Модель авторегрессии – скользящего среднего

В статистическом анализе временных рядов , авторегрессия-скользящее средняя ( ARMA ) модели обеспечивают экономное описание (слабо) стационарного случайного процесс в терминах двух полиномов, один для авторегрессии (AR) , а второй для скользящего среднего ( MA). Общая модель ARMA была описана в 1951 г. диссертации Питера Уиттла , тестирования гипотез в анализе временных рядов , и она стала популярной в 1970 книги Джорджа EP Box и Гвилим Дженкинс .

Учитывая временной ряд данных X _t , модель ARMA является инструментом для понимания и, возможно, прогнозирования будущих значений в этом ряду. Часть AR включает регрессию переменной по ее собственным запаздывающим (т. Е. Прошлым) значениям. Часть MA включает моделирование члена ошибки как линейной комбинации членов ошибки, возникающих одновременно и в разное время в прошлом. Модель обычно называется моделью ARMA ( p , q ), где p - это порядок части AR, а q - порядок части MA (как определено ниже).

Модели ARMA можно оценить с помощью метода Бокса – Дженкинса .

Модель авторегрессии [ править ]

Обозначение AR ( p ) относится к авторегрессионной модели порядка p . Модель AR ( p ) записывается

X_{t}=c+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}.\,

где - параметры , - постоянная, а случайная величина - белый шум . $\varphi _{1},\ldots ,\varphi _{p}$ $c$ $\varepsilon _{t}$

Необходимы некоторые ограничения на значения параметров, чтобы модель оставалась неподвижной . Например, процессы в модели AR (1) с не стационарны. $|\varphi _{1}|\geq 1$

Модель скользящего среднего [ править ]

Обозначение MA ( q ) относится к модели скользящего среднего порядка q :

X_{t}=\mu +\varepsilon _{t}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}\,

где θ ₁ , ..., θ _д являются параметрами модели, μ является ожидание (часто принимают равным 0), и , , ... снова, белый шум , векторы ошибок. $X_{t}$ $\varepsilon _{t}$ $\varepsilon _{t-1}$

Модель ARMA [ править ]

Обозначение ARMA ( p , q ) относится к модели с p членами авторегрессии и q членами скользящего среднего. Эта модель содержит модели AR ( p ) и MA ( q ),

X_{t}=c+\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.\,

Общая модель ARMA была описана в диссертации Питера Уиттла 1951 года , который использовал математический анализ ( ряд Лорана и анализ Фурье ) и статистический вывод. ^[1]^[2] Модели ARMA были популяризированы в 1970 году в книге Джорджа Е.П. Бокса и Дженкинса, которые изложили итерационный метод ( Бокса – Дженкинса ) для их выбора и оценки. Этот метод был полезен для полиномов низкого порядка (степени три или меньше). ^[3]

Модель ARMA - это, по сути, фильтр с бесконечной импульсной характеристикой, применяемый к белому шуму с некоторой дополнительной интерпретацией.

Примечание об условиях ошибки [ править ]

Члены ошибки обычно считаются независимыми одинаково распределенными случайными величинами (iid), выбранными из нормального распределения с нулевым средним: ~ N (0, σ ² ), где σ ² - дисперсия. Эти предположения могут быть ослаблены, но это изменит свойства модели. В частности, изменение предположения iid имело бы довольно фундаментальное значение. $\varepsilon _{t}$ $\varepsilon _{t}$

Спецификация в терминах оператора задержки [ править ]

В некоторых текстах модель будет определена в терминах оператора задержки L . В этих терминах модель AR ( p ) задается следующим образом:

\varepsilon _{t}=\left(1-\sum _{i=1}^{p}\varphi _{i}L^{i}\right)X_{t}=\varphi (L)X_{t}\,

где представляет собой многочлен $\varphi$

\varphi (L)=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^{i}\right)\varepsilon _{t}=\theta (L)\varepsilon _{t},\,

where θ represents the polynomial

\theta (L)=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^{i}\right)X_{t}=\left(1+\sum _{i=1}^{q}\theta _{i}L^{i}\right)\varepsilon _{t}\,,

or more concisely,

\varphi (L)X_{t}=\theta (L)\varepsilon _{t}\,

or

{\frac {\varphi (L)}{\theta (L)}}X_{t}=\varepsilon _{t}\,.

Alternative notation[edit]

Some authors, including Box, Jenkins & Reinsel use a different convention for the autoregression coefficients.^[4] This allows all the polynomials involving the lag operator to appear in a similar form throughout. Thus the ARMA model would be written as

\left(1-\sum _{i=1}^{p}\phi _{i}L^{i}\right)X_{t}=\left(1+\sum _{i=1}^{q}\theta _{i}L^{i}\right)\varepsilon _{t}\,.

Moreover, starting summations from $i=0$ and setting $\phi _{0}=-1$ and $\theta _{0}=1$ , then we get an even more elegant formulation: $-\sum _{i=0}^{p}\phi _{i}L^{i}\;X_{t}=\sum _{i=0}^{q}\theta _{i}L^{i}\;\varepsilon _{t}\,.$

Fitting models[edit]

Choosing p and q[edit]

Finding appropriate values of p and q in the ARMA(p,q) model can be facilitated by plotting the partial autocorrelation functions for an estimate of p, and likewise using the autocorrelation functions for an estimate of q. Extended autocorrelation functions (EACF) can be used to simultaneously determine p and q.^[5] Further information can be gleaned by considering the same functions for the residuals of a model fitted with an initial selection of p and q.

Brockwell & Davis recommend using Akaike information criterion (AIC) for finding p and q.^[6] Another possible choice for order determining is the BIC criterion.

Estimating coefficients[edit]

This section needs expansion. You can help by adding to it. (March 2017)

ARMA models in general can be, after choosing p and q, 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 the Yule-Walker equations may be used to provide a fit.

Implementations in statistics packages[edit]

In R, the arima function (in standard package stats) is documented in ARIMA Modelling of Time Series. Extension packages contain related and extended functionality, e.g., the tseries package includes an arma function, documented in "Fit ARMA Models to Time Series"; the fracdiff package contains fracdiff() for fractionally integrated ARMA processes; and the forecast package includes auto.arima for selecting a parsimonious set of p,q. The CRAN task view on Time Series contains links to most of these.
Mathematica has a complete library of time series functions including ARMA.^[7]
MATLAB includes functions such as arma and ar to estimate AR, ARX (autoregressive exogenous), and ARMAX models. See System Identification Toolbox and Econometrics Toolbox for more information.
Julia has some community driven packages that implement fitting with an ARMA model such as arma.jl.
Statsmodels Python module includes many models and functions for time series analysis, including ARMA. Formerly part of Scikit-learn it is now stand-alone and integrates well with Pandas. See here for more details.
PyFlux has a Python-based implementation of ARIMAX models, including Bayesian ARIMAX models.
IMSL Numerical Libraries are libraries of numerical analysis functionality including ARMA and ARIMA procedures implemented in standard programming languages like C, Java, C# .NET, and Fortran.
gretl can also estimate ARMA model, see here where it's mentioned.
GNU Octave can estimate AR models using functions from the extra package octave-forge.
Stata includes the function arima which can estimate ARMA and ARIMA models. See here for more details.
SuanShu is a Java library of numerical methods, including comprehensive statistics packages, in which univariate/multivariate ARMA, ARIMA, ARMAX, etc. models are implemented in an object-oriented approach. These implementations are documented in "SuanShu, a Java numerical and statistical library".
SAS has an econometric package, ETS, that estimates ARIMA models. See here for more details.

Applications[edit]

ARMA is appropriate when a system is a function of a series of unobserved shocks (the MA or moving average part) as well as its own behavior. For example, stock prices may be shocked by fundamental information as well as exhibiting technical trending and mean-reversion effects due to market participants.^{[citation needed]}

Generalizations[edit]

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 vector ARIMA (or VARIMA) model may be fitted. If the time-series in question exhibits long memory then fractional ARIMA (FARIMA, sometimes called ARFIMA) modelling may be appropriate: see Autoregressive fractionally integrated moving average. If the data is thought to contain seasonal effects, it may be modeled by a SARIMA (seasonal ARIMA) or a periodic 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.

Note that the ARMA model is a univariate model. Extensions for the multivariate case are the vector autoregression (VAR) and Vector Autoregression Moving-Average (VARMA).

Autoregressive–moving-average model with exogenous inputs model (ARMAX model)[edit]

The notation ARMAX(p, q, b) refers to the model with p autoregressive terms, q moving average terms and b exogenous inputs terms. This model contains the AR(p) and MA(q) models and a linear combination of the last b terms of a known and external time series $d_{t}$ . It is given by:

X_{t}=\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}+\sum _{i=1}^{b}\eta _{i}d_{t-i}.\,

where $\eta _{1},\ldots ,\eta _{b}$ are the parameters of the exogenous input $d_{t}$ .

Some nonlinear variants of models with exogenous variables have been defined: see for example Nonlinear autoregressive exogenous model.

Statistical packages implement the ARMAX model through the use of "exogenous" (that is, independent,) variables. Care must be taken when interpreting the output of those packages, because the estimated parameters usually (for example, in R^[8] and gretl) refer to the regression:

X_{t}-m_{t}=\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}(X_{t-i}-m_{t-i})+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.\,

where m_t incorporates all exogenous (or independent) variables:

m_{t}=c+\sum _{i=0}^{b}\eta _{i}d_{t-i}.\,

References[edit]

^ Hannan, Edward James (1970). Multiple time series. Wiley series in probability and mathematical statistics. New York: John Wiley and Sons.
^ Whittle, P. (1951). Hypothesis Testing in Time Series Analysis. Almquist and Wicksell.Whittle, P. (1963). Prediction and Regulation. English Universities Press. ISBN 0-8166-1147-5.
Republished as: Whittle, P. (1983). Prediction and Regulation by Linear Least-Square Methods. University of Minnesota Press. ISBN 0-8166-1148-3.
^ Hannan & Deistler (1988, p. 227): Hannan, E. J.; Deistler, Manfred (1988). Statistical theory of linear systems. Wiley series in probability and mathematical statistics. New York: John Wiley and Sons.
^ Box, George; Jenkins, Gwilym M.; Reinsel, Gregory C. (1994). Time Series Analysis: Forecasting and Control (Third ed.). Prentice-Hall. ISBN 0130607746.
^ Missouri State University. "Model Specification, Time Series Analysis" (PDF).
^ Brockwell, P. J.; Davis, R. A. (2009). Time Series: Theory and Methods (2nd ed.). New York: Springer. p. 273. ISBN 9781441903198.
^ Time series features in Mathematica Archived November 24, 2011, at the Wayback Machine
^ ARIMA Modelling of Time Series, R documentation

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