the methods of data analysis

..

2011

2

22.172.517.8:32.973:65.05 519.2:681.3

30

: .. , . . .,

; .. , . .-. .,

; .. , . . .,

.

30

.. : /

.. . : , 2011. 268 . ISBN 978-966-414-103-8 , -

. , , - , , , - , . ,

. 519.2:681.3

22.172.517.8:32.973:65.05

.., 2011 ISBN 978-966-414-103-8 , 2011

3

.5

1. . .8

1.1. .8 1.2. ..10 1.3. ..22 1.4. .34 ..40

2. 43 2.1. .43 2.2. 47 2.3. 52 2.4. .61 2.5. 65 2.6. 73 ..79

3. ..81 3.1. ...81 3.2. .90 3.3. .94 3.4. ..97 ...99

4. 101 4.1. ...102 4.2. .109 4.3. ...113 4.4. .116 4.5. 119 4.6. 123 .135

5. ...137 5.1. .139

4

5.2. 143 5.3. .148 5.4. 150 .155

6. 157 6.1. .158 6.2. 160 6.3. ...175 6.4. ..182 .191

7. .193

7.1. ...193 7.2. ..198 7.3. .205 7.4. ..209 7.5. 211 7.6. ..217 7.7. ...218 7.8. MS Excel.221 7.9. SPSS...224 7.10. MathCad..230 7.11. MS Excel.238 7.12. SPSS241 .247

249

..257

261

266

5

- , , -, , . - , - .

- . - ( , ), ( , , ), ( ); - ( -, , - ), ( ), - ( , ), - ( ), ( , , ), ( , ), ( , ). , XIX . - .

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- , (data mining), , , - .

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MS Excel, SPSS, STATISTICA, MatLab, MathCad . - - , - , . .

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

, - , , , , , -.

, ; ; - ; - ; - ; - . .

, - , :

1) http://datan.ucoz.ru ( , - , -);

2) http://www.basegroup.ru ( BaseGroup Labs - );

3) http://www.statsoft.ru/home/textbook/default.htm ( - StatSoft );

4) http://orlovs.pp.ru ( .. );

5) http://www.aup.ru/books/m163/ ( .. );

7

6) http://www.ami.nstu.ru/~headrd ( .. );

7) http://uk.wikipedia.org, http://ru.wikipedia.org, http://www.wikipedia. org (, -, , , -; );

8) http://www.biometrica.tomsk.ru ( - ; );

9) http://dvo.sut.ru/libr/opds/i130hodo_part1/index.htm; 10) http://www.dvo.sut.ru/libr/opds/i130hod2/index.htm ( -

.. ); 11) http://lib.socio.msu.ru/l/library?e=d-000-00---001ucheb--00-0-0-

0prompt-10---4------0-1l--1-ru-50---20-help---00031-001-1-0windowsZz-1251-10&a=d&c=01 ucheb &cl=CL1&d=HASHe10c3b36c7d751dd18704b (- SPSS);

12) http://www.gmdh.net/gmdh.htm ( , .. );

13) http://www.machinelearning.ru (, - , -);

14) http://riskcontrol.ru ( - );

15) http://attestatsoft.narod.ru/index.htm ( - AtteStatSoft);

16) http://www.medstatistica.com ( -- ).

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

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12

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13

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:

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n

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, - . 1.2.

1.2

9 12 7 5 6 8 11 9 1 12 7 20 2 6 4 1 8 13 15 3

(1.3, 1.51.7), : x = 7,95;

h = 4,144417; g = 6,188113; - w2 = 9,270922; w3 = 10,35166.

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n n, (1.9)

, 1, nt t-. 30

15

- , .

- m, :

1

0 0

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i ip p . (1.11)

:

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12+n n ,

. n - , :

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x

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0

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2

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18

. -, x

31/ x . . , , - . - .

() ():

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ii

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-

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2

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= ds , (1.21) d , . .

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19

1000 . - , - .

:

2 2

22 2

1, / 2 1,1 / 2

;

n nns ns , (1.22)

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1

1=

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n

i ji ji j

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. (1.25)

.

20

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

( ) . - . - , - . -

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= +Ass n . .

() . , - , [44]. - . , - . - :

63; 2 23

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21100%;

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xP

s PP s Pnx

. (1.27)

, 5%. - - .

21

: x ( k ); ( mk); ( bk). k,

. , , , - .

, - :

( ) ( )

1 1 1; ; = = =

= = = n n nk k ki i ii i i

k k k

x x x C xm b

n n n. (1.28)

b1 , 2 .

( )'M x , . :

( )' = xtxM t e . (1.29) :

( ) ( )2 21 2' 1 ... ... 1 ... ...2! ! 2! != + + + + + = + + + + +n n

x n

xt xt t tM xt b t b bn n

, (1.30)

, bn !

ntn

.

,

( )' 0= n xn nd Mb dt . (1.31)

:

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( ) ( )' = xtx xM t e M t . (1.33)

22

= kk kr s k. 3 , 4 . - 500, - . .

1.3.

- .

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

. (xmax) (xmin)

max min= R x x . (, ) k.

- . , - . -, -. , ( -, ). , [44].

- - . - , k 1020 915. - k. -, , 10 . , , . , . , - , 2 .

23

: 1,44ln 1= +k n , (1.34) n . , -.

.. , -, -:

0,4 0,41,56

+= =k n An , (1.35) , , - . . (1.35) , -, , . . -. (1.35) 0,55 () 1,25 ( -), , - .

1,52 .

. , . . - 2 n :

( )0,45 2 /= k n t , (1.36) 2 4; t , - 1= p ; .

, .

. - . /=d R k , d - ( , , ). - 1= i i id x x , , 1 . .

24

- , ( ) ( )= F x P X x . . 1.1 .

. 1.1. , -

Fn (x),

x X x :

( )0

1

0, ;/ , , 0 ;

1, ,+

kn x k k

n

x xF x n n x x x k n

x x

(1.37)

nx , (, ); n . ,

0 1, ,..., nx x x . .

kx

n () - , . .

: 1 2x x< , ( ) ( )1 2F x F x ; ( ) ( )lim 0, lim 1

x xF x F x += = .

25

, , .

( )S x , , , , x. : ( ) ( )1S x F x= . (1.38)

( ) ( )dF xf xdx

= , . . . 1.2 - .

. 1.2. ,

:

( ) 0f x ; ( ) 1f x dx

= ;

( ) ( )* *x f x dx F x

= .

26

( ) ( )( )x G G F x= = , x, , - , . - , .

( )Z , - , - . : ( ) ( )1Z G = . (1.39)

. , :

( ) { }* * * *, ,F x y P x x y y= . (1.40) :

( ) ( )2 ,, F x yf x yx y

= . (1.41) :

( ), 0f x y ; ( ), 1f x y dxdy

= ;

( ) ( )*** *, ,yxF x y f x y dxdy

= . ,

-:

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

= = f x f x y dy f y f x y dx (1.42) x y , :

( ) ( ) ( ),f x y f x f y= (1.43)

27

. 1.3 .

0,8 0,6 0,4 0,2

. 1.3. -

X :

( ) ( ) ( )* * *1 1 ,...,= < < =

28

1/4, 1/2 3/4 ; 10%, 20%, , 90% , 1%, 2%, , 99% ().

0,75 0,25X X . :

0,75 0,25

0,5

X XX

. 0,25 0,752

X X+

, - .

d Xq/2 X1 q/2 (0 < q < 1) - 1 q = . ,

0,95 0,975 0,025d X X= - 0,95. d0,9 , , 0,9 3,2d . [44]. . , - , - .

d , q , , - . , . [44], -:

max1 2

1n n

n + , (1.46)

n ; n . - , , (1.46).

:

0,75 0,25 0,5*0,75 0,25

2X X XAs

X X+ = . (1.47)

29

. :

( )0

r

ii

P X r p=

= ; ( )

1

s

ii r

P r X s p= +

< = ;

01i

ip

== .

