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n

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=

x

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, 01-01-00494.

.. -, .. , .. (, )

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7

. ,

, . . . , , .. . , , . , , . , () .

, , . . . , , , .. , . , .. , , - ( ), .

, .

1. .. // , , 1967. 2. Aida-zade K.R., Mustafayev E.E. On a hierarchical handwritten forms

recognition system on the basis of the neural networks // Conference material of TAINN 2003 (International XII Turkish Symposium on Artifical Intelligence and Neural Networks), Canakkale, Turkey, 2003.

8

..

()

.

[1,2] . , , () .

},...,{ 1 nSSM = - . M () )(MKi . )(M () M . : )(', MKK ,

')'( SKKS . tSSS ,...,, 10 .

' ,0 SSSS t == i t= 0 1 1, ,..., 1+iiRSS

1' +ii SRS ', RR - . ', KK .

)(Ml M

nll 1 , , ),( vuK )(1 Mn , vuSS vu ,, , . , uv > .

)},(),...,,({ 11 tt vuKvuKT = . 1. T , )()( jiji vvuu ,

i j i j t =, , , ,..., .1 2 K -

},,...,{ ),( 1 lKKKM = m ,...,1 - K . ,

9

. ,0 ,2 ,1,...,0 },,...,{ 011 1 nnnnlljSSK lnnj jj ==== +++

K

U Ul

j

K

ijj

j

ininKKT1

1

111 )}1,({)(

=

= +++= .

)},1),...(,2(),...,3,2(),,1(),...,3,1(),2,1{( nnnnV = .

.

NEK )(~ 2

1),1(),1()2,1( ,,...,,...,,...,)(~

nNnnn CNK =>>==< )(),...,()(~ 1 KKK N N . )(M ,

.NE d .

NE0 .

1. )(~ K , )(MKK l , 11 nl ,

nM = ln ( lnK =)(~ ). ,

mAA ,...,1 -

M )(,...,1 MKK m - , .

10

2. )(,...,1 MKK m . *K

,

=

=m

iiKKdK

1),()( , )(min)(

)(* KK

MKK=

.

>=< )(),...,()(~ 1 KKK Ni - iK iA , mi ,...,1= .

*K ,

))(~),(~(min))(~),(~( 01 )(

*0 i

m

i MKKi KKdKKd

= = , 0d -

.

NN E>=< ,...,~ 1* :

1=j , =

>m

iij mK

12/)( ; 0=j ,

Nj ,...,1= .

2. *~ NE0 ..

))(~),(~(min))(~,~( 01 )(

*0 i

m

i MKKi KKdKd

= = .

1. ..

// , , 1994. 56 . 2. M. B. Aidarkhanov Metric and Structural Approaches to the Construction of

Group Classifications // Pattern Recognition and Image Analysis. 1994 Vol.4, N 4. P. 372-389

11

.. , . , .. , .. ()

[1, 2]. , [2], , , .

() f(x) ,

( ) ( ) ( ) ( )=

=

k

iii xxf

1

1221 ,2exp2 (1)

2 k < , i i- , i , 2 , xR, R = (, +),

i (0, 1) =

k

ii

1 = 1.

1, 2, , k, 1, 2, , k, 2 .

1< 2<

12

1. () (1, k). , f(x).

2. , R,

xlim (x) = 1,

xlim (x) = k.

3. [1, k] . , [3, 4]

xn= (xn1), n = 1, 2, , x0 [1, k], (3) , f(x).

() f(x)

x = (x), (4a) (x) = (1 + )x (x), (4b)

=4( >is

si2 ) 1, s = 2, 3, , k, i = 1, 2, , k 1., (4c)

si , si = (s i) 1. (4) (2).

: 1. ,

R. 2. [x*1, x*m], m

f(x), 2 m k, x*1, x*m f(x), x*1< x*2< < x*m. 3.

xn= (xn1), n = 1, 2, , x0 [x*1, x*k], (5) , f(x).

f(x) (3) (5) (3) , .

, . 5

13

k = 4, 2 = 1 , . 1.

1.

N

1 2 3 4 1 2 3 4

1 2 3 4 5

0.00 0.00 0.00 0.00 0.00

5.00 3.00 2.00 2.00 2.00

11.00 6.00 4.00 4.00 4.00

16.00 9.00 8.00 6.00 6.00

0.15 0.25 0.25 0.45 0.10

0.45 0.25 0.05 0.25 0.70

0.25 0.25 0.45 0.05 0.15

0.15 0.25 0.25 0.25 0.05

, , . 2 ( f(x)).

2. f(x)

N

x*1 x*2 x*3 x*4 x1 x2 x3 1 2 3 4 5

0.00 0.04 0.06 0.20 2.02

5.00 3.00 3.97 5.94

11.00 6.00 8.00

16.00 8.96

2.24 1.50 1.76 4.00

8.11 4.50 6.20

13.62 7.50

1, 2 , k 2 m 1, 2, , k, 1, 2, , k, 1 m 4.

1. .., .., ..

, . , 1996.

2. Carreira-Perpon M.A. Mode-finding for Mixture of Gaussian Distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence. Vol. 22, 11, pp. 1318-1323, 2000.

3. .. . . .-.: 1936, . 1.

4. ., . . .: , 1983.

14

.. , ..

()

[1, 2]. , . , , [2].

, :

( ) ( ) ( ) ( )=

=

k

iii xxf

1

1221 ,2exp2 (1)

2 k < , i i- , i , 2 , xR, R = (, +),

i (0, 1) =

k

ii

1 = 1.

(1) 1, 2, , k, 1, 2, , k, 2 .

1< 2<

15

[1, k], x [1, k] :

1 (x) k, |(x2) (x1)| |x2 x1|, 0 < < 1, x1, x2 [1, k].

, f(x) , . , .

. 1. (1, k).

, - f(x).

2. , R, [1, k] .

3. [1, k] , f(x) ,

x (x) < 1 x [1, k].

4. k = 2, [1, k], f(x) ,

2 4, 2 , = 21,

21 = (2 1) 1.

5. k = 2, 2 > 4 1 2 [1, 2], f(x) ,

|ln(121)| 212+2 ln(21( + 42 )), 1 2. 6. k = 2 1 =2 x~ = 21(1+2)

. 2 4 , x~ f(x), 2 > 4, x~ f(x).

