A New Biased Estimator

8/13/2019 A New Biased Estimator

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k


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j, j = 1, . . . , p

XRnp

Xj, j = 1, . . . , p j x

Tj j

X j Xj Xj, j= 1, . . . , p


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A Rnn A

0

A> 0

i

j

A A{i,j} A{j} j A {0}

0 Rp1

j, j= 1, . . . , p


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yi= 0+1xi,1+. . .+pxi,p+i, i= 1, . . . , n, n N,

0, 1, . . . , p

R

yi

xi,j, j= 1, . . . p p

i E(i) = 0

y= X + ,

y, Rn1

X :=

1n X1 . . . X p

Rn(p+1)

1n := [1]1in X

Xj := [xi,j]1in, j= 1, . . . , p

X

X

XTX

p+ 1

n p + 1

i

2In N(0, 2In)

2

XTX

RSS() :=n

i=1

yi xTi

2= (y X)T (y X)=yTy + TXTX 2TXTy

xTi , i= 1, . . . , n i X

RSS()

= 2XTX

2XTy.


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XT

X = XT

y

:=

XTX1

XTy,

XTX

E() =

() =2

XTX

1

var(j) var(j), j= 0, 1, . . . , p

E

RSS()

= (n p 1)2,

RSS() =

ni=1(yi xTi )2

2

2 = RSS()

n p 1.


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2

:= y X

E() = 0

() =2

In X(XTX)1XT

.


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b R(p+1)1

y

b= C y + d.

C R(p+1)n d R(p+1)1

C=

XTX1

XT

d= 0.

b

d= 0

b

Bias(b) = E(b) = CE(y) + d = C X + d

(b) =C(y)CT =2CCT.

RSS

b

L(b) = (b )T W(b ) ,

W (p+ 1)(p+ 1)


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W (p + 1) (p + 1)

b

R(b) = E

(b )TW(b ) .

W = I(p+1)

b R(p+1)1

MSE(b) := E

(b )T(b )= E

(b E(b) + E(b) )T(b E(b) + E(b) )

= E

(b E(b))T(b E(b)) + (E(b) )T(E(b) )= tr((b)) + Bias(b)TBias(b),

tr

v v2 =

vTv MSE(b)

b

b

WMSE(b)

w R(p+1)1 MSE

wTb

= E

(wTb wT)T(wTb wT)

= E

(b )TwwT(b ) .

wTb

WMSE(b)

W =wwT


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(p+ 1) (p+ 1)

MtxMSE(b) := E (b )(b )T .

MtxMSE(b) = E

(b E(b) + E(b) )(b E(b) + E(b) )T= E

(b E(b))(b E(b))T + (E(b) )(E(b) )T

= (b) + Bias(b)Bias(b)T.

tr(MtxMSE(b)) = MSE(b) = E (b )T(b )

w R(p+1)1 wT(MtxMSE(b))w= tr

wTE

(b )(b )Tw

= E

tr

wT(b )(b )Tw= E

tr

(b )TwwT(b ) .

wT(MtxMSE(b))w 0

W =wwT

MSE

wTb 0

b1 b2

= MtxMSE(b2) MtxMSE(b1).

W

WMSE(b2) WMSE(b1) 0

W R(p+1)(p+1)

>

b1 b2

:= MSE(b2) MSE(b1) 0,

b1

b2 0


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bi= Ciy, i= 1, 2

= (b2) (b1) + Bias(b2)BiasT(b2) Bias(b1)BiasT(b1)=2S Bias(b1)BiasT(b1) + Bias(b2)BiasT(b2),

S= C2C

T2 C1CT1

2S Bias(b1)BiasT(b1)

Bias(b2)BiasT(b2)

A

(p + 1) (p + 1)

a

(p + 1)1

d

dAaaT

aTA1a d

dA aaT

Bias(bi) = (CiX Ip+1), i= 1, 2.

bi= Ciy, i= 1, 2

S

T(C1X Ip+1)TS1(C1X Ip+1)< 2.

