Imposing prior knowledge of symmetry

symmetry in time  reversing a signal 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 data samples are short, 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 (36), where the top half of the equations represent prediction-error predicting forward in time and the second half is prediction backward.  

$$\left[ \begin{array} {c} r_1 \\ r_2 \\ r_3 \\ r_4 \\ \h... ...\left[ \begin{array} {c} 1 \\ a_1 \\ a_2 \end{array} \right]$$ (36)

To get the bottom rows from the top rows, we simply reverse the order of all the components within each row. That reverses the input time function. (Reversing the order within a column would reverse the output time function.) Instead of the matrix being diagonals tipping $45^\circ$ to the right, they tip to the left. We could make this matrix from our old familiar convolution matrix and a time-reversal matrix

$$\left[ \begin{array} {cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$

It is interesting to notice how time-reversal symmetry applies to Figure . First of all, with time going both forward and backward the residual space gets twice as big. The time-reversal part gives a selector for Figure with a gap along the right edge instead of the left edge. Thus, we have acquired a few new regression equations.

Some of my research codes include these symmetries, but I excluded them here. Nowhere did I see that the reversal symmetry made noticable difference in results, but in coding, it makes a noticeable clutter by expanding the residual to a two-component residual array.

Where a data sample grows exponentially towards the boundary, I expect that extrapolated data would diverge too. You can force it to go to zero (or any specified value) at some distance from the body of the known data. To do so, surround the body of data by missing data and surround that by specification of ``enough'' zeros. ``Enough'' is defined by the filter length.