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ISSN 2072-9502. ¬естник ј√“”. —ер.: ?правление, вычислительна¤ техника и информатика. 2014. є 4
??? 519.722/.723:519.224
?. ?. ??????, ?. ?. ?????????
—ќќ“ЌќЎ≈Ќ»я ƒЋя јЋ№“≈–Ќј“»¬Ќќ√ќ ќѕ–≈ƒ≈Ћ≈Ќ»я
ќЋ»„≈—“¬ј »Ќ‘ќ–ћј÷»»
ѕ–» –ј«Ћ»„Ќџ’ «ј ќЌј’ –ј—ѕ–≈ƒ≈Ћ≈Ќ»я
??????????? ?????????? ???????????????? ???????? ???????, ?????????? ?????????????? ???? ??????????, ??????? ?? ???????? ?????????? ????????????, ?????????????
?????????? ????? ???????? ?????? ? ????????? ? ?????????? ??? ?????????????? ???????
??? ??????????? ????????? ??????? (???), ??? ? ??? ?????????? ????????? ???????
(???) ? ?????????? ?????????? Q. ????????? ?? ?????????????? ????????, ??????????
???????????? ???????????? ???? ?????????? ????? ????????? ???????. ??? ???????
????????????? ?????? ????????????? ????????? ????????, ??? ???????, ???????? ??????
?????? ????? ??????????????? ?????? ?????? ????????????? ????????????, ??????? ????????? ??????? ????????????? ??????????? ????????????????? ??????????. ???????????
????????? ??? ???????? ???????? ?????????? ??????? ????????? ????????, ?????????? ?
???????????? ?????????, ??????????? ????????? ??? ????????????. ?????? ???????? ????? ?????????????, ??????????? ??????????? ?????????? ????????????? ???????, ????????? ???????????, ???????? ?? ??? ??????? ??? ????????????????. ?? ????????? ??????
????????? ? ????????????? ??????????? ????? ?????? ????????? ????????????? ??????,
??????? ? ????????? ?????? ????????? ??????? ?????????? ???? ???????? ?????. ????
??????????? ?????? ?????????, ?. ?. ??????????? ???????? ?? ???? ????????? ??? ?? ??????????, ?? ?? ?????? ?????? ?????? ????? ????????? ??? ???????????? ??? ?????????????
?????????????? ????????? ???????????? ??????????? ?????????????? ????????, ???????
????? ?????????? ?? ???????? ???????? ?????? ?? ???? ????????????? ???????????????
(????????????? ??? ????????????????) ???????????? ??????????? ?????????. ?????????? ?????????? ????????? ?????????????? ??????????? ??? ??????? ????? ?????????????
????????????????? ??????? ????????????? ??? ??? ???. ????????? ???? ?? ??????? ??
???????? ????????????? ??? ??? ??? ? ??????????????? ????????, ?????????, ?????????????? ????????, ?????????, ???????? ? ?. ?. ????????? ?????????? ???????? ??????????
?????????? ??? ????????? ??????? ????????????? ??? ? ???. ? ???????? ??????? ?????????? ?????? ?????????? ?????????? ??? ??????????? ?????? ????????.
???????? ?????: ?????????? ??????????, ?????????? ????????? ????????, ??????????? ????????? ????????, ????????, ????????????? ?????? ?????????????.
????????
?????????? ??????????????? ???????? ?????? ?????????? ? ???????????? ?????????
?????????? ? ????????????? ????????? ? ?????? ??????????? ?????????? ?? ????? ??????????, ????????? ? ???????????????? ?????? ?????????? ?. ?. ????????.
?????????? ????????? ???????? ?????????? ?????? ??????????, ? ?????? ???????????
?????????????? ?????? ????????????????? ? ????????? ????????, ?????????????? ??????
???????? ? ?????????, ?? ????????, ?????????? ??? ???????? ?????????? ??? ??????????
? ??????? ?????, ????????????? ?????????????? ??????, ??????? ?????????????????? ? ?????? ???????? ? ?????????????? ?????? ?????????????????? ?????? ?????????? ??????????.
????? ?? ???????? ??????????? ? ??????? ?????? ?????????? ???????? ???????? ? ???????????? ?????????????? ???????, ??????? ?????? ? ??????, ??????? ?????? ?? ????????.
???????? ????? ? ???????? ????? ??????? ???? ?. ?. ??????, ??, ? ?????????, ??? ?????? ??
?????????? ?????????????? ??????? ?? ?? ???? ????????????? ???????????, ????????????? ? ???. ?????????? ?????? ??? ??????????:
? ?????????? ????????, ?? ???????, ?? ???????? ???????????, ?. ?. ?????????? ?????????? ???????? ??????????? ???????? ?????????, ??????? ????? ???????????? ?????????