= MtxMSE(b2) MtxMSE(b1)> 0,

S= C2CT2 C1CT1

opt= A0y

A0= T

XTXTXT +2In1 ,


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2

()


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yi= 0+1xi,1+. . .+pxi,p+i, i= 1, . . . , n, n p+ 1.

1

n

ni=1

yi= 0+11

n

ni=1

xi,1+. . .+p1

n

ni=1

xi,p+1

n

ni=1

i,

y = 0+1 X1+. . .+p Xp+ .

yci =1xci,1+. . .+px

ci,p+

ci , i= 1, . . . , n ,

yc =Xc{0}+ c,

T{0}:=

1, . . . , p

,

xci,j :=xi,j Xj,yci :=yi y,ci :=i , i= 1, . . . , n , j = 1, . . . , p .


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P :=In 1n 1n1Tn Rnn,

In n n 1n n 1

P

yc =P y,

c =P ,

Xc =P X{1},

X{1}:=

x1,1 . . . x1,p

xn,1 . . . xn,p

Rnp.

P

P

P2 =P

PT =P

P2 =PTP = (In 1n

1n1Tn )

T(In 1n

1n1Tn )

=In

1

n1n1

Tn

1

n1n1

Tn +

1

n21n1

Tn 1n1

Tn

=In 2n

1n1Tn +

1

n1n1

Tn

=In 1n

1n1Tn =P.

P y= y

y

y

=yc

c =P , Xc =P X{1}

P y= P X{1}{0}+ P .

RSS({0}) :=

P y P X{1}{0}T

P y P X{1}{0}

= y X{1}{0}T Py X{1}{0}


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c:=

c1, . . . ,

cp

c

=

(P X{1})TP X{1}1

(P X{1})TP y

=

(P X{1})TP X{1}1

(P X{1})T(P X{1}{0}+ P )

={0}+

(P X{1})TP X{1}1

(P X{1})TP

={0}+

XT{1}P X{1}1

XT{1}P ,

{0}=

1, . . . , p

T

E(c) ={0}+X{1}TP X{1}1 X{1}TPE() ={0}

(c) =

X{1}TP X{1}

1X{1}TP()

X{1}TP X{1}

1X{1}TP

T=2

X{1}TP X{1}

1=2

XcTXc

1.

0

c0:= y

pi=1

ci Xi= y

1

n 1Tn X{1}

c

.

y= X + ,

X=

1n X{1}

= 0

{0}

.

=

0

{0}

=

XTX1

XTy

=

1Tn 1n 1

Tn X{1}

X{1}T1n X{1}TX{1}

1 1Tn y

X{1}Ty


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= n 1Tn X{1}

X{1}T

1n X{1}T

X{1}1

1Tn y

X{1}T

y .

=

1n+

1n2

1Tn X{1}Q1X{1}T1n 1n 1Tn X{1}Q1

1n Q1X{1}T1n Q1

1Tn y

X{1}Ty

=

1n 1

Tn y +

1n2

1Tn X{1}Q1X{1}T1n1Tn y 1n 1Tn X{1}Q1X{1}Ty

1n Q1X{1}T1n1Tn y + Q1X{1}Ty

,

Q= X{1}TX{1} 1n(1Tn X{1})T(1Tn X{1})=X{1}TP X{1}=

P X{1}

TP X{1}.

{0}= 1

nQ1X{1}T1n1Tn y + Q

1X{1}Ty

= 1n

X{1}TP X{1}

1X{1}T1n1Tn y

+

X{1}TP X{1}

1

X{1}Ty

= X{1}TP X{1}1X{1}Ty 1n 1Tn X{1}T 1Tn y=

X{1}TP X{1}1

XT{1}P y

=

(P X{1})TP X{1}1

(P X{1})TP y

{0}= c.

0=

1

n1

T

n y +

1

n21

T

n X{1}Q1

X{1}T

1n

1T

n y 1

n1

T

n X{1}Q1

X{1}T

y

= y+1

n1

Tn X{1}

X{1}TP X{1}

1X{1}T1ny

1n

1Tn X{1}

X{1}TP X{1}

1X{1}Ty

= y 1n

1Tn X{1}

XT{1}P X{1}

1XT{1} (y 1ny)

= y 1n

1Tn X{1}

X{1}TP X{1}

1X{1}TP y

= y 1n

1Tn X{1}{0}


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0= c0.

