Time series
A time series is a sequence of data points, measured typically at successive points in time spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones Industrial Average and the annual flow volume of the Nile River at Aswan. Time series are very frequently plotted via line charts. Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, and communications engineering.
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. While regression analysis is often employed in such a way as to test theories that the current values of one or more independent time series affect the current value of another time series, this type of analysis of time series is not called "time series analysis", which focuses on comparing values of a single time series at different points in time.^{1}
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 respective education levels, 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 stochastic model for a time series 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.)
Time series analysis can be applied to realvalued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language.^{2}).
Contents
Methods for time series analyses
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. In time domain correlation analyses can be made in a filterlike manner using scaled correlation, thereby mitigating the need to operate in frequency domain.
Additionally, time series analysis techniques may be divided into parametric and nonparametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, nonparametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.
Methods of time series analysis may also be divided into linear and nonlinear, and univariate and multivariate.
Analysis
There are several types of motivation and data analysis available for time series which are appropriate for different purposes.
Motivation
In the context of statistics, econometrics, quantitative finance, seismology, meteorology, and geophysics the primary goal of time series analysis is forecasting. In the context of signal processing, control engineering and communication engineering it is used for signal detection and estimation, while in the context of data mining, pattern recognition and machine learning time series analysis can be used for clustering, classification, query by content, anomaly detection as well as forecasting.
Exploratory analysis
The clearest way to examine a regular time series manually is with a line chart such as the one shown for tuberculosis in the United States, made with a spreadsheet program. The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly steadily dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/ 10%, with 'surges' in 1975 and around the early 1990s. The use of both vertical axes allows the comparison of two time series in one graphic. Other techniques include:
 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.^{3}^{4} Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
 Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see trend estimation and decomposition of time series
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).
 Forecasting on time series is usually done using automated statistical software packages and programming languages, such as R, S, SAS, SPSS, Minitab, Pandas (Python) and many others.
Classification
 Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in sign language
See main article: Statistical classification
Regression analysis (method of prediction)
 Estimating future value of a signal based on its previous behavior, e.g. predict the price of AAPL stock based on its previous price movements for that hour, day or month, or predict position of Apollo 11 spacecraft at a certain future moment based on its current trajectory (i.e. time series of its previous locations).^{5}
 Regression analysis is usually based on statistical interpretation of time series properties in time domain, pioneered by statisticians George Box and Gwilym Jenkins in the 50s: see Box–Jenkins
Signal estimation
 This approach is based on harmonic analysis and filtering of signals in the frequency domain using the Fourier transform, and spectral density estimation, the development of which was significantly accelerated during World War II by mathematician Norbert Wiener, electrical engineers Rudolf E. Kálmán, Dennis Gabor and others for filtering signals from noise and predicting signal values at a certain point in time. See Kalman filter, Estimation theory, and digital signal processing
Segmentation
 Splitting a timeseries into a sequence of segments. It is often the case that a timeseries can be represented as a sequence of individual segments, each with its own characteristic properties. For example, the audio signal from a conference call can be partitioned into pieces corresponding to the times during which each person was speaking. In timeseries segmentation, the goal is to identify the segment boundary points in the timeseries, and to characterize the dynamical properties associated with each segment. One can approach this problem using changepoint detection, or by modeling the timeseries as a more sophisticated system, such as a Markov jump linear system.
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 on previous data points.^{6} 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", as in VAR for vector autoregression. 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 over 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. See also Markov switching multifractal (MSMF) techniques for modeling volatility evolution.
A Hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (hidden) states. An HMM can be considered as the simplest dynamic Bayesian network. HMM models are widely used in speech recognition, for translating a time series of spoken words into text.
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.^{7}
Models
The general representation of an autoregressive model, well known 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.
