Editing Autoregressive model (section)

==Graphs of AR(''p'') processes==

[[File:ArTimeSeries.svg|thumb|right|alt="Figure has 5 plots of AR processes. AR(0) and AR(0.3) are white noise or look like white noise. AR(0.9) has some large scale oscillating structure."|AR(0); AR(1) with AR parameter 0.3; AR(1) with AR parameter 0.9; AR(2) with AR parameters 0.3 and 0.3; and AR(2) with AR parameters 0.9 and −0.8]]
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 <math>\varphi</math>, only the previous term in the process and the noise term contribute to the output.  If <math>\varphi</math> is close to 0, then the process still looks like white noise, but as <math>\varphi</math> 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 <math>\varphi_1</math> and <math>\varphi_2</math> are positive, the output will resemble a low pass filter, with the high frequency part of the noise decreased. If <math>\varphi_1</math> is positive while <math>\varphi_2</math> is negative, then the process favors changes in sign between terms of the process.  The output oscillates. This can be linked to edge detection or detection of change in direction.