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| [[File:Random-data-plus-trend-r2.png|thumb|250px|Time series: random data plus trend, with best-fit line and different applied filters]]
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| A '''time series''' is a sequence of [[data point]]s, 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|Nile River]] at [[Aswan]]. Time series are very frequently plotted via [[line chart]]s. Time series are used in [[statistics]], [[signal processing]], [[pattern recognition]], [[econometrics]], [[mathematical finance]], [[weather forecasting]], [[earthquake prediction]], [[electroencephalography]], [[control engineering]], [[astronomy]], and [[communications engineering]] .
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| '''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 (abstract)|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 time series at different points in time.<ref>{{cite web|last=Imdadullah|title=Time Series Analysis|url=http://itfeature.com/time-series-analysis-and-forecasting/time-series-analysis-forecasting|work=Basic Statistics and Data Analysis|publisher=itfeature.com|accessdate=2 January 2014}}</ref>
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| 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 one-way 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]].)
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| Time series analysis can be applied to [[real number|real-valued]], continuous data, [[:wikt:discrete|discrete]] [[Data type#Numeric types|numeric]] data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the [[English language]].<ref>{{cite book |last=Lin |first=Jessica |last2=Keogh |first2=Eamonn |last3=Lonardi |first3=Stefano |last4=Chiu |first4=Bill |chapter=A symbolic representation of time series, with implications for streaming algorithms |title=Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery |year=2003 |location=New York |publisher=ACM Press |doi=10.1145/882082.882086 }}</ref>).
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| ==Methods for time series analyses==
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| Methods for time series analyses may be divided into two classes: [[frequency-domain]] methods and [[time-domain]] methods. The former include [[frequency spectrum#Spectrum analysis|spectral analysis]] and recently [[wavelet analysis]]; the latter include [[auto-correlation]] and [[cross-correlation]] analysis. In time domain correlation analyses can be made in a filter-like manner using [[scaled correlation]], thereby mitigating the need to operate in frequency domain.
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| Additionally, time series analysis techniques may be divided into [[Parametric estimation|parametric]] and [[Non-parametric statistics|non-parametric]] methods. The [[Parametric estimation|parametric approaches]] assume that the underlying [[stationary process|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, [[Non-parametric statistics|non-parametric approaches]] explicitly estimate the [[covariance]] or the [[spectrum]] of the process without assuming that the process has any particular structure.
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| Methods of time series analysis may also be divided into [[Linear regression|linear]] and [[Nonlinear regression|non-linear]], and [[Univariate analysis|univariate]] and [[Multivariate analysis|multivariate]].
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| ==Analysis==
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| There are several types of motivation and data analysis available for time series which are appropriate for different purposes. | |
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| ===Motivation===
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| 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 [[cluster analysis|clustering]], [[Statistical classification|classification]], query by content, [[anomaly detection]] as well as [[forecasting]].
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| ===Exploratory analysis===
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| [[File:Tuberculosis incidence US 1953-2009.png|thumb|Tuberculosis incidence US 1953-2009]]
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| 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.
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| Other techniques include:
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| * [[Autocorrelation]] analysis to examine [[serial dependence]]
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| * [[frequency spectrum#Spectrum analysis|Spectral analysis]] to examine cyclic behaviour which need not be related to [[seasonality]]. For example, sun spot activity varies over 11 year cycles.<ref>{{cite book |last=Bloomfield |first=P. |year=1976 |title=Fourier analysis of time series: An introduction |location=New York |publisher=Wiley |isbn=0471082562 }}</ref><ref>{{cite book |last=Shumway |first=R. H. |year=1988 |title=Applied statistical time series analysis |location=Englewood Cliffs, NJ |publisher=Prentice Hall |isbn=0130415006 }}</ref> Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity.
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| * Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see [[trend estimation]] and [[decomposition of time series]]
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| ===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-specific time-periods in the future
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| * 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).
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| * Forecasting on time series is usually done using automated statistical software packages and programming languages, such as [[R (programming language)|R]], [[S (programming language)|S]], [[SAS (software)|SAS]], [[SPSS]], [[Minitab]], [[Pandas (software)|Pandas (Python)]] and many others.
