## Copyright (C) 2012 Rik Wehbring
## Copyright (C) 1995-2015 Kurt Hornik
## Copyright (C) 2015 Lachlan Andrew
##
## 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} {} binoinv (@var{x}, @var{n}, @var{p})
## For each element of @var{x}, compute the quantile (the inverse of the CDF)
## at @var{x} of the binomial distribution with parameters
## @var{n} and @var{p}, where @var{n} is the number of trials and
## @var{p} is the probability of success.
##
## See also binopdf, binocdf
## @end deftypefn
## Author: Lachlan Andrew
## based on code by
## KH
## Description: Quantile function of the binomial distribution
function inv = binoinv_new (x, n, p)
if (nargin != 3)
print_usage ();
endif
if (! isscalar (n) || ! isscalar (p))
[retval, x, n, p] = common_size (x, n, p);
if (retval > 0)
error ("binoinv: X, N, and P must be of common size or scalars");
endif
endif
our_type = [x(1), n(1), p(1)]; # complex if any complex, single if any single
if (iscomplex (our_type))
error ("binoinv: X, N, and P must not be complex");
endif
if (isa (our_type, "single"))
inv = zeros (size (x), "single");
else
inv = zeros (size (x));
endif
k = (!(x >= 0) | !(x <= 1) | !(n >= 0) | (n != fix (n)) |
!(p >= 0) | !(p <= 1));
inv(k) = NaN;
k = find ((x >= 0) & (x <= 1) & (n >= 0) & (n == fix (n)
& (p >= 0) & (p <= 1)));
if (any (k))
x = x(k);
if (isscalar (n) && isscalar (p))
[inv(k), unfinished] = scalar_binoinv (x(:), n, p);
k = k(unfinished);
if (!isempty (k))
inv(k) = bin_search_binoinv (x(k), n, p);
endif
else
[inv(k), unfinished] = vector_binoinv (x(:), n(:), p(:));
k = k(unfinished);
if (!isempty (k))
inv(k) = bin_search_binoinv (x(k), n(k), p(k));
endif
endif
endif
endfunction
## "core" algorithm to calculate the inverse negative binomial, for n and p
## real scalars and y a column vector, for which the output is not NaN or Inf.
## Compute cdf in batches of doubling size until cdf > x, or answer>500
## Returns the locations of unfinished cases in k.
function [m, k] = scalar_binoinv (x, n, p)
k = 1:length(x);
m = zeros (size (x));
prev_limit = 0;
limit = 10;
cdf = 0;
v = 0;
while (!isempty (k) && limit < 1e3)
cdf = binocdf(prev_limit:limit-1, n, p);
r = bsxfun (@le, x(k), cdf);
[v, m(k)] = max (r, [], 2); # find first instance of x <= cdf
m(k) += prev_limit - 1;
k = k(v == 0);
prev_limit = limit;
limit += limit;
endwhile
endfunction
## "core" algorithm to calculate the inverse negative binomial, for n and p
## real scalars and y a column vector, for which the output is not NaN or Inf.
## Compute cdf in batches of doubling size until cdf > x, or answer>500
## Returns the locations of unfinished cases in k.
## Calculates cdf by summing pdf, which is faster than calls to binocdf.
function [m, k] = vector_binoinv (x, n, p)
k = 1:length(x);
m = zeros (size (x));
prev_limit = 0;
limit = 10;
cdf = 0;
v = 0;
while (!isempty (k) && limit < 1e3)
xx = repmat (prev_limit:limit-1, [length(k), 1]);
nn = kron(ones(1,limit-prev_limit), n(k));
pp = kron(ones(1,limit-prev_limit), p(k));
pdf = binopdf(xx, nn, pp);
pdf(:,1) += cdf(v==0,end);
cdf = cumsum(pdf, 2);
r = bsxfun (@le, x(k), cdf);
[v, m(k)] = max (r, [], 2); # find first instance of x <= cdf
m(k) += prev_limit - 1;
k = k(v == 0);
prev_limit = limit;
limit += min (limit, max (1e4/numel (k), 10)); % limit memory use
endwhile
endfunction
## vectorized binary search.
## Can handle vectors n and p, and is faster than the scalar case when the
## answer is large.
## Could be optimized to call binocdf only for a subset of the x at each stage,
## but care must be taken to handle both scalar and vector n,p. Bookkeeping
## may cost more than the extra computations.
function m = bin_search_binoinv (x, n, p)
k = 1:length(x);
lower = zeros (size (x));
limit = 500; # lower bound on point at which prev phase finished
while (any (k) && limit < 1e100)
cdf = binocdf(limit, n, p);
k = (x > cdf);
lower(k) = limit;
limit += limit;
end
upper = max (2*lower, 1);
k = find (lower != limit/2); # elements for which above loop finished
for i = 1:ceil (log2 (max (lower)))
mid = (upper + lower)/2;
cdf = binocdf(floor(mid(:)), n, p);
r = (x <= cdf);
upper(r) = mid(r);
lower(!r) = mid(!r);
endfor
m = ceil (lower);
m(x > binocdf (m(:), n, p))++; # fix off-by-one errors from binary search
endfunction
%!shared x
%! x = [-1 0 0.5 1 2];
%!assert (binoinv (x, 2*ones (1,5), 0.5*ones (1,5)), [NaN 0 1 2 NaN])
%!assert (binoinv (x, 2, 0.5*ones (1,5)), [NaN 0 1 2 NaN])
%!assert (binoinv (x, 2*ones (1,5), 0.5), [NaN 0 1 2 NaN])
%!assert (binoinv (x, 2*[0 -1 NaN 1.1 1], 0.5), [NaN NaN NaN NaN NaN])
%!assert (binoinv (x, 2, 0.5*[0 -1 NaN 3 1]), [NaN NaN NaN NaN NaN])
%!assert (binoinv ([x(1:2) NaN x(4:5)], 2, 0.5), [NaN 0 NaN 2 NaN])
## Test accuracy, to within +/- 1 since it is a discrete distribution
%!shared y tol
%! y = magic(3)+1;
%! tol = 1;
%!assert (binoinv (binocdf (1:10, 11, 0.1), 11, 0.1), 1:10, tol)
%!assert (binoinv (binocdf (1:10, 2*(1:10), 0.1), 2*(1:10), 0.1), 1:10, tol)
%!assert (binoinv (binocdf (y, 2*y, 1./y), 2*y, 1./y), y, tol)
## Test class of input preserved
%!assert (binoinv ([x, NaN], 2, 0.5), [NaN 0 1 2 NaN NaN])
%!assert (binoinv (single ([x, NaN]), 2, 0.5), single ([NaN 0 1 2 NaN NaN]))
%!assert (binoinv ([x, NaN], single (2), 0.5), single ([NaN 0 1 2 NaN NaN]))
%!assert (binoinv ([x, NaN], 2, single (0.5)), single ([NaN 0 1 2 NaN NaN]))
## Test input validation
%!error binoinv ()
%!error binoinv (1)
%!error binoinv (1,2)
%!error binoinv (1,2,3,4)
%!error binoinv (ones (3), ones (2), ones (2))
%!error binoinv (ones (2), ones (3), ones (2))
%!error binoinv (ones (2), ones (2), ones (3))
%!error binoinv (i, 2, 2)
%!error binoinv (2, i, 2)
%!error binoinv (2, 2, i)