The module "rand_idx" of the Mastrave modelling library
Copyright © 2009,2010,2011 Daniele de Rigo
The file rand_idx.m is part of Mastrave.
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[ random_vals, rnd ] = rand_idx( weights , siz = 1 , value_set =  , rnd =  )
Utility to generate random samples extracted from a given finite set of values ( value_set ) with a frequency proportional to a set of weights each of them associated to the corresponding element of value_set . The higher the i-th weight, the higher the probability for the corresponding i-th element of value_set to be randomly extracted. Multiple instances of the same element of value_set may be extracted. The size of the returned random-sample array random_vals is siz . To generate random_vals , a random-sample of real numbers uniformly distributed between 0 and 1 is required. The random-sample can be provided as an input with the array rnd , otherwise it is generated within the utility. In both cases, rnd is also returned as optional output argument.
weights ::nonnegative:: Array of non-negative elements each of them representing the weight (proportional to the frequency) with which the corresponding element of value_set has to be randomly sampled. siz ::numel:: Size of the output argument random_vals . If omitted, the default value is 1. value_set ::numstring:: Array providing the set of values to be randomly sampled. If empty, the default value is the sequence of position-indexes associated with weights i.e. 1:numel( weights ) . If omitted, the default value is  (empty). rnd ::probability:: Optional array of real numbers between 0 and 1 which can be passed as input to ensure the random-sampling is deterministically reproducible. If rnd is not uniformly distributed in [0 1], the average frequency of the value_set elements in the returned array random_vals will not be proportional to weights . If empty, an internal random-sampling ─ uniformly distributed ─ is generated. If omitted, the default value is  (empty).
% Motivational example: % Sensitivity analysis of a polynomial model of order 3 (cubic model) % against data generated with an higher-order polynomial. % Original (syntetic) data: xi = mean( sort( rand( 20, 5 ) ), 2 )* 8 + 1; yi = .01 * xi.^5 - xi.^3 + pi; % Cubic model: M = bsxfun( @power, xi, [0:3] ) th = M \ yi hold off; plot( xi , yi , 'o-k' , xi , M*th , '+k' ) legend( 'original data' , 'cubic model' ) % Fine-grid model (spatial resolution: 0.1 units). xii = [0:0.1:10].'; MM = bsxfun( @power, xii, [0:3] ); % Sensitivity analysis (simplified bootstrapping). % Bootstrapping extractions: 300. nb = 300 id = rand_idx( ones(20,1), [20,nb] ); Xi = xi( id ); Yi = yi( id ); thi = zeros( 4 , nb ); yii = zeros( numel(xii) , nb ); for i=1:nb thi(:,i) = bsxfun( @power, Xi(:,i), [0:3] ) \ Yi(:,i); yii(:,i) = MM * thi(:,i); end hold on plot( xii, quantile( yii.' , [.05 .5 .95] ).' ) legend( '5%' , '50%' , '95%' ); hold off; % Comparison between rand_idx( . ) and one of the possible % trivial implementations. The one being considered is: % O( numel( weights ) * numel( rnd ) ) % Its comparison with rand_idx( . ) is stopped when the size of % the involved temporary matrix to test weights against rnd % is approaching 50000 x 50000. This is done to prevent memory % exhaustion. Ni = floor( 10.^[3:.25:6] +.5 ) n = numel( Ni ); [ t1, t2 ] = deal( zeros( n, 1 ) * nan ); for i = 1:n N = Ni( i ); weights = rand( 1 , N ); rnd = rand( N , 1 ); if N < 50000 tic; w = cumsum([ 0 weights ]); w = w./w(end); id1 = sum( bsxfun( @le, w , rnd ), 2 ); t1( i ) = toc; end tic; id2 = rand_idx( weights, N, , rnd ); t2( i ) = toc; if N < 50000 assert( isequal( id1 , id2 ) ) end end hold off; loglog( Ni, [ t1, t2 ], 'o-' ) legend( 'trivial implementation' , 'rand_idx( . )' )
See also: create_contiguous_intervals Keywords: intervals, keys, indexes, random Version: 0.8.8
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