XTX

y := yc

zi,j =xi,j Xj

Sjj, i= 1, . . . , n , j = 1, . . . , p

Sjj =n

i=1

(xi,j Xj)2.

yi =1zi,1+2zi,2+ +pzi,p+i , i= 1, . . . , n ,

y = Z + ,

y = P y,

Z=:

Z1 . . . Z p

= P X{1}D1,

= D{0},

= P

D=

S11

Spp

.

Zj := [zi,j]1in, j = 1, . . . , p Z

=

ZTZ

1

ZTy

= D1XT{1}P X{1}D11 D1XT{1}P y


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=D XT{1}P X{1}

1XT{1}P y= D{0}

j = j

1

Sjj

12

, j= 1, . . . , p

0 = y p

j=1

j Xj .

E() =DE({0}) =D{0}.

() =

ZTZ1

ZT(y)Z

ZTZ1

=

ZTZ1

ZTP(y)PTZ

ZTZ1

=2

ZTZ1

ZTP Z

ZTZ1

=2 ZTZ1 2

n ZTZ1

ZT1n1Tn ZZTZ1

=2

ZTZ1

,

Z ZT1n= 0 Rp1

(XTX)1

E() = 0

() =P()PT =2P,

RSS() = (y Z)T (y Z)

=

P y P X{1}{0}T

P y P X{1}{0}

= P y P X{1}{0} P 0TP y P X{1}{0} P 0


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T0 :=

0, . . . , 0 R1n

P 0= 0

RSS() =

y XT

P

y X

= TP.

E() = 0,

() =2

In X(XTX)1XT

E (RSS()) = E

TP

= ET(In 1n 1n1Tn )= E

T

1n

E

T1n1Tn

= (n p 1)2 1n

tr(1n1Tn ())

= (n p 1)2 2

ntr

1n1Tn (In X(XTX)1XT)

= (n p 2)2 +

2

n 1

Tn X(X

TX)1XT1n.

E (RSS()) = ERSS() .

2

2 = RSS()

n p 2 + 1n 1Tn X(XTX)1XT1n.


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Z1 Z2

1,2

y,i

y Zi, i= 1, 2 =

ZTZ

1ZTy

ZTZ

1=

1 1,2

1,2 1

|ZTZ| ,

|ZTZ| = 1 21,2 21,2 1 |ZTZ| 0 var(i) , i= 1, 2 cov(i, j)

1,2

1

() =2

ZTZ1

.

Z1 Z2

=

y,1 1,2y,2y,2 1,2y,1

|ZTZ

|

.

1,2 = 1 |ZTZ| = 0

1+ 2= (y,1(1 1,2) +y,2(1 1,2))|ZTZ|1 =(y,1+y,2)

(1 +1,2)

1,2= 1

1 2= (y,2 y,1)(1 1,2)

(1+ 2)

(1

2)


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p

() =2

ZTZ1

.

ZTZ = VVT

pj=1

var(j) =2tr(V1VT) =2tr(VTV

=Ip

1) =2

pj=1

1

j,

j, j = 1, . . . , p Z

TZ

Zj, j = 1, . . . , p

var(j ) =

2p

k=1

v2j,k

k ,

vj,k (j, k) V =: [vj,k]1j,kp p

p

vj,k

p

(ZTZ)1

rj,j =

1

1 R2j , j = 1, . . . , p ,

R2j

Zj (p 1)

R2j , j = 1, . . . , p

rj,j j

var(j) = 2

1 R2j, j = 1, . . . , p ,

R2j

j


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Z y y yp y = Z = Z

ZTZ

1ZTy yp= Z

ZTZ

1ZTyp Z+yp2

2 cond(ZTZ)

y yp2y2

,

Z+ :=

ZTZ1

ZT 2 y

cond(ZTZ) ZTZ

cond(ZTZ) 1

(Z+ E)+yp22

cond(ZTZ) E12

+ cond(ZTZ)2 E22y y2

y2+ cond(ZTZ)3 E222,

E

E= E1+ E2 E1

E

Z

E2

Z

cond(ZTZ)

y

Z

2 0

L2 := [ ]T [ ] .

