octave-statistics 1.3.0-4 / usr / share / octave / packages / statistics-1.3.0 / jackknife.m

## 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.;")