??????? ????????????? ?????????? ? ??????????? ????????? ??????? (??? ? ???);
? ???????? ??? ?? ????? ??????? ??????????????? ???????????, ?. ?. ?????????, ? ?????
???????? ?????????? ??? ???????? ? ??? ???????? ????????????? ???????? ????????????????
???????? ?? ???????;
67
омпьютерное обеспечение и вычислительна¤ техника
? ???????? ???????? ??? ?? ?????????? ?? ??????, ?? ?????, ?. ?. ??????? ?? ?????????
??????????;
? ???????????? ?????????? ???? ????????? ???????? ???, ?. ?. ??????? ???? ? ???? ?????? ????????? ??????? ???????? ??? ?????????????? ???????, ?? ?? ????????? ? ???????????,
?. ?. ???????? ???????? ?? ??????????? ????? ???? ????????????? ?????????;
? ??? ??????? ?????????????? ??????? ?? ??????? ???????? ???????? ?????? ?????? ??????????? ????????. ??? ????????? ????? ???????????? ?????????? ??????? ? ?????????? ???????
???????? ? ??????????? ???????????? ???????????. ????? ?? ??????????? ???????? ???????.
??????? ????????????? ???????? ??????????? ?? ????? ??????? ??????????????? ????????, ??? ?????? ???????????, ??????? ?????, ????? ???? ?????? ?????????? ???????? ????????? ? ???????? ????????????.
?????????? ???????? ??????? ? ?????????????? ?? ???? ?????????? Q
??? ???????????? ?????????????-????????????? ? ?????????????-?????????????? ????????????? (??? ? ???) ??? ? ???????? ???????? ???? ?????????? ?????? ???????????
???????????????? ???????? H(X). ??????? ??? ?? ??????????? ????? ??? [1]:
?
H ? (X ) = ? ? f ( X ) log 2 f ( X )dx,
(1)
??
??? f (x) ? ????????? ??????????? ???.
???????? ?????????? ???????? H ? (X ) ??????????? ? ?????????:
1. H ? (X ) ????? ???? ? ??????, ? ?????? ????.
2. ?? ???? ???????, ? ??????? ?????????? ???????? H ? (X ) .
3. H ? (X ) ?????????? ?? ?????????? ??? ?????? ????????????? ???.
?????????? ?????????? ?????????? ????? ????????.
1. ? ????? ????, ????????, H ? (X ) ??? ???, ?????????????? ?? ???????????? ?????????????? ??????, ???????????? ?? ??????? H ?? ( X ) = log 2 (b ? a ), ??? b ? a ? ???????? ?????
????????????? ????????? ???????????. ??? (b ? a ) < 1 ????? H ?? < 0 , ??? (b ? a ) ? 0 H ?? ? 0 .
2. ????????? ?????? ???????? ????????? ???????. ???????????? f ( x)dx ???????? ???????????? ?????????, ? ?? ?? ????? f (x) ?????????? ? ?????????, ???????? ??????????? x,
? ??????? ????????? log 2 f ( x ) ??????. ????? ???????, ??? ???????????? f ( x ) log 2 f ( x ) ?????? ?????????? ??????? ?????????, ?????????????, ??? H ? (X ) ??? ???????? ???????.
3. ?????? ? ???? ?? ??????????? ???????? H ? (X ) ????? ??????????????? ????????? ?????? ????????????? ???. ????????, ??????? ???????? ??? ???????????? ?????????????? ??????: H ?? ( X ) = log 2 (b ? a) = 2 ????. ?? ????? ????? ?? ???????? ?? ??????? ? ??? ??????????? ?????? ?????????????: ??? ? = 4 / 6,28 ? 2,72 H ?? ( X ) = 2 ????. ????? ???????, ?????? ? ???? ??
???????? H ? (X ) ????? ??????????????? ????????? ?????? ????????????? ???.
??? ?????????? ?????????? ??????????? ? [2] ???? ??????? ???? ?????????? Q ??? ???
? ???:
?
Q? ( X ) = log 2
?
??
?
f ( X ) dx ? ? f ( X ) log 2 f ( X ) dx,
2
??
n
n
i =1
i =1
Q? ( X ) = log 2 ? pi2 ? ? pi log 2 pi ,
(2)
??? pi ? ??????????? ????????? ???????? ???; f(x) ? ????????? ??????????? ????????? ???????? ???; n ? ?????????? ???????????? ???.
