应用多元统计分析课后习题答案高惠璇(第六章习题解答)-文档资料_图文






6-1 :

(1) ;

(2)

;

(3)d,c>0, d * d

;

d c

(4)

;

: (1)d (1)d (2), d d (1) d (2).

d3.

2




(2) d,a >0.d*=ad,

di*j cdij 0,X (i) X ( j)di*j 0;



di*j

cdij

cd ji



d

* ji

, i,

j;

3



di*j cdij c(dik dkj ) cdik cdkj



di*k



d

* kj

, i, k,

j.

d*=ad.

(3) d,c>0,




4





di*j



dij dij c

1
1 c / dij

1
1 c /(dik

dkj )

dik dkj

dik



d kj

dik dkj c dik dkj c dik dkj c

dik dkj dik c dkj c

(dik 0, dkj 0)



d

* ik



d

* kj

i, k, j.

d*.

5


(4) d (1)d (2), d * d (1) d (2). d *2,. d *.
6



6-2 (6.2.2)
(6.2.3).

XiXjn xti, xtj (t=1,...,n). xti, xtj 01. 6.5Xi1 a+b,0c+d;XiXj1 a,0d

n

(xti xi )(xtj x j )

rij

t 1 n

n

(xti xi )2

(xtj x j )2

t 1

t 1

7



n
t 1

( xti



xi )(xtj

xj)



n t 1

xti xtj

nxi x j

an ab n

ac n

1 [an (a b)(a c)] 1 [a(a b c d ) (a b)(a c)]

n

n

ad bc n

n
t 1

( xti



xi )2



n t 1

xt2i

nxi2



a b n

a b 2
n

(a b) [n (a b)] 1 (a b)(c d )

n

n

8



n
t 1

( xtj



xj )2



n t 1

xt2j



nx

2 j



a c n

a c 2
n

(a c) [n (a c)] 1 (a c)(b d )

n

n



Cij (7)

n

(xti xi )(xtj x j )

t 1



n

n

(xti xi )2 (xtj x j )2

t 1

t 1

ad bc (a b)(c d) (a c)(b d)
(6.2.2)

9





n

xti xtj

cosij

t 1 n

n



xt2i

xt2j

t 1

t 1

n

n

n

xti xtj a, xt2i a b, xt2j a c

t 1

t 1

t 1

cij (9) cosij

a (a b)(a c)

(6.2.3)

10



6-3 5

0



D(0)



D(1)





4

6

0 9

0





1 6

7 3

10 5

0 8

0



.

::

{X(1),X(4)}=CL4,

D1=1.

D(2)





0 9 3 7

0 5 10

0 8

X (2)



X

(3)

0



X (5) CL4

11



{X(2),X(5)}=CL3, D2=3.

D(3) 100 9

0 8

0




X (3)
CL4 CL3

{CL3,CL4}=CL2, D3=8.

D(4) 100

0



X (3) CL2

CL1, D4=10.

12


:
Name of Observation or Cluster
X1

X4

X2

X5

X3

0

1

2

3

4

5

6

7

8

9

10

Maximum Distance Between Clusters
13



:

D(0)



D(1)





0 4 6

0 9

0



1 7 10 0

6 3 5 8 0

{X(1),X(4)}=CL4, D1=1.

0

D(2)







92 32

65 2

0 52 136
2

0 100
2

X (2)

X (3)

0




X (5) CL4

14



{X(2),X(5)}=CL3, D2=3.

D(3) 1306 2 106 2

0 165 4

0




X (3)
CL4 CL3

{CL3,CL4}=CL2, D3=(165/4)1/2.

D(4) 1201 2

0



X (3) CL2

CL1, D4=(121/2)1/2.

15


:
Name of Observation or Cluster
X1

X4

X2

X5

X3

0

1

2

3

4

5

6

7

8

Average Distance Between Clusters

16


6-4
Dk2r pDp2k qDq2k Dp2q | Dp2k Dq2k |
0,p0,q0,p+q+1, Ward.

LGpGqGr,DL Dpq,Dpq2Dij2 . GrGk ,0,p0,q0, p+q+ 1

Dk2r pDp2k qDq2k Dp2q ( p q )Dp2q Dp2q

Dpq DL

DL1 DL .

17









0, p



np nr



0, q



nq nr



0,

p

q







np nr



nq nr

0

11









0, p

(1 ) np
nr



0, q

(1 )

nq nr



0, (

1)

p

q







(1

)

np nr

(1

)

nq nr





11



18







0, p

1
2

0,q

1
2

0, (

1)

p

q



1
2

1
2



11







0, p



nk nr

np nk

0,q



nk nr

nq nk

0,

p q



nk np nr nk

nk nq nr nk

nk nr nk

11



19



6-5 .



L

D( L1)



D( L1) ij

D( L1) pq



min

D( L1) ij

GpGqGrL:

DL



D( L1) pq

GrGk

D(L) rk



min(Dp(Lk1)

,

D(L1) qk

)



D(L1) pq



D(L)

(k p,q)

L+1 D(L)



D(L) ij



20





D(L) rk



D ( L 1) pq



DL

(k p, q)

D(L) ij



D ( L 1) ij



DL

(i, j r, p, q)

L1:
DL1 min(Di(jL) ) DL ,

.

,.

21



6-6 A,B,C,

d

2 AB



d

2 AC

1.1,

d

2 BC

1.0

,

.

:, =-1/4,BC

,D1=1,AGr={B,C}



DA2r



1 2

( D A2 B



DA2C )

1 4

DB2C

0.5 (1.11.1) 0.25 1

1.1 0.25 0.85 22


A{BC}

D2 0.85 0.922 1 D1



,BC,

D1=1,AGr={B,C}

DA2r



nB nr

DA2B



nC nr

DA2C



nB nr

nC nr

DB2C

0.51.1 0.51.1 0.251

1.1 0.25 0.85

23


A{BC}

D2 0.85 0.922 1 D1



A



B

C

D (1)



0



1.1 0

110..01

A B C



D(2)



0

0.85 0



A Gr

D(3) 0

24



6-7 (6.3.2);

Dr2k



np nr

Dp2k



nq nr

Dq2k



n p nq nr2

Dp2q

:

X (r) 1 nr

np X ( p) nq X (q)

,

Dr2k ( X (k ) X (r) )'( X (k ) X (r) )





n

p

nr

nq

X (k)

np nr

X ( p)

nq nr

X (q) '



25



Dr2k





np nr

2 ( X (k)



X

(

p

)

)'()





nq nr

2 ( X (k)



X

(q) )'()



n p nq nr2

(X

(k)



X

( p) )'( X

(k)



X

(q) )



n p nq nr2

(X

(k)



X

(q) )'( X

(k)



X

( p) )



n2p nr2

D

2 pk



nq2 nr2

Dq2k



n p nq nr2

(X

(k)



X

( p) )'( X

(k)



X

( p)



X

( p)



X

(q) )



n p nq nr2

(X

(k)



X

(q) )'( X

(k)



X

(q)



X

(q)



X

( p) )

26



Dr2k



n

2 p

nr2

D

2 pk



nq2 nr2

Dq2k



n p nq nr2

D p2k



n p nq nr2

Dq2k



n p nq nr2

(X (k)



X ( p) )'(X

( p)



X (q) )

npnq ( X (k ) X (q) )'( X ( p) X (q) ) nr2



np nr

D p2k



nq nr

Dq2k



n p nq nr2

D

2 pq

27



:,

X (r) 1 nr

np X ( p) nq X (q)

Dr2k ( X (k ) X (r) )'( X (k ) X (r) )







X

(k)



1 nr

(np X ( p)

nq X

(q) )



X (k ) X (k ) 2 np X (k ) X ( p) 2 nq X (k ) X (q)

nr

nr



1 nr2

n

2 p

X

( p) X

( p)



2npnq X

( p) X

(q)



nq2 X

(q) X

(q)

28





X (k ) X (k )



1 nr



n

p

X

(k

)

X

(

k

)



nq X (k) X

(k)



nq2 nr2



1 nr2

(nq nr



nq

n

p

);

n

2 p

nr2



1 nr2

(n p nr

npnq );

Dr2k



np nr

( X (k) X (k)

2 X (k) X ( p)



X ( p) X ( p) )

nq ( X (k ) X (k ) 2 X (k ) X (q) X (q) X (q) ) nr



n p nq nr2

(X

( p) X

( p)

