Autoregressive model

In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, 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 (or recurrence relation) which should not be confused with a differential equation. 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. Another important extension is the time-varying autoregressive (TVAR) model, where the autoregressive coefficients are allowed to change over time to model evolving or non-stationary processes. TVAR models are widely applied in cases where the underlying dynamics of the system are not constant, such as in sensors time series modelling[1][2], finance[3], climate science[4], economics[5], signal processing[6] and telecommunications[7], radar systems[8], and biological signals[9].

Unlike the moving-average (MA) model, the autoregressive model is not always stationary; non-stationarity can arise either due to the presence of a unit root or due to time-varying model parameters, as in time-varying autoregressive (TVAR) models.

Large language models are called autoregressive, but they are not a classical autoregressive model in this sense because they are not linear.

  1. ^ Souza, Douglas Baptista de; Leao, Bruno Paes (26 October 2023). "Data Augmentation of Sensor Time Series using Time-varying Autoregressive Processes". Annual Conference of the PHM Society. 15 (1). doi:10.36001/phmconf.2023.v15i1.3565.
  2. ^ Souza, Douglas Baptista de; Leao, Bruno Paes (5 November 2024). "Data Augmentation of Multivariate Sensor Time Series using Autoregressive Models and Application to Failure Prognostics". Annual Conference of the PHM Society. 16 (1). arXiv:2410.16419. doi:10.36001/phmconf.2024.v16i1.4145.
  3. ^ Jia, Zhixuan; Li, Wang; Jiang, Yunlong; Liu, Xingshen (9 July 2025). "The Use of Minimization Solvers for Optimizing Time-Varying Autoregressive Models and Their Applications in Finance". Mathematics. 13 (14): 2230. doi:10.3390/math13142230.
  4. ^ Diodato, Nazzareno; Di Salvo, Cristina; Bellocchi, Gianni (18 March 2025). "Climate driven generative time-varying model for improved decadal storm power predictions in the Mediterranean". Communications Earth & Environment. 6 (1): 212. Bibcode:2025ComEE...6..212D. doi:10.1038/s43247-025-02196-2.
  5. ^ Inayati, Syarifah; Iriawan, Nur (31 December 2024). "Time-Varying Autoregressive Models for Economic Forecasting". Matematika: 131–142. doi:10.11113/matematika.v40.n3.1654.
  6. ^ Baptista de Souza, Douglas; Kuhn, Eduardo Vinicius; Seara, Rui (January 2019). "A Time-Varying Autoregressive Model for Characterizing Nonstationary Processes". IEEE Signal Processing Letters. 26 (1): 134–138. Bibcode:2019ISPL...26..134B. doi:10.1109/LSP.2018.2880086.
  7. ^ Wang, Shihan; Chen, Tao; Wang, Hongjian (17 March 2023). "IDBD-Based Beamforming Algorithm for Improving the Performance of Phased Array Radar in Nonstationary Environments". Sensors. 23 (6): 3211. Bibcode:2023Senso..23.3211W. doi:10.3390/s23063211. PMC 10052024. PMID 36991922.
  8. ^ Abramovich, Yuri I.; Spencer, Nicholas K.; Turley, Michael D. E. (April 2007). "Time-Varying Autoregressive (TVAR) Models for Multiple Radar Observations". IEEE Transactions on Signal Processing. 55 (4): 1298–1311. Bibcode:2007ITSP...55.1298A. doi:10.1109/TSP.2006.888064.
  9. ^ Gutierrez, D.; Salazar-Varas, R. (August 2011). "EEG signal classification using time-varying autoregressive models and common spatial patterns". 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society. pp. 6585–6588. doi:10.1109/IEMBS.2011.6091624. ISBN 978-1-4577-1589-1. PMID 22255848.
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