# Autoregressive model

autoregressiveautoregressionAutoregressive processautoregressive (AR) modelstochastic difference equationARAR(1)autoregressive modelsAR modelAR noise
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc.wikipedia
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### Autoregressive–moving-average model

ARMAautoregressive moving average modelautoregressive moving average
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).

### Time series

time series analysistime-seriestime-series analysis
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.
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).

### Autoregressive integrated moving average

ARIMAAutoregressive integrated moving average modelAutoregressive integrated moving average (ARIMA)
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.
Non-seasonal ARIMA models are generally denoted ARIMA(p,d,q) where parameters p, d, and q are non-negative integers, p is the order (number of time lags) of the autoregressive model, d is the degree of differencing (the number of times the data have had past values subtracted), and q is the order of the moving-average model.

### Autocorrelation

autocorrelation functionserial correlationautocorrelated
The autocorrelation function of an AR(p) process can be expressed as :
Unit root processes, trend stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation.

### Linear prediction

linearlySignal predictioncoefficient
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.
The above equations are called the normal equations or Yule-Walker equations.

### Unit root

difference stationarynon-stationary
Contrary to the moving-average model, the autoregressive model is not always stationary as it may contain a unit root.
Consider a discrete-time stochastic process, and suppose that it can be written as an autoregressive process of order p:

### Stationary process

stationarynon-stationarystationarity
Contrary to the moving-average model, the autoregressive model is not always stationary as it may contain a unit root.
Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model.

### Lag operator

backshift operatorlagLag or backshift operator
This can be equivalently written using the backshift operator B as

### Vector autoregression

VARvector autoregressive modelstructural VAR estimation
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.
VAR models generalize the univariate autoregressive model (AR model) by allowing for more than one evolving variable.

### Spectral density

frequency spectrumpower spectrumspectrum
The spectral density function is the Fourier transform of the autocovariance function.
For example, a common parametric technique involves fitting the observations to an autoregressive model.

### Colors of noise

blue noiseBlack noisecolored noise
AR noise or "autoregressive noise" is such a model, and generates simple examples of the above noise types, and more.

### Autocovariance

autocovariance functionautocovariance matrixmean and autocovariance
The autocovariance is given by

### Gilbert Walker

Gilbert Thomas WalkerSir Gilbert WalkerSir Gilbert Thomas Walker
The Yule–Walker equations, named for Udny Yule and Gilbert Walker, are the following set of equations.
Walker developed Blanford's idea with quantitative rigour and came up with correlation measures (with a lag) and regression equations (in time-series terminology, autoregression).

### Infinite impulse response

IIRIIR filterinfinite-impulse-response
An autoregressive model can thus be viewed as the output of an all-pole infinite impulse response filter whose input is white noise.

### Moving-average model

Moving average modelmoving averagemoving average process
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.
Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure.

### Ornstein–Uhlenbeck process

Ornstein-Uhlenbeck processOrnstein-Uhlenbeckmean reversion
The AR(1) model is the discrete time analogy of the continuous Ornstein-Uhlenbeck process.
The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.

### Linear difference equation

linear recurrencecharacteristic equationlinear recurrence relation
In econometric applications, linear difference equations are modeled with stochastic terms in the form of autoregressive (AR) models and in models such as vector autoregression (VAR) and autoregressive moving average (ARMA) models that combine AR with other features.

### Predictive analytics

predictiveCARTpredictive analysis
Two commonly used forms of these models are autoregressive models (AR) and moving-average (MA) models.

### Levinson recursion

Levinson-DurbinLevinson–Durbin algorithmLevinson algorithm

### Statistics

statisticalstatistical analysisstatistician
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.

### Econometrics

econometriceconometricianeconometric analysis
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.

### Signal processing

signal analysissignalsignal processor
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.

### Stochastic process

stochastic processesstochasticrandom process
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.

### Natural science

natural sciencesnaturalnatural scientist
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.

### Economics

economiceconomisteconomic theory
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.