The basic idea of least-squares fitting is that the residual is orthogonal to the fitting functions. Applied to the PE filter, this idea means that the output of a PE filter is orthogonal to lagged inputs. The orthogonality applies only for lags in the past, because prediction knows only the past while it aims to the future. What we want to show here is different, namely, that the output is uncorrelated with itself (as opposed to the input) for lags in both directions; hence the output spectrum is white.
In (21) are two separate and independent autoregressions, for finding the filter , and for finding the filter . By noticing that the two matrices are really the same (except a row of zeros on the bottom of is a row in the top of ) we realize that the two regressions must result in the same filters , and the residual is a shifted version of . In practice, I visualize the matrix being a thousand components tall (or a million) and a hundred components wide.
Our goal is a different theorem that is imprecise when applied to the three coefficient filters displayed in (21), but becomes valid as the filter length tends to infinity and the matrices become infinitely wide. Actually, all we require is the last component in , namely tend to zero. This generally happens because as increases, becomes a weaker and weaker predictor of .
Here's a mathematical fact we soon need: For any vectors and , if and , then and and for any and .
The matrix contains all of the columns that are found in except the last (and the last one is not important). This means that is not only orthogonal to all of 's columns (except the first) but is also orthogonal to all of 's columns except the last. Although isn't really perpendicular to the last column of , it doesn't matter because that column has hardly any contribution to since . Because is (effectively) orthogonal to all the components of , is also orthogonal to itself.
Here is a detail: In choosing the example of equation (21), I have shifted the two fitting problems by only one lag. We would like to shift by more lags and get the same result. For this we need more filter coefficients. By adding many more filter coefficients we are adding many more columns to the right side of . That's good because we'll be needing to neglect more columns as we shift further from . Neglecting these columns is commonly justified by the experience that ``after short range regressors have had their effect, long range regressors generally find little remaining to predict.'' (Recall that the damped harmonic oscillator from physics, the finite difference equation that predicts the future from the past, uses only two lags.)
Here is the main point: Since and both contain the same signal but time-shifted, the orthogonality at all shifts means that the autocorrelation of vanishes at all lags. An exception, of course, is at zero lag. The autocorrelation does not vanish there because is not orthogonal to its first column (because we did not minimize with respect to ).
As we redraw for various lags, we may shift the columns only downward because shifting them upward would bring in the first column of and the residual is not orthogonal to that. Thus we have only proven that one side of the autocorrelation of vanishes. That is enough however, because autocorrelation functions are symmetric, so if one side vanishes, the other must also.
If and were two-sided filters like the proof would break. If were two-sided, would catch the nonorthogonal column of . Not only is not proven to be perpendicular to the first column of , but it cannot be orthogonal to it because a signal cannot be orthogonal to itself.
The implications of this theorem are far reaching. The residual , a convolution of with has an autocorrelation that is an impulse function. The Fourier transform of an impulse is a constant. Thus the spectrum of the residual is ``white''. Thus and have mutually inverse spectra.
|Since the output of a PEF is white, the PEF itself has a spectrum inverse to its input.|
An important application of the PEF is in missing data interpolation. We'll see examples later in this chapter. My third book, PVIhas many examplesin one dimension with both synthetic data and field data including the gap parameter. Here we next extend these ideas to two (or more) dimensions.