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. This article is about the field of statistics. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... In statistics, a data point is a single typed measurement. ... Prediction of future events is an ancient human wish. ... An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of logical and quantitative relationships between them. ... Econometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. ... For other uses, see Stock (disambiguation). ...

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.) Spatial data analysis is a quantitative approach to geographical analysis that applies rigorous statistical techniques to geographic data, to ultimately analyze why phenomena occurs in particular places, and what dynamic factors are key. ...

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. Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... A spectrum analyzer is a device used to examine at the spectral composition of some electrical, acoustic or optical waveform. ... In mathematics, wavelets, wavelet analysis, and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). ... In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar...

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

A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ... A spectrum analyzer is a device used to examine at the spectral composition of some electrical, acoustic or optical waveform. ...

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).

Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems. ...

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". In econometrics, the Box-Jenkins methodology, named after the statisticians George Box and Gwilym Jenkins, applies autoregressive integrated moving average ARIMA models to find the best fit of a time series to past values of this time series, in order to make forecasts. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... The term moving average is used in different contexts. ... In statistics, autoregressive moving average (ARMA) models, sometimes called Box-Jenkins models after George Box and G. M. Jenkins, are typically applied to time series data. ... In statistics, an autoregressive integrated moving average (ARIMA) model is a generalisation of an autoregressive moving average or (ARMA) model. ...

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. For other uses, see Chaos Theory (disambiguation). ...

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 statistics, a sequence or a vector of random variables is heteroskedastic if the random variables in the sequence or vector may have different variances. ... In econometrics, an autoregressive conditional heteroscedasticity (ARCH, Engle (1982)) model considers the variance of the current error term to be a function of the variances of the previous time periods error terms. ...

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: In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

Y = {Y_t: t ∈ 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 the mathematical sciences, a stationary process (or strict(ly) stationary process) is a stochastic process in which the probability density function of some random variable X does not change over time or position. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... Stationary can mean: Look up stationary in Wiktionary, the free dictionary. ...

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

where the term ε_t is the source of randomness and is called white noise. It is assumed to have the following characteristics: Calculated spectrum of a generated approximation of white noise White noise is a random signal (or process) with a flat power spectral density. ...

1.

2.

3.

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: The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

.

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

A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ... In applied mathematics and physics, the spectral density is a general concept applied to a signal which may have any physical dimensions or none at all. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... An FIR filter In electronics,nirali a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ... Principal component analysis (PCA) is a technique used to reduce multidimensional data sets to lower dimensions for analysis. ... In statistics and signal processing, the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. ... An artificial neural network (ANN), often just called a neural network (NN), is a mathematical model or computational model based on biological neural networks. ... A time-frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. ... In mathematics and signal processing, the continuous wavelet transform (CWT) of a function is a wavelet transform defined by where represents translation, represents scale and is the mother wavelet. ... The short-time Fourier transform (STFT), or short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of a signal as it changes over time. ... Comparison of wave, wavelet, chirp, and chirplet In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. ... The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and it can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an intermediate domain between time and frequency. ... For other uses, see Chaos Theory (disambiguation). ... In chaos theory the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points. ... In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for a given moment in time, the times at which a phase space trajectory visits roughly the same area in the phase space. ... Recurrence quantification analysis (RQA) is a system for analyzing nonlinear systems that works by looking at the pattern and number of recurrences present in a data signal. ... In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. ...

See also

In statistics, analysis of rhythmic variance (ANORVA) is a new simple method for detecting rhythms in biological time series, published by Peter Celec (Biol Res. ... An anomaly time series is the time series of deviations of a quantity from some mean. ... A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ... In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. ... Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples. ... Longitudinal studies form a class of research methods that involve observations of the same items over a longer time. ... A model in macroeconomics is designed to simulate the operation of a national or international economy in terms of factors including the total amount of goods and services produced, total income earned, the level of employment of productive resources, and the general behavior of prices. ... A moving average, in finance and especially in technical analysis, is one of a family of similar statistical techniques used to analyze time series data. ... In time series modeling, a nonlinear autoregressive exogenous model (NARX) is a nonlinear autoregressive model which has exogenous inputs. ... In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter. ... Seasonal adjustment is a statistical method for time series analysis. ... System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. ... A time series database server (TSDS) is a software system that is optimized for handling a time series. ... A series of measurements of a process may be treated as a time series, and then trend estimation is the application of statistical techniques to make and justify statements about trends in the data. ...

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