Hello,
I have written a little script making the comparison between L1 and L2
norm. For the L1 case, I have used leastsq and cheated by returning the
square root of the abs of the residue and the 'nd' option.
function r=resid(coef,x,y,z)
r=sqrt(abs(z-(coef(1)+coef(2)*x+coef(3)*y)));
r=r(:);
endfunction
n=10;
[x,y]=meshgrid(linspace(-1,1,n));
z=1+x+y+rand(x,'normal')/10;
z(n,n)=20; // outlier
// L2 case
A=[x(:)^0 x(:) y(:)];
coefl2=A\z(:);
disp(coefl2);
// L1 case
[l1norm,coefl1]=leastsq(list(resid,x,y,z),coefl2,'nd');
disp(coefl1);
Things are not supposed to work so well because 'nd' is only meaning
that the function to minimize in non-differentiable at the optimum (and
only there....). However, it seems to work well (true value is [1,1,1]') :
L2 case :
1.165072
1.4380897
1.398408
L1 case :
0.9954939
1.0441875
0.9939874
S.
Le 04/03/13 18:51, Rafael Guerra a écrit :
Thanks Stéphane for the useful L1-references and for the insight on
iterative L2 methods and to the others for their repplies.
PS:
Strong outliers or spikes have infinite bandwidth and therefore bandpass
filtering/convolution does not seem, a priori, to be the most effective
method to remove them.
Regards,
Rafael
-----Original Message-----
From: users-boun...@lists.scilab.org [mailto:users-boun...@lists.scilab.org]
On Behalf Of Stéphane Mottelet
Sent: Monday, March 04, 2013 10:14 AM
To: users@lists.scilab.org
Subject: Re: [Scilab-users] Surface smoothing in Scilab, immune to outliers
Hello,
Replacing the squared L2 norm by the L1 norm in the linear regression gives
a good robustness to outliers (cf. Donoho and al. papers). The problem is
then non differentiable but you can implement it by iteratively reweighting
the classical L2 method (IRLS method), or by writing an equivalent linear
program.
S.
Le 04/03/13 13:23, Dang, Christophe a écrit :
Hello,
De la part de Rafael Guerra
Envoyé : lundi 4 mars 2013 04:37
Does somebody know if there are Scilab functions [...] that smooths
experimental data z=f(x,y) and is immune to strong outliers.
imho, the problem with smoothing and outliers is that the definition
of a outlier depends on the field.
How can Scilab know what a "strong outlier" is?
I personally would try Fourier filtering:
a strong outlier means a steep slope
and therefore correspond to a high frequency.
Thus fft2, set high frequencies to 0
(with possibly a smooth transition),
then inverse fft2 -- ifft2 does not exist, I never used 2-dimension
Fourier transform so I don't know if the inverse is easy to perform...
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