E(L2) = MSE() =2tr(ZTZ)1 =2p

j=1

1

j


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E(T) = E yTZ(ZTZ)2ZTy= E

(Z+ )TZ(ZTZ)2ZT(Z+ )

=T+ E

TZ(ZTZ)2ZT

,

E() = 0

E(T) =T+ tr

Z(ZTZ)2ZT()

=T+2tr

(ZTZ)1

,

ZT

1n =

0

T

T

j, j = 1, . . . , p

pT

p Rp1 ZTZ p

ZTZ

pT pTp= 1

pT = V1 V1

ZTZ

pT pT = Vp

t

R2


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j

i,j, i , j= 1, . . . , p , i =j X

Xi Xj, i ,j= 1, . . . , p, i =j

i,j

i,j i,j

rj,j, j = 1, . . . , p

VIFj :=rj,j = 1

1 R2j=

var(j)2

, j = 1, . . . , p .

ZTZ rj,j = 1

VIFj, j = 1, . . . , p

j

10

ZTZ

1, . . . , p ZTZ

Z

cond(ZTZ) =max(Z

TZ)

min(ZTZ)

.

ZTZ


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cond(Z

T

Z)

ZTZ

condj(ZTZ) =

max(ZTZ)

j(ZTZ)

, j= 1, . . . , p .

ZTZ

Z

n p

Z

Z=UVT,

U Rnp, V Rpp

Rpp

k, k= 1, . . . , p

Z

Z

condk(Z) =max(Z)

k(Z) , k= 1, . . . , p .

Z

() =2V2VT

j

var(j) =2

pk=1

v2j,k2k

=2VIFj, j = 1, . . . , p .

var(j)

p

k

2k

var(1) var(2) var(p)

1 1,1 1,2 1,p2 2,1 2,2 2,p

p p,1 p,2 p,p


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k

j,k :=v2j,k2k

VIFj =

p

k=1j,k, j, k= 1, . . . , p .

j,k := j,kVIFj

.

j,k

j

k

XTX

ZTZ

ZTZ


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y

X1

X2

X3


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y= 0+1X1+2X2+3X3+

y= X +

N(0, 2I17)

T

:=

0, 1, 2, 3

T :=

0, 1, 2, 3

R2 p t


39/163

y

X1

1

X

10

X

X

XTX

XTX

X

max

XTX j, j = 2, 3

X

cond(X) = maxmin = 3.936520.00003568 = 332.15196,


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95


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r

(ZTZ+kIp)r = ZTy,

r = (ZTZ+kIp)1ZTy,

k >0

k= 0

r =

ZTZ+kIp1

ZTy

=

ZTZ+kIp1

ZTZ

ZTZ1

ZTy

=

ZTZ+kIp

1

ZTZ

= Ip+k(ZTZ)11 =Kr,


42/163

Kr :=

Ip+k(ZTZ)1

1

=Ip k ZTZ+kIp

1

E(r) = E(Kr

) = Kr

r

Bias(r) = E(r) = (Kr Ip) = k ZTZ+kIp1

BiasT(r)Bias(r) =k2T

ZTZ+kIp

2.

r

(r) = (ZTZ+kIp)

1ZT(y)Z(ZTZ+kIp)1

= (ZTZ+kIp)1ZT()Z(ZTZ+kIp)1

= (ZTZ+kIp)1ZTP()PTZ(ZTZ+kIp)1

=2(ZTZ+kIp)1ZT

In 1

n1n1

Tn

Z(ZTZ+kIp)

1

=2(ZTZ+kIp)1ZTZ(ZTZ+kIp)1

=2Kr(ZTZ+kIp)

1 =:2KrZr,

ZT1n Z j, j =

1, . . . , p ZTZ

ZTZ=VVT

=

1

p

VTV =Ip.

