Daniel@0: function [F,H,Q,R,initx, initV] = AR_to_SS(coef, C, y)
Daniel@0: %
Daniel@0: % Convert a vector auto-regressive model of order k to state-space form.
Daniel@0: % [F,H,Q,R] = AR_to_SS(coef, C, y)
Daniel@0: % 
Daniel@0: % X(i) = A(1) X(i-1) + ... + A(k) X(i-k+1) + v, where v ~ N(0, C)
Daniel@0: % and A(i) = coef(:,:,i) is the weight matrix for i steps ago.
Daniel@0: % We initialize the state vector with [y(:,k)' ... y(:,1)']', since
Daniel@0: % the state vector stores [X(i) ... X(i-k+1)]' in order.
Daniel@0: 
Daniel@0: [s s2 k] = size(coef); % s is the size of the state vector
Daniel@0: bs = s * ones(1,k); % size of each block
Daniel@0: 
Daniel@0: F = zeros(s*k);
Daniel@0: for i=1:k
Daniel@0:    F(block(1,bs), block(i,bs)) = coef(:,:,i);
Daniel@0: end
Daniel@0: for i=1:k-1
Daniel@0:   F(block(i+1,bs), block(i,bs)) = eye(s);
Daniel@0: end
Daniel@0: 
Daniel@0: H = zeros(1*s, k*s);
Daniel@0: % we get to see the most recent component of the state vector 
Daniel@0: H(block(1,bs), block(1,bs)) = eye(s); 
Daniel@0: %for i=1:k
Daniel@0: %  H(block(1,bs), block(i,bs)) = eye(s);
Daniel@0: %end
Daniel@0: 
Daniel@0: Q = zeros(k*s);
Daniel@0: Q(block(1,bs), block(1,bs)) = C;
Daniel@0: 
Daniel@0: R = zeros(s);
Daniel@0: 
Daniel@0: initx = zeros(k*s, 1);
Daniel@0: for i=1:k
Daniel@0:   initx(block(i,bs)) = y(:, k-i+1); % concatenate the first k observation vectors
Daniel@0: end
Daniel@0: 
Daniel@0: initV = zeros(k*s); % no uncertainty about the state (since perfectly observable)