Editing Autoregressive model (section)

==Spectrum==
[[File:AutocorrTimeAr.svg|thumb|right]]
[[File:AutoCorrAR.svg|thumb|right]]

The [[Spectral density#Power spectral density|power spectral density]] (PSD) of an AR(''p'') process with noise variance <math>\mathrm{Var}(Z_t) = \sigma_Z^2</math> is<ref name=Storch/>
: <math>S(f) = \frac{\sigma_Z^2}{| 1-\sum_{k=1}^p \varphi_k e^{-i 2 \pi f k} |^2}.</math>

===AR(0)===
For white noise (AR(0))
: <math>S(f) = \sigma_Z^2.</math>

===AR(1)===
For AR(1)
: <math>S(f) = \frac{\sigma_Z^2}{| 1- \varphi_1 e^{-2 \pi i f} |^2}
     = \frac{\sigma_Z^2}{ 1 + \varphi_1^2 - 2 \varphi_1 \cos 2 \pi f }</math>
*If <math>\varphi_1 > 0</math>  there is a single spectral peak at <math>f=0</math>, often referred to as [[red noise]]. As <math>\varphi_1 </math>  becomes nearer 1, there is stronger power at low frequencies, i.e. larger time lags. This is then a low-pass filter, when applied to full spectrum light, everything except for the red light will be filtered.
*If <math>\varphi_1 < 0</math> there is a minimum at <math>f=0</math>, often referred to as [[blue noise]]. This similarly acts as a high-pass filter, everything except for blue light will be filtered.

===AR(2)===
The behavior of an AR(2) process is determined entirely by the roots of it [[characteristic equation (calculus)| characteristic equation]], which is expressed in terms of the [[lag operator]] as:

:<math> 1 - \varphi_1 B -\varphi_2 B^2 =0, </math>

or equivalently by the poles of its [[transfer function]], which is defined in the [[Z-transform|Z domain]] by:

:<math> H_z = (1 - \varphi_1 z^{-1} -\varphi_2 z^{-2})^{-1}. </math>

It follows that the poles are values of z satisfying:

:<math> 1 - \varphi_1 z^{-1} -\varphi_2 z^{-2} = 0 </math>,

which yields:

:<math> z_1,z_2 = \frac{1}{2\varphi_2}\left(\varphi_1 \pm \sqrt{\varphi_1^2 + 4\varphi_2}\right) </math>.

<math>z_1</math> and <math>z_2</math> are the reciprocals of the characteristic roots, as well as the eigenvalues of the temporal update matrix: 

:<math> \begin{bmatrix} \varphi_1 & \varphi_2 \\ 1 & 0 \end{bmatrix} </math>

AR(2) processes can be split into three groups depending on the characteristics of their roots/poles:

* When <math>\varphi_1^2 + 4\varphi_2 < 0</math>, the process has a pair of complex-conjugate poles, creating a mid-frequency peak at:
:<math>f^* = \frac{1}{2\pi}\cos^{-1}\left(\frac{\varphi_1}{2\sqrt{-\varphi_2}}\right),</math>

with bandwidth about the peak inversely proportional to the moduli of the poles:

:<math>|z_1|=|z_2|=\sqrt{-\varphi_2}.</math>

The terms involving square roots are all real in the case of complex poles since they exist only when <math>\varphi_2<0</math>.

Otherwise the process has real roots, and:
* When <math>\varphi_1 > 0</math> it acts as a low-pass filter on the white noise with a spectral peak at <math>f=0</math>
* When <math>\varphi_1 < 0</math> it acts as a high-pass filter on the white noise with a spectral peak at <math>f=1/2</math>.
The process is non-stationary when the poles are on or outside the unit circle, or equivalently when the characteristic roots are on or inside the unit circle.
The process is stable when the poles are strictly within the unit circle (roots strictly outside the unit circle), or equivalently when the coefficients are in the triangle <math>-1 \le \varphi_2 \le 1 - |\varphi_1|</math>.

The full PSD function can be expressed in real form as:
:<math>S(f) = \frac{\sigma_Z^2}{1 + \varphi_1^2 + \varphi_2^2 - 2\varphi_1(1-\varphi_2)\cos(2\pi f) - 2\varphi_2\cos(4\pi f)}</math>