7. k 3 [1, k], , f(x) ,

16

2k1 4, 2max si

Psi

2,

si = (s i) 1, P = {si, s > i} k1, s = 2, 3, , k, i = 1, 2, , k 1.

8. k 3 2k1 > 4 [1, k] , f(x) ,

21,max ss

s 2, s = 2, 3, , k, 2k1 < 4 + *,

* = (k1) 12max(k + k*(2

2min)),

max = max i, i = 1, 2, , k, min = min si, si P, k* 2si 2, si P.

8 , *k1 = (4 + *) k1, , k.

1. Titterington D.M., Smith A.F, Markov U.E. Statistical Analysis of Finite

Mixture Distributions. Chichester, New York, Brisbane, Toronto, Sin-gapore. 1987.

2. Carreira-Perpinan M.A. Mode-finding for mixtures of Gaussian distribu-tions. Techical Report CS-99-03, University of Sheffield UK, 1999.

3. .., .. . .: , 1972.

4. .. . , , . . , . . :, 1983, 5.

.. , ..

()

. , .

17

. . , .

1).

X ( ) ( )XAAP , . 2). X r

r - - ( ) ( ) ( )( )xhxhxH r,...,1= ( ( )xhi - i - ), : -,

( )xhi - P , -, Xx

( )xH ( ) ( ) 10,11

==

xhxh ir

ii .

3). , . , , - ( )xz ( kRZXz =: ). Z ,

). , 1) ( )( ) 0:0 =>> AxzPA , 2) kRc 1Rd ( )( )( ) 00, ==+ dxzcP

( )( ) ( )=X

ii xdPxhp , ( ) ( )( ) ( ) ==X

ii rixdPxhxzM ,...,1, .

( )h - - , [0,1] , ( ) 00 = ( ) 11 = . ( ( )h [1]).

( ) ( )rr MpMpH ,,...,, 11= - ( )1+kr - , .

,

( ) ( )( )HH = , - ( )1+kr - ( )H .

18

( )1+kr - ( ),,,...,, 11 rr cdcd=

riRcRd kii ,...,1,,1 = .

.

( )( )

( )( )( ) ( )=

=

=

+=r

iiii

r

iihihrhh

hdxzcxH1

11,0:,...,1

,maxarg

1. - ( )H , ( )( ) ( )( )HH . ,

.

- (1) { },...,...,1 nxxS = , P .

{ }nn xxS ,...,1= ( ) ( ) ( )( )xhxhxH r,...,1=

( )

( ) ( )( )=

=n

jjiji xhxzn

s1

,1 ( )( )=

==n

jjii rixhn 1

,...,1,1

( )H ( )1+kr - .

2. 0> 0> 1 H nS

( )( ) ( )( ) ( ) ( ) .2ln14ln2

12

+

+++

krkrDHH

- ( )H , .constD = nx

X

19

. .

( ) ( )( )n

xhn nni

nin

i

111 +=

( ) ( )( ) ( )n

xzxhsns nn

ni

nin

i

111 +=

, ( )nrnrnnn ss ,,...,, 11 = ,

n - n , nHH n =

2. - , :

( ) ( ) ( )SCH nn

n

n=

limlim , ( )SC - ,

S , - , - , ( )SC - { }n .

1. .., .. //

.: . 1. .: , 1999.

.. , .. , .. ()

, () , , .

y k- kXx R= n

, ( ) 1R~, += ktt Xxy . - ( )rHHH ,...,1= kR , k- i~ id ,

20

( )( )[ ] =

+=r

i

n

Hxitit

it

dxcyI1

2, (1) [1].

H - ( ) ( )( )xhxh r,...,1 .

, ( ) ( ) 1,101

= =

r

iii xhxh (2). (1)

: ( )( )[ ] ( )( ) = =

+=r

i

n

tiitit xhdxcyI

1 1

22 , (3), ( )h -

, . : 1) - ( ) hh =1 ; 2) - ( ) 1,)(2 >= thh

t ; 3)

- ( ) 1,)12(2

3 >= thttth . (3) [1]. [1].

, , . , (). , , - [2]. , X sZ R= , , X Z . 1++ sk

, .. ( )ttt zxy ,, . ,

[1]. Z :

( ) ( )( ) = =

=r

i

n

ttiit zhzJ

1 1

2 (4), r ,...,1 - .

Z, - ( )zH (2), 1 , 2 , 3 .

(4) ( )zH , ( )rA ,...,1= .

21

i- . (4)

[1]. ( )zH A , ( )rA ,...,1= .

( )rA ,...,1= . ( ) ( ) ( )( )zhzhzH ArAA ,...,1= ( )h

. Z. (3) (4) Z

( )xH . (3) :

( ) ( )( )[ ] ( )( )= =

+==r

i

n

tt

Aiititii zhdxcyridcAI

1 1

22 ,,...,1,,; (5).

(5) , .

- , . ( )rA ,...,1= ridc ii ,...,1,, = , (5). , , .

Z { }ppZ ,...,1= , . rp . -, pZ 100,

. pZ , , Z . pZ Z. .

Z. , ,

22

( ) ( ) ( )( )zhzhzH ArAA ,...,1= . r ...21 ( )1+r

, . , (5)

: ( )=

+=r

iiiiiSI

1112 ,, .

( )11,1 ,,)( + = rrrrrrr SF , ( )[ ])(,,)( 1,111

1,1 min +++

+ += iiiiiii

iiii FSF

, 1),...,1( = ri .

. ( )[ ])(,,)( 1,012101

100 min

FSF += , 1 ,

- 2 .. r .

1. .., .. //

.: . 1. .: , 1999.

2. .., .., .. // , , . 1985, 3.

.. ()

, [1-4]. , . 11, l , j K ,M( p ,..., p ,..., p ) = j- l- , l=1,,K, j=1,,M. . N , s={ l , jN } . s , ,

23

={}. , , p( ) .

,

: 11 l , ja

l , jl , j

p( ) ( p )Z

= , 0l , ja > - , Z .

1l , ja . Per .

Per, , , . - .

. Per

( t ) ( N A,N A;it ) = + +% ,

0

m( m )

m ( m )

b x( b,c; x )c m!

