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1 function [F,H,Q,R,initx, initV] = AR_to_SS(coef, C, y)
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2 %
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3 % Convert a vector auto-regressive model of order k to state-space form.
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4 % [F,H,Q,R] = AR_to_SS(coef, C, y)
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5 %
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6 % X(i) = A(1) X(i-1) + ... + A(k) X(i-k+1) + v, where v ~ N(0, C)
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7 % and A(i) = coef(:,:,i) is the weight matrix for i steps ago.
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8 % We initialize the state vector with [y(:,k)' ... y(:,1)']', since
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9 % the state vector stores [X(i) ... X(i-k+1)]' in order.
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10
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11 [s s2 k] = size(coef); % s is the size of the state vector
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12 bs = s * ones(1,k); % size of each block
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13
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14 F = zeros(s*k);
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15 for i=1:k
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16 F(block(1,bs), block(i,bs)) = coef(:,:,i);
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17 end
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18 for i=1:k-1
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19 F(block(i+1,bs), block(i,bs)) = eye(s);
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20 end
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21
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22 H = zeros(1*s, k*s);
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23 % we get to see the most recent component of the state vector
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24 H(block(1,bs), block(1,bs)) = eye(s);
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25 %for i=1:k
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26 % H(block(1,bs), block(i,bs)) = eye(s);
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27 %end
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28
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29 Q = zeros(k*s);
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30 Q(block(1,bs), block(1,bs)) = C;
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31
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32 R = zeros(s);
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33
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34 initx = zeros(k*s, 1);
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35 for i=1:k
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36 initx(block(i,bs)) = y(:, k-i+1); % concatenate the first k observation vectors
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37 end
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38
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39 initV = zeros(k*s); % no uncertainty about the state (since perfectly observable)
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