Computes least‑squares estimates of the multichannel Wiener filter coefficients for two mutually stationary multichannel time series.

Routine MFFE computes least‑squares estimates of the multichannel Wiener filter coefficients for two mutually stationary multichannel time series.

Define the multichannel time series X by

where

with p = NCHX. Similarly, define the multichannel time series Z by

where

with q = NCHZ. The columns of X and Z correspond to individual channels of multichannel time series and may be examined from a univariate perspective. The rows of X and Z correspond to observations of p‑variate and q‑variate time series and may be examined from a multivariate perspective. Note that an alternative characterization of a multivariate time series X considers the columns of X to be observations of a p‑variate time series with the rows of X containing univariate time series. For example, see Priestley (1981, page 692) and Fuller (1976, page 14).

Let

be the row vector containing the means of the channels of X. In particular,

where for j = 1, 2, …, p

Let

be similarly defined. In what follows, assume the channels of both X and Z have been centered about their respective means

Suppose the desired output is the multichannel time series Z defined by the model

where

and ɸk is the array of dimension p × q containing the Wiener filter coefficients

for k = 1, …, K. The array ɸk is the (k + 1)-st level of the 3‑dimensional array FIL.

The filter coefficients are computed by solving a set of normal equations. The algorithm utilizes the block Toeplitz (or Töplitz) matrix structure of these equations and is given by Robinson (1967, pages 238–246). In particular, the required input consists of the multichannel autocovariance matrices ΣX, ΣZ, and the multichannel cross‑covariance matrix ΣZX. The routine MCCF may be used to estimate these covariance matrices.

Note that successively longer filters are estimated until either the normalized mean square error is less than EPS or the filter length K = LFIL equals MLFIL. The normalized mean square error is defined by

where tr ΣZ(0) = TRACE is the trace of the multichannel autocorrelation coefficient of the desired output at lag zero. The values of Qk for the successive filters of length k = 1, 2, …, K are contained in ENMS.

Consider the Wölfer Sunspot Data (Box and Jenkins 1976, page 530) along with data on northern light activity and earthquake activity (Robinson 1967, page 204) to be a three‑channel time series. Routine MWFE applied to these data determines the following Wiener filter: