function result=qnm(x)
%QNM is a scale estimator which does not require an auxiliary location estimate.
% Essentially it is the first quartile of all pairwise distances between
% two data points. Its definition is given by
% qn(x)= c_n 2.2219{|x_i - x_j|; i<j}_(k)
% with c_n a small sample correction factor to make the qn unbiased at the
% normal distribution. It can resist 50% outliers.
% If x is a matrix, the scale estimate is computed on the columns of x. The
% result is then a row vector. If x is a row or a column vector,
% the output is a scalar.
%
% Reference:
% Rousseeuw P.J., Croux C. (1993),
% "Alternatives to the median absolute deviation",
% Journal of the American Statistical Association, 88, 1273-1283.
%
% Required input argument:
% x: either a data matrix with n observations in rows, p variables in columns
% or a vector of length n.
%
% I/O: result=qnm(x);
%
% This function is fully written in Matlab code. For a faster computation, use
% the qn.m function which calls qn.dll.
%
% This function is part of LIBRA: the Matlab Library for Robust Analysis,
% available at:
% http://wis.kuleuven.be/stat/robust.html
%
% Written by S. Verboven and Symen De Jong.
% Last Update: 03 April 2006
[n,p] = size(x);
if n==1 & p~=1
x=x';
[n,p]=size(x);
end
h = floor(n/2)+1;
Qn= zeros(1,p);
for j=1:p
dist = abs(repmat(x(:,j),1,n)-repmat(x(:,j)',n,1));
d = sort(dist(triu(ones(n),1)>0));
Qn(j) = 2.2219*d(h*(h-1)/2);
end
if n<=9
switch n
case 2
dn=0.399;
case 3
dn=0.994;
case 4
dn=0.512;
case 5
dn=0.844;
case 6
dn=0.611;
case 7
dn=0.857;
case 8
dn=0.669;
case 9
dn=0.872;
end
else
if mod(n,2)==1
dn=n/(n+1.4);
end
if mod(n,2)==0
dn=n/(n+3.8);
end
end
result=dn*Qn;