Content deleted Content added
→Autoregressive model: Links for acronyms, but citation still needed |
m →Software implementations: HTTP to HTTPS for SourceForge |
||
| (2 intermediate revisions by 2 users not shown) | |||
Line 16:
:<math> X_t = \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t</math>
where <math>\varphi_1, \ldots, \varphi_p</math> are [[parameter]]s and the random variable <math>\varepsilon_t</math> is [[white noise]], usually [[Independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) [[Normal distribution|normal random variables]].<ref>{{Cite book |last=Box |first=George E. P.
In order for the model to remain [[Stationary process|stationary]], the roots of its [[Autoregressive model#Characteristic polynomial|characteristic polynomial]] must lie outside the unit circle. For example, processes in the AR(1) model with <math>|\varphi_1| \ge 1</math> are not stationary because the root of <math>1 - \varphi_1B = 0</math> lies within the unit circle.<ref>{{Cite book |last1=Box |first1=George E. P.
The [[augmented Dickey–Fuller test]] can assesses the stability of an [[intrinsic mode function]] and trend components. For stationary time series, the ARMA models can be used, while for non-stationary series, [[Long short-term memory]] models can be used to derive abstract features. The final value is obtained by reconstructing the predicted outcomes of each time series.{{Citation needed|date=April 2025}}
Line 28:
:<math> X_t = \mu + \varepsilon_t + \sum_{i=1}^q \theta_i \varepsilon_{t-i}\,</math>
where the <math>\theta_1,...,\theta_q</math> are the parameters of the model, <math>\mu</math> is the expectation of <math>X_t</math> (often assumed to equal 0), and <math>\varepsilon_1</math>, ..., <math>\varepsilon_t</math> are i.i.d. white noise error terms that are commonly normal random variables.<ref>{{Cite book |last1=Box |first1=George E. P.
=== ARMA 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,<ref>{{Cite book |last=Shumway |first=Robert H.
:<math> X_t = \varepsilon_t + \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,</math>
Line 68:
== Spectrum ==
The [[spectral density]] of an ARMA process is<math display="block">S(f) = \frac{\sigma^2}{2\pi} \left\vert \frac{\theta(e^{-if})}{\phi(e^{-if})} \right\vert^2</math>where <math>\sigma^2</math> is the [[variance]] of the white noise, <math>\theta</math> is the characteristic polynomial of the moving average part of the ARMA model, and <math>\phi</math> is the characteristic polynomial of the autoregressive part of the ARMA model.<ref>{{Cite book |last=Rosenblatt |first=Murray
== Fitting models ==
Line 89:
* [[MATLAB]] includes functions such as [http://www.mathworks.com/help/econ/arma-model.html <code>arma</code>], [http://www.mathworks.com/help/ident/ref/ar.html <code>ar</code>] and [http://www.mathworks.com/help/ident/ref/arx.html <code>arx</code>] to estimate autoregressive, exogenous autoregressive and ARMAX models. See [http://www.mathworks.com/help/ident/ug/estimating-ar-and-arma-models.html System Identification Toolbox] and [http://www.mathworks.com/help/econ/arima.estimate.html Econometrics Toolbox] for details.
* [[Julia_(programming_language) | Julia]] has community-driven packages that implement fitting with an ARMA model such as [https://github.com/joefowler/ARMA.jl <code>arma.jl</code>].
* Python has the <code>statsmodels</code>[
* [[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 estimate ARMA models, as mentioned [http://constantdream.wordpress.com/2008/03/16/gnu-regression-econometrics-and-time-series-library-gretl/ here]
* [[GNU Octave]] extra package [
* [[Stata]] includes the function <code>arima</code>. for ARMA and [[Autoregressive integrated moving average|ARIMA]] models.
* [[SuanShu]] is a Java library of numerical methods that implements univariate/multivariate ARMA, ARIMA, ARMAX, etc models, documented in [http://www.numericalmethod.com/javadoc/suanshu/ "SuanShu, a Java numerical and statistical library"].
Line 144:
== Further reading ==
* {{cite book |last=Mills |first=Terence C. |title=Time Series Techniques for Economists |publisher=Cambridge University Press |year=1990 |isbn=0521343399 |url-access=registration |url=https://archive.org/details/timeseriestechni0000mill }}
* {{cite book |last1=Percival |first1=Donald B. |first2=Andrew T. |last2=Walden |title=Spectral Analysis for Physical Applications |url=https://archive.org/details/spectralanalysis0000perc |publisher=Cambridge University Press |year=1993 |isbn=052135532X }}
* {{citation| first1= C. | last1= Francq | first2= J.-M. | last2= Zakoïan | chapter= Recent results for linear time series models with non independent innovations | pages= 241–265 | title= Statistical Modeling and Analysis for Complex Data Problems | editor1-first= P. |editor1-last= Duchesne | editor2-first=B. | editor2-last= Remillard | publisher= Springer | year= 2005| citeseerx= 10.1.1.721.1754 }}.
* [https://link.springer.com/book/10.1007/978-3-319-52452-8 Shumway, R.H. and Stoffer, D.S. (2017). ''Time Series Analysis and Its Applications with R Examples''. Springer. DOI: 10.1007/978-3-319-52452-8]
| |||