NationMaster - Encyclopedia: Autoregressive moving average model

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. A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... George Edward Pelham Box, born 18 October 1919 in England, was one of the most influential statisticians of the 20th century and a pioneer in the area of design of experiments. ... Gwilym Meirion Jenkins (1933-82) was a British statistician. ... In statistics, signal processing, and econometrics, a time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals. ...

Given a time series of data X_t, the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the ARMA(p,q) model where p is the order of the autoregressive part and q is the order of the moving average part (as defined below).

1 Autoregressive model
- 1.1 Example: An AR(1)-process
- 1.2 Calculation of the AR parameters
  - 1.2.1 Derivation
2 Moving average model
3 Autoregressive moving average model
4 Note about the error terms
5 Specification in terms of lag operator
6 Fitting models
7 Generalizations
8 See also
9 References

Autoregressive model

The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is written

$X_t = c + sum_{i=1}^p varphi_i X_{t-i}+ varepsilon_t .,$

where $varphi_1, ldots, varphi_p$ are the parameters of the model, $c$ is a constant and $varepsilon_t$ is an error term (see below). The constant term is omitted by many authors for simplicity.

An autoregressive model is essentially an infinite impulse response filter with some additional interpretation placed on it. IIR (infinite impulse response) is a property of signal processing systems. ...

Some constraints are necessary on the values of the parameters of this model in order that the model remains stationary. For example, processes in the AR(1) model with |φ₁| > 1 are not stationary. 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. ...

Example: An AR(1)-process

An AR(1)-process is given by

$X_t = c + varphi X_{t-1}+varepsilon_t,,$

where $varepsilon_t$ is a white noise process with zero mean and variance $σ 2$ . (Note: The subscript on $varphi_1$ has been dropped.) The process is covariance-stationary if $|varphi|<1$ . If $varphi=1$ then $X t$ exhibits a unit root and can also be considered as a random walk, which is not covariance-stationary. Otherwise, the calculation of the expectation of $X t$ is straightforward. Assuming covariance-stationarity we get A process is said to be covariance-stationary (or weakly stationary), if , where denotes the autocovariance. ... A unit root is a concept from autoregressive models in econometrics. ... In mathematics, computer science, and physics, a random walk, sometimes called a drunkards walk, is a formalisation of the intuitive idea of taking successive steps, each in a random direction. ...

$mbox{E}(X_t)=mbox{E}(c)+varphimbox{E}(X_{t-1})+mbox{E}(varepsilon_t)Rightarrow mu=c+varphimu+0.$

thus:

$mu=frac{c}{1-varphi},$

where $μ$ is the mean. For c = 0, then the mean = 0 and the variance is found to be: In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

$textrm{var}(X_t)=E(X_t^2)-mu^2=frac{sigma^2}{1-varphi^2}.$

The autocovariance is given by Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ...

$B_n=E(X_{t+n}X_t)-mu^2=frac{sigma^2}{1-varphi^2},,varphi^{|n|}.$

It can be seen that the autocovariance function decays with a decay time of $tau=-1/ln(varphi)$ [to see this, write $B n = K φ | n |$ where $K$ is independent of $n$ . Then note that $φ | n | = e | n | lnφ$ and match this to the exponential decay law $e - n / τ$ ] The spectral density function is the Fourier transform of the autocovariance function. In discrete terms this will be the discrete-time Fourier transform: 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. ...

$Phi(omega)= frac{1}{sqrt{2pi}},sum_{n=-infty}^infty B_n e^{-iomega n} =frac{1}{sqrt{2pi}},left(frac{sigma^2}{1+varphi^2-2varphicos(omega)}right).$

This expression contains aliasing due to the discrete nature of the $X j$ , which is manifested as the cosine term in the denominator. If we assume that the sampling time ( $Δ t = 1$ ) is much smaller than the decay time ( $τ$ ), then we can use a continuum approximation to $B n$ : Properly sampled image of brick wall. ...

$B(t)approx frac{sigma^2}{1-varphi^2},,varphi^{|t|}$

which yields a Lorentzian profile for the spectral density: The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...

$Phi(omega)= =frac{1}{sqrt{2pi}},frac{sigma^2}{1-varphi^2},frac{gamma}{pi(gamma^2+omega^2)}$

where $γ = 1 / τ$ is the angular frequency associated with the decay time $τ$ .