:

( ) ( ) ( )aP X a F x f x dx

= = ;

( ) ( ) ( ) ( )ba

P a X b f x dx F b F a< = = .

: { } ( ), ,P x y f x y dxdy

= . (1.48)

, . - , .

(, ) i , - . -, , /i if n= , n . n - ().

p - , . :

( )11if

p p N nDn n = , (1.49)

n ; N . -

, , - .

30

- .

1. . , - . - , , . - , - ( 100 ). - . - . - . 200 , -, . 1.4.

. 1.4. 2. .

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

. 1.5 - , [3, 3]. . 1.6, , - .

31

. 1.5.

(- ). , -, -, , 0n d ( ) i if f x d .

. 1.6. ,

-

. k , . - . : 2 /md X k= ,

32

mX . min max

2x x+ .

max minx xd

k= .

( , ).

, .

, , - , - ( ) - ( ). - 200 , , . 1.7.

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1

i

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= 1

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f=

= , . - , , , ( ),i ix ( ),i ix . -

33

. . - 200 , - , . 1.8.

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y . 1 2,m m

n , , ,

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n. -

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. 1.9.

34

- . - .

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

:

( )21

1g

i ii

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= , (1.50) g ( ); ti - .

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; ; - . -, , .

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MS Excel, - (/ / )

35

. . 1.10 , .

- :

300

1

1 22,77671300 ii

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= = . :

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300

1

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:

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150 151 23,03882

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, (), (),

36

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

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1 20,82036300 1 =

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1 4,5629300 1 ii

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300

1

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= = MS Excel ,

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

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ii

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MS Excel , () ().

, MS Excel . , , - . 1.3.

1.3 ,

MS Excel 22,77671 0,263441 23,0388 19,92967

37

. 1.3 4,562934 20,82036 0,74392 0,09339 21,78896 11,7057 33,49466 6833,012 300 (1) 33,49466 (1) 11,7057 (95,0%) 0,518433

SPSS

Analyze/Descriptive Statistics/Frequencies. , -, , . , , . 1.4.

1.4 , SPSS

Valid 300 N Missing 0 Mean 22,7767 Std. Error of Mean ,26344 Median 23,0388 Mode 19,93 Std. Deviation 4,56293 Variance 20,82036 Skewness ,093 Std. Error of Skewness ,141 Kurtosis ,744 Std. Error of Kurtosis ,281 Range 21,79 Minimum 11,71 Maximum 33,49 Sum 6833,01

25 19,3176 50 23,0388 Percentiles 75 26,4383

. : 1,44ln 300 1 9k = + .

38

, max minx xdk=

. MS Excel (. 1.11).

. 1.11. SPSS -

Graphs/Interactive/Histogram (. 1.12). , , ( ); - Y (count) (ercent) -. Histogram (. 1.13) . , - SPSS, . 1.14.

. 1.12. SPSS

39

. 1.13.

15,00 20,00 25,00 30,00

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0%

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10%

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40

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43

2.

2.1.

- . - :

; ; ; .

. - .

, , (0). - : ; ; - . , - , , - (1). : ; - ; - . .

. -, . ( - ).

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44

. - .

(, ) (. 2.1, 2.2). .

. 2.1.

. 2.2.

45

, - 0, , , - 0 ( ) . ( )0| H = P x w , (2.1)

1p = .

, , . - : , - .

, :

-, ;

. -

, , .

. - 1 , - , , : { }1| H 1 = P x w . (2.2)

, . , - . , - , - , .

, , . :

[ ][ ]

0

1

/ H ;

/ H .

= = P Z w

P Z W (2.3)

. , , - .

46

-, . - 0,01; 0,05 0,1. -, , - , . , - , - .

. , . , , . .

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

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47

. , . :

( -);

, - ( );

( - ).

, -, . :

;

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2.2.

, , - : . - , - . - . - - . , , .

, -. , - , - , .

, - :

1 22 21 2

1 2

x xZ

n n

= +, (2.4)

n1, n2 . - .

48

Z- , - .

1 2x x , : ( ) ( ) ( ). .0 1 / 2 P Z z z< < = = , (2.5) (z) ( ), F (z) ( ) ( ) 1/ 2= z F z . . . z z= .

, , , , : 1 184x = , 2 181x = , 21 121 = , 22 107 = , 1 2 40n n= = . , , . :

184 181 1,26121 10740 40

Z = +

.

0,05 (5%), - :

( ) ( ). 1 / 2 0,475 z = = . . z

MS Excel, - (0,5 + 0,475). : . z = 1,96.

. Z z< , 0,05. MS Excel,

. . z = 1,26 0,209 . , , 21%. - . , - ( ) . .

: 1 2x x> , : ( ) ( )1 2 / 2z = , (2.6) , Z z< .

49

: 1 2x x< , z' - (2.6), , ' z z= . -, Z z> .

Z- - ( 1,2 30n ), , - , - .

t- . 1908 . - , - .

-, , .

:

2 2

A B

A B

A B

x xt

n n

= +, (2.7)

2 2, A B ; nA nB . m t- ( )1m n .

, , , - , :

( ) ( )2 21 1 2 221 2

1 12

s n s ns

n n + = + ; (2.8)

, :

2 2

2 1 1 2 2

1 2 2s n s nsn n

+= + , (2.9) . :

1 2

1 2

1 1

x xt

sn n

=+

. (2.10)

50

k = 1 2 2n n+ - .

(2.10) .

. -: ( )/ 2,< t t k 1 2x x ;

( ). , t t k> 1 2x x> ; ( ). ,< t t k 1 2x x< .

, , . - . - . :

12

2

1 1

1

n

ii

n n

i ii i

t

n

n

=

= =

=

, (2.11)

n ; i i ix y = , -.

-. 1n .

- , - , - . - ( ), . 1947 . :

1 2

2 21 2

1 2

x xd

s sn n

=+

, (2.12)

2 21 2, s s .

51

, - :

22 21 2

1 22 22 2

1 2

1 2

1 21 1

s sn n

s sn n

n n

+ = +

. (2.13)

(2.7) , .

F- - 1920 . . :

2 21 2/F s s= , (2.14) 2 21 2, s s , . -

1 1n 2 1n . , - . , 2 21 2s s> , , - . 2 21 2s s , - , / 2.

- , :

( ) 22 21n s = . (2.15) , 2 20 > ,

2 , 1k n= . , ( )2 2 , 1 < n .

2 20 < , - 2 , - 1 1k n= . -, ( )2 2 1 , 1 > n .

52

2 20 > , :

( )( )( )( )

2 2.

2 2.

/ 2, 1 / 2;

/ 2, 1 / 2.

< = > =

P n

P n (2.16)

, :

( ) ( )2 2 2. ./ 2, 1 / 2, 1 < < n n . , :

( ) ( )2 2 2 2. .1 1 / 2 P P < = > = . (2.17) 2

.. ( 1928 .):

2

2kR

k = . (2.18)

3R 2 , .

-, . , - . - , .

2.3.

. -. - , .

[45] - ( ) .

- ( ) - - ( - ). 19281930 .

53

, .

:

( ) ( ) ( )22 * *nF x F x dF x

= , (2.19)

Fn (x), F*(x) , .

1951 . - 1952 . - . - . [45] , - .

i, j . ri, sj . :

( ) ( ) ( ) ( )222 1 11 4 16m n

i ji j

mnA m m r i n s jmn m n m n= =

= = + + + , (2.20) m, n . , - .

1939 . - . - -. :

( ) ( ), supm n m nx

D F x G x= , (2.21) Fm(x), Gn(x) x.

:

( ) ( )' ', 1 1 1max maxm n m r n sr n s mr sD F y G xn m+ = = ; ( ) ( )' ', 1 11max maxm n m r n sr n s mr sD F y G xn m = = ; (2.22) { }, , ,max ;m n m n m nD D D+ = , ' ',s rx y .

