Multidimensional autoregression

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## Hexagonal coordinates

In a two-dimensional plane it seems that the one-sidedness of the PEF could point in any direction. Since we usually have a rectangular mesh, however, we can only do the calculations along the axes so we have only two possibilities, the helix can wrap around the 1-axis, or it can wrap around the 2-axis.

Suppose you acquire data on a hexagonal mesh as below

```                . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
```

and some of the data values are missing. How can we apply the methods of this chapter? The solution is to append the given data by more missing data shown by the commas below.

```                 . . . . . . . . . . . . . . . . , , , , , ,
. . . . . . . . . . . . . . . . , , , , , ,
, . . . . . . . . . . . . . . . . , , , , ,
, . . . . . . ._._._._._._. . . . , , , , ,
, , . ._._._._/_/ . . . . / . . . . , , , ,
, , . / . . . . . . . . . / . . . . , , , ,
, , , / . . . . . . . . . / . . . . . , , ,
, , , /_._._._._._._._._._/ . . . . . , , ,
, , , , . . . . . . . . . . . . . . . . , ,
, , , , . . . . . . . . . . . . . . . . , ,
, , , , , . . . . . . . . . . . . . . . . ,
, , , , , . . . . . . . . . . . . . . . . ,
, , , , , , . . . . . . . . . . . . . . . .
```

Now we have a familiar two-dimensional coordinate system in which we can find missing values, as well as perform signal and noise separations as described in a later chapter.

2011-08-27