=

= ,

( m )b : 1( m )b b ... ( b m )= + , 0 1( )b = , i

, 1

j

M

f ( c ), jj

A A - a=

= % , ljl , j

A a= .

. EPer=(N+ A~

)/(N+A). ,

, , , [3]. , . .

[5] . , () . .

24

[6] Q=(N+1)/(N+2), , , =2, =1 . , .

,

. . , .

, , . , UCI Machine Learning Database Repository (http://www.ics.uci.edu/~mlearn/ MLRepository.html) , , . .

, 01-01-00839.

1. Berikov, V.B. A priori estimates of recognition quality for discrete features. Pattern Recognition and Image Analysis, Vol. 12, N 3, 235-242. 2002.

2. Berikov, V.B. An approach to the evaluation of the performance of a discrete classifier. Pattern Recognition Letters. Vol. 23 (1-3), 227-233. 2002.

3. Berikov, V.B., Litvinenko A.G. The influence of prior knowledge on the expected performance of a classifier. Pattern Recognition Letters, 2003. ( )

4. .. . , 2003, 43, 9, .1448 1456 ( )

5. Breiman, L., Friedman, J., Olshen, R. and Stone, C. Classification and Regression Trees. Wadsworth International, California. 1984.

6. T. Niblett, I. Bratko. Learning decision rules in noisy domains. Expert Systems 86 Conf., Brighton, 15-18 Dec. 1986 In Developments in Expert Systems (ed. M. Bramer) Cambridge Univ. Press, 1986.

25

K

.. ()

K M N. ( ) ( ). . , , , . .

, N M . , , ( ) ( ), . , . , , . , , - , , , - .

. >< M1 xx ,, K

26

x )(][ , , xdPMK

1j1kkj == ,

)( jkj xdP - jx j , k ( j );

1N1xN

jkjk

jkxdP ++

=)(

)( kN -

k j , )( jk xN -

j jx ( j ). , , . j k ,

)( jkj xdP . -

>< M1 xx ,, K x , ""=jx ,

1xdP jkj =)( k .

:

( ) ( ) ( ) ])(...)()([lnmaxarg)(..

M21 MkM22k11kkK1k

xdPxdPxdPxd =

=

{ } M1j10j ..,, = , 0R kk > , K2k ..= 11 = . >>==

27

x , . , 11 = ( ). , >< K1 EE ,,K , }|)({ kk KxkxdPE = (c >< K1 EE ,,K ).

>< ; . , . d ; :

|},..:{||},..,)(:{| ;;

ki

kiiKxN1ii

KxN1ikxik

de

>=<

N 1iix =][ - . ,

, :

>< ; . : j , >==< )()()(,, KK

: >< =

=> , . , , . [3,7,9,10], NM + , K . 1K1KN ++ )( .

maxminmaxmaxmin ,..

>

29

[6], . :

++ = 156eEP N2

Nx N 1iiA )/(ln)]([ln

)]([ N 1iiA x = - , , d ( >< , - ).

maxminee MN 1iiA 2Nx = )]([ , M N NcNM ln 0c > ,

( 0e maxmin ), , E M N .

1. ..

// . 1999. 2. .. . //. , 1998. 3. ..

. // . , 2002. .2, .9, . 97-114.

4. .. // .

5. Vapnik V. The nature of Statistical Learning Theory. //Springer-Verlag. New York, 1995.

6. Boucheron S., Lugosi G., Massart P. A sharp concentration inequality with applications. //CNRS - Universite Paris-Sud. Orsay-Cedex, 1999.

7. Minoux M. Mathematical programming: Theory and algorithms. //Bordas. Paris, 1989.

8. Sill J. The capacity of monotonic functions. //Elsevier science. Berkeley, 1997.

9. Aardal K., Weismantel R., Wolsey L. //Non-standard approaches to integer programming. //DONET. 2000.

30

10. Mitchell J. Branch-and-cut algorithms for combinatorial optimization problems. //Mathematical sciences. NY, 1999.

..

() ,

, . . . , (, ). , , -, , , , -, , , . - ( ) , , [1,2]. . , -, , , , , , , . , , , , . , , (). , , . : 1) , 2) Xxi (i=1,,n; n=|X|; X , , 3) , , 4) / . , . : )

31

si Qx ; XQs ; Ss (S ); ) si Qx ; )

sQ ; )

si Qx ; ) . , [3].

sQ

, XQs , X: XQQ ss = .

Xxi ( / ) (, ). ( ) . . : ( ) , . BB 1 , : BB 2 - , - BB 3 - , .

XQs : ) / ; ) , . , . , , .

/ . , : 1min T , (1) : minT - , ;

1 - .

32

(1) , , , (, [4]). , , . [5]. , , ( ), , () , () .

. ( ) () . , : ; , . . [6].

1. .. . .:

, 1972. 208 . 2. .., .. . 3 . .:

. , 1989. 232 . 3. .. (

). .: , 1982. 168 ., ().

4. .. . . .: , 1974. 142 .

5. .., .. . . .. , 1958, . 51 .270-360.

6. .., .., .., .., ,., .. .//

33

. .: , , , 2001. .33-35.

.. , ..

() ,

[1]. , , , . , " " .

, , . , , , , . " " [2].

n- x, y, z, , w. , : x- , y-, , w-. : , 1, , 0. xyzw x z 1010, 0000.

: () (). , - -. n- ( ) n- , . , , - , , , . , ( ) , ,

34

. 1. , , , 0.

n- () (). , . 2n- , , n- . : - - . - -. - - , , , , - [3]. - -.

- - , ( , ) [4]. - - , , [5].

1. - / ..,

.., . , .. // , . .: - - , 1996. . 16-43.

2. .., .. // . .: -, 1998 . . 59-68.

3. .. // , . .: - - , 1996. . 16-43.

4. .., .. // . : , 2002. . 2. . 195-199.

5. .. // , 2'2002. , 2002. . 53-57.

35

.

.. ()

, , . , ( ) . ( 100-1000 ). , , . , , , .

( , ).

, .

() .

( 02-01-00326).

..

() , ,

, , , . , , , , . , , , .. , , .

36

, , . , , , , . , () .

, , () [1] () [2]. .

. () . . , . , , . , , . , .

() , . . , . - [2]. , . , , , , , . , . , - : , .

