sem_g.m

计量工具箱

function results = sem_g(y,x,W,ndraw,nomit,prior)
% PURPOSE: Bayesian estimates of the spatial error model
%          y = XB + u, u = rho*W + e
%          e = N(0,sige*V), V = diag(v1,v2,...vn) 
%          r/vi = ID chi(r)/r, r = Gamma(m,k)
%          B = N(c,T), 
%          1/sige = Gamma(nu,d0), 
%          rho = Uniform(rmin,rmax), or rho = beta(a1,a2); 
%-------------------------------------------------------------
% USAGE: results = sem_g(y,x,W,ndraw,nomit,prior)
% where: y = dependent variable vector (nobs x 1)
%        x = independent variables matrix (nobs x nvar)
%        W = spatial weight matrix (standardized, row-sums = 1)
%    ndraw = # of draws
%    nomit = # of initial draws omitted for burn-in            
%    prior = a structure variable with:
%            prior.beta  = prior means for beta,   c above (default 0)
%            priov.bcov  = prior beta covariance , T above (default 1e+12)
%            prior.novi  = 1 turns off sampling for vi, producing homoscedastic model            
%            prior.rval  = r prior hyperparameter, default=4
%            prior.m     = informative Gamma(m,k) prior on r
%            prior.k     = (default: not used)
%            prior.nu    = informative Gamma(nu,d0) prior on sige
%            prior.d0    = default: nu=0,d0=0 (diffuse prior)
%            prior.a1    = parameter for beta(a1,a2) prior on rho see: 'help beta_prior'
%            prior.a2    = (default = 1.0, a uniform prior on rmin,rmax) 
%            prior.rmin  = (optional) min rho used in sampling (default = -1)
%            prior.rmax  = (optional) max rho used in sampling (default = +1)  
%            prior.eigs  = 0 to compute rmin/rmax using eigenvalues, (1 = don't compute default)
%            prior.lflag = 0 for full lndet computation (default = 1, fastest)
%                        = 1 for MC approx (fast for large problems)
%                        = 2 for Spline approx (medium speed)
%            prior.dflag = 1 for Metropolis-Hastings sampling for rho (default)
%                        = 0 for griddy gibbs with univariate numerical integration
%            prior.order = order to use with prior.lflag = 1 option (default = 50)
%            prior.iter  = iters to use with prior.lflag = 1 option (default = 30)   
%            prior.lndet = a matrix returned by sar, sar_g, sarp_g, etc.
%                          containing log-determinant information to save time
%            prior.mlog  = 0 for no log-marginal likelihood, 
%                        = 1 for log-marginal likelihood, default = 1
%-------------------------------------------------------------
% RETURNS:  a structure:
%          results.meth   = 'sem_g'
%          results.beta   = posterior mean of bhat
%          results.rho    = posterior mean of rho
%          results.sige   = posterior mean of sige
%          results.bdraw  = bhat draws (ndraw-nomit x nvar)
%          results.pdraw  = rho  draws (ndraw-nomit x 1)
%          results.sdraw  = sige draws (ndraw-nomit x 1)
%          results.vmean  = mean of vi draws (nobs x 1) 
%          results.rdraw  = r draws (ndraw-nomit x 1) (if m,k input)
%          results.bmean  = b prior means, prior.beta from input
%          results.bstd   = b prior std deviations sqrt(diag(prior.bcov))
%          results.r      = value of hyperparameter r (if input)
%          results.nobs   = # of observations
%          results.nvar   = # of variables in x-matrix
%          results.ndraw  = # of draws
%          results.nomit  = # of initial draws omitted
%          results.y      = y-vector from input (nobs x 1)
%          results.yhat   = mean of posterior predicted (nobs x 1)
%          results.resid  = residuals, based on posterior means
%          results.rsqr   = r-squared based on posterior means
%          results.rbar   = adjusted r-squared
%          results.nu     = nu prior parameter
%          results.d0     = d0 prior parameter
%          results.a1     = a1 parameter for beta prior on rho from input, or default value
%          results.a2     = a2 parameter for beta prior on rho from input, or default value
%          results.time1  = time for eigenvalue calculation
%          results.time2  = time for log determinant calcluation
%          results.time3  = time for sampling
%          results.time   = total time taken  
%          results.rmax   = 1/max eigenvalue of W (or rmax if input)
%          results.rmin   = 1/min eigenvalue of W (or rmin if input)          
%          results.tflag  = 'plevel' (default) for printing p-levels
%                         = 'tstat' for printing bogus t-statistics 
%          results.lflag  = lflag from input
%          results.iter   = prior.iter option from input
%          results.order  = prior.order option from input
%          results.limit  = matrix of [rho lower95,logdet approx, upper95] 
%                           intervals for the case of lflag = 1
%          results.lndet = a matrix containing log-determinant information
%                          (for use in later function calls to save time)
%          results.mlike = log marginal likelihood for model comparisons,
%                          (a vector ranging over rho-values from rmin to rmax that can be
%                          integrated for model comparison)
%          results.acc   = acceptance rate for M-H sampling (ndraw x 1) vector
% --------------------------------------------------------------
% NOTES: - use either improper prior.rval 
%          or informative Gamma prior.m, prior.k, not both of them
% - for n < 1000 you should use lflag = 0 to get exact results  
% - use a1 = 1.0 and a2 = 1.0 for uniform prior on rho
% --------------------------------------------------------------
% SEE ALSO: (sem_gd, sem_gd2 demos) prt
% --------------------------------------------------------------
% REFERENCES: James P. LeSage, `Bayesian Estimation of Spatial Autoregressive
%             Models',  International Regional Science Review, 1997 
%             Volume 20, number 1\&2, pp. 113-129.
% For lndet information see: Ronald Barry and R. Kelley Pace, 
% "A Monte Carlo Estimator of the Log Determinant of Large Sparse Matrices", 
% Linear Algebra and its Applications", Volume 289, Number 1-3, 1999, pp. 41-54.
% and: R. Kelley Pace and Ronald P. Barry 
% "Simulating Mixed Regressive Spatially autoregressive Estimators", 
% Computational Statistics, 1998, Vol. 13, pp. 397-418.
%----------------------------------------------------------------