68
ISSN 2072-9502. ¬естник ј√“”. —ер.: ?правление, вычислительна¤ техника и информатика. 2014. є 4
?????????, ??? ?????????? ???????? Q? ( x) ? Q? ( x) ????? ? ???????? ?? 0 ?? 1, ?. ?.
? ???? ????? ??? ?????????? ???????? ???????????? ???. ???????????? ???????? (1) Q? ( x)
? Q? ( x) ????????????? ???????????? ??????? ??????????? ? ?????????? ????????? ???????,
? ?????? ?????? ???? ??? ??? ? ?????? ??????? ??? ???. ??????????? ???????? Q? ( x)
? Q? ( x) ????????????? ????? ???????, ??? ??????????? ????????????? ????? ? ??????????
??????????? ?????. ???????? Q? ( x) ? Q? ( x) ??? ?????? ??????? ????????????? ????? ? ???????? ?? 0 ?? 1 [3]. ??????? ????????, ??? ??? ??????? ?????? ????????????? ????????????? ?????????????????? ?????? ????????????? ??? ??? ??? ????? ????????? ???????? ???????????
???????? Q? ( x) ? Q? ( x) ?? ??????????-?????? ?????. ?????? ? ???? ?????? ?? ?????????? ??????, ?? ?????? ??????????? ?????? ?????????????? ????????????????? ??????.
??? ????? ???????????? ????????? ??? ???????? ??, ??? ??? ?? ??????? ?? ???????? ????????????? ??? ??? ??? ? ??????????????? ????????, ?????????, ?????????????? ????????,
?????????, ???????? ? ?. ?. ????? ???????, ???????? Q? ( x) ? Q? ( x) ???????? ???????, ?. ?. ???????????? ?????????? ?? ??????????, ??? ???? ?????????? ???????? Q ???????? ?????? ?????????? ? ? ?? ?? ????? ?????????? ???? ??????, ???????????? ?? 0 ?? 1. ??????? ????????
???? ?????? C s ???????? ??????? ??????? ???????? ?? ??????????? ???????; ??? ???????? C s
?????? ??????? ?? ????? ???????????? ???????? ?????? ?? ??????????? ???????.
? ????????? ????? ???????????? ?????? ???????? ?????? ?????? [4]:
1. ????????????? ??????.
2. ????????????? ????.
3. ????????? ??????.
4. ?????????? ?????????? ??????.
? ?????? ?????????? ????????? ?????????????? ???????, ?. ?. ???????, ???????????? ???????????? ?????????????-????????????? ????????????? ? ?????????????-?????????????? ?????????????,
?????????? ?? ???? ?????? ?s. ??? ???? ???????????? ?????????? ??????? ???????????, ?????????? ?? ???????. ????????: Cs (Y / X ) ? Cs (Y ), C s ( X / Y ) ? C s ( X ) ? ?. ?.
? ??????? ????????? ?????????? ???????? Q? ( x) ? Q? ( x) ??? ????????? ??????? ????????????? ??? ? ???.
?????????? ???????? Q? (x ) ? Q? ( x )
?????
????????????? ?????????
??????????
?????????? Q
?????????? ??????
????????????
?
?
m!
p (ti = k ) = ?
p k (1 ? p ) m ? k ?
k
!(
m
?
k
)!
?
?
0,19
???????????????????
?
?
m !( N ? m)! R !( N ? R )!
p (ti = k ) = ?
?
k
!(
m
?
k
)!(
R
?
k
)!(
N
?
m
?
R
+
k
)!
N
!
?
?
0,21
p (t i = k ) =
???????
????
????????
1
?
?
? k 1 (1+ ?)K[1+ (k ?1) ?] ?
p(ti = k) = ?(1+? ?) ? (
)
?
1+ ???
k!
?
?
ak ?a
p (ti = k ) =
e
k!
p(ti = k ) = p(1 ? p) k ?1
??????????????
?????????????
????????????
?????????????
???????????????????
ak
(1 + a ) k +1
p (t i = k ) =
p (ti = k ) =
( r + k ? 1)!
k !( r ? 1)!
p r (1 ? p ) k
( k + m ? 1)!( N ? m ? k )!M !( N ? M )!
k !( m ? 1)!( M ? m)!( N ? k ? M )! N !
0,408689
0,408689
0,11
0,40
0,35
0,06
69
омпьютерное обеспечение и вычислительна¤ техника
????????? ????.
?????
??????????
?????????? Q
????????????? ?????????
?????????? ??????
p (ti = k ) = ?
???????????????
??????????
???????????
pk
ln (1 ? p ) k
p (t i = k ) =
0,42
1
m
0
??????????? ??????
