By Wang P.
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Extra resources for A method for calling gains and losses in array CGH data (2005)(en)(14s)
We are particularly interested in an approximation to the number of long sequences. 3 Markov sources and Markov chains 19 Definition. 1) where the logarithm is to the base 2. The combinatorial entropy expresses the number of bits per symbol we can code using long sequences. Asymptotically the largest eigenvalue, , of T will dominate the expression, F(n), and thus determine the growth rate of the number of configurations. ) Assuming that T has s distinct eigenvalues, we can express Tn u = (αi λin ), where λi are the eigenvalues of T and αi are vectors given by the corresponding eigenvectors and u.
In this chapter we introduce the fundamentally important concept of channel capacity. It is defined in a straightforward way as the maximum of mutual information; however, the significance becomes clear only as we show how this is actually the amount of information that can be reliably transmitted through the channel. Reliable communication at rates approaching capacity requires the use of coding. For this reason we have chosen to present the basic concepts of channel coding in the same chapter and to emphasize the relation between codes and the information-theoretic quantities.
Let the vector fn represent the number of configurations after n transitions for each of the symbols as final state. After n transitions, we have fn = uTn , which thus expresses the number of sequences of length n having each of the final states. Summing over the states gives the total number of configurations of length n: F(n) = uTn u , where u denotes the transpose of u. With the string of n symbols we can code one out of F(n) messages or log2 F(n) bits. We are particularly interested in an approximation to the number of long sequences.