Vj V j, j = 1, . . . , p

ZTZ

(ZTZ)1 =V1VT.


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(ZTZ)1 ZTZ 1j

Vj, j= 1, . . . , p

j Z

TZ

Vj , j= 1, . . . , p

Zr = (Z

TZ+kIp)1

j =

1j+k

Vj

Kr =

k(ZTZ)1 + Ip

1

j = jj+k

Vj

j = 1, . . . , p

j Z

TZ

Vj

ZTZVj =jVj, j= 1, . . . , p .

ZTZ+kIp

Vj = (j+k) Vj , j= 1, . . . , p .

(ZTZ+ kIp) j+ k Vj

(ZTZ+ kIp) k >0

Kr

j := 1j+k

Vj

(ZTZ)1

1j

Vj, j = 1, . . . , p

(ZTZ)1Vj = 1

jVj

k(ZTZ)1Vj = kj

Vj

k(ZTZ)1 + IpVj = kj

+ 1

Vj, j = 1, . . . , p .

Kr j = kj + 11 = jj+k

Vj

pj=1

var(rj ) =2tr

V

1

p

VTV

1

p

VT

=2tr1

p 1

p = 2

p

j=1j

(j+ k)2

.


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r

k

r

MSE(r) =

pj=1

var(rj ) + biasT(r)bias(r)

=2p

j=1

j(j+k)2

+k2T(ZTZ+kI)2.

r

k

k

r

k >0

MSE(r)< MSE().

r

RSS(r) = (y Zr)T (y Zr)

= ((y Z) + (Z Zr))T ((y Z) + (Z Zr))= (y Z)T(y Z) + 2(y Z)T (Z Zr)

+ ( r)TZTZ( r)= (y Z)T(y Z) + ( r)TZTZ( r)= RSS() + ( r)TZTZ( r),

Z= ZZTZ1 ZTy 2(y Z)T (Z Zr) = 2yT

In Z

ZTZ

1ZT

Z( r) = 0.

X


45/163

g

RSS(g) = (y Zg)T(y Zg)= (y Z)T(y Z) + (g )TZTZ(g )= RSS() +(g).

RSS(g)

RSS(g) RSS()

(g ) g0

RSS(g0

) = RSS() +0

0

> 0

ZTZ

ZTZ

min gTg

(g )TZTZ(g ) =0.

F :=gTg +1

k

(g )TZTZ(g ) 0

,

1k

F

g = 2g +

1

k

2ZTZg 2ZTZ= 0.

r :=

Ip+

1

kZTZ

1 1k

ZTZ

=

ZTZ+kIp1

ZTy,

k


46/163

r

k

k

k

k

r k k (0, 1]

k

k

r

k

r

k

k

k

k

k= p2

T,

2 2

k= p2

T ZTZ= Ip


47/163

K

N(0, 2In) E (RSS()) =2(n p).

2 = RSS(

)np

2

k

k

k0

ki= p2p

j=1(j(ki1))2, i 1,

ki k

Q

Q:= T 2p

j=1

1j

,

j Z

TZ

k

r(k)Tr(k) =Q,

Q> 0

k= 0

k=

k

r k

k

E(T) =T+2tr

(ZTZ)1

.

k

T

E(r(k)Tr(k)) = E(

T) 2p

j=1

1

j

=T.


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Q

k= 0

k=

k

r(k)Tr(k) = abs(Q)

abs(Q)

Q

y= X + ,

N(0, 2

In)

2 =RSS()

n p

2

k

k

1.0

10

kI


49/163

k

k

k

k

k

ZTZ

ZTZ


50/163

ZTZ

kIp X

TX R(p+1)(p+1) X

bTb

min bTHb,

b

(b )TXTX(b ) =0.

H

R(p+1)(p+1)

Fg := bTHb +

1

k

(b )TXTX(b ) 0

.

XTX

g =

XTX+kH1

XTy.