N
p (ti = k ) = ? ? n ? n e ?? n k
?????????????????????
0,446116
n =1
p (t i = k ) =
????????????-?????????
???????
p(ti = k ) =
k m ?k
e
m!
1
?k?
? ?
b(a ? 1)! ? b ?
0,443396
a ?1
e
n
???????
?k?
?? ?
?b?
0,446316
?k?
(1 / 2) 2 n2 ?1 ??? 2 ??
p (ti = k ) =
x e
?n?
?? ?
?2?
0,448288
?k?
?????-?????????????
p (ti = k ) =
?? ?
1
k ? e ???
?+1
? ? ( ? + 1)
n? k ?
p (ti = k ) = ? ?
????
????????
n ?1
e
?
1
p (ti = k ) =
e
2??
??????????
?k?
?? ?
???
n
( k ?µ ) 2
2 ?2
p(ti = k ) =
( k +µ )
? ? ( k ?µ2)
?
2
p(ti = k ) =
?e 2? + e 2?
2?? ??
1
p(ti = k ) =
????????? ??????????
?-????????????? ??????
0,444997
2 ? 2 ?2
e
?? 2
2
t-?????????????
?????????
0,44046
k2
?????????????
??????????
????????????? ??????
?????????? ??
0,442637
A
?
e
0,219611
2
?
?
??
( k ?µ ) 2
2 ?2
2??
?(n + 1) / 2 ((1 + k 2 / n) ? ( n + 1) / 2 )
p (ti = k ) =
( n ?)1/2 ? ( n / 2 )
f x ( x) =
0,224997
1
? x ??1 (1 ? x )??1
? (a,?)!
0,224997
0,226204
0,22698
?????????? ?????? ??????? Q? ( x) ? ?????????????? ??????? MathCAD. ?????????? ?????????? ?????????? ??? ??????????? ?????? ????????. ??????? ????????????? ????? ???
p (ti = k ) =
ak ?a
e .
k!
??????? ????????????? ????? a > 0 . ????? k ?????????? ? ????????? k ? {1, 2, K, 11} .
??? ??????? k ????????? ???????? ???????????? p (k ) . ????? ?????????? ???????? p 2 ( k ) ,
log 2 p(k ) ? p(k ) ? log 2 p (k ) , ??????????? ??? ??????? Q? . ?? ??????? (2) ???????? ????????
Q? ??? ?????? ????????????? ????????.
70
ISSN 2072-9502. ¬естник ј√“”. —ер.: ?правление, вычислительна¤ техника и информатика. 2014. є 4
k :=1...11
a k e? a
p(k ) :=
k!
k=
a :=1
p(k ) =
1
2
3
4
5
6
7
8
9
10
11
? p(k ) = 0,9878
k
p(k ) 2 =
0,0011
0,0071
0,0197
0,0308
0,0308
0,0214
0,0109
0,0043
0,0013
3,288Ј10?4
6,7933Ј10?5
0,337
0,0842
0,1404
0,1755
0,1755
0,1462
0,1044
0,0653
0,0363
0,0181
0,0082
? p(k )
k
2
= 0,1278
? log p(k ),2) = ?43,7787
k
p(k ) log p(k ), 2)
?0,1648
?0,3006
?0,3976
?0,4406
?0,4406
?0,4056
?0,3404
?0,257
?0,1735
?0,1049
?0,0571
?
p (k )log p(k ), 2) = ?3,0827
k
?
?
Q := log ? ? p (k )2 , 2 ? ? ? p (k ) log p (k ), 2)) = 0,1144.
? k
? k
??????????
????????????? ???? ?????????? ? ???? ?????? ????? ????? ?????????? ??? ?????????
????????????? ?????????????? ?????? ? ?? ?????????.
??????????? ? ??????? ?????? ? ????????? Q? ( x ) ? Q? ( x) ????????? ??????????????
?? ????????? ??????????? ????????, ? ???????????? ?????????? ??????????. ????? ???????,
???????? Q (x ) ?????????? ?? 0 ?? 1, ?. ?. ?? ???????? ????????? ????????, ??????? ???????
????? ??? ??????? ????????? ?????????????? ?????? ? ?????????.
?????? ??????????
1. ?????? ?. ?. ?????? ?? ?????? ??????????? ? ??????????? / ?. ?. ??????. ?.: ??????. ???.,
1963. 832 ?.
2. ????????? ?. ?. ?????? ?????????? ? ???????????? ??????? ?????????????? ???????????? ? ???
?? ?????????????? ?????????: ???. ? ?-?? ????. ???? / ?. ?. ?????????. ?????, 2000. 234 ?.
