de Rigo, D. (2012). Weighting data for reducing data-transformation bias in linear regressions: the module "mwlin" of the Mastrave modelling library. In: Semantic Array Programming with Mastrave - Introduction to Semantic Computational Modelling. http://mastrave.org/doc/mtv_m/mwlin
Weighting data for reducing data-transformation bias in linear regressions: the module "mwlin" of the Mastrave modelling library
Copyright © 2008,2009,2010,2011,2012 Daniele de Rigo
The file mwlin.m is part of Mastrave.
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[x, transformed_M, transformed_y, transformed_weights] = mwlin( M , y , weights =  , func = @(x)x , is_nonneg = false )
Module for solving a linear system composed by a matrix of coefficients M and a vector of constant terms y , subject to a data-transformation func of both M and y . To each row of M and the corresponding element of y a given weight can be assigned so that the corresponding error is accordingly weighted. Weights can be provided within an array weights (with the same size of y ). If the flag is_nonneg is set to true, all elements of the solution x are constrained not to be negative.
The implemented algorithm is generally able to provide better solutions than the ones which can be achieved by solving the transformed system:
M1 = func( M )
y1 = func( y )
x1 = M1 \ y1
if weights are omitted or:
x1 = ( M1' * diag(weights) * M1 ) \ ( M1' * diag(weights) * y1 )
if them are passed as input argument. The improvement relies on complementing weights with the relative distance between the transformed values and the original ones so to take into account the modified effect of each error on the transformed values. The resulting transformed weights are returned within transformed_weights . In general, some original values might be tranformed in infinite or NaN values, which would numerically degrade the linear regression. Therefore, only transformed values which are @finite are considered when solving the transformed linear system.
M ::matrix,numeric:: Matrix of coefficients of the linear system whose weighted least squares solution x has to be computed after transforming both M and y with the transformation function y_func . y ::matrix,numeric:: Constant terms. weights ::matrix,nonnegative:: Weights associated to the constant terms y and the corresponding lines of M . If an empty matrix  is passed, then all weights are considered as having the same value. If omitted, default value is  . func ::function_handle:: Handle to a data-transformation function to be applied to both M and y . If omitted, the default value is the trivial function @(x)x . is_nonneg ::scalar,logical:: Flag describing whether the returne solution x must be of nonnegative elements, or not. If omitted, the default value is false .
[a, b] = deal( 0.01, 3 ); x = rand( 1000, 1 )*4; noise = randn(1000,1)/20; y_true = a*x.^b; % Observations affected by growing noise % ( noise also generates systematic overestimation for low values ) y1 = max( eps, y_true + noise/3 ); y2 = max( eps, y_true + noise ); M = [ x, exp(ones(1000,1)) ]; w1a = log( M ) \ log( y1 ) w1b = mwlin( M, y1, , @log ) w2a = log( M ) \ log( y2 ) w2b = mwlin( M, y2, , @log ) f = @(x,w) exp( w(2) ) * x.^w(1) figure( 1 ) plot( x, [ y1, y_true, f(x,w1a) f(x,w1b) ], '.' ) legend( 'noisy data', 'true data', 'log-regression', 'mwlin' ) figure( 2 ) plot( x, [ y2, y_true, f(x,w2a) f(x,w2b) ], '.' ) legend( 'noisy data', 'true data', 'log-regression', 'mwlin' )
See also: train_pca Keywords: linear system, regression, weighting, data-transformation, nonnegative Version: 0.2.9
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