comparison toolboxes/FullBNT-1.0.7/HMM/fwdback.m @ 0:e9a9cd732c1e tip

first hg version after svn

+function [alpha, beta, gamma, loglik, xi_summed, gamma2] = fwdback(init_state_distrib, ...
+   transmat, obslik, varargin)
+% FWDBACK Compute the posterior probs. in an HMM using the forwards backwards algo.
+%
+% [alpha, beta, gamma, loglik, xi, gamma2] = fwdback(init_state_distrib, transmat, obslik, ...)
+%
+% Notation:
+% Y(t) = observation, Q(t) = hidden state, M(t) = mixture variable (for MOG outputs)
+% A(t) = discrete input (action) (for POMDP models)
+%
+% INPUT:
+% init_state_distrib(i) = Pr(Q(1) = i)
+% transmat(i,j) = Pr(Q(t) = j | Q(t-1)=i)
+%  or transmat{a}(i,j) = Pr(Q(t) = j | Q(t-1)=i, A(t-1)=a) if there are discrete inputs
+% obslik(i,t) = Pr(Y(t)| Q(t)=i)
+%   (Compute obslik using eval_pdf_xxx on your data sequence first.)
+%
+% Optional parameters may be passed as 'param_name', param_value pairs.
+% Parameter names are shown below; default values in [] - if none, argument is mandatory.
+%
+% For HMMs with MOG outputs: if you want to compute gamma2, you must specify
+% 'obslik2' - obslik(i,j,t) = Pr(Y(t)| Q(t)=i,M(t)=j)  []
+% 'mixmat' - mixmat(i,j) = Pr(M(t) = j | Q(t)=i)  []
+%
+% For HMMs with discrete inputs:
+% 'act' - act(t) = action performed at step t
+%
+% Optional arguments:
+% 'fwd_only' - if 1, only do a forwards pass and set beta=[], gamma2=[]  [0]
+% 'scaled' - if 1,  normalize alphas and betas to prevent underflow [1]
+% 'maximize' - if 1, use max-product instead of sum-product [0]
+%
+% OUTPUTS:
+% alpha(i,t) = p(Q(t)=i | y(1:t)) (or p(Q(t)=i, y(1:t)) if scaled=0)
+% beta(i,t) = p(y(t+1:T) | Q(t)=i)*p(y(t+1:T)|y(1:t)) (or p(y(t+1:T) | Q(t)=i) if scaled=0)
+% gamma(i,t) = p(Q(t)=i | y(1:T))
+% loglik = log p(y(1:T))
+% xi(i,j,t-1)  = p(Q(t-1)=i, Q(t)=j | y(1:T))  - NO LONGER COMPUTED
+% xi_summed(i,j) = sum_{t=}^{T-1} xi(i,j,t)  - changed made by Herbert Jaeger
+% gamma2(j,k,t) = p(Q(t)=j, M(t)=k | y(1:T)) (only for MOG  outputs)
+%
+% If fwd_only = 1, these become
+% alpha(i,t) = p(Q(t)=i | y(1:t))
+% beta = []
+% gamma(i,t) = p(Q(t)=i | y(1:t))
+% xi(i,j,t-1)  = p(Q(t-1)=i, Q(t)=j | y(1:t))
+% gamma2 = []
+%
+% Note: we only compute xi if it is requested as a return argument, since it can be very large.
+% Similarly, we only compute gamma2 on request (and if using MOG outputs).
+%
+% Examples:
+%
+% [alpha, beta, gamma, loglik] = fwdback(pi, A, multinomial_prob(sequence, B));
+%
+% [B, B2] = mixgauss_prob(data, mu, Sigma, mixmat);
+% [alpha, beta, gamma, loglik, xi, gamma2] = fwdback(pi, A, B, 'obslik2', B2, 'mixmat', mixmat);
+
+if nargout >= 5, compute_xi = 1; else compute_xi = 0; end
+if nargout >= 6, compute_gamma2 = 1; else compute_gamma2 = 0; end
+
+[obslik2, mixmat, fwd_only, scaled, act, maximize, compute_xi, compute_gamma2] = ...
+   process_options(varargin, ...
+       'obslik2', [], 'mixmat', [], ...
+       'fwd_only', 0, 'scaled', 1, 'act', [], 'maximize', 0, ...
+                   'compute_xi', compute_xi, 'compute_gamma2', compute_gamma2);
+
+[Q T] = size(obslik);
+
+if isempty(obslik2)
+ compute_gamma2 = 0;
+end
+
+if isempty(act)
+ act = ones(1,T);
+ transmat = { transmat } ;
+end
+
+scale = ones(1,T);
+
+% scale(t) = Pr(O(t) | O(1:t-1)) = 1/c(t) as defined by Rabiner (1989).
+% Hence prod_t scale(t) = Pr(O(1)) Pr(O(2)|O(1)) Pr(O(3) | O(1:2)) ... = Pr(O(1), ... ,O(T))
+% or log P = sum_t log scale(t).