37

, , -, , -, , . -. -, , , .

, , , . , , , . , . , , . .

, - . - . , . . , . , , , . , -. - , .

1. ..

. ; , 1975 311. 2. .. //

. 1996. - 1,2 .239-281.

38

.. , .. ( )

. . [1] . . .

.

, () ( ) ( ). - . , .

, 1L . , . . , , . , . , , , 85-90% . , . , .

39

. , . , , , , , . ScX ScY [2]. . .

.

. . .

. , . . , , . . , - . , , . , , .

. .

. : . : , .

40

. , . , . ),,( NNN - . , . , , .

,

. , ScX . . 0.8 0.9. 1.0974 1.7172 . , , , 7.2584 4.6738 . , 2 4 , , .

, ,

. . , , , (- , ).

(00-15-96108) ( 0392).

1. .., .., .., ..

. // (-9): 9- . .: . . / .: - -, 1999. .149-151.

2. .., .., ..

41

. // (-10): 10- . .: . . / .: - -, 2001. .176-179.

.

.. ()

, . )(A }|)({ DA = . . : )(maxarg)(* AA

A=

. )(A

. , ( ) , . , , , , . , , , , . , . , , . (.. ) [1].

, .

42

},...,1{: lRK n , . },...,{ 1 qSS . n- :

nRX }|{)( XSSX iidf

=

: )(minarg)(* YA

A

= (1)

)}()(|{ iii SKSASY = . , .

, , . , . X :

)}()(,),(:),()(|{ iin

iii SASASSRSSKSASX

43

qqddXYAR ++++= ...))()(()( 11 (3) .

, 1) 2) , .. ,

3) ,

. 1. (3)

1= :

1 21 2

10 ,0 ,...,0 qd d

qR q q < < < < < <

),(min jiji SSR = .

, , (3) . , , . , (.. ) . ,

2 ( ). 0>C , C

44

, . , .

1. .., .. ., 2

., ., 1979. 2. .., .. // .:

, 1974.

.. ., .. ( )

, , . [1-3]. , .

[2-5].

() M Modi n ( ) , .

1 . PkMi

PjMi , M i ,

( ) ( ), 1/ 2( ( , ))i i i i i ii M M M M M MM k j k j k jP P P P D P P = +% % % % D , , . , D , , . , .

, Modn ( ) ,

45

:

( )( )

1 P P

P P

Modk j

MM Mod

kM

jM

n

ii n

i i

,

,

( )( )=

.

, ,

, , i ( 0.5 (0,1] D).

1 . ( ) , , i :

1. ( )0 1 i , 2. ( ) ( ) i i, ,= ().

( ) ( ) ( ) i i i, , , + ( ). ( ) = i , 0 (

,

Mi M Mi i )

5. ( ) = i , 1 6. ( ) ( ) ( ) i i i, , ,= = 1 . 7. ( ) ( ) i i, ,= . 8. ( ) ( ) ( ) i i i, , , = + . 2.

, , .

,

46

3. , , .

, ,

, ( ) , , ( , , ). [3] , , , .

4. (), .3 [ 2].

5. (), .2 [3].

. , . , . .. .

01-01-00839.

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47

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.. ()

- [1]. , (105107 ), . .

, , . , , [2]. , , ( ) . , , .

, ( ). , , , . , .

, , . . [4].

.

48

. .

, . . , , .

. , . , .

, , , , , . . .

. , .

. , . ( ), [3].

( 02-01-00325, 01-07-90242) . : www.ccas.ru/frc/papers/voron03qualdan.pdf.

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.. ()

, - . , . , 1 2 . () ( ) ( )1 2n i if x f x = , ix

1i ,n= .

nB A< < . , 1 , n A 2 , n B . A B . A B 1Q 2Q [1].

, . ix . , . :

( )( )

1 1

2 2

in n

i

f x ff x f

=

.

50

. , :

( )1

i i i

i i i

y f x , ,

x , i ,n

= +

= + =

i i ; i () ; .

, i i ,

( )2 i ( )2 ix 0i = . [2].

:

( )( )( )

( )( )

2 2

2 21

12

ni i i i

i ii

f x , xF

x

=

= +

.

( )D . , M , [3]:

2kt

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= =

1k , t ,S= ; S .

jf .

51

( ) ( )2

1 1 1 1G

Q Q f x, f x, dx = ,

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Q Q f x, f x, dx = ,

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A B , A A B B , .. . :

( ) ( )1 1 2 21A A Q Q Q Q ,

( ) ( )1 1 2 21B B Q Q Q Q .

, . . .

( 02-01-96023).

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. .: , 1990. 320 .

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52

nR .. ()

( ) sup | |Ey E

d x x y

= nE R

( . [1-3]). E . E , ( )Ed x . [1] k - k - ( 2k ). .

nE R , kE C , 2k , nU R - , a) x U E b) U E

( )kEd C U . : ( )f z - z U E . A U , ( )f A . E - , E , ( ) 0f A =

E 1

2 21 2 1 1 2 1

1( , ,..., , ( , ,..., )) |

nn n i

it t t t t t t r

=

, kC , (0) 0 = , grad (0) 0 = (.. n- E 0 ). , , n- E ( ). 1n .

, 2 (0)D i E 0

53

, 1.. 1i n= . B A ( )Ed A , S -

. E E B ( )f A E S I , S E ( )f A . , ( )A f A

E 0 , 0(0,0,....,0, )A y= 0 0y > . S E

0 E B , i S , 0

1y

.

, A - , 0

1i y

> , 1.. 1i n= .

, f U (., , 4 [3]). , V U , A V , 1( ,..., )nx x x V =

( )( ) ( ), ( ( ))f x t x t x= 1 2 1( ) ( ( ), ( ),..., ( ))nt x t x t x t x= , ( )t x r< . ( )t x

( ) ( )1 2 2

1 11

( ,..., )n

i i n ni

t x t t x

=

+

1( ,..., )nx x V ,

( ) ( ) ( ) ( )1 2 2

1( ) 2 2 ( ) ( ) 0

i

nj j n i i t n

i jt x t x t x t t x

t

=

+ = + =

( ), 1.. 1i it t x i n= = . , ( ) ( )( , ) 2 2 ( ) ( ) 0ii i i t nF x t t x t t x = + =

( ), 1.. 1i it t x i n= =

A . , ( )k rC B , rB - (n-1)- r 0 , iF rV B ( 1)k . , ( ,0) 0iF A = i .

