Autoregressive Process

Explained:

autoregressive process


 
   

An n-dimensional autoregressive process of order p, AR(p), is a stochastic process of the form

[1]

where a is an n-dimensional vector, the are nn matrices, and W is n-dimensional white noise (see the notation conventions documentation). The name "autoregressive" indicates that [1] defines a regression of on its own past values. In applications, AR(1) and AR(2) processes are popular.

Exhibit 1 indicates a realization of the univariate AR(2) process

[2]

where W is variance 1 Gaussian white noise.

Univariate Autoregressive Process
Exhibit 1

A realization of the AR(2) process [2]

Exercises

Below are indicated a realization of 50 consecutive terms of a variance 1 Gaussian white noise.

0.293 0.317 0.047 -0.286 -1.237
-0.554 0.535 -1.640 -0.899 -0.704
-1.886 0.271 0.418 1.651 0.078
0.528 1.013 2.296 0.086 1.471
-0.580 -1.776 -2.217 0.502 -1.104
-1.211 0.205 0.110 0.011 0.778
-1.036 1.195 -1.169 -0.162 -0.504
-0.679 -1.366 0.885 -0.476 1.644
-1.665 0.129 2.882 0.978 0.054
-0.396 0.685 1.403 -0.009 0.918

Realization of 50 consecutive terms of a variance 1 Gaussian white noise.

Use this to generate a corresponding realization of the AR(2) process

[e1]

where tW is a variance 1 Gaussian white noise. Initialize the realization with terms 0x = 1x = 0. [spreadsheet solution]

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