Авторегрессиялық модельдер: кездейсоқ процестерді бейнелеу

Representation of a type of random process

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, 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. Contrary to the moving average (MA) model, the autoregressive model is not always stationary as it may contain a unit root.

The simplest AR process is AR(0), which has no dependence between the terms. Only the error/innovation/noise term contributes to the output of the process, so in the figure, AR(0) corresponds to white noise. For an AR(1) process with a positive , only the previous term in the process and the noise term contribute to the output. If is close to 0, then the process still looks like white noise, but as approaches 1, the output gets a larger contribution from the previous term relative to the noise. This results in a "smoothing" or integration of the output, similar to a low pass filter. For an AR(2) process, the previous two terms and the noise term contribute to the output. If both and are positive, the output will resemble a low pass filter, with the high frequency part of the noise decreased. If is positive while is negative, then the process favors changes in sign between terms of the process. The output oscillates. This can be likened to edge detection or detection of change in direction.

The partial autocorrelation of an AR(p) process equals zero at lags larger than p, so the appropriate maximum lag p is the one after which the partial autocorrelations are all zero.

The impulse response of a system is the change in an evolving variable in response to a change in the value of a shock term k periods earlier, as a function of k. Since the AR model is a special case of the vector autoregressive model, the computation of the impulse response in vector autoregression#impulse response applies here.