54

, .

- . . 1920 . .

- , . - . ( ), 1, 2,...,=i ix y i n , n . :

( )1

1, 2,..., 2=

= =n nj ji j ji

s a x y j , (2.23)

( ) 1,...,2 ; i 1, ...,= =njia j n , . 2n n . - +1 1, 12 j . n = 3 :

1 1 11 1 11 1 11 1 11 1 11 1 11 1 11 1 1

A

+ + + + + + + + = + + + +

2

1

n

jj

S s=

= . -:

( )( )

2 j

1 j

0 s ;,

1, s .

n

j jj

SN n n

S=

p*, 1 p*. -

55

5 6 0,01 0,05, . , , 7 8. .

- - . : ( )1, 2,...,=i xx i n ( ) 1, 2,...,=j yy j n , nx, ny . . :

1 1

min ;= =

= yx nn

i ji j

S x y . (2.25)

:

( )( )1 j

0 ;2 ,

1 s ,

mnC j

j jj

s SN n n

S=

. -, . - , .

W- ( ) 1945 . . , -. , . , - .

56

, , , .

. , - . - . , , . , - . , . - :

1 2

1 1min ,

n n

i ii i

W R S= =

= , (2.27) Ri , , Si , . ( 25 ) W' -, , -, [25]. -, : ( ). 1 2' / 2, ,< W w n n ( ). 1 2' / 2, ,> W w n n ( ) ( )1 2F x F x ; ( ). 1 2' , ,< W w n n

( ) ( )1 2F x F x> ; ( ). 1 2' , ,> W w n n ( ) ( )1 2F x F x< . * W

W

WW = , ( )1 1

2Wn N + = -

, ( )1 22 112

+ =W n n N , 1 2N n n= + , N - . ( 25 ) - , * 1 / 2W z > . 0,02, , W + 0,5.

W- , [45] . , , .

, - . , , - , -

57

. - . - .

U- 1947 . - .. .. . - - , -, . . t- - . , t-.

:

( )( )

{ }1 1 2 1 1 1

2 1 2 2 2 2

1 2

1 / 2 ;

1 / 2 ;

max , ,

U n n n n R

U n n n n R

U U U

= + + = + + =

(2.28)

1 2, R R ; 1 2, n n . 1 2, 20>n n , U- . 1 2,U U :

1 2 1 2U U n n+ = . (2.29) U

U

U ,

1 2

2Un n = -

, ( )1 22 112U

n n N + = , 1 2N n n= + , - . -, W- . - .

- (, ) . 1945 . - . t- . - . W- , . , , . , (W+) (W) , * min( , )W W W += . -.

58

-:

( )12

N NW W+

++ = . (2.30)

*

*

*W

W

W ,

( )*

14W

N N + = ,

( )( )*

2 1 2 124W

N N N+ + = , N , - .

, - [23]. , - . W- - :

( )2 1 2 112 1Wn n TN

N N = +

, (2.31)

, - 1, 1 2N n n= + . U- - . - - :

( )( )*2 2 1 2 148WN N N T+ + = , (2.32)

N . -

, - . - . . .

2 (-) 1900 . - , . - . - . , . -

59

. , - 47, 2040.

, :

( )2

2

1

Ni i

i i i

f gf g=

= + , (2.33) ( ), = 1, 2,...,i if g i N ; N .

- :

( )2

2 12

11 2

1 N i ii i i

n f n gn n f g=

= + , (2.34) 1 2,n n .

2 . , . , - , , . - , - . :

( )22

1 1

r cij ij

i j ij

a ee= =

= , (2.35) ( ) 1, 2,..., ; 1, 2,...,= =ija i r j c ; r (); (); ije , ija .

- - j- -,

1 1

r c

iji j

a= = . -

- 2 ( )( )1 1r c . 1940 . -

. , -

60

n1 n2 .

, - , , 0 1. .

- - . , 00101111011 : 00, 1, 0, 1111, 0, 11. , N , . ( )1 2, 20>n n - :

( )( ) ( )

1 2

1 2

1 2 1 2 1 22

1 2 1 2

2 112

2 21

n nNn n

Zn n n n n nn n n n

+ + = + +

, (2.36)

. . . -

. y . -, i ix y -. , , . , . ( +) i ix y : 1/ 2; p m= , m , .

0H : 1/ 2=p . - : ( )11H : 1/ 2>p ; ( )21H : 1/ 2

61

( )21H :

0

12

mrim

iC

=

, (2.38) ( )31H :

0

12 2

mrim

iC

=

; (2.39)

12 2

mmim

i rC

=

. (2.40)

:

0

1 / 22 / 4

mrim

i

r mCm= , m > 50; (2.41)

0

1 / 2 0,52 / 4

mrim

i

r mCm=

+ , 50m , (2.42) ( ) x .

2.4.

- . - . , A - k , , - k = A , k < A . kA , k .

, - .

: 2, , 2, , .

2 ( ) 19281930 . . . . -, -

62

. - :

( ) 221

1 2 112 2

n

ii

in F xn n=

= + , (2.43) F (x) = , n .

40n > . 2.1 [37]. , .

2.1 2

0,900 0,950 0,990 0,995 0,999 n2() 0,3473 0,4614 0,7435 0,8694 1,1679

(2.21) , . , - ( - ), :

{ }

( ) ( )(1) (2)

1

(1) (2)

1 1

max , ;

1max ; max ,

n n nm n

n m n mm n m n

D D D

m mD F x D F xn n

= = =

(2.44)

(1)n nD D= . Dn

, nK D n= n - , . - - ( ).

: ; ( )200n > ; , ,

( ) .

35n > , - , , - :

ln

22

K

. (2.45)

63

- ( )1 2, 40>n n [19] :

0,05 = 1 21 2

1,36cn nD

n n= + ; (2.46)

0,10 = 1 21 2

1,22cn nD

n n= + . (2.47)

, . , , . , , . - .

2 - . , . . , 47, 20200.

:

( )2

2

1

''

ki i

i i

npnp=

= , (2.48) k ; 'i - ( - ); n ( - d. 1k r , r . , - 2k 3k .

2 . - . r , . , -

64

'i. 2 .

- . 350 W- . - 1965 . - . . :

( ) ( )22 2 2 1 11 1

/ ; ; ,n k

i n i n i ii i

W b S S x x b a x x + += =

= = = (2.49) ( ) =1, 2,...,ix i n ; n , k - n / 2 ( )1 / 2n n,

( )1 1, 2,..., ; 3, 4,..., 50 + = =n ia i k n . , .

- . -, :

2,5% 2 ; 16% ; 50% ; 84% + ; 97,5% 2 + .

. , , , 0,1A E < , - , , 0,5A E > .

- , . SPSS, Statistica - . . 2.3, 2.4 -, [3, 3] .

. 2.3 , - , . 2.4 - .

65

Normal P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b1,0

,8

,5

,3

0,0


Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.3. - ,

Uniform P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0


Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.4. - ,

2.5.

. .

1. - (SPSS, Statistica ) .

2. - - MS Excel, .

66

3. , - ( -), 2 ( ).

4. - [13].

SPSS (. 2.5) , - . Graphs / P-P Plots. (. 2.6).

. 2.5. SPSS , - -

, , . - : , 2, , , -, , , , , , , , .

. 2.6. -

67

. (, - , ). 2, , , , (. 2.72.12) .

Chi-square P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

Exponential P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.7. - 2

. 2.8. -

Half-Normal P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

Pareto P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.9. -

. 2.10. -

68

Student t P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b1,0

,8

,5

,3

0,0


Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.11. -

. 2.12. -

276,9 -

122,4 (. 2.13), (. 2.14), 2,26 0,0749 (. 2.15), 2,257 0,0627 (. 2.16), (. 2.17) 2,324 19,08 (. 2.18). .

