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![]() | Maximum entropy spectral analysis | ![]() |
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Suppose we have found an estimate of the prediction error filter of length using the
autocorrelation estimates
. In order to increase the operator length
to
, additional information is needed: namely,
. A quantitative measure of the
information in the operator is easily obtained from the average entropy, which we know is given
by
. Using (31), notice that
If the autocorrelation values
were known precisely, bound information would
continue to increase by using all the estimates and letting
. But the
's are not
precisely known. The finite number of measurements used to compute the
estimates means that
only
measurements of
were made, whereas
measurements of
were made. The quality of
information contained in
is correspondingly higher than that in
. A quantitative measure of
this change is therefore required.
For the moment, take Equation (19) as our estimate of the autocorrelation.
Then, assuming that the 's are normally distributed, () shows that
The probable error in increases like
as
. We imagine
that the factor
is proportional to the probability
that an operator
computed from
is a worse estimate of the true opertor than was the operator
computed using only
. Since we know empirically that the estimate
worsens as
with probability one, the
are normalized by writing:
The average entropy of measurement error associated with an operator of length is
Combining (38) and (43), the average information in the power spectrum can be quantitatively estimated using the expression
The values of (45) can be monitored continuously while the operator is being computed.
However, an approximate analytic solution for the maximum can be found without making very
restrictive assumptions on the behavior of . Numerical studies of the author on real
seismic data have shown that
can be represented approximately by
An analytic bound on can be obtained from (47) by noting that the right-hand
side of (47) increases with
, so its minimum value occurs when
.
Thus,
has the very simple bound:
Because the correspondence between and
has been established by
this heuristic argument, the results of this section of the paper should not be interpreted as
rigorous estimates of the optimum operator length. Nevertheless, I believe that
(47) and (49) are reasonable estimates of the operator length.
The derivation was not founded on any assumptions about the type of stochastic process
generating the time series. Hence, these estimates are definitely not intended to be
an estimate of the order of some underlying autoregressive process. Rather, (49) is
an upper bound on the operator length that will extract the most reliable information for
a data sample of length
. For example, suppose the time series
is
a representation of an autoregressive series of order
. Then computing the operator of length
should give the most efficient estimate of the spectrum; but computing the additional
terms should do little to alter that spectrum.
Next, suppose the time series is a representation of an AR series of order
. The
arguments above indicate that we probably cannot obtain a really good estimate of the operator
(or the spectrum), because our data sample is simply too small. The best we can hope to do is to
compute the operator of length
. In either case, when additional information about the underlying
stochastic process is lacking, the best operational decision that can be made appears to be
choosing
according to Equations (47) or (49).
FIG1
Figure 1. Operator length ![]() ![]() ![]() ![]() |
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![]() | Maximum entropy spectral analysis | ![]() |
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