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+++ b/main/statistics/inst/gevfit_lmom.m	Sun Dec 30 19:47:17 2012 +0000
@@ -0,0 +1,113 @@
+## Copyright (C) 2012 Nir Krakauer <nkrakauer@ccny.cuny.edu>
+##
+## This program is free software; you can redistribute it and/or modify it under
+## the terms of the GNU General Public License as published by the Free Software
+## Foundation; either version 3 of the License, or (at your option) any later
+## version.
+##
+## This program is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
+## details.
+##
+## You should have received a copy of the GNU General Public License along with
+## this program; if not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{paramhat}, @var{paramci} =} gevfit_lmom (@var{data})
+## Find an estimator (@var{paramhat}) of the generalized extreme value (GEV) distribution fitting @var{data} using the method of L-moments.
+##
+## @subheading Arguments
+##
+## @itemize @bullet
+## @item
+## @var{data} is the vector of given values.
+## @end itemize
+##
+## @subheading Return values
+##
+## @itemize @bullet
+## @item
+## @var{parmhat} is the 3-parameter maximum-likelihood parameter vector [@var{k}; @var{sigma}; @var{mu}], where @var{k} is the shape parameter of the GEV distribution, @var{sigma} is the scale parameter of the GEV distribution, and @var{mu} is the location parameter of the GEV distribution.
+## @item
+## @var{paramci} has the approximate 95% confidence intervals of the parameter values (currently not implemented).
+## 
+## @end itemize
+##
+## @subheading Examples
+##
+## @example
+## @group
+## data = gevrnd (0.1, 1, 0, 100, 1);
+## [pfit, pci] = gevfit_lmom (data);
+## p1 = gevcdf (data,pfit(1),pfit(2),pfit(3));
+## [f, x] = ecdf (data);
+## plot(data, p1, 's', x, f)
+## @end group
+## @end example
+## @seealso{gevfit}
+## @subheading References
+##
+## @enumerate
+## @item
+## Ailliot, P.; Thompson, C. & Thomson, P. Mixed methods for fitting the GEV distribution, Water Resources Research, 2011, 47, W05551
+##
+## @end enumerate
+## @end deftypefn
+
+## Author: Nir Krakauer <nkrakauer@ccny.cuny.edu>
+## Description: L-moments parameter estimation for the generalized extreme value distribution
+
+function [paramhat, paramci] = gevfit_lmom (data)
+
+  # Check arguments
+  if (nargin < 1)
+    print_usage;
+  endif
+  
+  # find the L-moments
+  data = sort (data(:))';
+  n = numel(data);
+  L1 = mean(data);
+  L2 = sum(data .* (2*(1:n) - n - 1)) / (2*nchoosek(n, 2)); # or mean(triu(data' - data, 1, 'pack')) / 2;
+  b = bincoeff((1:n) - 1, 2);
+  L3 = sum(data .* (b - 2 * ((1:n) - 1) .* (n - (1:n)) + fliplr(b))) / (3*nchoosek(n, 3));
+
+  #match the moments to the GEV distribution
+  #first find k based on L3/L2
+  f = @(k) (L3/L2 + 3)/2 - limdiv((1 - 3^(k)), (1 - 2^(k)));
+  k = fzero(f, 0);
+  
+  #next find sigma and mu given k
+  if abs(k) < 1E-8
+    sigma = L2 / log(2);
+    eg = 0.57721566490153286; %Euler-Mascheroni constant
+    mu = L1 - sigma * eg;
+  else
+    sigma = -k*L2 / (gamma(1 - k) * (1 - 2^(k)));
+    mu = L1 - sigma * ((gamma(1 - k) - 1) / k);
+  endif
+  
+  paramhat = [k; sigma; mu];
+  
+  if nargout > 1
+    paramci = NaN;
+  endif
+endfunction
+
+#internal function to accurately evaluate (1 - 3^k)/(1 - 2^k) in the limit as k --> 0
+function c = limdiv(a, b)
+  # c = ifelse (abs(b) < 1E-8, log(3)/log(2), a ./ b);
+  if abs(b) < 1E-8
+    c = log(3)/log(2);
+  else
+    c = a / b;
+  endif
+endfunction
+
+
+%!test
+%! data = 1:50;
+%! [pfit, pci] = gevfit_lmom (data);
+%! expected_p = [-0.28 15.01 20.22]';
+%! assert (pfit, expected_p, 0.1);