In order for the PEF to contain the inverse of the data spectrum,
it has to be causal.
The notion of causality is most obvious when there
is a single axis labeled ``time,'' but it extends readily
enough to two or more dimensions.
Your eye scans the page from left to right and top to bottom;
the words previous to the current one are to the left
on the same line and everywhere on the lines above.
This gives a sort of two-dimensional causal region.
Including all the words on previous pages gives a sort of
three-dimensional causal region.
This is somewhat arbitrary. In some other language the fast
and slow axes might be swapped or reversed.
The important thing is that along a line parallel to any axis
and going through the current point (the zero lag, the value 1*z ^{0}* in
equation pefdef), a causal region
lies only to one side of the current location.
Figure causal shows an illustration of a two-dimensional
causal region.
We can make a causal prediction of the data value in the
square labeled ``1'' from any or all of the other shaded squares.
The shaded squares are labeled to show the layout of
of a PEF with four adjustable coefficients.
The ``1'' is the value 1

Figure 1

Figure 2dpef shows a picture of a 2-D PEF, and Figure 3dpef
shows a picture of a 3-D PEF. In both cases, the dark shaded block
holds the 1, and the lighter blocks are the coefficients *a*(*n*)
calculated from the data.

2dpef
Form of a 2-D prediction error filter. The
shaded box holds the zero-lag coefficient, with a fixed value of 1.
Figure 2 |

3dpef
Form of a 3-D prediction error filter. The
shaded box holds the zero-lag coefficient, with a fixed value of 1.
Figure 3 |

1/18/2001