H=Ip+1


51/163

H = 1k G G R(p+1)(p+1)

g =

XTX+ G1

XTy.

XTX = VVT

V

H = 1k V KV

T

K

g = XTX+ V KVT1 XTy.

H=XTX

g =

(1 +k)XTX1

XTy

= 1

1 +k= , [0, 1].

s := 1 c2

T

Z

T

Z ,

c > 0

c R

2

k

(g) =

2 XTX+kH1 XTX XTX+kH1


52/163

Bias(g) = E(g) = XTX+kH1 XTX =

XTX+kH1

XTX Ip.

XTX+kH

1 XTX

Ip= XTX+kH1 kH

MtxMSE(g) =

XTX+kH1

k2HTH+2XTX

XTX+kH1

.

g

:= MtxMSE() MtxMSE(g)

T2

kH1 + (XTX)11 2,

H

=2

XTX1 XTX+kH1 k2HTH

+2XTX

XTX+kH

1 0.

= UT

U

U

p H

XTX+kH

pT

XTX+kH

XTX+kH

p

=pT

XTX+kH

UTU

XTX+kH

p

=pT

U

XTX+kHT

U

XTX+kH

p 0, p

R(p+1)1

(XTX+ kH)


53/163

2 XTX+kH XTX1 XTX+kH k2HTH 2XTX 0.

2

XTX+kH

XTX1

XTX+kH

=2

XTX+ 2kH+k2H

XTX1

H

2 2kH+k2HX

TX1

H k2HTH 0.

1k H

1

2

2

kH1 +

XTX

1 T 0.

pTg, p R(p+1)1

T

2

kIp+

XTX

1 2,

T2G1 + XTX1 2

G

T

2K1 +

XTX1

2,

T

XTX

k+ 2k

2.


54/163

bg =

XTX+ C+

XTy,

XTX+ C

+

XTX+ C+

C

XTX

C

Z

Z= UVT.

=

ZTZ

1

ZTy= V1VTV12 UTy

=V

12

UT

y,

ZTZ

r =V 1

2 UTy,

1, . . . , p R

W

j = jj+k

, j = 1, . . . , p

B

B

W


55/163

k

N(0, 2I17)

2

2 = RSS()

n

p

1

= 11.369

17

3

1

= 0.87453.

r

k

r =

ZTZ+kI31

ZTy.

r{0}

T :=

r1, r2,

r3

{0}T =

1, 2, 3

k

r{0}= D

1r = D1 ZTZ+kI31 ZTy,

D

r0 = y

3

j=1 r

jXj .


56/163


57/163

k

k

k

0.05

k

k

k

10

k 0.05

k

y = Z +

T = 19.106, 24.356, 6.4207 .


58/163

k

k

k = 42ZT

2Z=RSS()

n p =11.368

14 = 0.8120,

2

RSS()

k= 0.00325.


59/163

k

r

(k) =3.3497, 0.3787, 1.5042, 7.8142 104

T.

k

r(k)Tr(k) T 2tr

(ZTZ)1

.

2Z

2

QZ := T

2Ztr (ZTZ)1= 138.0


60/163

kZ= 0.0033.

r

(k) =3.3781, 0.3642, 1.4962, 7.80 104

T.

k

2

2 = RSS()1731 = 0.87453

k

MSEk(r) = 2

pj=1

j(j+ k)2

+k2T(ZTZ+kI)2.

r

k


61/163

kmin= 0.0012

kmin

MSE(kmin) = 517.06.

kmin= 0.0012

r

(k) =0.7410, 1.6031, 2.0893, 0.0012

T.

k

k

k


62/163


63/163

j

Zj, j = 1, . . . , p

Z

n p

n > p

Zj, j = 1, . . . , p

min

ni=1

(yi zTi T)2,

zTi i Z

T =

1, . . . , p

= 0

R

min (y (Z+

))

T

(y (Z+

)) ,


64/163

:=

1 . . . p

1 . . . p

Rnp, n > p.

(Z+)T (Z+) = (Z+)T y.

y

ni=1y

i = 0

Ty = 1

ni=1y

i

pn

i=1yi

= 0 Rp1.

