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########################################################################
##
## Copyright (C) 1995-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {[@var{d}, @var{dd}] =} diffpara (@var{x}, @var{a}, @var{b})
## Return the estimator @var{d} for the differencing parameter of an
## integrated time series.
##
## The frequencies from @math{[2*pi*a/t, 2*pi*b/T]} are used for the
## estimation. If @var{b} is omitted, the interval
## @math{[2*pi/T, 2*pi*a/T]} is used. If both @var{b} and @var{a} are omitted
## then @math{a = 0.5 * sqrt (T)} and @math{b = 1.5 * sqrt (T)} is used, where
## @math{T} is the sample size. If @var{x} is a matrix, the differencing
## parameter of each column is estimated.
##
## The estimators for all frequencies in the intervals described above is
## returned in @var{dd}.
##
## The value of @var{d} is simply the mean of @var{dd}.
##
## Reference: @nospell{P.J. Brockwell & R.A. Davis}. @cite{Time Series:
## Theory and Methods}. @nospell{Springer} 1987.
## @end deftypefn
function [d, dd] = diffpara (x, a, b)
if (nargin < 1)
print_usage ();
endif
if (isvector (x))
n = length (x);
k = 1;
x = reshape (x, n, 1);
else
[n, k] = size (x);
endif
if (nargin == 1)
a = 0.5 * sqrt (n);
b = 1.5 * sqrt (n);
elseif (nargin == 2)
b = a;
a = 1;
endif
if (! (isscalar (a) && isscalar (b)))
error ("diffpara: A and B must be scalars");
endif
dd = zeros (b - a + 1, k);
for l = 1:k
w = 2 * pi * (1 : n-1) / n;
x = 2 * log (abs (1 - exp (-i*w)));
y = log (periodogram (x(2:n,l)));
x = center (x);
y = center (y);
for m = a:b
dd(m-a+1) = - x(1:m) * y(1:m) / sumsq (x(1:m));
endfor
endfor
d = mean (dd);
endfunction
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