Next: MEDIANS IN AUTOREGRESSION Up: Claerbout: Medians in regression Previous: INTRODUCTION

# MEDIANS AND REGRESSION

Let ri 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.

Next: MEDIANS IN AUTOREGRESSION Up: Claerbout: Medians in regression Previous: INTRODUCTION
Stanford Exploration Project
11/12/1997