% written by:
% James P. LeSage, 12/2001, updated 7/2003
% Dept of Economics
% University of Toledo
% 2801 W. Bancroft St,
% Toledo, OH 43606
% [e-mail]


% NOTE: some of the speed for large problems comes from:
% the use of methods pioneered by Pace and Barry.
% R. Kelley Pace was kind enough to provide functions
% lndetmc, and lndetint from his spatial statistics toolbox
% for which I'm very grateful.

timet = clock;

% error checking on inputs
[n junk] = size(y);
results.y = y;
[n1 k] = size(x);
[n3 n4] = size(W);
time1 = 0;
time2 = 0;
time3 = 0;

if n1 ~= n
error('sem_g: x-matrix contains wrong # of observations');
elseif n3 ~= n4
error('sem_g: W matrix is not square');
elseif n3~= n
error('sem_g: W matrix is not the same size at y,x');
end;

if nargin == 5
    prior.lflag = 1;
end;

[nu,d0,rval,mm,kk,rho,sige,rmin,rmax,detval,ldetflag,eflag,order,iter,novi_flag,c,T,cc,metflag,a1,a2,inform_flag,mlog] = sem_parse(prior,k);

results.order = order;
results.iter = iter;

% error checking on prior information inputs
[checkk,junk] = size(c);
if checkk ~= k
error('sem_g: prior means are wrong');
elseif junk ~= 1
error('sem_g: prior means are wrong');
end;

[checkk junk] = size(T);
if checkk ~= k
error('sem_g: prior bcov is wrong');
elseif junk ~= k
error('sem_g: prior bcov is wrong');
end;

V = ones(n,1); in = ones(n,1); % initial value for V   
ys = y.*sqrt(V);
vi = in;
          
bsave = zeros(ndraw-nomit,1);    % allocate storage for results
ssave = zeros(ndraw-nomit,1);
vmean = zeros(n,1);
yhat = zeros(n,1);

if mm~= 0                        % storage for draws on rvalue
rsave = zeros(ndraw-nomit,1);
end;

[rmin,rmax,time1] = sem_eigs(eflag,W,rmin,rmax,n);

results.rmin = rmin;
results.rmax = rmax;
results.lflag = ldetflag;

[detval,time2] = sem_lndet(ldetflag,W,rmin,rmax,detval,order,iter);


% storage for draws
          bsave = zeros(ndraw-nomit,k);
          if mm~= 0
          rsave = zeros(ndraw-nomit,1);
          end;
          psave = zeros(ndraw-nomit,1);
          ssave = zeros(ndraw-nomit,1);
          vmean = zeros(n,1);
          acc_rate = zeros(ndraw,1);