2X

( p) X

(q)



X

(q) X

(q) )

29





Dr2k



np nr

(X (k)



X ( p) ) ( X (k )



X (p))

nq ( X (k ) X (q) ) ( X (k ) X (q) ) nr



n p nq nr2

(X

( p)



X

(q) )( X

( p)



X

(q) )



np nr

D

2 pk



nq nr

Dq2k



n p nq nr2

D p2q

30



6-8 Ward(6.3.3);

Ward

,GpGq
Dp2q Wr (Wp Wq ). Wr:

nr

Wr

(

X

(r) (t )



X

(r

)

)(

X

(r) (t )



X

(r)

)

t 1

np



(

X

(p (t )

)



X

(

r

)

)(

X

(p (t )

)



X

(r) )

t 1

nq



(

X

(q) (t )



X

(

r

)

)(

X

(q) (t )



X

(r) )

t 1

31



np

Wr

(

X

(p (t )

)



X

( p)



X

( p)



X

(r) )()

t 1

nq



(

X

(q) (t )



X

(q)



X

(q)



X

(r) )()

t 1

np

np



(

X

(p (t )

)



X

( p) )()



(X ( p) X (r) )() 0 0

t 1 nq

t 1 nq



(

X

(q) (t )



X

(q) )()



(X (q) X (r) )() 0 0

t 1

t 1

X (r) 1 nr

np X (p)

nq X (q)

: X ( p) X (r)



nq nr

(X (p)

X (q))

X (q) X (r) np (X (q) X (p))

nr

32



Wr

Wp

Wq





nq nr

2


np t 1

(X ( p)



X (q) )()





np nr

2

nq t 1

(X

(q)



X

( p) )()

Wp

Wq





nq nr

2
np ( X

( p)



X

(q) )( X

( p)



X

(q) )



np nr

2 nq ( X

( p)



X

(q) )( X

( p)



X

(q) )

Wp

Wq



n p nq nr

(X

( p)



X

(q) )( X

( p)



X

(q) )

33



D

2 pq

Wr

(Wp

Wq )



n p nq np nq

(X (p)



X

(q) )( X

( p)



X (q))



n p nq nr

Dp2q ()

(

Gr{Gp,Gq},GrGk

Dr2k



nr nk nr nk

(X (r)



X (k ) )( X (r)



X (k )



nr nk nr nk

Dr2k ()

(6-7)

34



Dr2k



nr nk nr nk

np



nr

Dp2k ()

nq nr

Dq2k ()

npnq nr2

Dp2q ()



nr nk nr nk

np



nr

(X (p)



X (k) )()

nq nr

(X (q)



X (k) )()

npnq nr2

(X (p)



X (q) )()

nk np ( X ( p) X (k) )() nk nq ( X (q) X (k) )()

nr nk

nr nk

nk npnq ( X ( p) X (k) )() nr nk nr



np nk nr nk

D

2 pk



nq nr

nk nk

Dq2k



nk nr nk

D

2 pq

35



6-9 5,1

2,5,7,10.5k(k5,4,3

2,1)bkW(k).

0

0



D(1)



D(1)



1 2

116 36

0 9 25

0 4

0







0.5 8

0 4.5

18 12.5

0 2

0



81 64 25 9 0 40.5 32 12.5 4.5 0

{1,2} CL4,D1=(0.5)1/2 =0.707

0

D(2)





49 121 289

6 6 2

0 2 12.5

0 4.5

0

CL4 5 7 10

36


{5,7} CL3,D2=(2)1/2 =1.414

0 D(3) 81 4
32 3

0 289 2

0

CL3 CL4 10

{CL3,10}={5,7,10} CL2, D3=(32/3)1/2 =3.266

D(4)





0 245

6

0

CL2 CL4

37



{CL4,CL2}={1,2,5,7,10} CL1,D4 =(245/6)1/2 = 6.39
D(5) 0 CL1

bkW(k):

k=5 {1},{2},{5},{7},{10} W(5)=0

k =4 {1,2}, {5},{7},{10} W(4)=0.5

k =3 {1,2}, {5,7},{10} W(3)=2.5

k =2 {1,2}, {5,7,10}

W(2)=13.666

k =1 {1,2,5,7,10}

W(1)=54

38


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