[Scilab-users] Surface smoothing in Scilab, immune to outliers

[Rafael Guerra]

Hello Stéphane,

 

Tried optim using a true L1-norm on your script.

It seems to work but results are not necessarily better than your previous
srqt trick:

 

function r=resid(coef, x, y, z)

    r = sum(abs(z-(coef(1)+coef(2)*x+coef(3)*y)));

endfunction

 

n=10;

[x,y]=meshgrid(linspace(-1,1,n));

z=1+x+y+rand(x,'normal')/10;

z(n,n)= 100; // outlier

z(5,5)= -200; // outlier

 

// L2 case

A=[x(:)^0 x(:) y(:)];

coefl2=A\z(:);

disp(coefl2);

 

// L1 case

coef0= [0; 0; 0];

[r,coefl1]= optim(list(NDcost,resid,x,y,z),coef0)

disp(coefl1);

 

Rgds

Rafael

 

From: users-bounces at lists.scilab.org [mailto:users-bounces at lists.scilab.org]
On Behalf Of Stéphane Mottelet
Sent: Monday, March 04, 2013 3:06 PM
To: users at lists.scilab.org
Subject: Re: [Scilab-users] Surface smoothing in Scilab, immune to outliers

 

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-bounces at lists.scilab.org [mailto:users-bounces at lists.scilab.org]
On Behalf Of Stéphane Mottelet
Sent: Monday, March 04, 2013 10:14 AM
To: users at 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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