Numerical recipes

15.7 Robust Estimation 701
where the function ρ is the negative logarithm of the probability density. Taking the
logarithm of (15.7.1) analogously with (15.1.4), we find that we want to minimize
the expression
N�
ρ(yi,y{xi; a}) (15.7.2)
i=1
Very often, it is the case that the function ρ depends not independently on its
two arguments, measured yi and predicted y(xi), but only on their difference, at least
if scaled by some weight factors σi which we are able to assign to each point. In this
case the M-estimate is said to be local, and we can replace (15.7.2) by the prescription
minimize over a
N�
� �
yi − y(xi; a)
ρ
where the function ρ(z) is a function of a single variable z ≡ [y i − y(xi)]/σi.
If we now define the derivative of ρ(z) to be a function ψ(z),
ψ(z) ≡ dρ(z)
dz
then the generalization of (15.1.7) to the case of a general M-estimate is
0=
N�
σi
i=1
i=1
� �� �
1 yi − y(xi) ∂y(xi; a)
ψ
σi ∂ak
σi
(15.7.3)
(15.7.4)
k =1,...,M (15.7.5)
If you compare (15.7.3) to (15.1.3), and (15.7.5) to (15.1.7), you see at once
that the specialization for normally distributed errors is
ρ(z) = 1
2 z2 ψ(z) =z (normal) (15.7.6)
If the errors are distributed as a double or two-sided exponential, namely
� � ��
�yi
Prob {yi − y(xi)} ∼exp − �
− y(xi) �
�
� �
(15.7.7)
then, by contrast,
ρ(x) =|z| ψ(z) =sgn(z) (double exponential) (15.7.8)
Comparing to equation (15.7.3), we see that in this case the maximum likelihood
estimator is obtained by minimizing the mean absolute deviation, rather than the
mean square deviation. Here the tails of the distribution, although exponentially
decreasing, are asymptotically much larger than any corresponding Gaussian.
A distribution with even more extensive — therefore sometimes even more
realistic — tails is the Cauchy or Lorentzian distribution,
Prob {yi − y(xi)} ∼
σi
1
1+ 1
� �2 yi − y(xi)
2
σi
(15.7.9)
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