An alternative expression for $X t$ can be derived by first substituting $c+varphi X_{t-2}+varepsilon_{t-1}$ for $X t - 1$ in the defining equation. Continuing this process N times yields

$X_t=csum_{k=0}^{N-1}varphi^k+varphi^NX_{t-N}+sum_{k=0}^{N-1}varphi^kvarepsilon_{t-k}.$

For N approaching infinity, $varphi^N$ will approach zero and:

$X_t=frac{c}{1-varphi}+sum_{k=0}^inftyvarphi^kvarepsilon_{t-k}$

It is seen that $X t$ is white noise convolved with the $varphi^k$ kernel plus the constant mean. By the central limit theorem, the $X t$ will be normally distributed as will any sample of $X t$ which is much longer than the decay time of the autocorrelation function. A central limit theorem is any of a set of weak-convergence results in probability theory. ...

Calculation of the AR parameters

The AR(p) model is given by the equation

$X_t = sum_{i=1}^p varphi_i X_{t-i}+ varepsilon_t.,$

It is based on parameters $varphi_i$ where i = 1, ..., p. Those parameters may be calculated using Yule-Walker equations:

$gamma_m = sum_{k=1}^p varphi_k gamma_{m-k} + sigma_varepsilon^2delta_m$

where m = 0, ... , p, yielding p + 1 equations. $γ m$ is the autocorrelation function of X, $sigma_varepsilon$ is the standard deviation of the input noise process, and δ_m is the Kronecker delta function. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

Because the last part of the equation is non-zero only if m = 0, the equation is usually solved by representing it as a matrix for m > 0, thus getting equation

$begin{bmatrix} gamma_1 gamma_2 gamma_3 vdots end{bmatrix} = begin{bmatrix} gamma_0 & gamma_{-1} & gamma_{-2} & dots gamma_1 & gamma_0 & gamma_{-1} & dots gamma_2 & gamma_{1} & gamma_{0} & dots dots & dots & dots & dots end{bmatrix} begin{bmatrix} varphi_{1} varphi_{2} varphi_{3} vdots end{bmatrix}$

solving all $varphi$ . For m = 0 have

$gamma_0 = sum_{k=1}^p varphi_k gamma_{-k} + sigma_varepsilon^2$

which allows us to solve $sigma_varepsilon^2$ .

Derivation

The equation defining the AR process is

$X_t = sum_{i=1}^p varphi_i,X_{t-i}+ varepsilon_t.,$

Multiplying both sides by X_t-m and taking expected value yields

$E[X_t X_{t-m}] = Eleft[sum_{i=1}^p varphi_i,X_{t-i} X_{t-m}right]+ E[varepsilon_t X_{t-m}].$

Now, $E [X t X t - m] = γ m$ by definition of the autocorrelation function. The values of the noise function are independent of each other, and X_t − m is independent of ε_t where m is greater than zero. For m ≠ 0, $E[varepsilon_t X_{t-m}] = 0$ . For m = 0,

$E[varepsilon_t X_{t}] = Eleft[varepsilon_t (sum_{i=1}^p varphi_i,X_{t-i}+ varepsilon_t)right] = sum_{i=1}^p varphi_i, E[varepsilon_t,X_{t-i}] + E[varepsilon_t^2] = 0 + sigma_varepsilon^2,$

Now we have

$gamma_m = Eleft[sum_{i=1}^p varphi_i,X_{t-i} X_{t-m}right] + sigma_varepsilon^2 delta_m.$

Furthermore,

$Eleft[sum_{i=1}^p varphi_i,X_{t-i} X_{t-m}right] = sum_{i=1}^p varphi_i,E[X_{t} X_{t-m+i}] = sum_{i=1}^p varphi_i,gamma_{m-i},$

which yields the Yule-Walker equations:

$gamma_m = sum_{i=1}^p varphi_i gamma_{m-i} + sigma_varepsilon^2 delta_m.$

Moving average model

The notation MA(q) refers to the moving average model of order q:

$X_t = varepsilon_t + sum_{i=1}^q theta_i varepsilon_{t-i},$

where the θ₁, ..., θ_q are the parameters of the model and the ε_t, ε_t-1,... are again, the error terms. The moving average model is essentially a finite impulse response filter with some additional interpretation placed on it. A finite impulse response (FIR) filter is a type of a digital filter. ...

