Given a
stationary time series *x*(*t*), and its first *M* auto-correlation
coefficients, the purpose of MEM is to obtain the power spectrum *P _{X}*
by determining the most random (i.e. with the fewest
assumptions) process, with the same auto-correlation coefficients as

The entropy *h* of a Gaussian process is given by

From the Wiener-Khintchin identity, the maximal entropy process and the
series** ***x*(*t*) will have the same power spectrum.

In practice, under the assumption that *x*(*t*) is generated by an autoregressive *AR(M)* process:

where* &xi(t)* is a white noise with a variance &sigma^{2}, one can obtain estimates of coefficients *a _{j }* by regression.The autocorrelation coefficients are computed first and used to form the same Toeplitz matrix as in SSA. This matrix is then inverted using standard numerical schemes to yield the estimated

where a_{0}=&sigma^{2}. Therefore, the knowledge of the coefficients *a _{j}* determined from the time series

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 *x*(*t*). The behavior of the spectral estimate depends on the choice of *M*: it
is clear that the number of its poles (or even maxima)
depends on the order of regression *M* and the
auto-regression coefficients *a _{j}*
, so that, for a given time
series, the number of peaks will increase with

The weaknesses can be remedied partly by (a)
determining which peaks survive reductions in *M*, (b) comparing MEM
spectra to those produced by Blackman Tukey 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).

1. Percival, D.B., and Walden, A.T.,
1993: *Spectral analysis for physical applications--Multitaper and
conventional univariate techniques*. Cambridge University, 580 pp.

2. Penland, C., Ghil, M., and
Weickmann, K., 1991: *Adaptive filtering and maximum entropy spectra
with application to changes in atmospheric angular momentum*:
J. Geophys. Res., **96**, 22659-22671.