = {Y_{t}: t ∈ T}
[edit] Conditions
There are two sets of conditions under which much of the theory is built:
However, ideas of stationarity must be expanded to consider two important ideas: strict stationarity and secondorder stationarity. Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified.
In addition, timeseries analysis can be applied where the series are seasonally stationary and nonstationary.
[edit] Models
The general representation of an autoregressive model, wellknown as AR(p), is
where the term ε_{t} is the source of randomness and is called white noise. It is assumed to have the following characteristics:
1.
2.
3.
With these assumptions, the process is specified up to secondorder moments and, subject to conditions on the coefficients, may be secondorder stationary.
If the noise also has a normal distribution, it is called normal white noise (denoted here by NormalWN):
In this case the AR process may be strictly stationary, again subject to conditions on the coefficients.
[edit] Notes
[edit] References
 Box, George; Jenkins, Gwilym (1976), Time series analysis: forecasting and control, rev. ed., Oakland, California: HoldenDay
 Gershenfeld, Neil (2000), The nature of mathematical modeling, Cambridge: Cambridge Univ. Press, ISBN 9780521570954, OCLC 174825352
[edit] Related tools
Tools for investigating timeseries data include:
 Consideration of the autocorrelation function and the spectral density function (also crosscorrelation functions and crossspectral density functions)
 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)
 Singular spectrum analysis
 Artificial neural networks
 timefrequency analysis techniques:
 Chaotic analysis
[edit] See also
 Analysis of rhythmic variance
 Anomaly time series
 Autocorrelation
 Partial autocorrelation
 Linear prediction
 Longitudinal study
 Model (macroeconomics)
 Moving average (finance)
 Nonlinear autoregressive exogenous model
 Prediction interval
 Seasonal adjustment
 System identification
 Time series database
 Trend estimation
[edit] External links
 A First Course on Time Series Analysis  an open source book on time series analysis with SAS
 Introduction to Time series Analysis (Engineering Statistics Handbook)  A practical guide to Time series analysis
 List of Free Software for Time Series Analysis
 Online Tutorial 'Recurrence Plot' (Flash animation); lots of examples
 Scientio's ChaosKit product performs online analysis and prediction of Chaotic time series. Access is provided free online via a web service and graphic interface.