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| ===Classification===
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| * Assigning time series pattern to a specific category, for example identify a word based on series of hand movements in [[sign language]]
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| See main article: [[Statistical classification]]
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| ===Regression analysis(method of prediction]===
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| * 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).<ref>{{cite book |last=Lawson |first=Charles L. |last2=Hanson |first2=Richard J. |year=1995 |title=Solving Least Squares Problems |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |isbn=0898713560 }}</ref>
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| * [[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]]
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| ===Signal estimation===
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| * 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]]
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| ===Segmentation===
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| * Splitting a time-series into a sequence of segments. It is often the case that a time-series 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 [[time-series segmentation]], the goal is to identify the segment boundary points in the time-series, and to characterize the dynamical properties associated with each segment. One can approach this problem using [[Change detection|change-point detection]], or by modeling the time-series as a more sophisticated system, such as a [[Markov jump linear system]].
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| ==Models==
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| 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 model|moving average]]'' (MA) models. These three classes depend linearly on previous data points.<ref name="linear time series">{{cite book |authorlink=Neil Gershenfeld |last=Gershenfeld |first=N. |year=1999 |title=The Nature of Mathematical Modeling |location=New York |publisher=Cambridge University Press |pages=205–208 |isbn=0521570956 }}</ref> 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 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", as in VAR for [[vector autoregression]]. 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".
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| Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a [[chaos theory|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, as for example in [[nonlinear autoregressive exogenous model]]s.
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| Among other types of non-linear 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 time-varying process, as in a [[doubly stochastic model]].
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| 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. See also [[Markov switching multifractal]] (MSMF) techniques for modeling volatility evolution.
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| 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.
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| ===Notation===
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| A number of different notations are in use for time-series analysis. A common notation specifying a time series ''X'' that is indexed by the [[natural number]]s is written
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| :''X'' = {''X''<sub>1</sub>, ''X''<sub>2</sub>, ...}.
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| Another common notation is
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| :''Y'' = {''Y<sub>''t''</sub>'': ''t'' ∈ ''T''},
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| where ''T'' is the [[index set]].
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| ===Conditions===
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| There are two sets of conditions under which much of the theory is built:
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| * [[Stationary process]]
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| * [[Ergodic process]]
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| However, ideas of stationarity must be expanded to consider two important ideas: [[strict stationarity]] and [[Stationary process#Weaker forms of stationarity|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.
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| In addition, time-series analysis can be applied where the series are [[Cyclostationary process|seasonally stationary]] or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in [[time-frequency analysis]] which makes use of a [[time–frequency representation]] of a time-series or signal.<ref>Boashash, B. (ed.), (2003) ''Time-Frequency Signal Analysis and Processing: A Comprehensive Reference'', Elsevier Science, Oxford, 2003 ISBN ISBN 0-08-044335-4</ref>
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| ===Models===
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| {{Main|Autoregressive model}}
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| The general representation of an autoregressive model, well known as AR(''p''), is
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| : <math> Y_t =\alpha_0+\alpha_1 Y_{t-1}+\alpha_2 Y_{t-2}+\cdots+\alpha_p Y_{t-p}+\varepsilon_t\, </math> | |
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| where the term ε<sub>''t''</sub> is the source of randomness and is called [[white noise]]. It is assumed to have the following characteristics:
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| :*<math> E[\varepsilon_t]=0 \, ,</math>
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| :*<math> E[\varepsilon^2_t]=\sigma^2 \, ,</math>
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| :*<math> E[\varepsilon_t\varepsilon_s]=0 \quad \text{ for all } t\not=s \, . </math>
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| With these assumptions, the process is specified up to second-order moments and, subject to conditions on the coefficients, may be [[Stationary process#Weaker forms of stationarity|second-order stationary]].
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| 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.