Gamma P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

Laplace P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.13. -

. 2.14. -

69

Logistic P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b1,0

,8

,5

,3

0,0

Lognormal P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.15. -

. 2.16. -


Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

Weibull P-P Plot of VAR00001

Observed Cum Prob

1,0,8,5,30,0

Expe

cted

Cum

Pro

b

1,0

,8

,5

,3

0,0

. 2.17. -

. 2.18. -

- ,

. ,

, MS Excel. - , - . . - . (. 2.19).

70

. 2.19. -

n / N, n , N . f (xi), .

- . - , : =(1EXP ( (A4/$N$20;$O$20))), (2.50) $N$20, $O$20 , - . , , - SPSS. . . - . - . (. 2.20).

. 2.20.

71

, , , , . . , - (. 2.21).

. 2.21. , , : ; ; ; ; ; ; . , -

. -

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

2,319 20,67. - , SPSS.

72

- , , - : 2,321 18,51. , - .

, 0,49. -, , [21] 0,895 0,05.

-. - - : = (A6;$M$3;$M$7; ), (2.51) $M$3, $M$7 , . = (A1: A33) = (A1: A33), A1: A33 , .

- 0,73, , , , .

. 2.22 , . , - , - .

. 2.22.

73

2.6.

- - - . - .

, () 2008 . [14]. . 2.23, 2.24 . - . - .

. 2.23. -2008

. 2.24. -2008

74

-:

( )=M i i ii

F N ,s , (2.52)

, 1ii = ,

, si .

MS Excel. - . , , - - . - - 2, , . , -. - . :

( ) ( )

( )0,204 3,49;1,63 0,507 10,60; 3,82

0,288 16,54; 5,57

= + ++

MF N N

N. (2.53)

0,03 , . (. 2.24). - (. 2.25).

. 2.25.

75

SPSS - . - , 6.

, , - a = 10 b = 15 20 s1,2 = 3, 100 .

SPSS, var0001 (. 2.26). (var0002), var0001. .

. 2.26. Analyze/Classify/K-means Cluster

Analysis. (. 2.27).

. 2.27.

76

, , , .

b = 15 : 84, 116; 9,16 15,15,

, , 8,4% 1%;

27, 11; 19%. b = 20 : 102, 98; 10,20 20,45,

, , 2% 2,25%;

2, 4; 3%. , -

, - .

. Analyze/Descriptive Statistics/Frequencies. (. 2.28).

Statistics (. 2.29) , . Charts (. 2.30) , - .

. 2.28.

77

. 2.29.

. 2.30. Charts . 2.31 -

, . 2.32, 2.33 . , , - .

: ( ) ( ) ( )0,51 10,2;2,86 0,49 20,45;2,93F x N N= + , (2.54) ( );N s s.

- 1 2 1 + = .

78

. 2.31.

VAR00006

1310853

VAR00006

Freq

uenc

y

16

14

12

10

8

6

4

2

0

Std. Dev = 2,86 Mean = 10

N = 102,00

. 2.32.

79

VAR00005

2724211815

VAR00005

Freq

uenc

y

20

10

0

Std. Dev = 2,93 Mean = 20

N = 98,00

. 2.33.

1. ?

2. ? -. - ?

3. ? . 4. ? -

? 5. ? 6. ?

? 7. -

? ? 8. ?

? 9. ? 10. ? 11. ?

. 12. ? -

?

80

13. Z-? ?

14. t- -? ?

15. ? ?

16. F- ? ?

17. - ? ?

18. W- -? ?

19. U- ? ?

20. - -? ?

21. 2? ?

22. ? ?

23. ? ?

24. ? .

25. 2 -? ?

26. -? ?

27. 2 ? - ?

28. W- ? ?

81

3.

, - - , , . - ANOVA ( Analysis of Variations). . 1925 .

, - . , , , . - . .

- . , . , . , - . - 1. .

- . :

; ; , . ,

, . 3.1.

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

82

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

, , - (. 3.1). - . -, - , . - , .

3.1

1 2 k 1 x11 x12 x1k 2 x21 x22 x2k ni 11nx 12nx 1n kx

, -

- ( ) :

( ) ( ) ( )2 221 1 1 1 1

j jn nk k k

ij j j ij jj i j j i

x x n x x x x= = = = =

= + , (3.1)

1 1

1 jnkij

j ix x

N = == ;

1

1 1, ...,=

= =jnj ijj i

x x j kn

;

1

k

jj

N n=

= ; k ; nj (j = 1, 2, , k) j- ; jx j- .

(3.1) (, )

83

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

,

( )2211 1

1 jnkij j

j ix x

N k = = = (3.2)

( )2221

11

k

j jj

n x xk =

= (3.3) 2 . ,

2 21 . -

. 2 22 , 22 ,

. :

( ) ( )( ) ( )

2

1

2

1 11

j

k

j jj

nk

ij jj i

N k n x xF

k x x

=

= =

=

. (3.4)

F- 1k N k. - , ( )*P F F , F* , (3.3) .

. - 1952 . - -, . . , k

1 2, , ..., kn n n .

84

U- -, 2k > .

, , - - , .

. 3.1 rij, - xij. . 3.2.

3.2

1 2 k

1 r11 r12 r1k 2 r21 r22 r2k ni 11nr 12nr 1n kr

, , -

1=

=jnj iji

R r 1

1 jnj ij

j iR r

n == .

:

( )1

12

k

ii

N NR

=

+= , (3.5)

1

k

ii

N n=

= . ,

, , ( )1 / 2N + . ( )1 / 2jR N + , .

:

( ) ( )2

1

12H 3 11 =

= ++ k

j

j j

RN

N N n, (3.6)

( )2

1

12 1H1 2=

+ = + k

j jj

Nn RN N

. (3.7)

85

5in 4k 2- k 1. - , 21H > , 21 1 2 k 1 . k = 2 - W .

, - . , . - , , , . , - :

31

HH '

1=

= g

j

j

TN N

, (3.8)

g , ; ( )3j j jT t t= ; tj , j- .

( ) - .. 1952 . .. 1954 . - , , - , . , - . , - , , - . - , -, .

u, v ( )1 u v k < , k , : ( ),

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

,uv

u v iu jvi nj n

U x y==

= , (3.9)

( )( )( )( )

1 ;

, 1/ 2 ;

0 .

i j

i j i j

i j

x y

x y x y

x y

(3.10)

86

:

,1

u vu v k

J U

= . (3.11) J .

:

( ) ( )2 2 2 21 1

1 1; 2 3 2 34 72

k k

j j jj j

MJ N n DJ N N n n= =

= = + + . (3.12) , -

, -

J MJDJ .

- 1937 . - - , , ( -). : / ,B V C= (3.13)

( )

2 2

1

1

ln ln ;

1 1 11 ,3 1

i ii

i i

V k s k s

Ck k

=

=

= = +

A

A

A

2

2 1i i

ik s

sk

==A

-

; 1

ii

k k=

=A ; ki i- ; ; 2is - . nj - 2- 1.

G- () 1941 .

87

k ( )2k - . :

2

1

2

1

max jj kk

jj

G

=

=

, (3.14)

2j j- . - . .

, - 1960 . , , - . :

( ) ( )

( ) ( )2*

1

2

1 11

j

k

j jjnk

jj ijj i

N k n Z ZW

k n Z Z

=

= =

=

, (3.15)

jij ijZ x x= ; xij i- j- ; jx - , j- ; Z - Z ; jZ - Z , j- . F- (k 1) (N k).

1974 . . . ( ), , jij ijZ x x= , jx - , j- .

, - , , - .

, , : ( ) 1, ..., ; 1, ..., ,= + = =ij j ijx a j k i n (3.16)

88

aj , - ; ij , - , .

(3.16) : ( ) 1, ..., ; 1, ..., ,= + + = =ij j ijx j k i n (3.17)

1

1 kj

ja

k = = ;

j ja = j- .

