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L1 or L2 or The easiest method of model fitting is linear least squares.
This means minimizing the sums of squares of residuals ().
On the other hand, we often encounter huge noises
and it is much safer to minimize
the sums of absolute values of residuals
().
(The problem with is that there are multiple minima,
so the gradient is not a sensible way to seek the deepest).
There exist specialized techniques for handling multivariate fitting problems.
They should work better than the simple
iterative reweighting outlined here.

A penalty function that ranges from to ,depending on the constant is

| |
(9) |

Where
is small, the terms in the sum amount to and where
is large, the terms in the sum amount to .We define the residual as
| |
(10) |

We will need
| |
(11) |

where we briefly used the notation that is 1 when
*j*=*k* and zero otherwise.
Now,
to let us find the descent direction ,we will compute
the *k*-th component *g*_{k} of the gradient .We have
| |
(12) |

| |
(13) |

where we have use the notation to designate
a diagonal matrix with its argument distributed along the diagonal.
Continuing, we notice that the new weighting of residuals
has nothing to do with the linear relation between model perturbation
and residual perturbation;
that is,
we retain the familiar relations,
and
.

In practice we have the question of how to choose .I suggest that be proportional to
or some other percentile.

** Next:** Nonlinear L.S. conjugate-direction template
** Up:** MEANS, MEDIANS, PERCENTILES AND
** Previous:** Weighted L.S. conjugate-direction template
Stanford Exploration Project

4/27/2004