Autoregressive (AR) Model
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An Autoregressive (AR) Model is a stochastic process model in which its outcome variable is based on a linear function on its previous values.
- Context:
- It can be trained by an AR Training System (that implements an autoregression algorithm).
- …
- Example(s):
- Autoregressive-Moving-Average (ARMA), and ARIMA.
- an Autoregressive Large Language Model, such as a GPT language model.
- …
- Counter-Example(s):
- See: Random Process, Stationary Process, Timeseries Modeling, Vector Autoregression.
References
2023
- https://venturebeat.com/ai/googles-muse-model-could-be-the-next-big-thing-for-generative-ai/
- QUOTE: Diffusion models use progressive denoising. Autoregressive models use serial decoding. The parallel decoding in Muse allows for very good efficiency without loss in visual quality.
2021
- (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/autoregressive_model Retrieved:2021-3-5.
- 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 (or recurrence relation which should not be confused with 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.
Contrary to the moving-average (MA) model, the autoregressive model is not always stationary as it may contain a unit root.
- 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 (or recurrence relation which should not be confused with 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.
2016
- (The Pennsylvania State University, 2016) ⇒ The Pennsylvania State University (2016). “Online Course Statistics 501" https://onlinecourses.science.psu.edu/stat501/node/358
- A time series is a sequence of measurements of the same variable(s) made over time. Usually the measurements are made at evenly spaced times - for example, monthly or yearly. Let us first consider the problem in which we have a y-variable measured as a time series. As an example, we might have y a measure of global temperature, with measurements observed each year. To emphasize that we have measured values over time, we use "t" as a subscript rather than the usual "i," i.e., [math]\displaystyle{ y_t }[/math] means [math]\displaystyle{ y }[/math] measured in time period t. An autoregressive model is when a value from a time series is regressed on previous values from that same time series. for example, [math]\displaystyle{ y_t }[/math] on [math]\displaystyle{ y_{t−1} }[/math]:
[math]\displaystyle{ y_t=\beta_0+\beta_1y_t−1+\varepsilon_t }[/math]
- In this regression model, the response variable in the previous time period has become the predictor and the errors have our usual assumptions about errors in a simple linear regression model. The order of an autoregression is the number of immediately preceding values in the series that are used to predict the value at the present time. So, the preceding model is a first-order autoregression, written as AR(1).
If we want to predict [math]\displaystyle{ y }[/math] this year ([math]\displaystyle{ y_t }[/math]) using measurements of global temperature in the previous two years ([math]\displaystyle{ y_{t−1},y_{t−2} }[/math]), then the autoregressive model for doing so would be:
- In this regression model, the response variable in the previous time period has become the predictor and the errors have our usual assumptions about errors in a simple linear regression model. The order of an autoregression is the number of immediately preceding values in the series that are used to predict the value at the present time. So, the preceding model is a first-order autoregression, written as AR(1).
- [math]\displaystyle{ y_t=\beta_0+\beta_1y_{t−1}+\beta_2y_{t−2}+\varepsilon_t. }[/math]
- This model is a second-order autoregression, written as AR(2), since the value at time tt is predicted from the values at times [math]\displaystyle{ t−1 }[/math] and [math]\displaystyle{ t−2 }[/math]. More generally, a kth-order autoregression, written as AR(k), is a multiple linear regression in which the value of the series at any time [math]\displaystyle{ t }[/math] is a (linear) function of the values at times [math]\displaystyle{ t−1,t−2,…,t−k }[/math].
2016
- (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/Autoregressive_model Retrieved 2016-07-10
- Definition
- The notation [math]\displaystyle{ AR(p) }[/math] indicates an autoregressive model of order p. The AR(p) model is defined as
- [math]\displaystyle{ X_t = c + \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t \, }[/math]
- where [math]\displaystyle{ \varphi_1, \ldots, \varphi_p }[/math] are the parameters of the model, [math]\displaystyle{ c }[/math] is a constant, and [math]\displaystyle{ \varepsilon_t }[/math] is white noise. This can be equivalently written using the backshift operator B as
- [math]\displaystyle{ X_t = c + \sum_{i=1}^p \varphi_i B^i X_t + \varepsilon_t }[/math]
- so that, moving the summation term to the left side and using polynomial notation, we have
- [math]\displaystyle{ \phi (B)X_t= c + \varepsilon_t \, . }[/math]
- An autoregressive model can thus be viewed as the output of an all-pole infinite impulse response filter whose input is white noise.
Some parameter constraints are necessary for the model to remain wide-sense stationary. For example, processes in the AR(1) model with [math]\displaystyle{ |\varphi_1 | \geq 1 }[/math] are not stationary. More generally, for an AR(p) model to be wide-sense stationary, the roots of the polynomial [math]\displaystyle{ \textstyle z^p - \sum_{i=1}^p \varphi_i z^{p-i} }[/math] must lie within the unit circle, i.e., each root [math]\displaystyle{ z_i }[/math] must satisfy [math]\displaystyle{ |z_i|\lt 1 }[/math]
- An autoregressive model can thus be viewed as the output of an all-pole infinite impulse response filter whose input is white noise.
2008
- (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised.” Oxford University Press. ISBN:0199541450
- QUOTE: ... autoregressive model; auto regressive process: A model for a time series having no trend (the constant mean is taken as 0). Let [math]\displaystyle{ X_l, X_2, ..., }[/math] be successive instances of the random variable X measured at regular intervals of time. Let [math]\displaystyle{ a_j }[/math] be the random variable denoting the random error at time j. A pth-order autoregressive model (or autoregressive process) relates the value at time j to the preceding p values by :[math]\displaystyle{ X_j = \alpha_1X_{j-1} + \alpha_2X{j-2} + … + \alpha_pX_{j-p} + \varepsilon_j }[/math] ...