## 2. THEORY

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*_{o}.
And hence *P*_{X} 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 *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.

This file last modified: February 1, 2000