+% Rabiner suggests multiplying beta(t) by scale(t), but we can instead
+% normalise beta(t) - the constants will cancel when we compute gamma.
+
+loglik = 0;
+
+alpha = zeros(Q,T);
+gamma = zeros(Q,T);
+if compute_xi
+ xi_summed = zeros(Q,Q);
+else
+ xi_summed = [];
+end
+
+%%%%%%%%% Forwards %%%%%%%%%%
+
+t = 1;
+alpha(:,1) = init_state_distrib(:) .* obslik(:,t);
+if scaled
+ %[alpha(:,t), scale(t)] = normaliseC(alpha(:,t));
+ [alpha(:,t), scale(t)] = normalise(alpha(:,t));
+end
+assert(approxeq(sum(alpha(:,t)),1))
+for t=2:T
+ %trans = transmat(:,:,act(t-1))';
+ trans = transmat{act(t-1)};
+ if maximize
+   m = max_mult(trans', alpha(:,t-1));
+   %A = repmat(alpha(:,t-1), [1 Q]);
+   %m = max(trans .* A, [], 1);
+ else
+   m = trans' * alpha(:,t-1);
+ end
+ alpha(:,t) = m(:) .* obslik(:,t);
+ if scaled
+   %[alpha(:,t), scale(t)] = normaliseC(alpha(:,t));
+   [alpha(:,t), scale(t)] = normalise(alpha(:,t));
+ end
+ if compute_xi & fwd_only  % useful for online EM
+   %xi(:,:,t-1) = normaliseC((alpha(:,t-1) * obslik(:,t)') .* trans);
+   xi_summed = xi_summed + normalise((alpha(:,t-1) * obslik(:,t)') .* trans);
+ end
+ assert(approxeq(sum(alpha(:,t)),1))
+end
+if scaled
+ if any(scale==0)
+   loglik = -inf;
+ else
+   loglik = sum(log(scale));
+ end
+else
+ loglik = log(sum(alpha(:,T)));
+end
+
+if fwd_only
+ gamma = alpha;
+ beta = [];
+ gamma2 = [];
+ return;
+end
+
+%%%%%%%%% Backwards %%%%%%%%%%
+
+beta = zeros(Q,T);
+if compute_gamma2
+ M = size(mixmat, 2);
+ gamma2 = zeros(Q,M,T);
+else
+ gamma2 = [];
+end
+
+beta(:,T) = ones(Q,1);
+%gamma(:,T) = normaliseC(alpha(:,T) .* beta(:,T));
+gamma(:,T) = normalise(alpha(:,T) .* beta(:,T));
+t=T;
+if compute_gamma2
+ denom = obslik(:,t) + (obslik(:,t)==0); % replace 0s with 1s before dividing
+ gamma2(:,:,t) = obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M]) ./ repmat(denom, [1 M]);
+ %gamma2(:,:,t) = normaliseC(obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M])); % wrong!
+end
+for t=T-1:-1:1
+ b = beta(:,t+1) .* obslik(:,t+1);
+ %trans = transmat(:,:,act(t));
+ trans = transmat{act(t)};
+ if maximize
+   B = repmat(b(:)', Q, 1);
+   beta(:,t) = max(trans .* B, [], 2);
+ else
+   beta(:,t) = trans * b;
+ end
+ if scaled
+   %beta(:,t) = normaliseC(beta(:,t));
+   beta(:,t) = normalise(beta(:,t));
+ end
+ %gamma(:,t) = normaliseC(alpha(:,t) .* beta(:,t));
+ gamma(:,t) = normalise(alpha(:,t) .* beta(:,t));
+ if compute_xi
+   %xi(:,:,t) = normaliseC((trans .* (alpha(:,t) * b')));
+   xi_summed = xi_summed + normalise((trans .* (alpha(:,t) * b')));
+ end
+ if compute_gamma2
+   denom = obslik(:,t) + (obslik(:,t)==0); % replace 0s with 1s before dividing
+   gamma2(:,:,t) = obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M]) ./ repmat(denom, [1 M]);
+   %gamma2(:,:,t) = normaliseC(obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M]));
+ end
+end
+
+% We now explain the equation for gamma2
+% Let zt=y(1:t-1,t+1:T) be all observations except y(t)
+% gamma2(Q,M,t) = P(Qt,Mt|yt,zt) = P(yt|Qt,Mt,zt) P(Qt,Mt|zt) / P(yt|zt)
+%                = P(yt|Qt,Mt) P(Mt|Qt) P(Qt|zt) / P(yt|zt)
+% Now gamma(Q,t) = P(Qt|yt,zt) = P(yt|Qt) P(Qt|zt) / P(yt|zt)
+% hence
+% P(Qt,Mt|yt,zt) = P(yt|Qt,Mt) P(Mt|Qt) [P(Qt|yt,zt) P(yt|zt) / P(yt|Qt)] / P(yt|zt)
+%                = P(yt|Qt,Mt) P(Mt|Qt) P(Qt|yt,zt) / P(yt|Qt)