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An n-dimensional
autoregressive process of order p, AR(p), is a
stochastic process of the form
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[1] |
where a is an n-dimensional vector,
the
are n n
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
common.
Exhibit 1 indicates a realization of the univariate AR(2)
process
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[2] |
where W is
variance 1 Gaussian white noise.
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A realization of
the AR(2) process [2] |
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 Below
are indicated a realization of 50
consecutive terms of a variance 1
Gaussian white noise.
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0.293 |
0.317 |
0.047 |
-0.286 |
-1.237 |
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-0.554 |
0.535 |
-1.640 |
-0.899 |
-0.704 |
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-1.886 |
0.271 |
0.418 |
1.651 |
0.078 |
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0.528 |
1.013 |
2.296 |
0.086 |
1.471 |
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-0.580 |
-1.776 |
-2.217 |
0.502 |
-1.104 |
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-1.211 |
0.205 |
0.110 |
0.011 |
0.778 |
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-1.036 |
1.195 |
-1.169 |
-0.162 |
-0.504 |
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-0.679 |
-1.366 |
0.885 |
-0.476 |
1.644 |
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-1.665 |
0.129 |
2.882 |
0.978 |
0.054 |
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-0.396 |
0.685 |
1.403 |
-0.009 |
0.918 |
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Realization of 50 consecutive terms of a variance 1
Gaussian white noise. |
Use this to generate a
corresponding realization of the AR(2) process
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[e1] |
where tW is a variance 1 Gaussian white
noise. Initialize the realization with terms 0x
= 1x = 0.
[spreadsheet solution] |
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