comparison toolboxes/FullBNT-1.0.7/Kalman/learn_kalman.m @ 0:e9a9cd732c1e tip

first hg version after svn

+function [A, C, Q, R, initx, initV, LL] = ...
+    learn_kalman(data, A, C, Q, R, initx, initV, max_iter, diagQ, diagR, ARmode, constr_fun, varargin)
+% LEARN_KALMAN Find the ML parameters of a stochastic Linear Dynamical System using EM.
+%
+% [A, C, Q, R, INITX, INITV, LL] = LEARN_KALMAN(DATA, A0, C0, Q0, R0, INITX0, INITV0) fits
+% the parameters which are defined as follows
+%   x(t+1) = A*x(t) + w(t),  w ~ N(0, Q),  x(0) ~ N(init_x, init_V)
+%   y(t)   = C*x(t) + v(t),  v ~ N(0, R)
+% A0 is the initial value, A is the final value, etc.
+% DATA(:,t,l) is the observation vector at time t for sequence l. If the sequences are of
+% different lengths, you can pass in a cell array, so DATA{l} is an O*T matrix.
+% LL is the "learning curve": a vector of the log lik. values at each iteration.
+% LL might go positive, since prob. densities can exceed 1, although this probably
+% indicates that something has gone wrong e.g., a variance has collapsed to 0.
+%
+% There are several optional arguments, that should be passed in the following order.
+% LEARN_KALMAN(DATA, A0, C0, Q0, R0, INITX0, INITV0, MAX_ITER, DIAGQ, DIAGR, ARmode)
+% MAX_ITER specifies the maximum number of EM iterations (default 10).
+% DIAGQ=1 specifies that the Q matrix should be diagonal. (Default 0).
+% DIAGR=1 specifies that the R matrix should also be diagonal. (Default 0).
+% ARMODE=1 specifies that C=I, R=0. i.e., a Gauss-Markov process. (Default 0).
+% This problem has a global MLE. Hence the initial parameter values are not important.
+% 
+% LEARN_KALMAN(DATA, A0, C0, Q0, R0, INITX0, INITV0, MAX_ITER, DIAGQ, DIAGR, F, P1, P2, ...)
+% calls [A,C,Q,R,initx,initV] = f(A,C,Q,R,initx,initV,P1,P2,...) after every M step. f can be
+% used to enforce any constraints on the params. 
+%
+% For details, see
+% - Ghahramani and Hinton, "Parameter Estimation for LDS", U. Toronto tech. report, 1996
+% - Digalakis, Rohlicek and Ostendorf, "ML Estimation of a stochastic linear system with the EM
+%      algorithm and its application to speech recognition",
+%       IEEE Trans. Speech and Audio Proc., 1(4):431--442, 1993.
+
+
+%    learn_kalman(data, A, C, Q, R, initx, initV, max_iter, diagQ, diagR, ARmode, constr_fun, varargin)
+if nargin < 8, max_iter = 10; end
+if nargin < 9, diagQ = 0; end
+if nargin < 10, diagR = 0; end
+if nargin < 11, ARmode = 0; end
+if nargin < 12, constr_fun = []; end
+verbose = 1;
+thresh = 1e-4;
+
+
+if ~iscell(data)
+  N = size(data, 3);
+  data = num2cell(data, [1 2]); % each elt of the 3rd dim gets its own cell
+else
+  N = length(data);
+end
+
+N = length(data);
+ss = size(A, 1);
+os = size(C,1);
+
+alpha = zeros(os, os);
+Tsum = 0;
+for ex = 1:N
+  %y = data(:,:,ex);
+  y = data{ex};
+  T = length(y);
+  Tsum = Tsum + T;
+  alpha_temp = zeros(os, os);
+  for t=1:T
+    alpha_temp = alpha_temp + y(:,t)*y(:,t)';
+  end
+  alpha = alpha + alpha_temp;
+end
+
+previous_loglik = -inf;
+loglik = 0;
+converged = 0;
+num_iter = 1;
+LL = [];
+
+% Convert to inline function as needed.
