Time series

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 space-time models (often called spatial-temporal 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 one-way 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: frequency-domain methods and time-domain methods. The former centre around spectral analysis and recently wavelet analysis, and can be regarded as model-free analyses well-suited to exploratory investigations. Time-domain methods have a model-free subset consisting of the examination of auto-correlation and cross-correlation analysis, but it is here that partly and fully-specified time series models make their appearance.

Time Series Analyses
There are several types of data analysis available for time series which are appropriate for different purposes.

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

Description

 * Separation into components representing trend, seasonality, slow and fast variation, Cyclical irregular
 * Simple properties of marginal distributions

Prediction and Forecasting

 * Fully-formed statistical models for stochastic simulation purposes, so as to generate alternative versions of the time series, representing what might happen over non-specfic time-periods in the future (prediction).
 * Simple or fully-formed statistical models to describe the likely outcome of the time series in the immediate future, given knowledge of the most recent outcomes (forecasting).

Time Series Models
As shown by Box and Jenkins in their 1976 book, Time Series Analysis: Forecasting and Control, 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 (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. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series 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 time-series is driven by some "forcing" time-series (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".

Non-linear 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 non-linear models, over those from linear models.

Among other types of non-linear 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 locally-varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a doubly stochastic model.

In recent work on model-free 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.

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


 * X = {X1, X2, ...}

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


 * Y = {Yt: t &isin; T}

Conditions
There are two sets of conditions under which much of the theory is built:


 * Stationary process
 * Ergodicity

However, ideas of stationarity must be expanded to consider two important ideas: strict stationarity and second-order 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, time-series analysis can be applied where the series are seasonally stationary and non-stationary.

Models
The general representation of an autoregressive model, well-known as AR(p), is


 * $$ Y_t =\alpha_0+\alpha_1 Y_{t-1}+\alpha_2 Y_{t-2}+\cdots+\alpha_p Y_{t-p}+\varepsilon_t\, $$

where the term εt is the source of randomness and is called white noise. It is assumed to have the following characteristics:

1. $$ E[\varepsilon_t]=0 \,$$

2. $$ E[\varepsilon^2_t]=\sigma^2 \, $$

3. $$ E[\varepsilon_t\varepsilon_s]=0 \quad\forall t\not=s \, $$

With these assumptions, the process is specified up to second-order moments and, subject to conditions on the coefficients, may be second-order stationary.

If the noise also has a normal distribution, it is called normal white noise:


 * $$ \{\varepsilon_t\}_{(t \in T)} : \mbox{Normal-WN} $$.

In this case the AR process may be strictly stationary, again subject to conditions on the coefficients.

Related tools
Tools for investigating time-series data include:


 * Consideration of the autocorrelation function and the spectral density function (also cross-correlation functions and cross-spectral 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
 * 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