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![]() | Maximum entropy spectral analysis | ![]() |
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When the autocorrelation values
are known, Equation (10)
is a linear set of
equations for the
unknown
's. On the other hand, if a prediction error
filter
and prediction error
are known, Equation (10) together
with (2) forms a linear set of equations that could be solved for the
's.
Thus, there exists a one-to-one correspondence between the prediction error filter and the
autocorrelation function. This relationship is exploited by () in his algorithm for
computing the minimum phase operator.
The autocorrelation function defined by (2) requires an infinite series, yet it can
only be estimated from a series of finite length . Given the data set
,
a reasonable estimate of
for large
is given by
We conclude that, if an autocorrelation must be computed, then Equation (18) should be used. However, () has observed that, in order to compute the maximum entropy spectrum, all that is required is an estimate of the minimum phase deconvolution operator. If this estimate can be computed without first estimating the autocorrelation values, then so much the better.
Suppose an estimate of the operator length is known
(
for
). How can the operator length be
increased from
to
? Note that by definition the forward
prediction error is given by
() suggests that one reasonable procedure is to choose the
that minimizes the total power of the prediction errors.
Setting
Finally, the algorithm for computing the set
is this: (a) Compute the
's using
Equation (30). (b) Store the
's until the desired operator length
is attained.
(c) Compute the
's from the
's using the recursion (28). This algorithm (simplified for
real data) is the one used in the maximum entropy processor for MESA that I developed.
It is important to notice before proceeding further that the Burg algorithm has been constructed
to remove the first difficulty discussed earlier in computing . All the information
has
been used; the operator is minimum phase; but no explicit averaging of lag products was required.
On the other hand, this algorithm does nothing to alleviate the second problem we discussed. It is still inherent
in the finite time series problem that the numbers we compute become less reliable as the operator length increases.
A major difficulty in applying MESA is that there is no built-in mechanism for choosing the operator length.
From the derivation of (5), it is clear the operator length should be if the first
autocorrelation values are known precisely and unknown otherwise. However, the autocorrelation function
has (normally) been estimated from the time series data and its estimated values are inaccurate for
close
to
. How to choose a practical operator length
satisfying
is therefore
the subject of the next section.
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![]() | Maximum entropy spectral analysis | ![]() |
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