Maximum-Entropy spectral estimates

The toolkit also provides spectral estimation by MEM. Given a stationary time series X, and its first M auto-correlation coefficients  , the purpose of MEM is to obtain the spectral density PX by determining the most random (i.e. with the fewest assumptions) process, with the same auto-correlation coefficients as X. In terms of information theory, this is the notion of maximal entropy, hence the name of the method.

The entropy h of a Gaussian process is given by

From the Wiener-Khintchin identity, the maximal entropy process and the series X will have the same spectral density. Some algebra (Percival and Walden, 1993) shows that under the constraints of  , h is maximized by an autoregressive process Y of size M-1:

where bn is a Gaussian white process with variance ao. And hence PX is


In summary, the method boils down to looking for an auto-regressive process that ``mimics'' the original time series. This is why it is a called parametric method.

The MEM is very efficient for detecting frequency lines in stationary time series. However, if this time sereis is not-stationary, misleading results can occur, with little chance of being detected otherwise than by cross-checking with other techniques.

The art of using MEM resides in the appropriate choice of M, i.e. the order of regression of Y. The behavior of the spectral estimate depends on the choice of M: it is clear that the number of poles (or even maxima) of Eq. (3) depends on the order of regression M and the auto-regression coefficients ak , so that, for a given time series, the number of peaks will increase with M! Therefore, a trade-off between a good resolution (high M) and few spurious peaks (low M) has to be found. A few guides are provided by the default values of the toolkit (i.e. M should not exceed half the length of the time series).

The weaknesses can be remedied partly by (a) determining which peaks survive reductions in M, (b) comparing MEM spectra to those produced by correlogram and MTM which generally should not share spurious peaks with MEM, and (c) using SSA to pre-filter the series and thus to decompose the original series into several components, each of which contains only a few harmonics (so that small M values can be chosen; see Penland et al., 1991). The ease with which these various analyses can be interwoven in the Toolkit was a major motivation for its development.