462
M, AR(M). Exhaustive details can be found in Burg (1967) or Childers
(1978).
Given a stationary time series {X(t), t = 1, ... , N} of zero mean, M' + 1
auto-correlation coefficients {
N-k
In the absence of the knowledge about the process that engenders the
time series X(t), M' is arbitrary and has to be optimized. The purpose
of evaluating (14) is to determine the spectral density Px , given by (12),
which is equivalent to the most random, or least predictable process with
the same auto-correlation coefficients
the name of the method.
In practice, one determines the coefficients {ak' k = 0, ... , M'} from
X(t) by assuming that it is generated by an AR process and that its order
M = M'. The autocorrelation coefficients
Penland et aL 1991, Vautard et aL 1992) and this matrix is then inverted
using standard numerical schemes (Press et aL 1988) to yield the {ak}.
The spectral density Px of the (true) AR process with coefficients {ak' k =
° ... , M} is given by
(15)
where ao is the variance of the residual noise e in Eq. (1). Therefore, the
knowledge of the {ak' k = 0, ... , M'} coefficients, determined from the
time series X(t), also yields an estimate of the power spectrum Px .
An example of MEM estimates is given for the SOl time series in Fig. 8,
using a number of lags M = 10, 20 and 40. It is clear that the number
of peaks increases with M. A separation of two interannual peaks only
occurs for M = 40, but is accompanied, unfortunately, by many spurious
peaks at higher frequencies.
In general, if the time series is not stationary or otherwise not close to
auto-regressive, great care in applying MEM - as well as cross-testing with
the application of other techniques - is necessary. As the number of peaks
in the spectrum increases with M, regardless of the spectral content of the
M, AR(M). Exhaustive details can be found in Burg (1967) or Childers
(1978).
Given a stationary time series {X(t), t = 1, ... , N} of zero mean, M' + 1
auto-correlation coefficients {
N-k
In the absence of the knowledge about the process that engenders the
time series X(t), M' is arbitrary and has to be optimized. The purpose
of evaluating (14) is to determine the spectral density Px , given by (12),
which is equivalent to the most random, or least predictable process with
the same auto-correlation coefficients
the name of the method.
In practice, one determines the coefficients {ak' k = 0, ... , M'} from
X(t) by assuming that it is generated by an AR process and that its order
M = M'. The autocorrelation coefficients
Penland et aL 1991, Vautard et aL 1992) and this matrix is then inverted
using standard numerical schemes (Press et aL 1988) to yield the {ak}.
The spectral density Px of the (true) AR process with coefficients {ak' k =
° ... , M} is given by
(15)
where ao is the variance of the residual noise e in Eq. (1). Therefore, the
knowledge of the {ak' k = 0, ... , M'} coefficients, determined from the
time series X(t), also yields an estimate of the power spectrum Px .
An example of MEM estimates is given for the SOl time series in Fig. 8,
using a number of lags M = 10, 20 and 40. It is clear that the number
of peaks increases with M. A separation of two interannual peaks only
occurs for M = 40, but is accompanied, unfortunately, by many spurious
peaks at higher frequencies.
In general, if the time series is not stationary or otherwise not close to
auto-regressive, great care in applying MEM - as well as cross-testing with
the application of other techniques - is necessary. As the number of peaks
in the spectrum increases with M, regardless of the spectral content of the