FEATURES OF 1-D THAT APPLY TO MANY DIMENSIONS

Time-series analysis is rich with concepts that the helix now allows us to apply to many dimensions. First is the notion of an impulse function. Observe that an impulse function on the surface of the helical cylinder maps to an impulse function on the line of the unwound coil. An autocorrelation function that is an impulse corresponds both to a white spectrum in 2-D and to a white spectrum in 1-D.

The prediction-error-filter (PEF) unites many well established concepts in time-series analysis:

• Subtracting a time series from its prediction yields the prediction error.
• The method of least-squares is used to find the prediction filter. This is also called ``autoregression''.
• Textbooks such as PVI show that the spectrum of the output of the PEF tends towards whiteness (as the filter length increases). Thus the spectrum of the PEF tends to the inverse of that of the input.
• A time series can be decomposed into random impulses (white spectrum) convolved with a natural wavelet that is the inverse of the PEF.
• For any power spectrum, there is a causal wavelet (with that spectrum) that can be found by ``spectral factorization''. In the frequency domain this is known as the Kolmogoroff method.
• The PEF has the property of ``minimum phase'' which means that both it and its convolutional inverse are causal, and this means we have stable recursions.
• Stable filters can be modeled as layered media where waves resonate among reflection coefficients bounded by unity. Such models help in PEF estimation (Burg spectral method).

In summary, the (one-sided) PEF has magical mathematical properties and stable recursions. Symmetrical filters cannot be used recursively and do not have white outputs (which limits their usefulness). Therefore, let us use the helix idea to examine the two-dimensional manifestation of a PEF. For clarity, I adopt the convention that the zero-lag response of the one-dimensional PEF has the value ``1''. In one dimension, there are zeros before the ``1'' and adjustable values after it. Figure 2 shows such a filter wrapped on a helix. For most cases of interest, the significant filter coefficients cluster near the ``1'' and decay with distance (something like most autocorrelation functions). Supposing that nonzero filter coefficients lie within a short distance (two lags) from the ``1'', we can extract and display the coefficients of the 2-D PEF like this:  

$$\begin{array} {ccc} h & c & 0 \\ p & d & 0 \\ q & e & \bold 1 \\ s & f & a \\ u & g & b \end{array}$$ (4)

The adjustable values in the PEF, $a,b,c,\cdots,u$ are generally found by the method of least squares to minimize the power out of the PEF. Observe that the PEF has the ``$\bold 1$'' along its side, not at a corner (as might be guessed by the analogy of a one-dimensional PEF being one-sided). That a 2-D PEF has its ``1'' along its side is not a new result. It is in my textbook PVI, but its proof there is abstract and subtle so it is little known. The helix idea makes it far more clear that a 2-D PEF has its ``1'' along one of its sides.