By L. P. Hyvärinen (auth.)
This publication is predicated on lectures given by way of the writer on the IBM eu platforms study Institute (ESRI) in Geneva. info concept at the syntactic point, as brought via Claude Shannon in 1949, has many barriers whilst utilized to info processing by way of pcs. yet regardless of a few noticeable shortcomings, the underlyin~ ideas are of basic value for structures engineers in knowing the character of the issues of dealing with details, its acquisition, garage, processing, and interpretation. The lectures, as awarded during this booklet, try to supply an exposition of the lovical beginning and uncomplicated ideas, and to supply whilst a foundation for additional examine in additional particular parts of this expan1in~ conception, equivalent to coding, detection, trend reputation, and filtering. lots of the difficulties in Appendix C are meant as extensions of the textual content, whereas calling for actjve participation through the stu1ent. another difficulties are direct functions of the idea to precise events. a few difficulties require vast numerical calculations. it truly is assumed in these instances that the scholar has entry to a working laptop or computer and that he's in a position to writing the required courses. The stu1ent is thought to have an excellent command of the calculus, and of the idea of likelihood in addition to facts. accordingly no simple mathematical innovations are mentioned during this IV ebook. The Fourier remodel and a few comparable mathematical options are brought in Appendix A.
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Additional resources for Information Theory for Systems Engineers
1 = the double'summation n (41 ) n L: L,j=1 i=1 p(i,j) == 1 As in the Shannon-Fano method (see page 44) the ordered pairs SiSj may be considered as a new set of n 2 characters with probabilities p(i,j). Accordingly, if the statistical dependence does not extend further than to the next symbol, the information rate I(i,j) of the n 2 symbols can be given by the entropy (42) I(i,j) = - ~ p(i,j) log p(i,j) i,j The summation variables i and j do not, of course, appear as true variables in I(i,j), which is a soalar quantity; they are included in the notation only to remind that I(i,j) comes from the matrix p(i,j).
Thus, if that is, the two symbols Si and Sj are statistically independent, (44) is equal to (42). In all other cases by inequality (30) sum (44) is greater. (45) I ( i) + I (j ) ~ I (i, j ) The statistical dependence introduces an additional constraint to the system, and this has a decreasing effect on the information rate. The same argument can be carried over to ordered groups of more than two symbols. If the statistical depen- dence extends over more than two consequtive symbols, as is the case in natural languages, the information rate is further reduced.
22 - logarithm, it is valid for binary logarithms, too. By applying again equation m = ~+m2 and dividing through by m (15 ) If~e I = let m grow indefinitely, the ratios m1 /m and m2/m can be interpreted as the probabilities P1 and P2 at which the two characters S1 and S2 occur in a typical message. Further, the assumption that every sequence with m1 A's and m2 B's occur at equal probabilities implies that successive symbols are statistically independent. Hence, (15) may be written The same approach applied to an arbitrary number n of characters S,1 results in similar expression \ n (1,) I Pj log P~1 = H j=1 =-1: This expression is called the entropy of the probability distribution Pj and denoted by H.