+if ~isempty(constr_fun)
+  constr_fun = fcnchk(constr_fun,length(varargin));
+end
+
+
+while ~converged & (num_iter <= max_iter) 
+
+  %%% E step
+  
+  delta = zeros(os, ss);
+  gamma = zeros(ss, ss);
+  gamma1 = zeros(ss, ss);
+  gamma2 = zeros(ss, ss);
+  beta = zeros(ss, ss);
+  P1sum = zeros(ss, ss);
+  x1sum = zeros(ss, 1);
+  loglik = 0;
+  
+  for ex = 1:N
+    y = data{ex};
+    T = length(y);
+    [beta_t, gamma_t, delta_t, gamma1_t, gamma2_t, x1, V1, loglik_t] = ...
+	Estep(y, A, C, Q, R, initx, initV, ARmode);
+    beta = beta + beta_t;
+    gamma = gamma + gamma_t;
+    delta = delta + delta_t;
+    gamma1 = gamma1 + gamma1_t;
+    gamma2 = gamma2 + gamma2_t;
+    P1sum = P1sum + V1 + x1*x1';
+    x1sum = x1sum + x1;
+    %fprintf(1, 'example %d, ll/T %5.3f\n', ex, loglik_t/T);
+    loglik = loglik + loglik_t;
+  end
+  LL = [LL loglik];
+  if verbose, fprintf(1, 'iteration %d, loglik = %f\n', num_iter, loglik); end
+  %fprintf(1, 'iteration %d, loglik/NT = %f\n', num_iter, loglik/Tsum);
+  num_iter =  num_iter + 1;
+  
+  %%% M step
+  
+  % Tsum =  N*T
+  % Tsum1 = N*(T-1);
+  Tsum1 = Tsum - N;
+  A = beta * inv(gamma1);
+  %A = (gamma1' \ beta')';
+  Q = (gamma2 - A*beta') / Tsum1;
+  if diagQ
+    Q = diag(diag(Q));
+  end
+  if ~ARmode
+    C = delta * inv(gamma);
+    %C = (gamma' \ delta')';
+    R = (alpha - C*delta') / Tsum;
+    if diagR
+      R = diag(diag(R));
+    end
+  end
+  initx = x1sum / N;
+  initV = P1sum/N - initx*initx';
+
+  if ~isempty(constr_fun)
+    [A,C,Q,R,initx,initV] = feval(constr_fun, A, C, Q, R, initx, initV, varargin{:});
+  end
+  
+  converged = em_converged(loglik, previous_loglik, thresh);
+  previous_loglik = loglik;
+end
+
+
+
+%%%%%%%%%
+
+function [beta, gamma, delta, gamma1, gamma2, x1, V1, loglik] = ...
+    Estep(y, A, C, Q, R, initx, initV, ARmode)
+%
+% Compute the (expected) sufficient statistics for a single Kalman filter sequence.
+%
+
+[os T] = size(y);
+ss = length(A);
+
+if ARmode
+  xsmooth = y;
+  Vsmooth = zeros(ss, ss, T); % no uncertainty about the hidden states
+  VVsmooth = zeros(ss, ss, T);
+  loglik = 0;
+else
+  [xsmooth, Vsmooth, VVsmooth, loglik] = kalman_smoother(y, A, C, Q, R, initx, initV);
+end
+
+delta = zeros(os, ss);
+gamma = zeros(ss, ss);
+beta = zeros(ss, ss);
+for t=1:T
+  delta = delta + y(:,t)*xsmooth(:,t)';
+  gamma = gamma + xsmooth(:,t)*xsmooth(:,t)' + Vsmooth(:,:,t);
+  if t>1 beta = beta + xsmooth(:,t)*xsmooth(:,t-1)' + VVsmooth(:,:,t); end
+end
+gamma1 = gamma - xsmooth(:,T)*xsmooth(:,T)' - Vsmooth(:,:,T);
+gamma2 = gamma - xsmooth(:,1)*xsmooth(:,1)' - Vsmooth(:,:,1);
+
+x1 = xsmooth(:,1);
+V1 = Vsmooth(:,:,1);
+
+
+