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 *P _{X}* 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 *b _{n}* is a Gaussian white process with variance

*a*. And hence

_{o}*P*is

_{X}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 *a _{k}* , 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.