3. ????????? ?. ?. ??????????? ? ???????????? ?????????? ?????? ??? ????????? ??????? ????????????? ????????? ??????? ? ????????? ????????? / ?. ?. ?????????. ?????, ???????, 2014. 80 ?.
4. ???????? ?. ?. ???? ?????? ???????????????? ?????? / ?. ?. ????????, ?. ?. ???????????.
???.: ?????, 2001. 384 ?.
?????? ????????? ? ???????? 21.08.2014,
? ????????????? ???????? ? 10.10.2014
»Ќ‘ќ–ћј÷»я ќЅ ј¬“ќ–ј’
озлов јртем Ћеонидович ? –осси¤, 440039, ѕенза; ѕензенский государственный
технологический университет; аспирант кафедры Ђѕрикладна¤ информатикаї; temakozlov.pnz@gmail.com.
—еливанов ≈вгений ѕавлович ? –осси¤, 440039, ѕенза; ѕензенский государственный
технологический университет; д-р техн. наук, доцент; профессор кафедры Ђѕрикладна¤
информатикаї; pvv@pgta.ru.
71
омпьютерное обеспечение и вычислительна¤ техника
A. L. Kozlov, E. P. Selivanov
RATIOS FOR ALTERNATIVE DETERMINATION
OF THE INFORMATION AMOUNT
AT VARIOUS DISTRIBUTION LAWS
Abstract. The article discusses the shortcomings of the differential Shannon entropy and proposes an alternative measure of information, which does not have these disadvantages, quantitatively expressed through the integral Lebesgue ? Stieltjes and exists for mathematical models as
continuous random variables (CRV) and for discrete random variables (DRV) ? the amount of information Q. Its mathematical description is given, potential advantages of the proposed measures
information before the Shannon entropy are justified. Under the identification problem of the distribution law of a random variable, as a rule, the task of choosing a parametric model of the probability distribution, which best fits to the experimental results, is understood. Measurement errors
as values are influenced by many factors, random and non-random origin acting continuously or
episodically. However, the true distribution law describing the uncertainty of the particular measurement system remain unknown, in spite of all the attempts to identify it. On the basis of the
measured data and theoretical considerations you can just pick up a probabilistic model, which in
some sense makes this law best approximated. If the designed model is adequate, that is the used
criteria do not give grounds for its rejection, on the basis of this model, we can calculate all the
probabilistic characteristics of the random component of the error measuring device, which will
differ from the true values only at the expense of non-excluded systematic (unobserved or unrecorded) component of the measurement error. The amount of information minimizes the error in
the solution of the problems of identification of the experimental distribution laws DRV or CRV.
The proposed measure does not depend on numerical characteristics of DRV or CRV ? mathematical expectation, variance, correlation moments, incidents, variations, etc. The results of the calculations of the amount of information for different distribution laws DRV and CRV are given. As an
example, the calculation of the amount of information for the discrete Poisson law is considered.
Key words: amount of information, discrete random variables, continuous random variables,
entropy, identification of the distribution law.
REFERENCES
1. Shennon K. E. Raboty po teorii veroiatnosti i kibernetike [Materials on theory of probability and cybernetics]. Moscow, Inostrannaia literatura Publ., 1963. 832 p.
2. Selivanov E. P. Metody sistemnogo i strukturnogo analiza statisticheskikh analizatorov i IIS po informatsionnym kriteriiam. Dissertatsiia dok. tekhn. nauk [Methods of system and structural analysis of static analyzers and intelligent system on information criteria. Dis. doc. tech. sci.]. Penza, 2000. 234 p.
3. Selivanov E. P. Opredelenie i issledovanie kolichestva znanii dlia razlichnykh zakonov raspredeleniia
sluchainykh velichin i sluchainykh protsessov [Determination and research of the information amount for different laws of distribution of random values and random processes]. Penza, PenzGTU, 2014. 80 p.
4. Gavrilov T. A., Khoroshevskii V. F. Bazy znanii intellektual'nykh sistem [Bases of intellectual system
data]. Saint-Petersburg, Piter Publ., 2001. 384 p.
The article submitted to the editors 21.08.2014,
in the final version ? 10.10.2014
INFORMATION ABOUT THE AUTHORS
Kozlov Artem Leonidovich ? Russia, 440039, Penza; Penza State Technological University;
Postgraduate Student of the Department "Applied Informatics"; temakozlov.pnz@gmail.com.
Selivanov Eugeniy Pavlovich ? Russia, 440039, Penza; Penza State Technological University; Doctor of Technical Sciences, Assistant Professor; Professor of the Department "Applied
Informatics"; pvv@pgta.ru.
72
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