54

, 1 1

1 1

( ,..., ) 0( ,..., )

n

n

D F FD t t

( ,0)A .

( )( )2( ) ( , ) 2 1 ( ) ( ) ( )i i i ii t t t n tF x t t t x t = + + ,

( ) 20( ) ( ,0)

1 (0) (0) (0)2i

i i ii t

t t tF A

y

= + + =

0 01 (0) 1 0i it t iy y = = < , . ,

( )( )( ) ( , ) 2 ( ) ( ) ( ) ( )j i j i ji t t t n t tF x t t t x t t = + ( )( )( ) ( ,0) 2 (0) (0) (0) (0) 0j i j i ji t t t n t tF A x = + = i j .

, . , W A

1( ) kt x C . 2.4 [1], ( )( ), ( ( ))( )grad ( )

( ) ( )E E E

x t x t xx f xd xd x d x

= = 1C -

, , , 1grad ( ) ( )Ed x C W , .. 2( ) ( )Ed x C W .

2k , ( ) ( )kEd x C W .

A U , ( ) ( )kEd x C U . Q.E.D. ( 03-0100241)

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, , . . , , . , .

, , , (), . , .. , (, ), .

fa_pri (; *) * , . L (, x) , x . fa_post (; *, x). *,

57

L : L = arg max L (, x). B * a_post , .. B = a_post [2]. , L B x (, x) . L = B . b = b(x), *. b c = c(), . fa_pri (; ) . ( c, D (c)) fa_post (; , x) u * [3].

p*, p* , . m > 0 ( - ) n > 1. , m , , n . pL p* m / n. [2] pB p* (m+1)/(n+b+1), b > 1 ( , = b), , b.

- Be (m+1, nm+b). pL = pB b = (nm)/m. L = pm(1p)n-m p* Be (1, b) [3, 4]. , , .

, , , [1], , (1) b = (nm)/m p = 1/(b+1) b* p* , (2) D (p) b/[(b+1)2(n1)].

, a_post = arg max L = p = 1/(b+1) = a_pri .

, b

58

1: 20% ( ) b = 4.

, b p

Ip (m+1, nm+b) = , Ip -, , p+, (0, p+), p*.

b [1, 3] : p+ p- ( 0,5 P < 1)

Ip (m+a1, nm+b) = 1 P = (1)/2 Ip (m+a, nm+b1) = P = (1+)/2 .

, , , Be (1, b) - Be (a, b) 1 a, b (.. ), . n, m ( = 0,900,99).

, n / .

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.. ()

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, .

, , . 1.

, . , ,

, , 500 . , - . 3000 4000 . 2000 3000 . , .

2. .

, . ,

60

, . .

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. . , , , .

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62

1. ,

, [2], . .

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[5] , , . ( ) , . .

C++ x86. mxn 0 1. (m, n) 10 . m=n, mn. 1-3 1-3.

1. ( m=n).

()

mxn 1 2 2/1 /1 /2 20x20 48 20 20 0,417 0,416 1 22x22 97 32 24 0,330 0,247 0,75 24x24 215 72 31 0,335 0,144 0,430 26x26 389 126 48 0,324 0,123 0,380 28x28 610 252 70 0,413 0,114 0,277 30x30 1 456 485 121 0,333 0,083 0,249 32x32 2 289 849 192 0,371 0,083 0,226 34x34 4 323 1 310 343 0,303 0,079 0,261 36x36 7 080 2 148 526 0,303 0,074 0,244 38x38 12 334 3 805 919 0,308 0,074 0,241 40x40 19 879 6 545 1566 0,329 0,078 0,239 45x45 81 083 21 177 4374 0,261 0,053 0,206 50x50 264 800 68 679 13440 0,259 0,050 0,195

63

050 000

100 000150 000200 000250 000300 000

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64

3. ( m>n).

()

Mxn 1 2 2/1 /1 /2 30x30 1 456 485 150 0,333 0,103 0,309 40x30 5 734 1 078 275 0,188 0,047 0,255 50x30 18 847 2 583 426 0,137 0,022 0,164 60x30 42 757 4 236 695 0,099 0,017 0,164 70x30 125 420 5 655 945 0,045 0,008 0,167 80x30 235 072 9 732 1 579 0,041 0,006 0,162 90x30 13 178 2 030 0,154 100x30 17 497 2 714 0,155

0

50 000

100 000

150 000

200 000

250 000

30x30 40x30 50x30 60x30 70x30 80x30 90x30 100x30

()

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3.

(m>500, n

65

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, .. , ..

() -

, [1]. . . .

, . [2]:

( ) = =

+=N

nnm

n

mnmnmN PmBmAF

0 0)(cos)sincos(, ,

)12()!()!(

)()(1

1 ++

= nmn

mndttPtP mlnkklnm

m 0.

N.

66

. SO(3) [3] .

.

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, ( ). - : 148- 148 , 500 . , .. 148 148- . , , . , , . , , , 1/3. , . . .

67

. Drug design

. . - . .

. , . , , . , (.1).

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1 A00 = ()

B00 2

2 {A00, A10, A11} =

()

{B00, B10, B11}

3

. . . . .

.

( 01-02-16127, 01-07-90317, 00-01-05000), 107 6- .

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, , , [4].

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, ,

69

. , , .

1M , , , 0M . (i,j), , * , :

*00,

~~),(),(}{

|))()((min))()((minmax

=

= uvi

jMvu

uv

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SSSS .

: .

: 1. .

, . , ( , ). , , .

2. , , . , , .

, , .. ],[...],[ 11 nnE = , . , , , , . , .

: 1.

))()((minmax))()((minmax00 ),(),(),0[

uv

ijMvuP

uv

ijMvu

SSSSn

=

70

, ],[...],[ 11 nnE = , , , , .

. ,

0)()()()( 1001

1 +=+=iiii

j SSxSxS jjj

=j A

jKS

iii SSBpSQ

S~

1

1 ),()()(1)(

.

