Imposing prior knowledge of symmetry

  Reversing a signal in time does not change its autocorrelation. In the analysis of stationary time series, it is well known (FGDP) that the filter for predicting forward in time should be the same as that for ``predicting'' backward in time (except for time reversal). When the short data are samples, however, a different filter may be found for predicting forward than for backward. Rather than average the two filters directly, the better procedure is to find the filter that minimizes the sum of power in two residuals. One is a filtering of the original signal, and the other is a filtering of a time-reversed signal, as in equation (29), where the top half of the equations represent prediction-error predicting forward in time and the second half is prediction backward.  

$$\left[ \begin{array} {c} r_1 \\ r_2 \\ r_3 \\ r_4 \\ \h... ...\left[ \begin{array} {c} 1 \\ a_1 \\ a_2 \end{array} \right]$$ (29)

In an (x,y) space, you might also expect rotational symmetry. In an (x,t) space, dips are preserved if t-axis reversal is accompanied by x-axis reversal. Many of my research codes include these symmetries, but I excluded them here. Nowhere did I prove that the symmetry presumption made a large difference, but in Fortran it always adds noticeable clutter, expanding the residual to a two-component residual array, which as usual needs to have three copies allocated.

Where a data sample grows exponentially towards the boundary, I expect that extrapolated data would diverge too. This is prevented, however, by basing missing-data estimation on transient convolution (instead of internal convolution). When transient convolution

is used, extrapolations are forced to zero beyond a known specified distance.