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An n-dimensional
autoregressive
moving-average process of orders p and q, ARMA(p,q),
is a stochastic
process of the form
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[1] |
where a is an n-dimensional vector,
the
and
are n n
matrices, and W is n-dimensional
white noise (see the
notation conventions documentation). As the name
suggests, this combines an
AR(p) model
with an MA(q)
model of the same dimension n. In applications, ARMA(1,1)
processes are common.
Exhibit 1 indicates a realization of the univariate
ARMA(1,1) process
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[2] |
where W is variance
1 Gaussian white noise.
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A realization of
the ARMA(1,1) 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 ARMA(1,1) process
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[e1] |
where tW is a variance 1 Gaussian white
noise. Initialize the realization with term 0x
= 0.
[spreadsheet solution] |
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