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A number of fairly simple criteria are commonly discussed in the literature. Some of these will be reviewed here.
() suggest monitoring the magnitude of to determine the
operator length empirically. Their criterion is to choose that
value for which
first satisfies
. The argument is that
computed from (30) is
``a partial correlation coefficient,'' measuring the correlation between the forward and backward
prediction errors. When
, the correlation is high. When
,
the correlation is low -- presumably because most of the predictable information in the data
has been removed by the filter. However, they point out that this procedure fails to produce
reliable results for series not purely autoregressive in character. Numerical studies of the author
on real seismic traces have shown the fluctuations in
to be too great for this approach
to give a reliable criterion.
() review a number of possible approaches. The two
which are probably easiest to apply are the -test and the relative error
coefficient test: (a) The
-test monitors
and checks whether the
change in going from
to
is statistically significant
according to some predetermined criterion.
This method is limited by computer round off error for large data samples. It is also limited for small
data samples because the predetermined criterion of statistically significant change may very well be met
for all
. (b) The relative error coefficient test amounts to finding the minimum of the modified
prediction error
() review a number of alternatives and conclude that the final prediction error (FPE) criterion of () is an objective basis for choosing the operator length. This criterion monitors
Although each of these criteria has its merits, none of them is really satisfactory for a data sample of arbitrary length. Furthermore, none of them has been derived in the spirit of MESA, i.e., with no assumptions about the data off the ends of the sample. Much has been said about the application of optimum criteria from autoregressive analysis to MESA (, ). But an important point should be made: The fact that an autoregressive process has the maximum entropy is interesting but irrelevant. The spectrum of an arbitrary time series (whether MA, AR, or ARMA of any order) can be estimated using MESA. But, making any assumption about the nature of the stochastic process that generated the series is contrary to the spirit of MESA.
Thus, it seems that the choice of operator length should be made without assumptions concerning the nature of the stochastic process involved. The argument in the next subsection is based only on information theory, and measurement theory. It is believed to free of these inconsistencies.
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![]() | Maximum entropy spectral analysis | ![]() |
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