## Copyright (C) 2008, 2009 Ben Abbott and Jaroslav Hajek
##
## This file is part of Octave.
##
## 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
## .
## -*- texinfo -*-
## @deftypefn {Function File} {@var{q} =} __quantile__ (@var{x}, @var{p})
## @deftypefnx {Function File} {@var{q} =} __quantile__ (@var{x}, @var{p}, @var{method})
## Undocumented internal function.
## @end deftypefn
## For the cumulative probability values in @var{p}, compute the
## quantiles, @var{q} (the inverse of the cdf), for the sample, @var{x}.
##
## The optional input, @var{method}, refers to nine methods available in R
## (http://www.r-project.org/). The default is @var{method} = 7. For more
## detail, see `help quantile'.
## @seealso{prctile, quantile, statistics}
## Author: Ben Abbott
## Vectorized version: Jaroslav Hajek
## Description: Quantile function of a empirical samples
function inv = __quantile__ (x, p, method = 5)
if (nargin < 2 || nargin > 3)
print_usage ();
endif
if (! ismatrix (x))
error ("quantile: x must be a matrix");
endif
## Save length and set shape of quantiles.
n = numel (p);
p = p(:);
## Save length and set shape of samples.
## FIXME: does sort guarantee that NaN's come at the end?
x = sort (x);
m = sum (! isnan (x));
mx = size (x, 1);
nx = size (x, 2);
## Initialize output values.
inv = Inf*(-(p < 0) + (p > 1));
inv = repmat (inv, 1, nx);
## Do the work.
if (any(k = find((p >= 0) & (p <= 1))))
n = length (k);
p = p (k);
## Special case.
if (mx == 1)
inv(k,:) = repmat (x, n, 1);
return
endif
## The column-distribution indices.
pcd = kron (ones (n, 1), mx*(0:nx-1));
mm = kron (ones (n, 1), m);
switch method
case {1, 2, 3}
switch method
case 1
p = max (ceil (kron (p, m)), 1);
inv(k,:) = x(p + pcd);
case 2
p = kron (p, m);
p_lr = max (ceil (p), 1);
p_rl = min (floor (p + 1), mm);
inv(k,:) = (x(p_lr + pcd) + x(p_rl + pcd))/2;
case 3
## Used by SAS, method PCTLDEF=2.
## http://support.sas.com/onlinedoc/913/getDoc/en/statug.hlp/stdize_sect14.htm
t = max (kron (p, m), 1);
t = roundb (t);
inv(k,:) = x(t + pcd);
endswitch
otherwise
switch method
case 4
p = kron (p, m);
case 5
## Used by Matlab.
p = kron (p, m) + 0.5;
case 6
## Used by Minitab and SPSS.
p = kron (p, m+1);
case 7
## Used by S and R.
p = kron (p, m-1) + 1;
case 8
## Median unbiased .
p = kron (p, m+1/3) + 1/3;
case 9
## Approximately unbiased respecting order statistics.
p = kron (p, m+0.25) + 0.375;
otherwise
error ("quantile: Unknown method, '%d'", method);
endswitch
## Duplicate single values.
imm1 = mm == 1;
x(2,imm1) = x(1,imm1);
## Interval indices.
pi = max (min (floor (p), mm-1), 1);
pr = max (min (p - pi, 1), 0);
pi += pcd;
inv(k,:) = (1-pr) .* x(pi) + pr .* x(pi+1);
endswitch
endif
endfunction
%!test
%! p = 0.5;
%! x = sort (rand (11));
%! q = __quantile__ (x, p);
%! assert (q, x(6,:))
%!test
%! p = [0.00, 0.25, 0.50, 0.75, 1.00];
%! x = [1; 2; 3; 4];
%! a = [1.0000 1.0000 2.0000 3.0000 4.0000
%! 1.0000 1.5000 2.5000 3.5000 4.0000
%! 1.0000 1.0000 2.0000 3.0000 4.0000
%! 1.0000 1.0000 2.0000 3.0000 4.0000
%! 1.0000 1.5000 2.5000 3.5000 4.0000
%! 1.0000 1.2500 2.5000 3.7500 4.0000
%! 1.0000 1.7500 2.5000 3.2500 4.0000
%! 1.0000 1.4167 2.5000 3.5833 4.0000
%! 1.0000 1.4375 2.5000 3.5625 4.0000];
%! for m = (1:9)
%! q = __quantile__ (x, p, m).';
%! assert (q, a(m,:), 0.0001)
%! endfor
%!test
%! p = [0.00, 0.25, 0.50, 0.75, 1.00];
