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Let *r*_{i} be components of a residual vector and consider perturbation components .We study the properties of the new residual

| |
(1) |

where is a scaling factor.
Consider the implications of
| |
(2) |

| (3) |

Thus choosing
causes
the median of to vanish
which means has as many positive terms
as negative ones.
Consequently, the number of polarity agreements
in the component pairs of the new residual and the regressor equals their polarity disagreements.
In other words,
| |
(4) |

In what sense have we created the smallest new residual ?
It is smallest in the sense that if we change by adding or subtracting some of the perturbation ,the count of growing components less the decreasing components
of is
| |
(5) |

The best residual is one in which any perturbation
serves only to increase the polarity agreements with the regressor.
We chose to reduce as many components
of as we could.

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Stanford Exploration Project

11/12/1997