--- /dev/null
+## Copyright (C) 2011 Alexander Klein <alexander.klein@math.uni-giessen.de>
+##
+## This program 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.
+##
+## This program 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
+## this program; if not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn{Function File} {@var{jackstat} =} jackknife (@var{E}, @var{x}, @dots{})
+## Compute jackknife estimates of a parameter taking one or more given samples as parameters.
+## In particular, @var{E} is the estimator to be jackknifed as a function name, handle,
+## or inline function, and @var{x} is the sample for which the estimate is to be taken.
+## The @var{i}-th entry of @var{jackstat} will contain the value of the estimator
+## on the sample @var{x} with its @var{i}-th row omitted.
+##
+## @example
+## @group
+## jackstat(@var{i}) = @var{E}(@var{x}(1 : @var{i} - 1, @var{i} + 1 : length(@var{x})))
+## @end group
+## @end example
+##
+## Depending on the number of samples to be used, the estimator must have the appropriate form:
+## If only one sample is used, then the estimator need not be concerned with cell arrays,
+## for example jackknifing the standard deviation of a sample can be performed with
+## @code{@var{jackstat} = jackknife (@@std, rand (100, 1))}.
+## If, however, more than one sample is to be used, the samples must all be of equal size,
+## and the estimator must address them as elements of a cell-array,
+## in which they are aggregated in their order of appearance:
+##
+## @example
+## @group
+## @var{jackstat} = jackknife(@@(x) std(x@{1@})/var(x@{2@}), rand (100, 1), randn (100, 1)
+## @end group
+## @end example
+##
+## If all goes well, a theoretical value @var{P} for the parameter is already known,
+## @var{n} is the sample size,
+## @code{@var{t} = @var{n} * @var{E}(@var{x}) - (@var{n} - 1) * mean(@var{jackstat})}, and
+## @code{@var{v} = sumsq(@var{n} * @var{E}(@var{x}) - (@var{n} - 1) * @var{jackstat} - @var{t}) / (@var{n} * (@var{n} - 1))}, then
+## @code{(@var{t}-@var{P})/sqrt(@var{v})} should follow a t-distribution with @var{n}-1 degrees of freedom.
+##
+## Jackknifing is a well known method to reduce bias; further details can be found in:
+## @itemize @bullet
+## @item Rupert G. Miller: The jackknife-a review; Biometrika (1974) 61(1): 1-15; doi:10.1093/biomet/61.1.1
+## @item Rupert G. Miller: Jackknifing Variances; Ann. Math. Statist. Volume 39, Number 2 (1968), 567-582; doi:10.1214/aoms/1177698418
+## @item M. H. Quenouille: Notes on Bias in Estimation; Biometrika Vol. 43, No. 3/4 (Dec., 1956), pp. 353-360; doi:10.1093/biomet/43.3-4.353
+## @end itemize
+## @end deftypefn
+
+## Author: Alexander Klein <alexander.klein@math.uni-giessen.de>
+## Created: 2011-11-25
+
+function jackstat = jackknife ( anEstimator, varargin )
+
+
+ ## Convert function name to handle if necessary, or throw
+ ## an error.
+ if ( !strcmp ( typeinfo ( anEstimator ), "function handle" ) )
+
+ if ( isascii ( anEstimator ) )
+
+ anEstimator = str2func ( anEstimator );
+
+ else
+
+ error ( "Estimators must be passed as function names or handles!" );
+ end
+ end
+
+
+ ## Simple jackknifing can be done with a single vector argument, and
+ ## first and foremost with a function that does not care about
+ ## cell-arrays.
+ if ( length ( varargin ) == 1 && isnumeric ( varargin { 1 } ) )
+
+ aSample = varargin { 1 };
+
+ g = length ( aSample );
+
+ jackstat = zeros ( 1, g );
+
+ for k = 1 : g
+ jackstat ( k ) = anEstimator ( aSample ( [ 1 : k - 1, k + 1 : g ] ) );
+ end
+
+ ## More complicated input requires more work, however.
+ else
+
+ g = cellfun ( @(x) length ( x ), varargin );
+
+ if ( any ( g - g ( 1 ) ) )
+
+ error ( "All passed data must be of equal length!" );
+ end
+
+ g = g ( 1 );
+
+ jackstat = zeros ( 1, g );
+
+ for k = 1 : g
+
+ jackstat ( k ) = anEstimator ( cellfun ( @(x) x( [ 1 : k - 1, k + 1 : g ] ), varargin, "UniformOutput", false ) );
+ end
+
+ end
+endfunction
+
+
+%!test
+%! ##Example from Quenouille, Table 1
+%! d=[0.18 4.00 1.04 0.85 2.14 1.01 3.01 2.33 1.57 2.19];
+%! jackstat = jackknife ( @(x) 1/mean(x), d );
+%! assert ( 10 / mean(d) - 9 * mean(jackstat), 0.5240, 1e-5 );
+
+%!demo
+%! for k = 1:1000
+%! x=rand(10,1);
+%! s(k)=std(x);
+%! jackstat=jackknife(@std,x);
+%! j(k)=10*std(x) - 9*mean(jackstat);
+%! end
+%! figure();hist([s',j'], 0:sqrt(1/12)/10:2*sqrt(1/12))
+
+%!demo
+%! for k = 1:1000
+%! x=randn(1,50);
+%! y=rand(1,50);
+%! jackstat=jackknife(@(x) std(x{1})/std(x{2}),y,x);
+%! j(k)=50*std(y)/std(x) - 49*mean(jackstat);
+%! v(k)=sumsq((50*std(y)/std(x) - 49*jackstat) - j(k)) / (50 * 49);
+%! end
+%! t=(j-sqrt(1/12))./sqrt(v);
+%! figure();plot(sort(tcdf(t,49)),"-;Almost linear mapping indicates good fit with t-distribution.;")
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff