comparison toolboxes/FullBNT-1.0.7/KPMstats/clg_Mstep.m @ 0:e9a9cd732c1e tip

first hg version after svn

+function [mu, Sigma, B] = clg_Mstep(w, Y, YY, YTY, X, XX, XY, varargin)
+% MSTEP_CLG Compute ML/MAP estimates for a conditional linear Gaussian
+% [mu, Sigma, B] = Mstep_clg(w, Y, YY, YTY, X, XX, XY, varargin)
+%
+% We fit P(Y|X,Q=i) = N(Y; B_i X + mu_i, Sigma_i) 
+% and w(i,t) = p(M(t)=i|y(t)) = posterior responsibility
+% See www.ai.mit.edu/~murphyk/Papers/learncg.pdf.
+%
+% See process_options for how to specify the input arguments.
+%
+% INPUTS:
+% w(i) = sum_t w(i,t) = responsibilities for each mixture component
+%  If there is only one mixture component (i.e., Q does not exist),
+%  then w(i) = N = nsamples,  and 
+%  all references to i can be replaced by 1.
+% Y(:,i) = sum_t w(i,t) y(:,t) = weighted observations
+% YY(:,:,i) = sum_t w(i,t) y(:,t) y(:,t)' = weighted outer product
+% YTY(i) = sum_t w(i,t) y(:,t)' y(:,t) = weighted inner product
+%   You only need to pass in YTY if Sigma is to be estimated as spherical.
+%
+% In the regression context, we must also pass in the following
+% X(:,i) = sum_t w(i,t) x(:,t) = weighted inputs
+% XX(:,:,i) = sum_t w(i,t) x(:,t) x(:,t)' = weighted outer product
+% XY(i) = sum_t w(i,t) x(:,t) y(:,t)' = weighted outer product
+%
+% Optional inputs (default values in [])
+%
+% 'cov_type' - 'full', 'diag' or 'spherical' ['full']
+% 'tied_cov' - 1 (Sigma) or 0 (Sigma_i) [0]
+% 'clamped_cov' - pass in clamped value, or [] if unclamped [ [] ]
+% 'clamped_mean' - pass in clamped value, or [] if unclamped [ [] ]
+% 'clamped_weights' - pass in clamped value, or [] if unclamped [ [] ]
+% 'cov_prior' - added to Sigma(:,:,i) to ensure psd [0.01*eye(d,d,Q)]
+%
+% If cov is tied, Sigma has size d*d.
+% But diagonal and spherical covariances are represented in full size.
+
+[cov_type, tied_cov, ...
+ clamped_cov, clamped_mean, clamped_weights,  cov_prior, ...
+ xs, ys, post] = ...
+    process_options(varargin, ...
+		    'cov_type', 'full', 'tied_cov', 0,  'clamped_cov', [], 'clamped_mean', [], ...
+		    'clamped_weights', [], 'cov_prior', [], ...
+		    'xs', [], 'ys', [], 'post', []);
+
+[Ysz Q] = size(Y);
+
+if isempty(X) % no regression
+  %B = [];
+  B2 = zeros(Ysz, 1, Q);
+  for i=1:Q
+    B(:,:,i) = B2(:,1:0,i); % make an empty array of size Ysz x 0 x Q
+  end
+  [mu, Sigma] = mixgauss_Mstep(w, Y, YY, YTY, varargin{:});
+  return;
+end
+
+
+N = sum(w);
+if isempty(cov_prior)
+  cov_prior = 0.01*repmat(eye(Ysz,Ysz), [1 1 Q]);
+end
+%YY = YY + cov_prior; % regularize the scatter matrix
+
+% Set any zero weights to one before dividing
+% This is valid because w(i)=0 => Y(:,i)=0, etc
+w = w + (w==0);
+
+Xsz = size(X,1);
+% Append 1 to X to get Z
+ZZ = zeros(Xsz+1, Xsz+1, Q);
+ZY = zeros(Xsz+1, Ysz, Q);
+for i=1:Q
+  ZZ(:,:,i) = [XX(:,:,i)  X(:,i);
+	       X(:,i)'    w(i)];
+  ZY(:,:,i) = [XY(:,:,i);
+	       Y(:,i)'];
+end
+
+
+%%% Estimate mean and regression 
+
+if ~isempty(clamped_weights) & ~isempty(clamped_mean)
+  B = clamped_weights;
+  mu = clamped_mean;
+end
+if ~isempty(clamped_weights) & isempty(clamped_mean)
+  B = clamped_weights;
+  % eqn 5
+  mu = zeros(Ysz, Q);
+  for i=1:Q
+    mu(:,i) = (Y(:,i) - B(:,:,i)*X(:,i)) / w(i);
+  end
+end
+if isempty(clamped_weights) & ~isempty(clamped_mean)
+  mu = clamped_mean;
+  % eqn 3
+  B = zeros(Ysz, Xsz, Q);
+  for i=1:Q
+    tmp = XY(:,:,i)' - mu(:,i)*X(:,i)';
+    %B(:,:,i) = tmp * inv(XX(:,:,i));
+    B(:,:,i) = (XX(:,:,i) \ tmp')';
+  end
+end
+if isempty(clamped_weights) & isempty(clamped_mean)
+  mu = zeros(Ysz, Q);
+  B = zeros(Ysz, Xsz, Q);
+  % Nothing is clamped, so we must estimate B and mu jointly
+  for i=1:Q
+    % eqn 9
+    if rcond(ZZ(:,:,i)) < 1e-10
+      sprintf('clg_Mstep warning: ZZ(:,:,%d) is ill-conditioned', i);
+      % probably because there are too few cases for a high-dimensional input
+      ZZ(:,:,i) = ZZ(:,:,i) + 1e-5*eye(Xsz+1);
+    end
+    %A = ZY(:,:,i)' * inv(ZZ(:,:,i));
+    A = (ZZ(:,:,i) \ ZY(:,:,i))';
+    B(:,:,i) = A(:, 1:Xsz);
+    mu(:,i) = A(:, Xsz+1);
+  end
+end
+
+if ~isempty(clamped_cov)
+  Sigma = clamped_cov;
+  return;
+end
+
+
+%%% Estimate covariance
+
+% Spherical
+if cov_type(1)=='s'
+  if ~tied_cov
+    Sigma = zeros(Ysz, Ysz, Q);
+    for i=1:Q
+      % eqn 16
+      A = [B(:,:,i) mu(:,i)];
+      %s = trace(YTY(i) + A'*A*ZZ(:,:,i) - 2*A*ZY(:,:,i)) / (Ysz*w(i)); % wrong!
+      s = (YTY(i) + trace(A'*A*ZZ(:,:,i)) - trace(2*A*ZY(:,:,i))) / (Ysz*w(i));
+      Sigma(:,:,i) = s*eye(Ysz,Ysz);
+
+      %%%%%%%%%%%%%%%%%%% debug
+      if ~isempty(xs)
+	[nx T] = size(xs);
+	zs = [xs; ones(1,T)];
+	yty = 0;
+	zAAz = 0;
+	yAz = 0;
+	for t=1:T
+	  yty = yty + ys(:,t)'*ys(:,t) * post(i,t);
+	  zAAz = zAAz + zs(:,t)'*A'*A*zs(:,t)*post(i,t);
+	  yAz = yAz + ys(:,t)'*A*zs(:,t)*post(i,t);
+	end
+	assert(approxeq(yty, YTY(i)))
+	assert(approxeq(zAAz, trace(A'*A*ZZ(:,:,i))))
+	assert(approxeq(yAz, trace(A*ZY(:,:,i))))
+	s2 = (yty + zAAz - 2*yAz) / (Ysz*w(i));
+	assert(approxeq(s,s2))
+      end
+      %%%%%%%%%%%%%%% end debug
+      
+    end
+  else
+    S = 0;
+    for i=1:Q
+      % eqn 18
+      A = [B(:,:,i) mu(:,i)];
+      S = S + trace(YTY(i) + A'*A*ZZ(:,:,i) - 2*A*ZY(:,:,i));
+    end
+    Sigma = repmat(S / (N*Ysz), [1 1 Q]);
+  end
+else % Full/diagonal
+  if ~tied_cov
+    Sigma = zeros(Ysz, Ysz, Q);
+    for i=1:Q
+      A = [B(:,:,i) mu(:,i)];
+      % eqn 10
+      SS = (YY(:,:,i) - ZY(:,:,i)'*A' - A*ZY(:,:,i) + A*ZZ(:,:,i)*A') / w(i);
+      if cov_type(1)=='d'
+	Sigma(:,:,i) = diag(diag(SS));
+      else
+	Sigma(:,:,i) = SS;
+      end
+    end
+  else % tied
+    SS = zeros(Ysz, Ysz);
+    for i=1:Q
+      A = [B(:,:,i) mu(:,i)];
+      % eqn 13
+      SS = SS + (YY(:,:,i) - ZY(:,:,i)'*A' - A*ZY(:,:,i) + A*ZZ(:,:,i)*A');
+    end
+    SS = SS / N;
+    if cov_type(1)=='d'
+      Sigma = diag(diag(SS));
+    else
+      Sigma = SS;
+    end
+    Sigma = repmat(Sigma, [1 1 Q]);
+  end
+end
+
+Sigma = Sigma + cov_prior;
+