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Consider four numerical values, say (5,7,98,100).
The median value is the middle value.
Since this is an even number of points, there is no middle.
A way to define the median *m* mathematically is to choose *m*
to minimize
| |
(1) |

If you plot the function *q*(*m*) you find it has a flat spot between
*m*=7 and *m*=98.
This illustrates a principle of *L*_{1} optimizations:
The minimum residual is unique.
The model, however, is not unique, but is a range.
This is an interesting feature of *L*_{1}
which differs from our usual least squares *L*_{2} experience
where we never get intervals.
With *L*_{2}, solutions are unique, except for a possible null space,
an infinite family of added solutions.
Having a 5th data point, even if very weakly weighted,
would resolve the ambiguity
so we might be on the track of a Busch-like solution.

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Stanford Exploration Project

4/27/2000