(- k ), - k,

1

0k

jj=

= . ,

. :

{ } ( ) 1, ..., ; 1, ..., .= = =ij ui vj i jz med x x u n v n (3.18) : ij jiz z= . zij

i j . -: ij ik kjz z z= + . ( ):

ij i jW = , (3.19)

( )1 1, ...,= = =k

u iuu

i

n zi k

N (3.20)

- , - 1, ..., kn n . - ij ik kjW W W= + . : .

89

, - , . .

- k - :

1

k

j jj

L c m=

= , (3.21) ,

10

k

ii

c=

= ; mj j- , - .

:

2

2 21

1

kj

Lj j

cs

n== , (3.22)

21 (3.2). :

( ) ( )11 1; LL s k F k N k . (3.23) i jm m ,

: = 1, j = 1, cu = 0 ,u i j . , , , i jm m= - . , .

, , - , - (), -. L = L0 - :

0

12

1

k

i ii

ki

i i

c x Lt

cMn

=

=

=

, (3.24)

N ; ni - ; ix - ;

90

( )21 1

1 inkij i

i jM x x

N k = == . -

.

3.2.

- . . 3.3, , , - , . n k , n k , .

3.3

1 2 k 1 2 n

x11 x21 xn1

x12 x22 xn2

x1k x2k xnk

, . - , , - ( ), - .

. 3.3 -, , bi tj, ij: ij i j ijx b t= + + . (3.25)

(3.24) : ij i j ijx = + + + , (3.26)

1

1 kij

jx

kn = = ;

j , , -.

, ij - i, j 2 ( ).

91

01 1 2H : ... 0 = = = =k ,

02 1 2H : ... 0 = = = =n , - , .

. 01 :

( )( ) ( )221 1 111 1n k

ij i ji j

x x x xn k = =

= + , (3.27)

1 1

1 n kij

i jx x

nk = == ;

1

1 ki ij

jx x

k == - ;

1

1 nj ij

ix x

n == j- . 21 -

. :

( )22211

k

jj

n x xk =

= . (3.28) .

, -, (3.28).

01 :

( )( ) ( )

( ) ( )2

212

2 21

1 1

1 1

1

k

jj

n k

ij i ji j

n n k x xF

k x x x x

=

= =

= = +

. (3.29)

F- (k 1) ( )( )1 1n k . ,

1F F < , 1F - F- .

92

02 - (3.28), n k, - j.

, - , , (, ), - 1930 . . -, ij , - .

01 1 2H : ... 0 = = = =k , n - , k . () . . - - . , xij , k - rij. , .

:

( ) ( )2

1 1

12 3 11

k n

ijj i

S r n knk k = =

= + + . (3.30)

02 1 2H : ... 0 = = = =n , - n k .

n S- - 2 k 1 , ,

( )21 1S k< . (L-), -

.. 1963 ., - 01 1 2H : ... 0 = = = =k 02 1 2H : ... 0 = = = =n , - 1 2 ... k , , 1 2 ... n ,

93

. . :

11

k

jj

L jr=

= ; 21

n

ii

L ir=

= , (3.31)

1

n

j iji

r r=

= , 1

k

i ijj

r r=

= . :

( )( ) ( )( )

( ) ( )2 2

* *1 22 23 3

1 / 4 1 / 4;

/144 1 /144 1

L nk k L nk nL L

n k k k k n n n

+ += =

, (3.32)

- .

, , - . , .

Q- . 1937 . - , -, , - (0) - (1). .

:

( )2

2

1 1

2

1 1

1c c

j jj j

r r

i ii i

c c T T

Qc T T

= =

= =

=

, (3.33)

( )1

1, 2,...,=

= =rj iji

T x j c ; ( )1

1, 2,...,=

= =ci ijj

T x i r

; (); r (). 2 , 1.

- , , - .

94

- . :

( )

2

1

2

1

1

r

i ii

r

ii

c xt

r Sc

c

=

=

=

, (3.34)

( ) 1, 2, ...,=ic i r ;

22

1 1

r c

iji j

TS xrc= =

= ;

1 1

r c

iji j

T x= =

= ; (); r (). - (r 1) (r 1)(c 1) - ( )1c (r 1)(c 1) .

3.3.

MS Excel - 200 : 1 150; 5x s= = ; 2 249; 5= =x s ;

3 351; 6x s= = ; 4 449,5; 5,5x s= = .

. , , , . .

(3.4) : n1 = n2 = n3 = n4 = 200; N = 800; k = 4; F = 1,544. k 1 = 3; N k = 796. =F (0,05;3;796), - MS Excel, - 0,05 F = 2,62. , -, . = F (F7;3;796) , - F* = 0,202. ,

95

. .

MS Excel - . . 3.1 - , .

. 3.1. MS Excel , -

, - (MS) (SS), -, (3.13.3), .

- . - , . MS Excel . (3.14). 0,293G = .

0,05 . , - 0,05cG < ( 0,0495cG , 120), , . - , .

SPSS - . Analyze/Compare means/One way ANOVA. -

96

, VAR00001. VAR00002 , VAR00001. ; ( ); , ( ), . , , Options - , Post Hoc - , , . 3.2.

. 3.2. , SPSS

97

, - , MS Excel, - - . , , , .

3.4.

, MS Excel 100 20x = , 3s = 100 , , : [17; 22], [18; 22], [18; 22] [17; 23]. MS Excel -. . - -. (). - . 3.3.

. 3.3. () -

. - , ( ). : 0 , . (3.6) (3.7) . - 7,22. 2, = 2 (0,05;3) = 7,82. (0,05) ( 3). , , . - .

98

SPSS - , , - .

Analyze/Nonparametric Tests/K Independent Samples . - . . 3.43.6. , , , , 0,05. 0,05.

. 3.4.

. 3.5.

99

. 3.6.

1. ? 2. ?

. 3. ? . 4.

? 5. -

? 6. -

? 7. . 8.

? 9. -

? 10. ,

? 11. -

? 12.

? 13.

? 14. ?

?

100

15. ? ?

16. G- ? ?

17. ?

18. ? - ?

19. ? .

20. ? 21. -

? 22. -

? ? 23. -

? ? 24. -

? ? 25. Q- -

? ?

101

4.

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

1963 . [58]:

, , , ( );

( );

. .

, , .

-. r 1, 2, r, s 1, 2, s. , j , j, :

( ) ( ) ( ),i j i jP A B P A P B= . (4.1) : , -

- :

( ) ( ) ( )/ , ,= j i j i jP B A P B A B . (4.2)

102

[4]. - . - . , . - , .

: ; - , , - . -, .

-. , 1 +1 0 1. , , - . - , ( ), . : ( ) ( ).

4.1.

, 1888 .

- . - y ( ) ( ) ( )( )1 2, ,...,=X px x x :

( ) 22; 1dy

sK ys= X , (4.3)

( )221

1 ny k

ks y y

n == , (4.4)

103

n , - :

( )( )221

1 ?n

i ii

s y fn =

= X , (4.5) ( )? if X f (X) , :

( )22 *1 1

1 1 jmij j

jj is y y

m

= == , (4.6)

j , j- ; yji - , j- ;

1*

j

jii

jj

yy

==

,

j- ; m .

(4.5) , - , ( ) 2|D const = =X , - . (4.6) , -, , - . . ( ). . - , ( )j , - Y y *jy - (4.3, 4.4, 4.6).

- -, f (X). - - , - . , - , . .

, - .

104

. - : - .

, .

. - - . . , , , .

. ( - , , - , ) - . . 1896 . , - . , 1846 . , - ( Bravais, - ). :

( )( )

( ) ( )1

2 2

1 1

n

i ii

n n

i ii i

x x y yr

x x y y

=

= =

=

. (4.7)

y y , - y ax b= + , a b , :

( )21

1 /y yr

=+

. (4.8)

( )30N - :

21

rrN = . (4.9)

105

0,01 , / 2,6rr .

, . . , - .