, , ,

nkgggS knni

j ,...,1},1,0{,...)( 11 =++= ,

nsgggS vusnvu

nvuu

v ,...,1},1,0{,...)(,,

1,

1 =++= . : 2.

nsgs

s ,...,1,,01,1

:~ ** = =

= .

, , .

, , ,B , .

, , , . .

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() ()

, , . ( ). , . , . VCD () , .

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r - , DFBSA - Decision Forest Building Sequencing Algorithm. .

1. () nmT , , m n .

2. 0> nm, r , r [1,2].

72

3. ( ) . r , r , , , . -

nB , nmT , , . () , . , , .

4. 1k . () , USED. , k - ( ), , , qk = , q , . ( nmT , ), . , . nmT , , , , . nmml TT ,, , ml < . mlT , . , USED, . , USED. , .

DFBSA q r2 . rq 2 ().

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73

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DMFSA . , , . . r .

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)(,...),( 1 njj txtx ,

nttt ,...,, 21 , , j- . )1( +tx j (

1+nt ). . , , 1+nt . [1]. 1t () N X (3-5) r, ( ). () .

75

( ) . , .

() . 2t . , [2]. , , (, 2N ). , ( ). n . () n . r+1 , , , ( ) . ( ), 1+nt .

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.. , .. ()

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. : , .

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( ) ( ){ }xhxhxH r,...,)( 1= , ( )xhi - x i- ,

( ) ( ) 1,101

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r

iii xhxh , .. - H(x) r-

. H(x). , x H(x) V , .

77

. , i- () i . A. X A ),( axK .

( )( ) = =

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( )h V , , .

, . , . , , . ( )h V, [2].

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rr

rr

rr

1321

1321

1321

.....................

...

...

,

pp rp ,,2,1 K= . -.

( )0,...,0

81

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{ }1,,1,0 kK , 2k ; rkE , 2k , nr , ( )r ,,1 K , { }1,,1,0 ki K .

kmnML , rkE .

),( LC - - L , ),( LS - - L , ),( LB - - L ,

),()(1

LCLCrkE

n

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= , ),()(1

LSLSrkE

n

rUU=

= ,

),()(1

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n

rUU=

= ,

A - A .

[1,2] , ( )LB ( )LS kmnM

n mknm , 1> , 1 , 1

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I

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V1 , .

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87

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*(g(.)): L(X),

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: ?

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92

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

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94

.

),,( 1 paaA K= p n (+ 1, 1) , A . , .

[2]:

=

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kj

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95

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. , , . n- N- , N >> . , N exp(n) .

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f Ti

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, [3], . T , . ; . [3].

. n = 6. 32 , 26 6. 32 3 , .

96

; ijijw = , I,

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H , ),( xud p pu

, ipz pu , :

( ) .,,0,,,,, 11 nruuuuz pnprprppr KKK =+= + ,

, . , , T.

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.. , .. ()

,

97

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NNxxX = )...,,( 10 L

LUUw = )...,,( 1 , MMnn = )...,,( 1 , = )...,,( 1 L ,

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ii j

iux 1 11)( ,

1,0 = Nn , 0)( =iu j , 1...,,1,0 qj ,

iq Uiuiu = ))(...,),(( 10 , Li ,1= ; = :{)(kU

,)...,,( )( 1)(

0)( qk

qkk uuU = ,||||0

2)(

98

. Mnn ...,,1 , M X .

,max1min TnnT mm Mm ,2= , ; minT maxT ,

minM maxM .

, w , , j , i . Li ,1= 1 ii , i - , 11 +i , i - , i , . , w . X , w .

, )...,,( 1 LUUw = . ),,( 10 = NyyY K : EwXY += )|,( , INE 2,010 ),,( = K ;

I2,0 ),0(2I .

, Y ,

)|,( wX . N , minT , maxT , M q .

. 1 , 3 w 4; , 9

99

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, 03-01-00036.

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100

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. , . , : 1) ; , , ( ) ; 2) / ; , , ; ; 3) .

, , () , [1,2]. . .

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quuU = )...,,( 10 = ),...,( 1 Mnn , .. ),( UXX = ,

, = =Mm nnn mux 1 , 1,,0 = Nn K , 0=ju ,

1,,0 qj K ,

101

xn , 1,,0 = Nn K . . Mnn ,,1 K , M X . max1min TnnT mm ,

Mm ,,2 K= , ; minT Tmax .

+ qq uu ,,K

10 ,,,,,, + qqq uuuu KKK , + ),( qq :),{( += qqQ

}10 + qqq ,

, q +q .

+==

===

+

++

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,,,,),,(~~

qqjqj

qqjuuqquu jjjj

KK

K

,

),,(~)~,,~(~ 10 UqqUuuU q+

== K .

)~,,~(~ 10 = NxxX K

= +

=Mm nnmmnnn mm uqqux 1 ),,(

~~ ,

1,,0 = Nn K , ,0~ =ju 1,,0 qj K ,

)),(,,),(( 11++MM qqqq K

MQ= . =X~ ),(~ UX , ))(,(~ UXX ,, X ,

U , . ,

EUXY +|= ),(~ , IE 2,0 , ,

M , ),(~ UX | )( UX | .

102

maxmin ,, TTN , q U . .

. , . , X~ Y ; , , . , . .

,

)= =+ =)

+

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2 min}2{,(M

m

q

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m

m

muyuS ,

MQ M . . , . , . , . , .

, 03-01-00036.

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103

.. , .. ()

, , , , , , . , , , , , , . [1].

. . . , . , . , : , .

,

, [2]:

=

YNaN

dtdN

, 0 , >a , (1)

ANY = , 0>A , 10

104

.

, (1)

),;()1()1( 11 axFxxax tttt =+=

+ . (2)

, ,

),( xf . , , , . . [3]

(2) :

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: . .[4]

: , .

. , , . )(wd = , d - ; )(w - ,

2 [5]. ,

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105

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106

.. , ..

()

- [1].

( ), . ( , , , -) . , [2] , [3] - . , . . , . - .

:

= .,0

,1),,(

2121 jjj

jjjcxc

xccP

njcc jj ,...,2,1,

21 = , - .

{ }liiii

SSSN ,...,,21

= - iK . njxccPSP jjjNi ,...,2,1),,,()(

21& == , ),...,,( 21 nxxxS = , :

107

iN ,

)(),( 21 jNS

jjNS

j xMaxcxMincii

== .

