X-Git-Url: https://git.creatis.insa-lyon.fr/pubgit/?a=blobdiff_plain;f=octave_packages%2Fm%2Fstatistics%2Fbase%2Fquantile.m;fp=octave_packages%2Fm%2Fstatistics%2Fbase%2Fquantile.m;h=bb6353e7729ceb61d7c5e694a86eff00ebc43d14;hb=1c0469ada9531828709108a4882a751d2816994a;hp=0000000000000000000000000000000000000000;hpb=63de9f36673d49121015e3695f2c336ea92bc278;p=CreaPhase.git
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+## Copyright (C) 2008-2012 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{dim})
+## @deftypefnx {Function File} {@var{q} =} quantile (@var{x}, @var{p}, @var{dim}, @var{method})
+## For a sample, @var{x}, calculate the quantiles, @var{q}, corresponding to
+## the cumulative probability values in @var{p}. All non-numeric values (NaNs)
+## of @var{x} are ignored.
+##
+## If @var{x} is a matrix, compute the quantiles for each column and
+## return them in a matrix, such that the i-th row of @var{q} contains
+## the @var{p}(i)th quantiles of each column of @var{x}.
+##
+## The optional argument @var{dim} determines the dimension along which
+## the quantiles are calculated. If @var{dim} is omitted, and @var{x} is
+## a vector or matrix, it defaults to 1 (column-wise quantiles). If
+## @var{x} is an N-D array, @var{dim} defaults to the first non-singleton
+## dimension.
+##
+## The methods available to calculate sample quantiles are the nine methods
+## used by R (http://www.r-project.org/). The default value is METHOD = 5.
+##
+## Discontinuous sample quantile methods 1, 2, and 3
+##
+## @enumerate 1
+## @item Method 1: Inverse of empirical distribution function.
+##
+## @item Method 2: Similar to method 1 but with averaging at discontinuities.
+##
+## @item Method 3: SAS definition: nearest even order statistic.
+## @end enumerate
+##
+## Continuous sample quantile methods 4 through 9, where p(k) is the linear
+## interpolation function respecting each methods' representative cdf.
+##
+## @enumerate 4
+## @item Method 4: p(k) = k / n. That is, linear interpolation of the
+## empirical cdf.
+##
+## @item Method 5: p(k) = (k - 0.5) / n. That is a piecewise linear function
+## where the knots are the values midway through the steps of the empirical
+## cdf.
+##
+## @item Method 6: p(k) = k / (n + 1).
+##
+## @item Method 7: p(k) = (k - 1) / (n - 1).
+##
+## @item Method 8: p(k) = (k - 1/3) / (n + 1/3). The resulting quantile
+## estimates are approximately median-unbiased regardless of the distribution
+## of @var{x}.
+##
+## @item Method 9: p(k) = (k - 3/8) / (n + 1/4). The resulting quantile
+## estimates are approximately unbiased for the expected order statistics if
+## @var{x} is normally distributed.
+## @end enumerate
+##
+## Hyndman and Fan (1996) recommend method 8. Maxima, S, and R
+## (versions prior to 2.0.0) use 7 as their default. Minitab and SPSS
+## use method 6. @sc{matlab} uses method 5.
+##
+## References:
+##
+## @itemize @bullet
+## @item Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New
+## S Language. Wadsworth & Brooks/Cole.
+##
+## @item Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in
+## statistical packages, American Statistician, 50, 361--365.
+##
+## @item R: A Language and Environment for Statistical Computing;
+## @url{http://cran.r-project.org/doc/manuals/fullrefman.pdf}.