1 +1. - 1 +1 , , . y const= , , , . 1 +1, - . - , - .

. 4.1 , .

x

y

12

. 4.1.

. , , 1 0,995r = 2 0,006r = . - , -. , 0,98 1,00.

, - - .

106

, - . , , ( ),x y -, 0r = x y.

. , MS Excel, -. , , - , , .

, , - . , . - - :

( ) ( )1 12 2tanh ; tanh3 3

p pN Nr z r z r

n n

+ + + , (4.10)

n ; 1

2pN + ;

; z (r) z- ( ) r.

- . - - , :

( ) 12 21 2 /r n t > + , (4.11) t t- ( )2n .

1n r r- :

( ) ( ) ( ) ( )42 21

1 2 1 1 12

2

n

r n

n

r r rn

= <

107

1n r .

, , , . . , - - - . - :

( )2 2 2/yx yy xs s = , (4.13)

( ) ( )22 *1

1 sj jy x

js y y

n == ;

( )221 1

1 jsy ji

j is y y

n

= == ;

*1

1 sj j

jy y

n == ;

*1

/j

ij jji

y y

=

= ,

n ; s ; j , j- . , , .

. 1yx = , , , 1yx = .

0yx = , , 0yx = , , - *jy y= , *jy .

, : yx xy . , -, , . -

108

, , . , . 4.1, - 0,98 - . 0,63yx 0,72xy .

, , . . - . :

( )( )( )( )

2 22

22 1yx

yx

n k rv

k

= , (4.14)

n ; k . - F- s 2 n s. (4.13) 2v ( )2, F s n s , . -, - , , .

:

2 2 1FC H C n CrC H n n = = = + , (4.15)

; , . - 1860 .

1 +1. , : , . - , - - . , . 4.1 - , ,

1 0,941Fr = 2 0,010Fr = . , .

109

, , , - .

. - :

( ) ( )( )1

1,1

n

i ii

Cov X Y x x y yn =

= . (4.16) , -

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

. -

, ( ),X Y

Cov X Yr = ,

2 2, X Y . -

:

11 12 1

21 22 2

1 2

...

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

...

n

n

n n nn

c c cc c c

C

c c c

= , (4.17)

( )cov ,=ij i jc x x . (4.17) ( )2ii ic x= . -

, ij jic c= . 4.2.

- . . 4.1, xik - k- .

110

4.1

0 1 2 k p

1 x10 x11 x12 x1k x1p 2 x20 x21 x22 x2k x2p i xi0 xi1 xi2 xik xip

n xn0 xn1 xn2 xnk xnp

: -

; ; ( ).

k- - n ( ) ( ) ( ) ( )( )1 2, , ..., ,k kk knx x x=X 0, 1, ..., k p= n- . .

1. - . - - .

2. () , , . .

3. , . , , , , .

. .

, . - . - , .

( - , )

111

1904 . , , . , , , . . , , - . , - , ri si. ,

1 2, , ..., nr r r 1 2, , ..., ns s s . :

( )21

n

i ii

S r s=

= . (4.18)

, ri, (4.18) - :

( )21

n

kk

S k s=

= . (4.19) S S = 0

, . -

( )313S n n = , - , - i, j i jr r> - i js s< , { }1, 2, ..., ir n= { }, 1, ..., 1is n n= . S - , n.

, , :

( )

3

61 x ys

S B Bn n

+ + = , (4.20) , , - :

( )21

1 112

m

i i ii

B n n=

= , (4.21) m ; ni - .

112

1 +1, 1 , +1 .

- . :

2

21

sp

s

t n = . (4.22)

(n > 50) ( )2n . - 1 sn - .

, - . , , . , , n = 4, r , { }4, 3, 1, 2is = . - : 4 3; 4 1; 4 2; 3 1; 3 2> > > > > . 5K = . - = 0 -

, ( )12

n nK

= . ,

. ( , ). - 1938 . - :

( ) ( )21

1 12 2x y

Kn n n n

B B

=

, (4.23)

, j ir s ; n . , , :

( )1

1 12

m

i i ii

B n n=

= , (4.24) m ; ni - .

113

- :

( )( )9 12 2 5

n nn + (4.25)

, . -

, . -, 1 +1, 1 , +1 - . , , , - . [23]:

, - ;

; .

4.3. , -

, , - , , , , , .

(. 4.2). - , , .

fij , , i, j; mj -, ni . . 4.2 , (4.1):

4.2

2 1 1 2 r

1 f11 f12 f1r n1 2 f21 f22 f2r n2 fc1 fc2 fcr nc

m1 m2 mr S

114

( ) ( )

( )1

1

; ;

.

=

=

= = = = =

= = =

rij i

ij i j i i ijj

cj

j j iji

f np P A B p P A pS S

mp P B p

S

(4.26)

: /ij i jf n m S . (4.27)

/ij i jn m S = . , , , . - :

( )2 22

1 1 1 11

c r c rij ij ij

ij i ji j i j

f fS

n m= = = =

= = , (4.28)

2 ( )( )1 1r c . - , fij . - fij .

- , - 2 2 / S = , { }min 1, 1 r c .

, , -, .

.. - 1946 . :

( ) ( )

1/ 22

1/ 221 1

1

min 1, 1 min 1, 1= =

= =

c r iji ji j

fn m

Cc r c r

. (4.29)

. = 0 , - = 1

115

. :

( )21

min 1, 1 C n c r . (4.30)

: [ ]1 1; + C CC u C u , (4.31) uq q- .

r c . .. 1926 . - :

( )( )2

1 1

1 ; 1 1

c rij

i j i j

fJT Jn mr c = =

= = . (4.32) ,

, -, . 2 2 : 11 12 21 22, , , a f b f c f d f= = = = .

( ) , 1901 . , :

aJa b c

= + + . (4.33)

. (

) . 1958 . :

a d a dJn a b c d+ += = + + + . (4.34)

, - .

1940 . - . - . . :

a aJn a b c d

= = + + + . (4.35)

116

. ( -

) .. 1923 . :

( )( )( )( )ad bcC

a b a c d b d c= + + + + . (4.36)

1 +1. , , .

, - 22.

- 1900 . :

ad bcQad bc

= + . (4.37) ,

.. 1912 ., :

ad bcYad bc

= + . (4.38) .

1 +1. ( ) H a d= + -

. , , - - (- n). 1950 .

4.4.

1971 . -, . - :

( )11

, 1, ..., ; 1, ...,=

=

= = =

p

ijkk

ij p

ijkk

Ss i n j n

W, (4.39)

( ) , 1, ..., ; 1, ...,= =ijkS i j n k p ;

117

ijkW ; , ; n .

- . - , , . :

1 ik jkijkk

x xS

R= , (4.40)

,ik jkx x k- j; Rk k- , .

. . 2n , - , q . .

:

( )1 1

bx k

x x nr

ns z

= , (4.41)

1x ; x ; sx ; n1 ; n ; zk k, k : ( ) 11 /F k n n = . (4.42)

1 +1. :

1b

br

rmn

= . (4.43)

t- ( )2n .

118

- . (Louis L. Thurstone) 1928 . :

1 2bx

x x pqr

s

= , (4.44)

1 /p n n= , p q> ; q ; , . :

2b

r

pq rm

n

= . (4.45) -

:

22

1

2

1

1

11

qyi

ii i y

qi

ii i

nn

r

nn

=

=

= +

, (4.46)

n ; ni - ; mi/si , - ; my/sy , - .

, , , - -, :

p ppbx q

x x nr

s n

= , (4.47)

px , - () ; x ;

119

sx ; np ( 1) ; nq ( 2) .

, : 1 ( ) 0 ( ). - - .

- , :

22

1pb pb

nr tr , (4.48)

t t- ( )2n . 4.5.

, - , .

, , . - , - .

-, . .. 1896 ., . -.

- , j :

( ), = i j ijijXii jj

Rr

R R, (4.49)

Rkl rkl . . 1897 .

