function?left_variables?=?backward_elimination（XyAfoldmethodvarnumber）

%?Inputs:
%+++?X:?The?data?matrix?of?size?m?x?p
%+++?y:?The?reponse?vector?of?size?m?x?1
%+++?A:?the?maximal?principle?to?extract.
%+++?fold:?the?group?number?for?cross?validation.
%+++?method?:?pretreatment?method.
%
%?Outputs:
%???left_variables?:?the?variables?that?finally?left.



CV=plscvfold（XyAfoldmethod）;
RMSECV0?=?CV.RMSECV;
screenProcess=nan（size（X2）-1size（X2）+2）;

jj=1;
di=1:size（X2）;
index=1:size（X2）;
while?size（X2）>1
????parc=size（X2）;
?%???fprintf（‘Cross-validation?with?whole?variables?of?this?turn:?%g\n‘RMSECV0）;
????screenProcess（jj1）=RMSECV0;
????RMSECV_temp=nan（1parc）;????
????i=1;
????for?k=1:parc
????????cX=X;
????????cX（:k）=[];
????????CV1=plscvfold（cXyAfoldmethod0）;
????????RMSECV_temp（k）?=?CV1.RMSECV;
???%??????fprintf（‘Cross-validation??without?variables?#%g:?%g\n‘index（k）RMSECV_temp（k））;??
????????if?di（index（k））==0
????????????i=i+1;
????????else
????????????%???screenProcess:?the?screening?process?of?backward_elimination
????????????screenProcess（jjdi（index（k））+i）=RMSECV_temp（k）;
????????end
????end?
????[minRMSECVminRMSECVIndex]=min（RMSECV_temp）;
????jj=jj+1;
????if?minRMSECV<=RMSECV0?
????????fprintf（‘Variable?#%g?has?been?washed?out?%d?variables?have?been?deleted.\n‘varnumber（minRMSECVIndex）jj-1）;
????????screenProcess（jj-1end）=index（minRMSECVIndex）;
????????di（index（minRMSECVIndex））=0;
????????X（:minRMSECVIndex）=[];
????????index（minRMSECVIndex）=[];
????????varnumber（minRMSECVIndex）=[];
????????RMSECV0=minRMSECV;
????else
????????disp（‘No?any?variable-deleting?is?necessary?screenning?is?finished.‘）;
???%?????disp（[‘Left?variables:?‘sprintf（‘%g‘varnumber）]）;
????????break
????end
end

left_variables=varnumber;