+## @end itemize
+##
+## Examples:
+## @c Set example in small font to prevent overfull line
+##
+## @smallexample
+## @group
+## x = randi (1000, [10, 1]); # Create empirical data in range 1-1000
+## q = quantile (x, [0, 1]); # Return minimum, maximum of distribution
+## q = quantile (x, [0.25 0.5 0.75]); # Return quartiles of distribution
+## @end group
+## @end smallexample
+## @seealso{prctile}
+## @end deftypefn
+
+## Author: Ben Abbott
+## Description: Matlab style quantile function of a discrete/continuous distribution
+
+function q = quantile (x, p = [], dim = 1, method = 5)
+
+ if (nargin < 1 || nargin > 4)
+ print_usage ();
+ endif
+
+ if (! (isnumeric (x) || islogical (x)))
+ error ("quantile: X must be a numeric vector or matrix");
+ endif
+
+ if (isempty (p))
+ p = [0.00 0.25, 0.50, 0.75, 1.00];
+ endif
+
+ if (! (isnumeric (p) && isvector (p)))
+ error ("quantile: P must be a numeric vector");
+ endif
+
+ if (!(isscalar (dim) && dim == fix (dim))
+ || !(1 <= dim && dim <= ndims (x)))
+ error ("quantile: DIM must be an integer and a valid dimension");
+ endif
+
+ ## Set the permutation vector.
+ perm = 1:ndims(x);
+ perm(1) = dim;
+ perm(dim) = 1;
+
+ ## Permute dim to the 1st index.
+ x = permute (x, perm);
+
+ ## Save the size of the permuted x N-d array.
+ sx = size (x);
+
+ ## Reshape to a 2-d array.
+ x = reshape (x, [sx(1), prod(sx(2:end))]);
+
+ ## Calculate the quantiles.
+ q = __quantile__ (x, p, method);
+
+ ## Return the shape to the original N-d array.
+ q = reshape (q, [numel(p), sx(2:end)]);
+
+ ## Permute the 1st index back to dim.
+ q = ipermute (q, perm);
+
+endfunction
+
+
+%!test
+%! p = 0.5;
+%! x = sort (rand (11));
+%! q = quantile (x, p);
+%! assert (q, x(6,:))
+%! x = x.';
+%! q = quantile (x, p, 2);
+%! 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, 1, 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, 1, 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, 1, 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, 1, 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, 1, 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, 1, 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, 1), [0.27950, 0.79780, 0.32960, 0.55670, 0.44460], tol);
+%! x(1,1) = NaN;
+%! assert (quantile(x, p, 1), [0.35415, 0.79780, 0.32960, 0.55670, 0.44460], tol);
+%! x(3,3) = NaN;
+%! assert (quantile(x, p, 1), [0.35415, 0.79780, 0.42590, 0.55670, 0.44460], tol);
+
+%!test
+%! sx = [2, 3, 4];
+%! x = rand (sx);
+%! dim = 2;
+%! p = 0.5;
+%! yobs = quantile (x, p, dim);
+%! yexp = median (x, dim);
+%! assert (yobs, yexp);
+
+%% Test input validation
+%!error quantile ()
+%!error quantile (1, 2, 3, 4, 5)
+%!error quantile (['A'; 'B'], 10)
+%!error quantile (1:10, [true, false])
+%!error quantile (1:10, ones (2,2))
+%!error quantile (1, 1, 1.5)
+%!error quantile (1, 1, 0)
+%!error quantile (1, 1, 3)
+%!error quantile ((1:5)', 0.5, 1, 0)
+%!error quantile ((1:5)', 0.5, 1, 10)
+
+## 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 empirical samples
+
+function inv = __quantile__ (x, p, method = 5)
+
+ if (nargin < 2 || nargin > 3)
+ print_usage ();
+ endif
+
+ if (isinteger (x) || islogical (x))
+ x = double (x);
+ endif
+
+ ## set shape of quantiles to column vector.
+ 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));
+ [xr, xc] = size (x);
+
+ ## Initialize output values.
+ inv = Inf (class (x)) * (-(p < 0) + (p > 1));
+ inv = repmat (inv, 1, xc);
+
+ ## Do the work.
+ if (any (k = find ((p >= 0) & (p <= 1))))
+ n = length (k);
+ p = p(k);
+ ## Special case of 1 row.
+ if (xr == 1)
+ inv(k,:) = repmat (x, n, 1);
+ return;
+ endif
+
+ ## The column-distribution indices.
+ pcd = kron (ones (n, 1), xr*(0:xc-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
+