1 . , .

. 1. .

, ( 1), . , , .

, 0 , iK , iP , , .

, , . iK

108

k , k - . : j- - , j - iK iP . , , . [4-5]. .

()

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, . 2-

, .

3- , .

( 02-01-08007, 02-01-00558, 02-07-90134, 02-07-90137, 03-01-00580), 7, 17 ,

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109

.. ( )

, , ()

{ }MixxxxX iiniiii ,1),,,,,,,( 21 = KK , iX - i , , ,1, nxi = -

n iX , i - , iX , ,2},,,2,1{ KKK K - , M - . , . . , . , , . .

[1], , . , , . nks

},,2,1{ ,,1 , nikx si K=

siii kbxa ,1 , = , (1)

- ,

ia ib - ,

110

. n -

sk - skP ,

. . , , . , , . [2], (1), . , , , , , , , . , , n -

ii ba = (1) n .

iX jX x [1]

Dxxxji // = ,

D - x , ,

= x/

>>

jiij

ijij

jiji

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0 /)( /)(

.

: , ,

111

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.

.. ()

, , , , . , , , . , . , ( ) , . , , , .

( ) i

ii xy min (1)

x , y (), , .

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. -, [5,6] , -,

( ) ( )( )tt 21 = , ( ) 21 uu = , 2 , 1 (-). ( )t2 ,

113

1, . [7]

( ) ( )2/exp 2xxx = , > 0 . ( )t = 1 . .. [4].

. , . ,

( )( ) min, i

ii xy , (2)

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( )[ ]{ } min2/,exp 2 i

ii xy , (3)

, , Neural Network Constructor [8-10].

, 20 , 8

. ijx : i = 1,2,...,20; j = 1,2,...,8 , N(0;3). :

i

j

ij

i xy += =

8

1, (4)

i N(0;3). . 17 = 30 18 = 15.

3 : ( ), ,

114

[6] , . , , 17 18 . 2 (,) . 1 .

1. . 2R 2 2s R eff stb 0,97 3,29 8,79 5,50 1,00 0,00 0,97 3,29 8,67 5,37 1,00 0,02 0,92 6,56 6,65 0,09 0,50 1,00

: 2R , 2 , 2s , R , eff stb . [2-4].

8- 2 , . , , 17 , 18 . , , .

-40

-30

-20

-10

0

10

20

30

40

50

-40 -30 -20 -10 0 10 20 30 40 50

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Control

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18

-40

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-10

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10

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40

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-40 -30 -20 -10 0 10 20 30 40 50

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17

18

. 2. . ,

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[1] ,

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))(exp()( += tit ,

0, , q : { } .1qr N : ))(,),(),(( 21 ttt N K= , p p -, :

1 2( ( ), ( ), , ( )), 1,.., .Nt t t p = =K

m -

1 1 1 2 2 2 ( ( ), ( ), , ( )),m m m N N mNa b t a b t a b t = K

{ }Nia { } Nib - : ia - , -1 +1 a

a1 ; ib - , b mi - , b1 . , [0,1]a ,

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118

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2errNq a b

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01-07-90134 ( 2.45).

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.. , .. ()

1 . ,

[1], [2] N is :

i,j 1min ( ) ( , ) , , 1.

Nij i j ij ji iE J s s J J s

=

= = = =

s

s Js srr r r

(1)

),...,,( 21 Nsss=sr

- , - N (, ..), )(s

rE

, 1( )N

ij i jJ ==J ( , ) .

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rE ,

1( 1) sgn ( ) ,

Ni ij j

js t J s t

=

+ = (2)

. . , , , N2 sr 40N > .

, [3]. . )( 3NO .

120

2 . , *sr , ,

, J , .

, J

)(ifr

, i :

( ) ( ) ( ) ( )1 2, , ( , ) , , 1, ;

i i i ji N ij i j N = = =J f f f f

r r r rK K

ij - . :

( )(1) 2 (2) 2 ( ) 21 2( ) ( , ) ( , ) ( , ) .NNE = + + +s s f s f s fr r rr r r rK (3) , 1 -

1 , 2, ,i i N >> = K (1)

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. )1(f

r:

( )(1) (1)1 11

* sgn ( *, ) | | | ( , ) | . N

( ) ( )i i i

is f i f

== = s f s f s

r rr r r

, (1) . , J

. , . , )( 3NO , .

1500 J -

500 (15x15), (16x16) (17x17) . [-4,+4], 0. , ,

121

(1); - . , 7- (1) (7), ,f f

r rK 3

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. , 21 (2), ? ( .)

, ( 1500 ) , , 3- , , 0.8. , , , 7- , 0.97.

, ( )E sr , , 1, 21- .

( ) )( 3NO . , (1). J ( ).

2 310 10 , , , , ? .

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122

.. , ..

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N- {Y}, (=1,2,,p), . Y RN - X , {X} (). ; X. : YRN , Y , X , . N- . , RN , .

{Y}

{X}. N- Y=(y1, y2 ,, yN), { }0 /1iy = . n , m+1

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123

x Rq ( mq 2= ) : x=ek , ek - k- - Rq. , YRN 2m- , Rq: X=(x1, x2,, xn).

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{Y}, p {X}, [1],[2].

m . , , .. X Y. , m, -, ( ); , -, 2m - . , , , .

. Y=(s1y1, s2y2, , snyn) - - , si - , s yi , 1s . , s . , 50% - 0.5s , - 50% ( ) 50%, .

- :

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2m

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(1)

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124

[ ]2 2(1 2 ) 2(1 )

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, . . , , , . 20% .

( 03-01-00355 01-07-90308).

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.. (. )

,

- ; , , , . . , , ,

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; , , . [1,2,3], , . , x(t) )(tn ( An). :

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dttttxA nT

n = (1)

)(t - .

An .

, . , , . . .

, [4] An, , , . , , .

1 , , [0,T] ( 14). 2 , .

126

( ) .

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0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7t

. -

.1.

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0102030

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.. , ..

()

. , , .. , .. , .. , .. . [1,2,3,4].

[5,6], . , .

, a nXXX ,...,, 21 ( ) () mYYY ,...,, 21 ( ).