:

( )( )01 02 1201(2) 2 202 121 1r r rr

r r

=

. (4.50)

120

k, , - k , - k 1, :

( )( ) ( ) ( )

( )( ) ( )( )101 2, ..., 0k+1 2, ..., k+1 2, ...,

01 2, 3, ..., 12 2

0k+1 2, ..., 1k+1 2, ...,1 1+

=

k k kk

k k

r r rr

r r. (4.51)

- . , .

- :

( )( )

2

12

1

n

iin

ii

Y YR

y y

=

=

=

, (4.52)

Yi , ; yi , .

, , y 1, 2, , n y - 1, , n. , - .

, 1935 . . , . - :

200

1yXR

RR

= , (4.53)

R . :

( ) ( )( ) ( )( ) ( )( )2 2 2 2 201 02 1 03 12 0 1, 2, ..., 11 1 1 1 ... 1 = yX n nR r r r r . (4.54) z

(x, y) :

2 2

/ 2

21

zx zy xy zx zyz z xy

xy

r r r r rR R

r+ = = .

121

- - , - . (4.52), .

:

( ) 2,d yXK y X R= . (4.55) ,

. () . p q , , , - . ( , q) , - , . :

11 1221 22

R RR

R R =

, (4.56)

R11 p p ; R22 q q q ; R12 p q .

:

112 11 12 22 = TR R R v R v , (4.57)

q. -

. n ( )1p + -

, :

pFRpF

+ , (4.58)

F F- ( )1 ( )1n p .

, , . -

122

() :

( )

2

12

1 1

1

1

;

; ,

n

in ni

W i ii i

n

i ki

i ijj

SS S S S

n

SS S R

n

== =

==

= =

= =

(4.59)

Rij - j- . (W- ) -

:

( )2 312=

WSWk n n

. (4.60)

. 1939 . - , - , -. , k 3 - .

- :

11s

kWk

= . (4.61)

( )11

WkW

F- - ( )1n ( )( )( )1 1 2n k . F- - .

n > 7 ( )1k n W , 2 ( 1)n . : ( ) ( )21 1k n W n > , (4.62) .

123

4.6. . -

2 0,05. : 2i i iy x= + , i - MS Excel , [ ]0,5; 0,5 .

- SPSS (Analyze/Correlate/Bivariate). : , -; , ; - ; , , - . . 4.2 , - SPSS, . 4.3 , - .

. 4.2. , SPSS,

124

, , , - .

MS Excel . () - , / . , , - /.

. 4.3. ,

, .

, , . - (), , - .

, , 0,991.

. : ( )22i i iy x= + , i - MS Excel - , [ ]0,5; 0,5 .

. 4.4 , , , SPSS, . 4.5.

125

. 4.4.

. 4.5. , SPSS,

126

, , , - SPSS, . - .

, SPSS MS Excel - . MS Excel, - . y

( ) ( ) ( )( )1 2, , ...,=X px x x (4.3; 4.4; 4.6). -

0,946, .

MS Excel , . .

. - MS Excel , - (. 4.6).

. 4.6.

127

- 100 , [5; 5]. - , - . - (. 4.7).

. 4.7. : ( , -

); ( ); ; , , .

, : =2*A2+3+C3, 2 , - , 3 , [b; b].

, - . : = (A2;$A$2:$A$101;1), 2 , , ; $A$2:$A$101 , ; 1 , .

128

, - . , : =16*H102/(K1^3-K1), 102 , , 1 , - . . 4.8, 4.9 - b, - , . 4.10 b.

. 4.8. b = 1

, SPSS. , -.

(. 4.11). Analyze Correlate/ Bivariate Cor-relations. (. 4.12). , , , ( ) .

129

. 4.9. b = 15

. 4.10. b

130

. 4.11. SPSS

. 4.12. Options (. 4.13) , -

, . . 4.14. , ,

. , .

. 4.13.

131

. 4.14. . -

, . , : =2*A2*A2+3+C3, 2 , - , 3 , [b; b].

. 4.15, 4.16 - b, - . -

132

: S - ; S 0x < , S+ 0x > .

. 4.15. b = 1

SPSS - . -, - . - .

. 4.16. b = 20

133

- SPSS. Analyze Regression/Curve Estimation. - (. 4.17).

. 4.17. , ,

ANOVA, , - . - b . 4.184.21.

X

531-1-3-5

Y 60

50

40

30

20

10

0

. 4.18. b = 1

134

MODEL: MOD_1. Dependent variable.. VAR00003 Method.. QUADRATI Listwise Deletion of Missing Data Multiple R ,99933 R Square ,99867 Adjusted R Square ,99864 Standard Error ,57182 Analysis of Variance: DF Sum of Squares Mean Square Regression 2 23761,560 11880,77995 Residuals 97 31,717 ,32698 F = 36334,75101 Signif F = ,0000 -------------------- Variables in the Equation ------------------- Variable B SE B Beta T Sig T VAR00001 4,24885409E-05 ,018585 8,475E-06 ,002 ,9982 VAR00001**2 1,995714 ,007403 ,999333 269,570 ,0000 (Constant) 3,016919 ,090626 33,290 ,0000

. 4.19. b = 1

X

531-1-3-5

Y 80

60

40

20

0

-20

. 4.20. b = 20

135

MODEL: MOD_2. Dependent variable.. VAR00009 Method.. QUADRATI Listwise Deletion of Missing Data Multiple R ,79548 R Square ,63278 Adjusted R Square ,62521 Standard Error 11,43645 Analysis of Variance: DF Sum of Squares Mean Square Regression 2 21861,932 10930,966 Residuals 97 12686,870 130,792 F = 83,57488 Signif F = ,0000 -------------------- Variables in the Equation ------------------- Variable B SE B Beta T Sig T VAR00008 ,000850 ,371697 ,000141 ,002 ,9982 VAR00008**2 1,914277 ,148066 ,795477 12,928 ,0000 (Constant) 3,338370 1,812522 1,842 ,0686

. 4.21. b = 20

, -

, , . - , , - , .

1. ? 2. ? 3.

? 4. -

? 5. -

? 6.

?

136

7. ? 8.

. 9. -

? 10. ? 11. -

.

12. -? ?

13. ? 14. -

. 15. ? -

? 16. -

? ? 17. ? 18.

? 19.

? 20.

? 21.

.

22. - ? ?

23. 2, - .

24. , ? -?

25. ? ?

137

5.

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

. XIX . . , . , . , . , . .

-. - :

1

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

= =

i

ii

ip

xx

X i n

x

(5.1)

n (, ); p , .

:

1

2

'

, = 1, ...,...

i

ii

ip

zz

Z i n

z

=

, (5.2)

, p' , . Z ' . , : ' (0,1...0,25)p p= ' 1...3p = .

(), -

138

. - , . .

, , , . - . , . .

: , -

;

;

, (), - .

R- Q- . . - (), . - . Q- . 19351936 . .. 1946 . . - . - (. 5.1), .

5.1

() () 1 2 ... p

1 x11 x12 x1p 2 x21 x22 x2p ... n xn1 xn2 xnp

, P-, -

, - , -, S- -, - . 1940 . 50% , - , R-.

139

-, , . , -. - , .

- ( ). - -. , n - p'- . - . , - .

, -, . , . . - .

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

: ; ,

; ; .

5.1.

, . 1901 ., ( ) . - 1933 . , - .

140

. - , . , , , . - .

, - , - , , - .

:

iZ LX= , (5.3) X - ( )1, ..., pa a a=G ( )ij = , i, j = 1, , p;

11 1

'1 '

...

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

p

p p p

L =

A A

A A (5.4)

, .

:

( ) 1 '1

'

...

...p

p

z zp

x x

D DI Z

D D

+ += + + , (5.5)

D . :

i( ) ( ) ( )' 'maxp pZ X FI Z I Z= , (5.6) F(X) .

i ( )1Z X ( )1, ...,= TpX x x - , - 1, , .

