## The module "rand_idx" of the Mastrave modelling library

**Daniele de Rigo**

#### Copyright and license notice of the function rand_idx

Copyright © 2009,2010,2011 Daniele de Rigo

The file rand_idx.m is part of Mastrave.

Mastrave 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.

Mastrave 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.

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#### Function declaration

[random_vals,rnd] = rand_idx(weights,siz= 1 ,value_set= [] ,rnd= [] )

#### Description

Utility to generate random samples extracted from a given finite set of
values (` value_set `) with a frequency proportional to a set of

`each of them associated to the corresponding element of`

**weights**`. 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`

**value_set**`is`

**random_vals**`. To generate`

**siz**`, 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`

**random_vals**`, otherwise it is generated within the utility. In both cases,`

**rnd**`is also returned as optional output argument.`

**rnd**

#### Input arguments

weightsArray of non-negative elements each of them representing the weight (proportional to the frequency) with which the corresponding element of::nonnegative::has to be randomly sampled.value_setsizSize of the output argument::numel::. If omitted, the default value is 1.random_valsvalue_setArray providing the set of values to be randomly sampled. If empty, the default value is the sequence of position-indexes associated with::numstring::i.e. 1:numel(weights) . If omitted, the default value is [] (empty).weightsrndOptional array of real numbers between 0 and 1 which can be passed as input to ensure the random-sampling is deterministically reproducible. If::probability::is not uniformly distributed in [0 1], the average frequency of therndelements in the returned arrayvalue_setwill not be proportional torandom_vals. If empty, an internal random-sampling ─ uniformly distributed ─ is generated. If omitted, the default value is [] (empty).weights

#### Example of usage

% 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 testagainstweights% 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( . )' )rnd

See also: create_contiguous_intervals Keywords: intervals, keys, indexes, random Version: 0.8.8

#### Support

The Mastrave modelling library is committed to provide reusable and general - but also robust and scalable - modules for research modellers dealing with computational science. You can help the Mastrave project by providing feedbacks on unexpected behaviours of this module. Despite all efforts, all of us - either developers or users - (should) know that errors are unavoidable. However, the free software paradigm successfully highlights that scientific knowledge freedom also implies an impressive opportunity for collectively evolve the tools and ideas upon which our daily work is based. Reporting a problem that you found using Mastrave may help the developer team to find a possible bug. Please, be aware that Mastrave is entirely based on voluntary efforts: in order for your help to be as effective as possible, please read carefully the section on reporting problems. Thank you for your collaboration.