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t=1. using barcode generator for software control to generate, create ean13+2 image in software applications. USS-128 Ot =ek T (15.71). t (j). is computed. The basic MA TLAB code is for j = 1:states i = find(out==0); % find indices of correct bits for k = 1:length(i) sum gamma = sum gamma +gamma(i(k),j); end b(1,j) = sum gamma/sum(gamma(:,j)); sum gamma = 0; end for j = 1:states ii = find(out==1); % find the indices of errors for k = 1:length(ii) sum gamma = sum gamma +gamma(ii(k),j); end b(2,j) = sum gamma/sum(gamma(:,j)); sum gamma = 0; end for i = 1:states b(:,i) = b(:,i)/sum(b(:,i)); end. i i i i i TranterBook 2003/ 11/18 16:12 page 611 #629. Section 15.5. Estimation of Markov Model Parameters Using more e cient comput UPC-13 for None ations, the output symbol probability matrix can be estimated as % find correct bits indices out 0 = find(out == 0); out 1 = find(out == 1); % find indices of errors sum 0 = zeros(1,states); sum 1 = zeros(1,states); gamma sum = sum(gamma); for i = 1:length(out 0) sum 0 = sum 0 + gamma(out 0(i),:); % adds correct bits end for i = 1:length(out 1) sum 1 = sum 1 + gamma(out 1(i),:); % adds error bits end for i = 1:states for j = 1:2 if j == 1 b(j,i) = sum 0(i)/gamma sum(i); % elements b correct bits end if j == 2 b(j,i) = sum 1(i)/gamma sum(i); % elements b error bits end end end for i = 1:states b(:,i) = b(:,i)/sum(b(:,i)); % normalize the b matrix end We can also compute i = (expected number of times in state Si at time t = 1) = 1 (i) 1 (i). (15.72). Step 3: Go back to Step 1 GTIN-13 for None with the new values of = A, B, , or equivalently = , obtained in Step 2 and repeat until the desired level of convergence, as discussed in a section to follow, is reached.. Scaling The forward and backward Software EAN-13 Supplement 5 vectors tend to zero exponentially for large data size and must be scaled properly in order to prevent numerical under ow. The scaling. i i i i i TranterBook 2003/ Software GTIN-13 11/18 16:12 page 612 #630. Discrete Channel Models 15 . constant, the use of whic h is implemented in the MATLAB code given in Appendix B, is rst de ned as . Ct = t (i). (15.73). The scaled values of t ( j), denoted t (j), are given by t (j) = t (j)/Ct This, of course, implies that. (15.74). t (i) = 1. (15.75). The values of Ct are save d and used to scale the backward variables. The scaled values of t (i), denoted t , are given by t (i) = i (i)/Ct with the initialization T = 1 CT (15.76).

where 1 denotes the colum Software ean13+5 n vector containing all ones. The gamma variable can also be normalized if desired, but scaling the gamma variables is not necessary. Turin [1] discusses scaling in more detail.

. Convergence and Stopping Criteria Since the Baum-Welch algo Software European Article Number 13 rithm is iterative, the number of iterations to be performed for a required level of model accuracy must be determnined. Perhaps the best way to accomplish this is to display the estimates of A and B as the algorithm is executing. If one desires each element of A and B to be accurate to a given number of signi cant gures, execution of the algorithm is allowed to continue until the elements of A and B no longer change, within the given accuracy, from iteration to iteration.

The algorithm is then terminated manually. This technique has considerable appeal, since the level of accuracy is known. Also, based on previous knowledge, one may simply perform a given number of iterations.

Another commonly used method for determining convergence is to continue the iterations until successive values of Pr O . di er very little. (Th e Baum-Welch algorithm is guaranteed to converge to a maximum likelihood solution. A proof of this statement is given in [1].

) The value of Pr O . is determined in terms of the scaling constant Ct in (15.73). Speci cally
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