Let's consider two different phenomena: Now suppose that you have some data on these two cases: in the first case, you have everyone's location and several peoples' temperatures at the given time; in the second case, you have "stain index" for an equivalent proportion of the carpet.

Ask yourself this: if you want to estimate (guess) the temperature of a guest Bob, do you

  1. use the average temperature of the guests in the data group?
  2. use 98.6 degrees fahrenheit? or
  3. use the known guests' temperatures, weighting them according to their actual distance from Bob at that time?
In most cases (barring snuggling, for example) neighboring guests' temperatures don't have anything more to do with Bob's temperature than any other guest's temperature does, so you would probably elect for either 1) or 2) above.

On the other hand, if you were trying to guess how stained a particular patch of carpet is, sight unseen, you might be most interested in knowing about the carpet in the vicinity of the patch. That is, the closer a neighboring patch of carpet, the more it has to do with the patch of interest.

The difference in the two cases is that the first statistic of interest is spatially random, and the second shows spatial correlation. Mappers faced with the first case would probably use the mean temperature (or some other measure of central tendency) of the group. In the second case, they would try to quantify the spatial correlation, and use it to improve their estimates based on how helpful they expect a given neighbor to be.

We quantify that correlation with a function called the "variogram". Here's an example (click on it!): . The data themselves give us some idea of what the variogram looks like, then we model it with a function which allows us to do our estimation by quantifying the relative confidence (trust) we have in neighbors as a function of distance. We would like to see a variogram that increases smoothly as a function of distance, which says that trust falls off smoothly with distance. This one, a variogram of an image of Abe Lincoln's face, shows a degree of trust that is lost by about eight pixel lengths.

Here are some two-dimensional variograms (showing variation as a function of distance AND direction/angle). Here's one of a random scattering of data , and one of Abe Lincoln's face. Obviously the random variogram looks random too. In fact, all the values are roughly the same, which is to say that in inspecting the variogram we find that we can't trust a close neighbor any more than one at great distance.

In the case of Abe, however, we see that the variogram is very smooth. It is zero top-center, and increases from the white to the red (highest values). We trust neighbors more that are located in the yellow and green directions - that is, vertically more than horizontally - as there is more variation in his face in a horizontal direction.

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