K- i ( )kZ X - ( )1, ...,= TpX x x - -

141

, k 1 - - k 1 1, , .

:

j j j j ji i ix x x x a= = , , 1, ...,=i j p . (5.7) :

*ivi

vii

xx =

, ( )( )

1 ; , 1, ..., ; 1, ...,=

= = =n k k j jv vv

kj

x x x xj k p v n

n. (5.8)

, : 0I = , (5.9) .

p p 1, ..., p , . , i1 1z

D = , 1 max jj = . p p :

* *11

TR X Xp

= . (5.10) R ,

. :

( )1 1 0TI =A , (5.11) 1. .

, - :

1 ''1

...,

...p

pp

I + + = + + (5.12)

1,..., p , . -

,

142

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

-, .

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

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

, , , -, .

, - , . , , , . - , , 100% . 100%, , , - , - . . . - , .

143

5.2.

. , , - , . -, .

, , , . -, , - .

-, , -, - .

:

*X MF= . (5.13) X* -

, n p . , -

X* m - .

, - . - , . , - , .

, . 1900 . . 1920 . - , , , . - . .

M : M A D= + , (5.14) D , - , F.

144

A ( ) - :

11 12 1

21 22 2

1 2

... 0 0 ... 0

... 0 0 ... 0... ... ... ... ... ... ... ...

... 0 0 ... 0

m

m

p p pm

a a aa a a

A

a a a

=

. (5.15)

n m -. . (5.13), A - .

1

2

0 0 ... 0 0 ... 00 0 ... 0 0 ... 0... ... ... ... ... ... ... ...0 0 ... 0 0 0 ... p

dd

D

d

=

(5.16)

. - .

F ( ) . :

11 12 1

21 22 2

1 2

11 12 1

21 22 2

1 2

...

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

...

...

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

...

n

n

m m mn

n

n

p p pn

f f ff f f

f f fF

=

. (5.17)

, - . , j- :

2 2 21j j js d h= = + , (5.18)

2 21

m

j jrr

h a=

= , m - ;

145

2jd , - j *jx . , :

2 2 2j j jd b c= + , (5.19) 2jb , ; 2jc . :

2 2 2 21j j j jr h b c= + = , (5.20) , .

:

0 1

m

rrVk

p p=

= =

. (5.21)

X* R:

1R X Xn

= . (5.22) (5.9), :

2T T T hR MM AA DD R D= = + = + . (5.23) R . -

, . - R.

Rh () . , ,

2 ThR R D ACA= = , (5.24) 1 TC FF

n= .

. - ,

ThR AA= .

. (5.24) , 2 1

146

, - , . -: m, - , Rh m.

- j- , . : 2 2 2j j jR h r . (5.25)

- . . , , - .

. ,

(5.25) . :

2 11j jjh r , (5.26)

jjr j- R1. ,

( ) - , . -, , 20 , .

-, () :

21

11

p

j jkkk j

h rp =

= . (5.27) , ( ) j- (-

) ( ) jkr jlr ( ), k l j . - :

2 jk jljkl

r rh

r= . (5.28)

147

R, . Rh - .

, , 1 , : 21 1

1max

p

jj

a=

= . (5.29) :

( )1

, 1, ...,=

= =mjk jr krr

r a a j k p . (5.30)

- , :

1 1 11

0p

jk k jk

r a a=

= , (5.31) 21 1

1

p

jj

a=

= . , 2jj jr h= , :

( )( )

( )

21 11 12 21 1 1

221 11 2 21 2 1

21 11 2 21 1

... 0;

0;

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

... .

p p

p p

p p p p

h a r a r a

r a h a r a

r a r a h a

+ + + = + + = + + +

(5.32)

(5.32):

21 12 1

221 2 2

21 2

...

...0

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

p

p

p p n

h r rr h r

r r h

=

. (5.33)

(5.33) (5.32), p- . , (5.33) .

148

1 (5.33) (5.32). ( )11 21 1, , ..., pa a a . - :

11 1 2 2 211 21 1...

j j

p

a aa a a

= + + + . (5.34)

k- -, k- - Rh, , . , Rh.

- . , m - -, . :

( )2 1 22 5 ln6 3

TAAn p m

R = + . (5.35)

, - ( )20,5 p m p m = .

5.3.

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

. , ,

149

1930 ., - . -. , . - . - . - , .

, ( - ). . - R - ( ). ( ) :

2

12

1 1

p

jkk

j p p

klk l

rh

r

=

= =

=

. (5.36)

, , .

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

. , , n :

2 1 8 1

2p p

n+ += . (5.37)

:

( )2p n p n+ < . (5.38)

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

150

1/4 1/3 . , - [29] - 2 / N = , (5.39) 2c -, N . c < , -. , . .

, , . , :

. . , - ;

. ; - , .

. 1965 .; . ; , .

- .

5.4.

. , 100 x1, , x6. - MS Excel [0; 10] . , : 4 1 2 42x x x= + + ; 5 1 2 3 53x x x x= + + ; 6 2 3 62x x x= + , - 0 5, MS Excel.

SPSS x1, , x6 . , Analyze/Data Reduction/ Factor, . x1, , x6, . -

151

Descriptives , . , . - Exstruction (Principle Components), . -, , . , -. , . Rotation , (Varimax) - . Scores -, . Options . , , . 1719.

. 17.

. 18.

152

. 19. , . 17 , . 18

. - , , - . 1 3, - 0,013 0,451 , - 0,854 2 4, 0,000 . 2 4, 5 6, 5 6, . .

. 19 , - . , , , 70% , 90%. - .

. 20 - .

. 20.

153

, . -, , - .

. 21 . - .

. 21. , -

, -. . .

MS Excel , .

, , MathCad.

: : 0,98.I =

. -, .

154

:

-

:

, -

- ( - ):

155

:

1. ? 2. ? 3. ? 4. -

? 5. ? -

? 6. ? 7. ? 8. ? 9. ? 10. ? 11. ? 12.

? 13. ? 14.

? 15. -

? 16. k- ? 17. -

?

156

18. -?

19. ? 20. ? 21. ? 22.

? 23. ? 24.

? ? 25. ? 26. ? 27. ? 28. ? 29. -

? ? 30. ?

?

158

- :

( )0 1

1 |k

ii

C P i ic =

= . (6.4) . -

[39; 45]. 1. , .

, - , .

2. , . , - .

3. > 0, xi, - xj dij < , - xk, - , dij < .

4. - R x , - R x , .

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

6.1.

(). , -.

1, 2, , n, - ( ),j jf X 1, ...,=j k .

159

f(X) , , :

( ) ( )( )1

,k

j jj

f X f X j=

= (6.5)

( ) ( )( ) ( ),f X f X d= . (6.6) . -

() k, ( ) j ( ),j jf X - 1 2, , ..., nX X X .

- , - . , ( ),j jf X , - j. , , . , - - , . - - . , . .

- EM (Expectation Maximization) , - 1977 . , . , . . (Expectation) - , - . (Maximization) , . - . (

160

). , - . - EM : -, (SEM), (CEM), (GEM), .

. 6.1 - 2008 . http://www.ege.ru. - , : ( ) ( ) ( )( ) 0,061 11, 9; 3, 6 0,278 24, 2; 6, 9 0,661 39, 3;10, 0= + +f n N N N , ( ),N m m .

. 6.1. 2008 .

6.2.

( ),j jf X . : , - . - , , , .

161

, - . - . -, .

- . - (. 6.2). (. 6.3), - - ( ).

. 6.2. ,

-

, - , , , . , -.

, . - (, - ).

162

. 6.3. . . (http://uk.wikipedia.org/wiki/ : H-R_diagram.svg)

(). (, ) , , , - , . , , . , , , f(X) , , . - , .

- , :

() ( - , - );

the methods of data analysis

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