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=

=n

jX

defX jDD

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m

jY

defY jDD

1,

128

jXD , jYD - jX jY

. f ,

XDx YDy , , f , - .

, , Mf [6], M - , ( ). [7].

F c f( , ) c , ),/()(),({ xypxpyxpc == ,XDx }YDy . )(K . c , )|( xyP .

, ( , )t t tx y X Yp P x E y E= x tXE

y tYE , | ( | )t t ty x Y Xp P y E x E= , ( )

t tx Xp P x E= ,

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tYE .

=

==M

t

tx

t pKfcF1

1)(1),( , |max( )tY Y

t t ty x y

E Wp p

= [7].

c f x0 ( ) , :

),(inf),(0

0 fcFfcF f = , 0 - .

, , , V x yi i

i N=

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129

. Q EF c fN ( , ) , N , f .

c ( ) ( , ) ( , )c F c f F c f= 0 , f

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( , ) inf ( , )

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.

, () .

Q N ( , ) ( , ) ( , )c N EF c f F c fN=

, f f, . :

( , )c N ( ), N ( , )c N 0 . , Q c N, , :

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),( fcEFN ( 3212 N ) ( 5,4,3=M

5=M ) 3=n 2=m . , , ),(30 fcEF 4=

M , ),(30 fcEF 5=M ,

23 ),( fcEFN

3=M . , 01-01-00839

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.. , ..

(-) 1.

s ( ){ } ( ) ( ) ( )( ){ } 1,03211,0 ;; == ss nqnqnqn (1)

(). ( )nq1 , ( )nq2 , ( )nq3 - n - , 1,...,1,0 = sn .

n - n -

( ) ( ) ( ) ( ) knqjnqinqnq ++= 321 , (2) i , j k - . () [1], ,

( ){ } ( ) ( ) ( ){ } 1,03211,0 ++== ss knqjnqinqnqQ , (3) ( ){ } 1,0 = snpP

Q . ( )( ) 1,0 = snN P .

()

131

. . [2,3].

2.

=+++=+= kbjbibbrb 3210sincos ( ) sincos 321 krjrir +++= , (4)

12322

21

20 =+++= bbbbb , 1

23

22

21 =++= rrrr , 2 -

, 1b , 2b 3b . b .

1. Qg Ng , Q N :

( ) ( ) ( )

( ) ( ) ( )21

03

21

02

21

01

1

03

1

02

1

01

+

+

++=

=

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=

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n

s

n

s

n

s

n

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n

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s

n

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knjning

N .

2. kbjbibbb 3210 QQQQQ +++= kbjbibbb 3210 NNNNN +++= ,

Qg Ng OZ ( k ):

( )jgigb 12sin2sin

cos QQQ

QQQ +=

, (6)

( )jgigb 12sin2sincos NN

N

NNN +=

,

3arccos2 QQ g= , 3arccos2 NN g=

132

Qg Ng OZ . 3. Q N :

1= QQ QQ bbz , 1= NN NN bbz . (7)

4. zQ OZ zN 2 .

kbz = sincos , (8)

mz

z

ml

ml

=

Q

N

,

,arg2

, llm ,...1,1,...,= , (9)

( ) ( )( ) ( )( )

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n

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==

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( ) , , Q ; ( ) ( )( )nnY lm , - l m ; ( )( )( )nPml cos - ; ( )n , ( )n - n - Q .

5. :

kbjbibbbbbb z +++== 3210*

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=1322022310 QNQNQN bbbbbbbbbb zzz

000101 QNQN bbbbbb zz ,

0320011002301 QNQNQNQN bbbbbbbbbbbbb zzzz ++= ,

2001300310022 QNQNQNQN bbbbbbbbbbbbb zzzz = ,

++=1312011022 QNQNQN bbbbbbbbbb zzz

232030 QNQN bbbbbb zz + .

3.

133

.

01-01-00298.

1. / .., .., .., .. , .., .., ... .: , 2002. 592 .

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:

134

1). , N . N , , , .

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N ;

N ;

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N (, N 108109) / . [1] , , N , M , . , ( ) N.

3). , , : , N . N . , , .

4). , - , : M , (, , , ).

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.. , .. , ..

() [1] , q,

, : q f (n, , ), f , : .

Z (m, q) U LP {M} , Z, f (n, , ) = 4/ (1 ln /4 ln /3), , , , Z , Z ( ),

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n, 1 , 1 .

. . , > 0 1 -

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138

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, , .

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.. , ..

(, )

= Ai ii zxy )()( , A )(iz , { }Nty tt ,...,1);,( = . , || An = , , N . , , ( ) AA = Ai ii zx )()( , , .

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RVM (Relevance Vector Machines) [1], () ,

139

, . SVM (Support Vector Machines) [1], , , kernel function ),( K ( [1] ), . SVM RVM = Ai ii Kxy ),()( ,

},...,1,{ niAA i == , , , , .

, ,

= ),(K = n

i iixx

1)()(

, , , , .

y

==n

i iiT zx

1xz

2 : += xzTy , 0)(M = , 1)(M 2 = . (1) nTn Rxx = ),...,( 1x , , , , ir ,

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140

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.

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.. , .. (, )

, , , . , , )(f , , ,

)( f \ , [1]. , )(f , ,

142

.

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, ,

nR )(),( xx , , , )(f , , .

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[3], [3], ),( K , ( kernel function ), , , .

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143

, .

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)|( , , . )|( , .

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= dpK )|()|()(),( (1)

)|( )|( (Conditionally Symmetrically Independent Distributions), [5].

, )|( )|()( =x , , . (1) )(p , , . )(p ,

(1), )|( . ,

144

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= d)()|()( , (2)

, )|()()|()( = , . (3) ,

...),3,2,1,( = ss , )|( 1 ss , (3) , )( . (3) , .. .

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)|(]2[

)|()()|()(),( ]2[]2[ ==K ,

=== dd )|()|()|()|()|()|( ]2[]2[ .

, (1), .

, .

1. .. .

.: , 1979. 2. Duin R.P.W, De Ridder D., Tax D.M.J. Experiments with a featureless

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MIT Press, 2000, pp. 39-50.

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3. .. // . . .-. ; 400; , 1982. - 33 .

148

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1. . ., . . . // .:

, 1974. 415 . 2. . .

. // " ", - , , 1999.

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