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## Copyright (C) 2009 Esteban Cervetto <estebancster@gmail.com>
##
## Octave 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.
##
## Octave 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 Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{quotas} =} quotapanning (@var{s})
## Calculate the cumulative quotas by the Panning method.
##
## @var{s} is a mxn matrix that contains the run-off triangle, where m is the number of accident-years
## and n is the number of periods to final development. @var{s} may contain u = m-n complete years.
## The value @var{s}(i,k), 1<=i<=m, 0<=k<=n-1 represents the cumulative losses from accident-period i
## settled with a delay of at most k years.
## The values @var{s}(i,k) with i + k > m must be zero because is future time.
##
## The Panning method asumes that exists a development pattern on the incremental ratios.
## This means that the identity
##
## @verbatim
## E[Z(i,k) ]
## B(k) = ------------
## E[Z(i,0) ]
## @end verbatim
##
## holds for all k = @{0, @dots{}, n-1@} and for all i = @{1, @dots{}, m@}.
## Z represents the incremental losses; then losses satisfy
## Z(k) = (S(k) - S(k-1) ),Z(0) = S(0) for all i = @{1, @dots{}, m@}.
##
## @var{quotas} returns a row vector with the cumulative quotas. The transformation
## from incremental ratios to cumulative quotas is:
##
## @verbatim
## l=k
## E B(l)
## l=0
## quotas(k) = -----------
## l=n-1
## E B(l)
## l=0
## @end verbatim
##
## @seealso {bferguson, ultimatepanning, quotald, quotaad, quotamack}
## @end deftypefn
function quotas = quotapanning (S)
[m,n] = size (S); #triangle with m years (i=1,2,u,...u+1,u+2,....m) and n periods (k=0,1,2,...n-1)
u = m - n; #rows of the upper square
S = fliplr(triu(fliplr(S),-u)); #ensure S is triangular
# calculate Z
Z = [S(:,1), S(:,2:n)-S(:,1:n-1)];
Z = fliplr(triu(fliplr(Z),-u)); #clean Z
# calc empirical values of the incremental factors
B = Z ./ (Z * [ones(n,1),zeros(n,n-1)]');
# weights Z(i,0)^2/Z(0)^2
W = repmat((Z(:,1).^2),1,n); #numerator
W =fliplr(triu(fliplr(W),-u)); #clean low triangle
a = repmat(sum(W),m,1); #denominator
a = fliplr(triu(fliplr(a),-u)); #clean low triangle
W = W./a; #divido
W = fliplr(triu(fliplr(W),-u)); #clean low triangle
# Pannings incremental factors
B_Pan = diag(B' * W)'; #weighted product
quotas = cumsum(porcentual(B_Pan)); #cumulated quota
end