By Mary Smith, Chris Bunyan
Книга A Shetland Knitters computing device посвящена вязанию, приведены описания, схемы узоров
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5 display a declining trend in the numerical values of three entropy measures corresponding to the logical order of fuzzy sets. 5 clearly depicts the similarity between the proposed entropy measures and the existing one. An inequality among two proposed measures of fuzzy information and the existing one may also be seen from Fig. 5. That is: HR ! HRa ! HRm . 5777 2 Parametric Generalized R-norm Fuzzy Information … 36 Fig. 4 Generalized Measures of R-norm Fuzzy Divergence As mentioned in Chap. 1, in the recent years, many authors have introduced various measures of divergence for probability distributions and their corresponding divergence measures between fuzzy sets.
1 1 nðR À m þ 1Þ nðR À m þ 1Þ À HRm ðAÞ À IRm A; ¼ 2 2 RÀm RÀm ! 11 holds. 8 Concluding Remarks In this chapter, we have proposed two new parametric generalizations of one of the existing R-norm fuzzy information measures along with proof of their validity. It is noted that the proposed generalized fuzzy measures of information are valid measures which reduce to the known measure on substituting the particular values of parameters. Some of the interesting properties of these measures have also been studied.
7 For fuzzy sets A and B (a) IRa ðA [ B; A \ BÞ þ IRa ðA \ B; A [ BÞ ¼ IRa ðA; BÞ þ IRa ðB; AÞ. (b) IRm ðA [ B; A \ BÞ þ IRm ðA \ B; A [ BÞ ¼ IRm ðA; BÞ þ IRm ðB; AÞ. 7(a) Applying Eqs. 6 Properties of New Generalized R-norm Fuzzy Divergence Measures 45 Adding Eqs. 7(a) holds. 7(b) IRm ðA [ B; A \ BÞ 2 3 þ1 þ1 n flARÀm ðxi ÞlRÀm ðxi Þ [B A\B R À mþ1X 4 5 ¼ 1 R À m i¼1 þ ð1 À lA [ B ðxi ÞÞRÀm þ 1 ð1 À lA \ B ðxi ÞÞRÀm þ 1 gRÀm þ 1 À 1 2 3 RÀm þ 1 þ1 ðxi ÞlRÀm ðxi Þ B R À m þ 1 X 4 flA 5 ¼ 1 R À m X1 þ ð1 À lA ðxi ÞÞRÀm þ 1 ð1 À lB ðxi ÞÞRÀm þ 1 gRÀm þ 1 À 1 2 3 RÀm þ 1 þ1 ðxi ÞlRÀm ðxi Þ A R À m þ 1 X 4 flB 5 þ 1 R À m X2 þ ð1 À l ðx ÞÞRÀm þ 1 ð1 À l ðx ÞÞRÀm þ 1 gRÀm þ 1 À 1 B i A i ð2:46Þ Now IRm ðA \ B; A [ BÞ 2 3 þ1 þ1 n flARÀm ðxi ÞlRÀm ðxi Þ \B A[B R À mþ1X 4 5 ¼ 1 R À m i¼1 þ ð1 À lA \ B ðxi ÞÞRÀm þ 1 ð1 À lA [ B ðxi ÞÞRÀm þ 1 gRÀm þ 1 À 1 2 3 RÀm þ 1 þ1 ðxi ÞlRÀm ðxi Þ A R À m þ 1 X 4 flB 5 ¼ 1 R À m X1 þ ð1 À lB ðxi ÞÞRÀm þ 1 ð1 À lA ðxi ÞÞRÀm þ 1 gRÀm þ 1 À 1 2 3 RÀm þ 1 þ1 ðxi ÞlRÀm ðxi Þ B R À m þ 1 X 4 flA 5 þ 1 R À m X2 þ ð1 À l ðx ÞÞRÀm þ 1 ð1 À l ðx ÞÞRÀm þ 1 gRÀm þ 1 À 1 A i B i ð2:47Þ 2 Parametric Generalized R-norm Fuzzy Information … 46 Adding Eqs.
A Shetland Knitter's Notebook by Mary Smith, Chris Bunyan