%! x = [1; 2; 3; 4; 5];
%! a = [1.0000 2.0000 3.0000 4.0000 5.0000
%! 1.0000 2.0000 3.0000 4.0000 5.0000
%! 1.0000 1.0000 2.0000 4.0000 5.0000
%! 1.0000 1.2500 2.5000 3.7500 5.0000
%! 1.0000 1.7500 3.0000 4.2500 5.0000
%! 1.0000 1.5000 3.0000 4.5000 5.0000
%! 1.0000 2.0000 3.0000 4.0000 5.0000
%! 1.0000 1.6667 3.0000 4.3333 5.0000
%! 1.0000 1.6875 3.0000 4.3125 5.0000];
%! for m = (1:9)
%! q = __quantile__ (x, p, m).';
%! assert (q, a(m,:), 0.0001)
%! endfor
%!test
%! p = [0.00, 0.25, 0.50, 0.75, 1.00];
%! x = [1; 2; 5; 9];
%! a = [1.0000 1.0000 2.0000 5.0000 9.0000
%! 1.0000 1.5000 3.5000 7.0000 9.0000
%! 1.0000 1.0000 2.0000 5.0000 9.0000
%! 1.0000 1.0000 2.0000 5.0000 9.0000
%! 1.0000 1.5000 3.5000 7.0000 9.0000
%! 1.0000 1.2500 3.5000 8.0000 9.0000
%! 1.0000 1.7500 3.5000 6.0000 9.0000
%! 1.0000 1.4167 3.5000 7.3333 9.0000
%! 1.0000 1.4375 3.5000 7.2500 9.0000];
%! for m = (1:9)
%! q = __quantile__ (x, p, m).';
%! assert (q, a(m,:), 0.0001)
%! endfor
%!test
%! p = [0.00, 0.25, 0.50, 0.75, 1.00];
%! x = [1; 2; 5; 9; 11];
%! a = [1.0000 2.0000 5.0000 9.0000 11.0000
%! 1.0000 2.0000 5.0000 9.0000 11.0000
%! 1.0000 1.0000 2.0000 9.0000 11.0000
%! 1.0000 1.2500 3.5000 8.0000 11.0000
%! 1.0000 1.7500 5.0000 9.5000 11.0000
%! 1.0000 1.5000 5.0000 10.0000 11.0000
%! 1.0000 2.0000 5.0000 9.0000 11.0000
%! 1.0000 1.6667 5.0000 9.6667 11.0000
%! 1.0000 1.6875 5.0000 9.6250 11.0000];
%! for m = (1:9)
%! q = __quantile__ (x, p, m).';
%! assert (q, a(m,:), 0.0001)
%! endfor
%!test
%! p = [0.00, 0.25, 0.50, 0.75, 1.00];
%! x = [16; 11; 15; 12; 15; 8; 11; 12; 6; 10];
%! a = [6.0000 10.0000 11.0000 15.0000 16.0000
%! 6.0000 10.0000 11.5000 15.0000 16.0000
%! 6.0000 8.0000 11.0000 15.0000 16.0000
%! 6.0000 9.0000 11.0000 13.5000 16.0000
%! 6.0000 10.0000 11.5000 15.0000 16.0000
%! 6.0000 9.5000 11.5000 15.0000 16.0000
%! 6.0000 10.2500 11.5000 14.2500 16.0000
%! 6.0000 9.8333 11.5000 15.0000 16.0000
%! 6.0000 9.8750 11.5000 15.0000 16.0000];
%! for m = (1:9)
%! q = __quantile__ (x, p, m).';
%! assert (q, a(m,:), 0.0001)
%! endfor
%!test
%! p = [0.00, 0.25, 0.50, 0.75, 1.00];
%! x = [-0.58851; 0.40048; 0.49527; -2.551500; -0.52057; ...
%! -0.17841; 0.057322; -0.62523; 0.042906; 0.12337];
%! a = [-2.551474 -0.588505 -0.178409 0.123366 0.495271
%! -2.551474 -0.588505 -0.067751 0.123366 0.495271
%! -2.551474 -0.625231 -0.178409 0.123366 0.495271
%! -2.551474 -0.606868 -0.178409 0.090344 0.495271
%! -2.551474 -0.588505 -0.067751 0.123366 0.495271
%! -2.551474 -0.597687 -0.067751 0.192645 0.495271
%! -2.551474 -0.571522 -0.067751 0.106855 0.495271
%! -2.551474 -0.591566 -0.067751 0.146459 0.495271
%! -2.551474 -0.590801 -0.067751 0.140686 0.495271];
%! for m = (1:9)
%! q = __quantile__ (x, p, m).';
%! assert (q, a(m,:), 0.0001)
%! endfor
%!test
%! p = 0.5;
%! x = [0.112600, 0.114800, 0.052100, 0.236400, 0.139300
%! 0.171800, 0.727300, 0.204100, 0.453100, 0.158500
%! 0.279500, 0.797800, 0.329600, 0.556700, 0.730700
%! 0.428800, 0.875300, 0.647700, 0.628700, 0.816500
%! 0.933100, 0.931200, 0.963500, 0.779600, 0.846100];
%! tol = 0.00001;
%! x(5,5) = NaN;
%! assert (__quantile__ (x, p), [0.27950, 0.79780, 0.32960, 0.55670, 0.44460], tol);
%! x(1,1) = NaN;
%! assert (__quantile__ (x, p), [0.35415, 0.79780, 0.32960, 0.55670, 0.44460], tol);
%! x(3,3) = NaN;
%! assert (__quantile__ (x, p), [0.35415, 0.79780, 0.42590, 0.55670, 0.44460], tol);