Autoregressive moving average model

The notation ARMA(p, q) refers to the model with p autoregressive terms and q moving average terms. This model contains the AR(p) and MA(q) models,

$X_t = varepsilon_t + sum_{i=1}^p varphi_i X_{t-i} + sum_{i=1}^q theta_i varepsilon_{t-i}.,$

Note about the error terms

The error terms ε_t are generally assumed to be independent identically-distributed random variables (i.i.d.) sampled from a normal distribution with zero mean: ε_t ~ N(0,σ²) where σ² is the variance. These assumptions may be weakened but doing so will change the properties of the model. In particular, a change to the i.i.d. assumption would make a rather fundamental difference. In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ...

Specification in terms of lag operator

In some texts the models will be specified in terms of the lag operator L. In these terms then the AR(p) model is given by In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element. ...

$varepsilon_t = left(1 - sum_{i=1}^p varphi_i L^iright) X_t = varphi X_t,$

where φ represents polynomial

$varphi = 1 - sum_{i=1}^p varphi_i L^i.,$

The MA(q) model is given by

$X_t = left(1 + sum_{i=1}^q theta_i L^iright) varepsilon_t = theta varepsilon_t,$

where θ represents the polynomial

$theta= 1 + sum_{i=1}^q theta_i L^i.,$

Finally, the combined ARMA(p, q) model is given by

$left(1 - sum_{i=1}^p varphi_i L^iright) X_t = left(1 + sum_{i=1}^q theta_i L^iright) varepsilon_t,$

or more concisely,

$varphi X_t = theta varepsilon_t.,$

Fitting models

ARMA models in general can, after choosing p and q, be fitted by least squares regression to find the values of the parameters which minimize the error term. It is generally considered good practice to find the smallest values of p and q which provide an acceptable fit to the data. For a pure AR model then the Yule-Walker equations may be used to provide a fit. Least squares or ordinary least squares (OLS) is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a best fit). It attempts to minimize the sum of the squares of the ordinate differences (called residuals) between points...

Generalizations

The dependence of X_t on past values and the error terms ε_t is assumed to be linear unless specified otherwise. If the dependence is nonlinear, the model is specifically called a nonlinear moving average (NMA), nonlinear autoregressive (NAR), or nonlinear autoregressive moving average (NARMA) model.

Autoregressive moving average models can be generalized in other ways. See also autoregressive conditional heteroskedasticity (ARCH) models and autoregressive integrated moving average (ARIMA) models. If multiple time series are to be fitted then a vectored ARIMA (or VARIMA) model may be fitted. If the time-series in question exhibits long memory then fractional ARIMA (FARIMA, sometimes called ARFIMA) modelling is appropriate. If the data is thought to contain seasonal effects, it may be modeled by a SARIMA (seasonal ARIMA) model. In econometrics, an autoregressive conditional heteroskedasticity (ARCH) model considers the variance of the current error term to be a function of the variances of the previous time periods error terms. ... In statistics, an autoregressive integrated moving average (ARIMA) model is a generalisation of an autoregressive moving average or (ARMA) model. ...

Another generalization is the multiscale autoregressive (MAR) model. A MAR model is indexed by the nodes of a tree, whereas a standard (discrete time) autoregressive model is indexed by integers. See multiscale autoregressive model for a list of references.

References

George Box and Gwilym M. Jenkins. Time Series Analysis: Forecasting and Control, second edition. Oakland, CA: Holden-Day, 1976.
Mills, Terence C. Time Series Techniques for Economists. Cambridge University Press, 1990.
Percival, Donald B. and Andrew T. Walden. Spectral Analysis for Physical Applications. Cambridge University Press, 1993.

Categories: Noise | Time series analysis George Edward Pelham Box, born 18 October 1919 in England, was one of the most influential statisticians of the 20th century and a pioneer in the area of design of experiments. ... Gwilym Meirion Jenkins (1933 â€“ July 10, 1982) was a British statistician and systems engineer . ...

Results from FactBites:

Autoregressive moving average model - Wikipedia, the free encyclopedia (815 words)
	are the parameters of the model and the ε
	The moving average model is essentially a finite impulse response filter with some additional interpretation placed on it.
	ARMA models in general can, after choosing p and q, be fitted by least squares regression to find the values of the parameters which minimise the error term.

Autoregressive conditional heteroskedasticity - Wikipedia, the free encyclopedia (274 words)
	In econometrics, an autoregressive conditional heteroskedasticity (ARCH, Engle (1982)) model considers the variance of the current error term to be a function of the variances of the previous time period's error terms.
	If an autoregressive moving average model(ARMA model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH, Bollerslev(1986)) model.
	IGARCH or Integrated Generalized Autoregressive Conditional Heteroskedasticity is a restricted version of the GARCH model, where the sum of the persistent parameters sum up to one.

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