This section is in a list format that may be better presented using prose. (February 2012) 
Tools for investigating timeseries data include:
 Consideration of the autocorrelation function and the spectral density function (also crosscorrelation functions and crossspectral density functions)
 Scaled cross and autocorrelation functions to remove contributions of slow components^{8}
 Performing a Fourier transform to investigate the series in the frequency domain
 Principal component analysis (or empirical orthogonal function analysis)
 Singular spectrum analysis
 "Structural" models:
 General State Space Models
 Unobserved Components Models
Measures
Time series metrics or features that can be used for time series classification or regression analysis:^{9}
 Univariate linear measures
 Moment (mathematics)
 Spectral band power
 Spectral edge frequency
 Accumulated Energy (signal processing)
 Characteristics of the autocorrelation function
 Hjorth parameters
 FFT parameters
 Autoregressive model parameters
 Mann–Kendall test
 Univariate nonlinear measures
 Measures based on the correlation sum
 Correlation dimension
 Correlation integral
 Correlation density
 Correlation entropy
 Approximate entropy^{10}
 Sample entropy
 Fourier entropy
 Wavelet entropy
 Rényi entropy
 Higherorder methods
 Marginal predictability
 Dynamical similarity index
 State space dissimilarity measures
 Lyapunov exponent
 Permutation methods
 Local flow
 Other univariate measures
 Algorithmic complexity
 Kolmogorov complexity estimates
 Hidden Markov Model states
 Surrogate time series and surrogate correction
 Loss of recurrence (degree of nonstationarity)
 Bivariate linear measures
 Maximum linear crosscorrelation
 Linear Coherence (signal processing)
 Bivariate nonlinear measures
 Nonlinear interdependence
 Dynamical Entrainment (physics)
 Measures for Phase synchronization
 Similarity measures:^{11}
 Dynamic Time Warping
 Hidden Markov Models
 Edit distance
 Total correlation
 Newey–West estimator
 Prais–Winsten transformation
 Data as Vectors in a Metrizable Space
 Data as Time Series with Envelopes
 Global Standard Deviation
 Local Standard Deviation
 Windowed Standard Deviation
 Data Interpreted as Stochastic Series
 Data Interpreted as a Probability Distribution Function
See also
 Anomaly time series
 Decomposition of time series
 Detrended fluctuation analysis
 Digital signal processing
 Distributed lag
 Estimation theory
 Forecasting
 Hurst exponent
 Monte Carlo method
 Random walk
 Scaled correlation
 Seasonal adjustment
 Sequence analysis
 Signal processing
 Stringology
 Trend estimation
 Unevenly spaced time series
References
 ^ Imdadullah. "Time Series Analysis". Basic Statistics and Data Analysis. itfeature.com. Retrieved 2 January 2014.
 ^ Lin, Jessica; Keogh, Eamonn; Lonardi, Stefano; Chiu, Bill (2003). "A symbolic representation of time series, with implications for streaming algorithms". Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery. New York: ACM Press. doi:10.1145/882082.882086.
 ^ Bloomfield, P. (1976). Fourier analysis of time series: An introduction. New York: Wiley. ISBN 0471082562.
 ^ Shumway, R. H. (1988). Applied statistical time series analysis. Englewood Cliffs, NJ: Prentice Hall. ISBN 0130415006.
 ^ Lawson, Charles L.; Hanson, Richard J. (1995). Solving Least Squares Problems. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0898713560.
 ^ Gershenfeld, N. (1999). The Nature of Mathematical Modeling. New York: Cambridge University Press. pp. 205–208. ISBN 0521570956.
 ^ Boashash, B. (ed.), (2003) TimeFrequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003 ISBN ISBN 0080443354
 ^ Nikolić, D.; Muresan, R. C.; Feng, W.; Singer, W. (2012). "Scaled correlation analysis: a better way to compute a crosscorrelogram". European Journal of Neuroscience 35 (5): 742–762. doi:10.1111/j.14609568.2011.07987.x.
 ^ Mormann, Florian; Andrzejak, Ralph G.; Elger, Christian E.; Lehnertz, Klaus (2007). "Seizure prediction: the long and winding road". Brain 130 (2): 314–333. doi:10.1093/brain/awl241.
 ^ Land, Bruce; Elias, Damian. "Measuring the ‘Complexity’ of a time series".
 ^ Ropella, G. E. P.; Nag, D. A.; Hunt, C. A. (2003). "Similarity measures for automated comparison of in silico and in vitro experimental results". Engineering in Medicine and Biology Society 3: 2933–2936. doi:10.1109/IEMBS.2003.1280532.
Further reading
 Box, George; Jenkins, Gwilym (1976), Time Series Analysis: forecasting and control, rev. ed., Oakland, California: HoldenDay
 Cowpertwait P.S.P., Metcalfe A.V. (2009), Introductory Time Series with R, Springer.
 Durbin J., Koopman S.J. (2001), Time Series Analysis by State Space Methods, Oxford University Press.
 Gershenfeld, Neil (2000), The Nature of Mathematical Modeling, Cambridge University Press, ISBN 9780521570954, OCLC 174825352
 Hamilton, James (1994), Time Series Analysis, Princeton University Press, ISBN 0691042896
 Priestley, M. B. (1981), Spectral Analysis and Time Series, Academic Press. ISBN 9780125649018
 Shasha, D. (2004), High Performance Discovery in Time Series, Springer, ISBN 0387008578
 Shumway R. H., Stoffer (2011), Time Series Analysis and its Applications, Springer.
 Weigend A. S., Gershenfeld N. A. (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), AddisonWesley.
 Wiener, N. (1949), Extrapolation, Interpolation, and Smoothing of Stationary Time Series, MIT Press.
 Woodward, W. A., Gray, H. L. & Elliott, A. C. (2012), Applied Time Series Analysis, CRC Press.
External links
 Time series at Encyclopaedia of Mathematics.
 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.
 MATLAB Toolkit for Computation of Multiple Measures on Time Series Data Bases.
 A Matlab tutorial on power spectra, wavelet analysis, and coherence on website with many other tutorials.
 OpenNN: Open Neural Networks Library

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