Eje 3: papel de la ciudadanía no afectada en los procesos de rememoración en el

In this appendix we show Matlab functions Nglmf it and Sglmf it , coded by the au­ thor, for fitting a generalized linear model with canonical link using the new algorithm and the scoring method respectively.

function [b , f it , iter , flop] =Nglmf it (x , y , distr)

Nglmf it fits a generalized linear model with canonical link using the new algorithm.

[b , Fit , iter , f lop] =Nglmf it (x , y , distr) fits a generalized linear model using the design matrix x, response y, and distribution distr. The result b is a vector of coefficient estimates. The result f it is a vector of fitted value of the response y. The result iter is the number of iterations needed for convergence. The result f lop is the approximate number of floating point operations when the algorithm converges. Acceptable values for distr are ' normal ' ,

' binomial ' , ' poisson ' , ' gamma ' , and ' inverse gaussian ' . The distribution parameter is fit as a function of the x columns using the canonical link.

Example:

b = Nglmf it (x , [y N] , ' binomial ' )

This example fits a logistic regression model for y on x. Each y (i) is the number of successes in N ( i) trials.

f lops ( O ) ;

if (nargin< 2 ) , error ( ' At least two arguments are required ' ) ; end if (nargin<3 I isempt y ( distr) ) , distr = ' normal ' ; end

xx= [ones ( size (x , 1 ) , 1) x] ; [n p] =size (xx) ;

%Drthonomaliz the des ign matrix x A=orth( [xx eye (n) ] ) ; x=A ( : , 1 : p) ; convcrit = 1 e - 6 ; b = zeros ( p , 1 ) ; bO = b+1 ; iter = 0 ; iterlim = 200 ; switch (distr) case ' normal ' yO y ; eta yO ; case ' binomial '

if ( size (y , 2) =2) , error ( ' Y must have two c olumns . ' ) ; end N y ( : , 2 ) ;

y = y ( : , 1 ) . / N ;

yO = (N . *y + 0 . 5) . / (N + 1 ) ; eta = log (y0 . / ( 1-yO) ) ; case ' poisson '

yO = y + 0 . 25 ; eta = log (yO ) ;

case ' gamma '

i f ( any (y ( : ) <=O) )

delta = min (abs ( y (y =0) ) ) * . 00 1 ; yO = max(delt a , y ) ;

else yO=y ; end

eta = 1 . /yO ; case ' inverse gauss ian '

if ( any (y ( : ) <=O) )

delta = min (abs (y ( y =0) ) ) * . 00 1 ; yO = max (delta, y) ;

else yO=y ; end

eta = 1 . / (y0 . A2) ;

otherwis e , error ( ' Distribut ion name is invalid . ' ) ; end

while ( 1 )

iter = i t er+ 1 ;

% Compute parameter by using inverse link function switch (distr)

cas e ' normal ' t = eta ;

mu= (t+ (x*x ' ) * (y-t) ) ; case ' binomial '

end

p = 1 . / ( 1 + exp ( - eta) ) ; t= ( p . *N+(x*x ' ) * (y . *N-p . *N ) ) ; mu=max ( eps ,min ( 1- eps , t . /N) ) ; case ' poiss on '

t = exp (eta) ;

mu= (t+ ( x*x ' ) * (y-t ) ) ; mu = max (eps ,mu) ; case ' gamma '

t = 1 . /eta ;

mu= (t+ ( x*x ' ) * (y-t ) ) ; mu = max (eps ,mu) ; case ' inverse gaussian '

t = 1 . / ( sqrt (eta) ) ; mu= (t+ (x*x ' ) * (y-t) ) ; mu = max (eps ,mu) ;

% Compute adj usted dependent variable f o r l e ast squares f it switch ( distr)

case ' normal ' z = mu ; case ' binomial '

z=log (mu . / ( 1-mu) ) ; case ' po is s on '

z = l og (mu) ; case ' gamma '

end

case ' invers e gaussian ' z = 1 . / (mu . A2) ; end

% Check stopping c onditions

if (norm (b-bO) < convcrit) , break ; end

if ( i ter>iterlim) , warning ( ' Iteration limit reached . ' ) ; break ; end % Compute weight f unction as the inverse of the variance function switch ( distr) end case ' normal ' w =ones ( s ize (y , 1) , 1 ) ; case ' binomial ' w=N . * (mu · * ( 1 -mu) ) ; case ' poisson ' w =mu ; case ' gamma ' w =mu . /\2 ;

case ' inverse gaussian ' w =mu . /\3 ;

% Compute coef f ic i ent estimates for this iteration bO = b ;

[b , R] = wf it (z , _{x ,} w) ;

% Form current linear combination eta = x * b ;

% Calculate the f itted value switch (distr)

case 'normal ' f it = X*b ; case ' binomial ' fit = N . / ( 1 + exp ( -x*b) ) ; c ase ' poisson ' f it = exp (x*b) ; case ' gamma ' f it = 1 . /x*b ; case ' inverse gaussian '

f it = 1 . / sqrt ( x*b) ; end

% Compute coeff i c i ent estimate s based on the unorthonomalized design matrix xx b=wfit (z , xx , w) ;

f l op=flops ;

% Perform a weighted least squares f it function [b , R] =wfit (y , x , w) sw = s qrt (w) ; [ r c] = size (x) ; yw y · * sw ; xw = x · * sw ( : , ones ( 1 , c) ) ; [Q , R] =qr (xw , O) ; b = R\(Q ' *yw) ;

funct ion [b , f it , iter , f lop] =Sglmfit (x , y , distr)

Sglmf it fits a generalized linear model with canonical link using the scoring method.

