 Time series

In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the Nile River at Aswan. Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. Time series are very frequently plotted via line charts.
Time series data have a natural temporal ordering. This makes time series analysis distinct from other common data analysis problems, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their education level, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). 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 for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility.)
Methods for time series analyses may be divided into two classes: frequencydomain methods and timedomain methods. The former include spectral analysis and recently wavelet analysis; the latter include autocorrelation and crosscorrelation analysis.
Contents
Analysis
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. For example, sun spot activity varies over 11 year cycles.^{[1]}^{[2]} Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
Description
 Separation into components representing trend, seasonality, slow and fast variation, cyclical irregular: see decomposition of time series
 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 nonspecific timeperiods in the future
 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).
Models
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^{[3]} 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, as for example in nonlinear autoregressive exogenous models.
Among other types of nonlinear time series models, there are models to represent the changes of variance along time (heteroskedasticity). These models represent 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 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.
Notation
A number of different notations are in use for timeseries analysis. A common notation specifying a time series X that is indexed by the natural numbers is written
 X = {X_{1}, X_{2}, ...}.
Another common notation is
 Y = {Y_{t}: t ∈ T},
where T is the index set.
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 or nonstationary. Situations where the amplitudes of frequency components change with time can be dealt with in timefrequency analysis which makes use of a time–frequency representation of a timeseries or signal.^{[4]}
Models
Main article: Autoregressive modelThe 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:
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 or Gaussian white noise. In this case, the AR process may be strictly stationary, again subject to conditions on the coefficients.
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)
See also
References
 ^ Bloomfield, P. (1976). Fourier analysis of time series: An introduction. New York: Wiley.
 ^ Shumway, R. H. (1988). Applied statistical time series analysis. Englewood Cliffs, NJ: Prentice Hall.
 ^ Gershenfeld, N. (1999). The nature of mathematical modeling. p.20508
 ^ Boashash, B. (ed.), (2003) TimeFrequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003 ISBN ISBN 0080443354
Further reading
 Bloomfield, P. (1976). Fourier analysis of time series: An introduction. New York: Wiley.
 Box, George; Jenkins, Gwilym (1976), Time series analysis: forecasting and control, rev. ed., Oakland, California: HoldenDay
 Brillinger, D. R. (1975). Time series: Data analysis and theory. New York: Holt, Rinehart. & Winston.
 Brigham, E. O. (1974). The fast Fourier transform. Englewood Cliffs, NJ: PrenticeHall.
 Elliott, D. F., & Rao, K. R. (1982). Fast transforms: Algorithms, analyses, applications. New York: Academic Press.
 Gershenfeld, Neil (2000), The nature of mathematical modeling, Cambridge: Cambridge Univ. Press, ISBN 9780521570954, OCLC 174825352
 Hamilton, James (1994), Time Series Analysis, Princeton: Princeton Univ. Press, ISBN 0691042896
 Jenkins, G. M., & Watts, D. G. (1968). Spectral analysis and its applications. San Francisco: HoldenDay.
 Priestley, M. B. (1981). Spectral analysis and time series. New York: Academic Press.
 Shasha, D. (2004), High Performance Discovery in Time Series, Berlin: Springer, ISBN 0387008578
 Shumway, R. H. (1988). Applied statistical time series analysis. Englewood Cliffs, NJ: Prentice Hall.
 Wiener, N.(1964). Extrapolation, Interpolation, and Smoothing of Stationary Time Series.The MIT Press.
 Wei, W. W. (1989). Time series analysis: Univariate and multivariate methods. New York: AddisonWesley.
 Weigend, A. S., and N. A. Gershenfeld (Eds.) (1994) Time Series Prediction: Forecasting the Future and Understanding the Past. Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis (Santa Fe, May 1992) MA: AddisonWesley.
 Durbin J., and Koopman S.J. (2001) Time Series Analysis by State Space Methods. Oxford University Press.
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
 Measuring the "Complexity" of a time series
 Measures of Analysis of Time Series toolkit (MATLAB)
 iSAX: Indexing and Mining Terabyte Sized Time Series
 Harvard Time Series Center
 STATStream: a Toolkit for High Speed Statistical Time Series Analysis
 Data Stream Algorithms
 Bibliography of symbolic timeseries analysis
 The Santa Fe Time Series Competition Data
Categories: Time series analysis
 Formal sciences
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