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function [f,g,h,s,k] = studentst(r,k,s)
% Students T penalty with 'auto-tuning'
%
% use:
% [f,g,h,{k,{s}}] = studentst(r) - automatically fits s and k
% [f,g,h,{k,{s}}] = studentst(r,k) - automatically fits s
% [f,g,h,{k,{s}}] = studentst(r,k,s) - use given s and k
%
% input:
% r - residual as column vector
% s - scale (optional)
% k - degrees of freedom (optional)
%
% output:
% f - misfit (scalar)
% g - gradient (column vector)
% h - positive approximation of the Hessian (column vector, Hessian is a diagonal matrix)
% s,k - scale and degrees of freedom
%
% Tristan van Leeuwen, 2012.
% tleeuwen@eos.ubc.ca
% fit both s and k
if nargin == 1
opts = optimset('maxFunEvals',1e2);
tmp = fminsearch(@(x)st(r,x(1),x(2)),[1;2],opts);
s = tmp(1);
k = tmp(2);
end
if nargin == 2
opts = optimset('maxFunEvals',1e2);
tmp = fminsearch(@(x)st(r,x,k),[1],opts);
s = tmp(1);
end
% evaulate penalty
[f,g,h] = st(r,s,k);
function [f,g,h] = st(r,s,k)
n = length(r);
c = -n*(gammaln((k+1)/2) - gammaln(k/2) - .5*log(pi*s*k));
f = c + .5*(k+1)*sum(log(1 + conj(r).*r/(s*k)));
g = (k+1)*r./(s*k + conj(r).*r);
h = (k+1)./(s*k + conj(r).*r);
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