[b , Fit , iter , flop] =Sglmfit (x , y , distr) fits a generalized linear model using the design matrix x, response y, and distribution distr. The result b is a vector of coefficient estimates. The result f it is a vector of fitted value of the response y. The result iter is the number of iterations needed for convergence. The result flop is the approximate number of floating point operations when the algorithm converges. Acceptable values for distr are ' normal ' ,

' binomial ' , ' poisson ' , ' gamma ' , and ' inverse gauss i an 1 • The distribution parameter is

fit as a function of the x columns using the canonical link.

Example:

b = Sglmfit (x , [y N] , ' binomial ' )

This example fits a logistic regression model for y on x. Each y (i ) is the number of successes in N ( i ) trials.

f lops ( O ) ;

if (nargin<2) , error ( ' At least two arguments are requ ired ' ) ; end if (nargin<3 I isempt y ( distr) ) , distr = ' normal ' ; end

xx= [one s ( size (x , 1 ) , 1) x] ; [n p] =size ( xx) ;

% Orthonormalize the design matrix x A=orth( [xx eye (n) ] ) ; x=A ( : , 1 : p) ; convcrit = 1 e-6 ; b = zero s (p , 1 ) ; bO = b+1 ; iter = 0 ; iterlim = 200 ; switch (distr)

case ' normal ' yO = y ; eta = yO ; case ' binomial '

if ( size (y , 2) =2) , error ( ' Y must have two c olumns . ' ) ; end N y ( : , 2) ;

y y ( : , 1 ) . / N ;

y O = (N . *y + 0 . 5) . / ( N + 1 ) ; eta = log (y0 . / ( 1 -yO ) ) ; case ' poisson '

yO = y + 0 . 25 ; eta = log (yO) ; case ' gamma '

if ( any (y ( : ) <=O) )

delta = min (abs ( y ( y =0) ) ) * . 00 1 ; yO = max (delt a , y) ;

else yO=y ; end

eta = 1 . /yO ; case ' inverse gaussian '

if ( any ( y ( : ) <=O) )

delta = min (abs ( y ( y =0) ) ) * . 00 1 ; yO = max ( delt a , y) ;

else yO=y ;

end

eta = 1 . / (y0 . A2 ) ;

otherwise , error ( ' Distribut ion name is invalid . ' ) ; end

while ( 1 )

iter = iter+1 ;

% Compute parameter by using inverse link f uncti on switch(distr) end case ' normal ' mu=eta ; case ' binomial ' mu 1 . / ( 1 + exp ( -eta) ) ; mu max (eps , min( 1-eps , mu) ) ; case ' poisson ' mu exp (eta) ; mu max (O , mu) ; case ' gamma ' mu 1 . /et a ; mu max (O , mu) ; case ' inverse gaussian '

mu = 1 . /sqrt ( eta) ; mu = max ( O , mu) ;

% Compute adjusted dependent variable f or l east squares f it switch ( distr)

case ' normal ' z = eta ; case ' binomial '

z = eta + (y - mu) . / ( mu · * ( 1-mu) ) ; case ' poisson '

z = eta + ( y - mu) . /mu ; case ' gamma '

z = eta - ( y - mu) . / (mu . A2 ) ; case ' inverse gaussian '

z = eta - 2 * ( (y - mu) . / (mu . A3) ) ; end

% Check stopping conditions

if (norm (b-bO) < convcrit) , break ; end

if ( iter>iterlim) , warning ( ' Iterati on limit reached . ' ) ; break ; end % Compute weight fun ction as the inverse of the variance funct ion switch( di str) case ' normal ' w =ones ( s i ze (y , 1) , 1 ) ; case ' binomial ' w=N . * (mu · * ( 1 -mu) ) ; case ' poisson' w =mu ; case ' gamma ' w =mu . A2 ;

end

w =mu . !\3 ; end

% Compute coef f ic ient estimates for this iteration bO = b ;

[b , R] = wf it (z , x , w) ;

% Form current linear combination eta = x * b ;

% Calculate the f itted value switch (distr) case ' normal ' f it = X*b ; case ' binomial ' f it = N . / ( 1 + exp (-x*b) ) ; case ' po i s son ' f it = exp (x*b) ; case ' gamma ' f it = 1 . /X*b ; case ' inverse gaussian '

f it = 1 . /sqrt (x*b) ; end

% Compute coefficient estimates based on the unorthonomalized design matrix xx b=wf it (z , xx , w) ;

% Perform a weighted least squares f it function [b , R] =wfit ( y , x , w) sw = sqrt (w) ; [r c] = s ize (x) ; yw = y · * sw ; xw = x · * sw ( : , ones ( 1 , c) ) ; [Q , R] =qr ( xw , O) ; b = R\ ( Q ' *yw) ;

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