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| {{Prose|section|date=February 2012}}
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| Tools for investigating time-series data include:
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| * Consideration of the [[autocorrelation|autocorrelation function]] and the [[Spectral density|spectral density function]] (also [[cross-correlation function]]s and cross-spectral density functions)
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| * [[Scaled correlation|Scaled]] cross- and auto-correlation functions to remove contributions of slow components<ref name="Nikolicetal">{{cite journal |last=Nikolić |first=D. |last2=Muresan |first2=R. C. |last3=Feng |first3=W. |last4=Singer |first4=W. |year=2012 |title=Scaled correlation analysis: a better way to compute a cross-correlogram |journal=European Journal of Neuroscience |volume=35 |issue=5 |pages=742–762 |doi=10.1111/j.1460-9568.2011.07987.x }}</ref>
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| * Performing a [[Fourier transform]] to investigate the series in the [[frequency domain]]
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| * Use of a [[digital filter|filter]] to remove unwanted [[noise (physics)|noise]]
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| * [[Principal component analysis]] (or [[empirical orthogonal function]] analysis)
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| * [[Singular spectrum analysis]]
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| * "Structural" models:
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| **General [[State Space Model]]s
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| **Unobserved Components Models
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| * [[Machine Learning]]
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| ** [[Artificial neural network]]s
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| ** [[Support Vector Machine]]
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| ** [[Fuzzy Logic]]
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| * [[Hidden Markov model]]
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| * [[Control chart]]
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| ** [[Shewhart individuals control chart]]
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| ** [[CUSUM]] chart
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| ** [[EWMA chart]]
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| * [[Detrended fluctuation analysis]]
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| * [[Dynamic time warping]]
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| * [[Dynamic Bayesian network]]
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| * [[Time-frequency representation|Time-frequency analysis techniques:]]
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| ** [[Fast Fourier Transform]]
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| ** [[Continuous wavelet transform]]
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| ** [[Short-time Fourier transform]]
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| ** [[Chirplet transform]]
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| ** [[Fractional Fourier transform]]
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| * [[Chaos theory|Chaotic analysis]]
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| ** [[Correlation dimension]]
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| ** [[Recurrence plot]]s
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| ** [[Recurrence quantification analysis]]
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| ** [[Lyapunov exponent]]s
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| ** [[Entropy encoding]]
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| ===Measures===
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| Time series metrics or [[Features (pattern recognition)|features]] that can be used for time series [[Classification (machine learning)|classification]] or [[Regression analysis|regression]] analysis:<ref>{{cite journal |last=Mormann |first=Florian |last2=Andrzejak |first2=Ralph G. |last3=Elger |first3=Christian E. |last4=Lehnertz |first4=Klaus |title=Seizure prediction: the long and winding road |journal=[[Brain (journal)|Brain]] |year=2007 |volume=130 |issue=2 |pages=314–333 |doi=10.1093/brain/awl241 }}</ref>
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| *'''Univariate linear measures'''
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| **[[Moment (mathematics)]]
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| **[[Spectral band power]]
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| **[[Spectral edge frequency]]
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| **Accumulated [[Energy (signal processing)]]
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| **Characteristics of the [[autocorrelation]] function
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| **[[Hjorth parameters]]
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| **[[Fast Fourier transform|FFT]] parameters
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| **[[Autoregressive model]] parameters
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| **[[Mann–Kendall test]]
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| *'''Univariate non-linear measures'''
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| **Measures based on the [[correlation]] sum
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| **[[Correlation dimension]]
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| **[[Correlation integral]]
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| **[[Correlation density]]
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| **[[Correlation entropy]]
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| **[[Approximate entropy]]<ref>{{cite web |last=Land |first=Bruce |last2=Elias |first2=Damian |title=Measuring the ‘Complexity’ of a time series |url=http://www.nbb.cornell.edu/neurobio/land/PROJECTS/Complexity/ }}</ref>
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| **[[Sample entropy]]
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| **[[Fourier entropy]]
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| **[[Wavelet entropy]]
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| **[[Rényi entropy]]
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| **Higher-order methods
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| **[[Marginal predictability]]
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| **[[Dynamical similarity]] index
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| **[[State space]] dissimilarity measures
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| **[[Lyapunov exponent]]
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| **Permutation methods
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| **[[Local flow]]
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| *'''Other univariate measures'''
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| **[[Algorithmic complexity]]
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| **[[Kolmogorov complexity]] estimates
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| **[[Hidden Markov Model]] states
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| **Surrogate time series and surrogate correction
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| **Loss of recurrence (degree of non-stationarity)
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| *'''Bivariate linear measures'''
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| **Maximum linear [[cross-correlation]]
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| **Linear [[Coherence (signal processing)]]
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| *'''Bivariate non-linear measures'''
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| **Non-linear interdependence
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| **Dynamical Entrainment (physics)
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| **Measures for [[Phase synchronization]]
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| *'''Similarity measures''':<ref>{{cite journal |last=Ropella |first=G. E. P. |last2=Nag |first2=D. A. |last3=Hunt |first3=C. A. |title=Similarity measures for automated comparison of in silico and in vitro experimental results |journal=Engineering in Medicine and Biology Society |year=2003 |volume=3 |issue= |pages=2933–2936 |doi=10.1109/IEMBS.2003.1280532 }}</ref>
