function [inmodel,gcv,history,RSS,DOF,inmodelHistory] = backwardsStepwisePrune(X,inpe,y,lambda,varargin)


[N,T]=size(y);

nargin
okargs   = {'mvweights'};
defaults = {ones(1,T) };
[eid,emsg,mvweights] = mygetargs(okargs,defaults,varargin{:});
eid
emsg




Mmax = length(inpe)+1;
inmodel = ones(1,Mmax);
Kstar = inmodel;


xin = X(:,find(inmodel));     
B= (xin'*xin)\(xin')*y;
%S = xin*((xin'*xin)\(xin'));
yhat = xin*B;
res = y-yhat;
temprss=0;
rss0=0;
for j=1:T
    temprss =temprss+mvweights(j)*res(:,j)'*res(:,j);
    rss0=rss0+sum((y(:,j)-mean(y(:,j))).^2)*mvweights(j);
end
rss = temprss;


[N,ninmodel]=size(xin);
[D,nbetas]=calculateEffectiveParameters(xin,inpe(find(inmodel(2:length(inmodel)))),N,T,lambda);
gcv(1)=calculateLOF(rss,N,D,T,rss0,nbetas);




lofstar = gcv(1);

Mrange = Mmax-[0:1:(Mmax-2)];
counter = 1;
history = cell(1,Mmax);
inmodelHistory = cell(1,Mmax);
RSS = zeros(1,Mmax);
DOF = zeros(1,Mmax);
for M = Mrange
    counter = counter+1;
    b = Inf; % keeps track of min{GCV) for all models with M dof.
    L = Kstar; % keeps track of the variables left in the model at this stage.
    Linmodel = find(L);
    for m = Linmodel  % TRY DELETING EACH VARIABLE IN MODEL ONE BY ONE.
%        fprintf('M=%d,m=%d\n',M,m);
        if m ~= 1 % never delete the intercept term.
            K = L;
            K(m) = 0;
            xin=X(:,find(K));
            % CALCULATE THE LOF
            B= (xin'*xin)\(xin')*y;
            %S = xin*((xin'*xin)\(xin'));
            yhat = xin*B;
            res = y-yhat;

            temprss=0;
            for j=1:T
                temprss =temprss+mvweights(j)*res(:,j)'*res(:,j);
            end
            rss = temprss;

            [N,ninmodel]=size(xin);
            
            [D,nbetas]=calculateEffectiveParameters(xin,inpe(find(K(2:length(K)))),N,T,lambda);
            lof=calculateLOF(rss,N,D,T,rss0,nbetas);
 
            if lof < b
                b = lof;
                Kstar = K;
                inmodelHistory{counter}= Kstar;
                history{counter} = inpe{m-1};
                RSS(counter-1) = rss;
                DOF(counter-1) = D;
            end
            if lof < lofstar
                lofstar = lof;
                inmodel = K;
            end
        end
    end
    gcv(counter) = b;
    fprintf('At step %d, %s would be deleted.\n', counter-1, char(history{counter}));
end

RSS=RSS(1:counter-1);
DOF=DOF(1:counter-1);