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Next: An information theory criterion Up: Choosing the Operator Length Previous: A mean square error

The common criteria

A number of fairly simple criteria are commonly discussed in the literature. Some of these will be reviewed here.

() suggest monitoring the magnitude of $ C_M$ to determine the operator length empirically. Their criterion is to choose that $ M$ value for which $ C_M$ first satisfies $ \vert C_M\vert << 1$. The argument is that $ C_M$ computed from (30) is ``a partial correlation coefficient,'' measuring the correlation between the forward and backward prediction errors. When $ \vert C_M\vert \simeq 1$, the correlation is high. When $ \vert C_M\vert << 1$, 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 $ \vert C_M\vert$ 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 $ F$-test and the relative error coefficient test: (a) The $ F$-test monitors $ E_M$ and checks whether the change in going from $ E_M$ to $ E_{M+1}$ 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 $ M \le N$. (b) The relative error coefficient test amounts to finding the minimum of the modified prediction error

$\displaystyle E_M^\prime = \frac{N}{N-M}E_M.$ (A-35)

The prediction error is modified in this manner to account for the decrease in the degrees of statistical freedom for the time series as the operator length increases. Clearly, $ E_M$ decreases whereas the multiplicative factor increases as $ M$ increases. $ E_M^\prime$ will therefore exhibit a minimum. A number of such minima can (and do) occur in practice. The parameter $ E_M^\prime$ is easily monitored while computing the $ C_M$'s using the Burg algorithm. Results obtained using this approach have been found satisfactory for moderate to large data samples. For small $ N$, the variations in both factors in (35) can be dramatic and the results become less reliable.

() 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

$\displaystyle (FPE)_M = \frac{N+M}{N-M}E_M.$ (A-36)

Like (35), this expression has a minimum since $ E_M$ decreases monotonically while the multiplicative factor increases monotonically with $ M$. In fact, (35) and (36) have very similar behavior, the principal difference being that ``when $ E_M$ is sufficiently smoothly varying,'' the minimum of (36) always occurs for smaller $ M$ values than that of (35). For short time series with sharp spectral lines, () found that FPE did not give a clear minimum. Both (35) and (36) suffer from this same ambiguity. For data samples of length $ 20 \le N \le 40$ in their work, they found that $ M = N/2$ was a satisfactory choice. This choice is also confirmed for short time series by the work of ().

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.


next up previous [pdf]

Next: An information theory criterion Up: Choosing the Operator Length Previous: A mean square error

2009-04-13