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| **[[Dynamic Time Warping]]
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| **[[Hidden Markov Models]]
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| **[[Edit distance]]
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| **[[Total correlation]]
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| **[[Newey–West estimator]]
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| **[[Prais-Winsten transformation]]
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| **Data as Vectors in a Metrizable Space
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| ***[[Minkowski distance]]
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| ***[[Mahalanobis distance]]
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| **Data as Time Series with Envelopes
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| ***Global [[Standard Deviation]]
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| ***Local [[Standard Deviation]]
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| ***Windowed [[Standard Deviation]]
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| **Data Interpreted as Stochastic Series
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| ***[[Pearson product-moment correlation coefficient]]
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| ***[[Spearman's rank correlation coefficient]]
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| **Data Interpreted as a [[Probability Distribution]] Function
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| ***[[Kolmogorov–Smirnov test]]
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| ***[[Cramér–von Mises criterion]]
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| ==See also==
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| {{Columns-list|2|
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| * [[Anomaly time series]]
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| * [[Decomposition of time series]]
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| * [[Detrended fluctuation analysis]]
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| * [[Digital signal processing]]
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| * [[Distributed lag]]
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| * [[Estimation theory]]
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| * [[Forecasting]]
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| * [[Hurst exponent]]
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| * [[Monte Carlo method]]
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| * [[Random walk]]
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| * [[Scaled correlation]]
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| * [[Seasonal adjustment]]
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| * [[Sequence analysis]]
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| * [[Signal processing]]
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| * [[Stringology]]
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| * [[Trend estimation]]
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| * [[Unevenly spaced time series]]
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| }}
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| ==References==
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| {{Reflist|2}}
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| ==Further reading==
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| *{{Citation
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| | authorlink = George E. P. Box
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| | last1 = Box | first1 = George
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| | last2 = Jenkins | first2 = Gwilym
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| | title = Time Series Analysis: forecasting and control, rev. ed.
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| | publisher = Holden-Day
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| | location = Oakland, California
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| | year = 1976
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| }}
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| *Cowpertwait P.S.P., Metcalfe A.V. (2009), ''Introductory Time Series with R'', [[Springer Science+Business Media|Springer]].
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| *[[James Durbin|Durbin J.]], Koopman S.J. (2001), ''Time Series Analysis by State Space Methods'', [[Oxford University Press]].
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| *{{Citation
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| | last = Gershenfeld | first = Neil
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| | year = 2000
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| | title = The Nature of Mathematical Modeling
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| | isbn = 978-0-521-57095-4
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| | publisher = [[Cambridge University Press]]
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| | oclc = 174825352
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| }}
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| *{{Citation
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| | last = Hamilton | first = James
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| | year = 1994
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| | title = Time Series Analysis
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| | isbn = 0-691-04289-6
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| | publisher = [[Princeton University Press]]
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| }}
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| *[[Maurice Priestley|Priestley, M. B.]] (1981), ''Spectral Analysis and Time Series'', [[Academic Press]]. ISBN 978-0-12-564901-8
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| *{{Citation | last = Shasha | first = D. | title = High Performance Discovery in Time Series | publisher = [[Springer Science+Business Media|Springer]] | year = 2004 | isbn = 0-387-00857-8 }}
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| *Shumway R. H., Stoffer (2011), ''Time Series Analysis and its Applications'', Springer.
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| *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), [[Addison-Wesley]].
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| *[[Norbert Wiener|Wiener, N.]] (1949), ''Extrapolation, Interpolation, and Smoothing of Stationary Time Series'', [[MIT Press]].
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| *Woodward, W. A., Gray, H. L. & Elliott, A. C. (2012), ''Applied Time Series Analysis'', [[CRC Press]].
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Time_series ''Time series''] at Encyclopaedia of Mathematics.
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| * [http://statistik.mathematik.uni-wuerzburg.de/timeseries/ A First Course on Time Series Analysis] — An open source book on time series analysis with [[SAS (software)|SAS]].
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| * [http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm Introduction to Time series Analysis (Engineering Statistics Handbook)] — A practical guide to Time series analysis.
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| * [http://www.jstatsoft.org/v33/i05/paper MATLAB Toolkit for Computation of Multiple Measures on Time Series Data Bases].
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| * [http://www.nbtwiki.net/doku.php?id=tutorial:power_spectra_wavelet_analysis_and_coherence A Matlab tutorial on power spectra, wavelet analysis, and coherence] on website with many other tutorials.
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| * [http://www.intelnics.com/opennn OpenNN: Open Neural Networks Library]
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| {{Statistics}}
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| {{Portal bar|Statistics}}
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| {{DEFAULTSORT:Time Series}}
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| [[Category:Time series analysis| ]]
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