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## -*- texinfo -*-
## @deftypefn {} {@var{q} =} quantile (@var{x})
## @deftypefnx {} {@var{q} =} quantile (@var{x}, @var{p})
## @deftypefnx {} {@var{q} =} quantile (@var{x}, @var{p}, @var{dim})
## @deftypefnx {} {@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}.
##
## If @var{p} is unspecified, return the quantiles for
## @code{[0.00 0.25 0.50 0.75 1.00]}.
## The optional argument @var{dim} determines the dimension along which
## the quantiles are calculated. If @var{dim} is omitted it defaults to
## the first non-singleton dimension.
##
## The methods available to calculate sample quantiles are the nine methods
## used by R (@url{https://www.r-project.org/}). The default value is
## @w{@var{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
## @tex
## $p(k)$
## @end tex
## @ifnottex
## @var{p}(k)
## @end ifnottex
## is the linear
## interpolation function respecting each method's representative cdf.
##
## @enumerate 4
## @item Method 4:
## @tex
## $p(k) = k / N$.
## @end tex
## @ifnottex
## @var{p}(k) = k / N.
## @end ifnottex
## That is, linear interpolation of the empirical cdf, where @math{N} is the
## length of @var{P}.
##
## @item Method 5:
## @tex
## $p(k) = (k - 0.5) / N$.
## @end tex
## @ifnottex
## @var{p}(k) = (k - 0.5) / N.
## @end ifnottex
## That is, a piecewise linear function where the knots are the values midway
## through the steps of the empirical cdf.
##
## @item Method 6:
## @tex
## $p(k) = k / (N + 1)$.
## @end tex
## @ifnottex
## @var{p}(k) = k / (N + 1).
## @end ifnottex
##
## @item Method 7:
## @tex
## $p(k) = (k - 1) / (N - 1)$.
## @end tex
## @ifnottex
## @var{p}(k) = (k - 1) / (N - 1).
## @end ifnottex
##
## @item Method 8:
## @tex
## $p(k) = (k - 1/3) / (N + 1/3)$.
## @end tex
## @ifnottex
## @var{p}(k) = (k - 1/3) / (N + 1/3).
## @end ifnottex
## The resulting quantile estimates are approximately median-unbiased
## regardless of the distribution of @var{x}.
##
## @item Method 9:
## @tex
## $p(k) = (k - 3/8) / (N + 1/4)$.
## @end tex
## @ifnottex
## @var{p}(k) = (k - 3/8) / (N + 1/4).
## @end ifnottex
## The resulting quantile estimates are approximately unbiased for the
## expected order statistics if @var{x} is normally distributed.
## @end enumerate
##
## @nospell{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 @nospell{Becker, R. A., Chambers, J. M. and Wilks, A. R.} (1988)
## The New S Language. @nospell{Wadsworth & Brooks/Cole}.
##
## @item @nospell{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{https://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
function q = quantile (x, p = [], dim, method = 5)
if (nargin < 1)
print_usage ();
endif
if (! (isnumeric (x) || islogical (x)) || isempty (x))
error ("quantile: X must be a non-empty 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 (nargin < 3)
## Find the first non-singleton dimension.
(dim = find (size (x) > 1, 1)) || (dim = 1);
else
if (!(isscalar (dim) && dim == fix (dim)) || !(1 <= dim))
error ("quantile: DIM must be an integer and a valid dimension");
endif
endif
## Set the permutation vector.
perm = 1:(max (ndims (x), dim));
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), []);
## 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);
## For Matlab compatibility, return vectors with the same orientation as p
if (isvector (q) && ! isscalar (q) && ! isscalar (p))
if (isrow (p))
q = reshape (q, 1, []);
else
q = reshape (q, [], 1);
endif
endif
endfunction
%!test
%! p = 0.50;
%! q = quantile (1:4, p);
%! qa = 2.5;
%! assert (q, qa);
%! q = quantile (1:4, p, 1);
%! qa = [1, 2, 3, 4];
%! assert (q, qa);
%! q = quantile (1:4, p, 2);
%! qa = 2.5;
%! assert (q, qa);
%!test
%! p = [0.50 0.75];
%! q = quantile (1:4, p);
%! qa = [2.5 3.5];
%! assert (q, qa);
%! q = quantile (1:4, p, 1);
%! qa = [1, 2, 3, 4; 1, 2, 3, 4];
%! assert (q, qa);
%! q = quantile (1:4, p, 2);
%! qa = [2.5 3.5];
%! assert (q, qa);
%!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);
%!assert <*45455> (quantile ([1 3 2], 0.5, 1), [1 3 2])
%!assert <*54421> (quantile ([1:10], 0.5, 1), 1:10)
%!assert <*54421> (quantile ([1:10]', 0.5, 2), [1:10]')
%!assert <*54421> (quantile ([1:10], [0.25, 0.75]), [3, 8])
%!assert <*54421> (quantile ([1:10], [0.25, 0.75]'), [3; 8])
%!assert (quantile ([1:10], 1, 3), [1:10])
## Test input validation
%!error quantile ()
%!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 ((1:5)', 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
## (https://www.r-project.org/). The default is @var{method} = 7.
## @seealso{prctile, quantile, statistics}
## Description: Quantile function of empirical samples
function inv = __quantile__ (x, p, method = 5)
if (nargin < 2)
print_usage ("quantile");
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.
x = sort (x, 1);
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,:) == 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