% ====== initializations
% compute this stuff once to save time
TI = inv(T);
TIc = TI*c;
iter = 1;
in = ones(n,1);
V = in;
Wy = sparse(W)*y;
Wx = sparse(W)*x;
vi = in;
V = vi;

switch (novi_flag) 
    
case{0} % we do heteroscedastic model
    
hwait = waitbar(0,'sem\_g: MCMC sampling ...');
t0 = clock;                  
iter = 1;
acc = 0;
          while (iter <= ndraw); % start sampling;
                  
          % update beta   
          xs = matmul(sqrt(V),x);
          ys = sqrt(V).*y;;
          Wxs = W*xs;
          Wys = W*ys;
          xss = xs - rho*Wxs;
          AI = inv(xss'*xss + sige*TI);
          yss = ys - rho*Wys;
          b = xss'*yss + sige*TIc;
          b0 = AI*b;
          bhat = norm_rnd(sige*AI) + b0; 
            
          % update sige
          nu1 = n + 2*nu; 
          e = yss-xss*bhat;
          ed = e - rho*sparse(W)*e;
          d1 = 2*d0 + ed'*ed;
          chi = chis_rnd(1,nu1);
          sige = d1/chi;

          % update vi
          ev = ys - xs*bhat; 
          %chiv = chis_rnd(n,rval+1);   
          chiv = chi2rnd(rval+1,n,1);
          vi = ((ev.*ev/sige) + in*rval)./chiv;
          V = in./vi; 
              
          % update rval
          if mm ~= 0           
          rval = gamm_rnd(1,1,mm,kk);  
          end;

if metflag == 0
          % update rho using numerical integration          
          rho = draw_rho(detval,y,x,Wy,Wx,V,n,k,rmin,rmax,rho);
else
          % update rho using metropolis-hastings
          % numerical integration is too slow here
          xb = x*bhat;
          rhox = c_sem(rho,y,x,bhat,sige,W,detval,V,a1,a2);
          accept = 0;
          rho2 = rho + cc*randn(1,1);
          while accept == 0
           if ((rho2 > rmin) & (rho2 < rmax)); 
           accept = 1;  
           else
           rho2 = rho + cc*randn(1,1);
           end; 
          end; 
           rhoy = c_sem(rho2,y,x,bhat,sige,W,detval,V,a1,a2);
          ru = unif_rnd(1,0,1);
          if ((rhoy - rhox) > exp(1)),
          p = 1;
          else,          
          ratio = exp(rhoy-rhox);
          p = min(1,ratio);
          end;
              if (ru < p)
              rho = rho2;
              acc = acc + 1;
              end;
      acc_rate(iter,1) = acc/iter;
      % update cc based on std of rho draws
       if acc_rate(iter,1) < 0.4
       cc = cc/1.1;
       end;
       if acc_rate(iter,1) > 0.6
       cc = cc*1.1;
       end;
end; % end of if metflag
                                                         
    if iter > nomit % if we are past burn-in, save the draws
    bsave(iter-nomit,1:k) = bhat';
    ssave(iter-nomit,1) = sige;
    psave(iter-nomit,1) = rho;
    vmean = vmean + vi;

    if mm~= 0
        rsave(iter-nomit,1) = rval;
    end;         
    end;
                    

iter = iter + 1; 
waitbar(iter/ndraw);         
end; % end of sampling loop
close(hwait);

time3 = etime(clock,t0);

% compute posterior means and evaluate the log-marginal
vmean = vmean/(ndraw-nomit);
bmean = mean(bsave);
bmean = bmean';
rho = mean(psave);

V = in./vmean;
ys = y.*sqrt(V);
xs = matmul(x,sqrt(V));
Wys = sparse(W)*ys;
Wxs = sparse(W)*xs;
[nobs,nvar] = size(xs);
if mlog == 1
    % compute log marginal likelihood for model comparisions
    if inform_flag == 0
    mlike = sem_marginal(detval,ys,xs,Wys,Wxs,nobs,nvar,a1,a2);
    else
    mlike = sem_marginal2(detval,ys,xs,Wys,Wxs,nobs,nvar,a1,a2,c,TI,sige);
    end;
end;

[n nvar] = size(x);
yhat = x*bmean;
y = results.y;
n = length(y);
e = y-yhat;
eD = e - rho*sparse(W)*e;
epe = eD'*eD;

sigu = epe;
sige = sigu/(n-nvar);