Z

TZ=ZT= 0.

M= [mu,v]1u,vp :=

Z+T

Z+

= ZTZ+2T.

M ZTZ

T

pTMp= pT

ZTZ

p +2pT

T

p> 0

p Rp1

=

(Z+)T (Z+)1

(Z+)T y

=

ZTZ+2T1

ZTy.

2T

M

M{u,v} (p 1) (p 1)


65/163

M u v

M{u,v}=

m1,1 . . . m1,v1 m1,v+1 . . . m1,p

mu1,1 . . . mu1,v1 mu1,v+1 . . . mu1,pmu+1,1 . . . mu+1,v1 mu+1,v+1 . . . mu+1,p

mp,1 . . . mp,v1 mp,v+1 . . . mp,p

, u , v= 1, . . . , p .

M

M1 =M

|M

|,

M:= [mu,v]1u,vp Rpp M mu,v := (1)u+v

M{v,u} .

M1x :=(X+)T(X+)

1 X Rnp

X

n p

n p

n p

n p j , j = 1, . . . , p Mx = (X+)

T(X+)

|Mx| =2TAx + 2bxT + |XTX|,

T :=

1, . . . , p

,

bxT :=

|XTX[1]|, . . . , |XTX[p]|

Ax:=

|X[1]TX[1]| . . . |X[1]TX[p]|

|X[p]TX[1]| . . . |X[p]TX[p]|

,

X[j], j = 1, . . . , p n p X j

X


66/163

|Mx| =

1j1...jpn

(X+)T

1 . . . p

j1 . . . jp

2

=

1j1...jpn

XT

1 . . . p

j1 . . . jp

+T

1 . . . p

j1 . . . jp

2

.

p p

XT

1 . . . p

j1 . . . jp

T

1 . . . p

j1 . . . jp

s=

n

p

{j1, . . . , jp} 1j1 . . .jp n p {1, . . . , n} {j1,k, . . . , jp,k}, k= 1, . . . , s

XT k :=XT

1 . . . p

j1,k . . . jp,k

T k := T

1 . . . p

j1,k . . . jp,k

k

{j1,k, . . . , jp,k

}

{1, . . . , n

}

XT k +T k , k = 1, . . . , s

{i1, . . . , ir} {1, . . . , n} |C{i1,...,ir}p |k p p i1, . . . , ir i1, . . . , ir

T k XT k

k = 1, . . . , s

r 2 C{i1,...,ir}p

T k |C{i1,...,ir}p |k = 0, r 2 r= 1 ir =r

C{r}p k =r|X[r]T

k

|,

X[r]

T k

XT k

r

XT k

|C{}p|k =|XT k |

XT k +T k= {i1,...,ir}

C{i1,...,ir}p k

= |XT k | +p

r=1

C{r}p k

= |XT k | +1|X[1]T k | +. . .+p|X[p]T k |, X[j]

T k , j = 1, . . . , p XT k

j

XT

k


67/163

|Mx| =s

k=1

|XT k | +

pr=1

C{r}p k

2

=s

k=1

|XT k |2 + 2|XT k |

pr=1

r|X[r]T k |

+2p

t=1

pr=1

rt|X[r]T k ||X[t]T k |

=s

k=1|XT k |2 + 2

p

r=1r

s

k=1|XT k ||X[r]T k |

+2p

t=1

pr=1

rt

sk=1

|X[r]T k ||X[t]T k |.

|XTX| =s

k=1

|XT k |2,

|XTX[r]| =s

k=1

|XT k ||X[r]T k |,

|X[r]TX[t]| =s

k=1

|X[r]T k ||X[t]T k |

|Mx| =2p

t=1

pr=1

rt|X[r]TX[t]| + 2p

r=1

r|XTX[r]| + |XTX|.

|Mx| =2TAx + 2bxT + |XTX|.

ARpn B Rnp np

|AB

|=

1j1...jpn A1 . . . p

j1 . . . jp B j1 . . . jp

1 . . . p


68/163


69/163

|Mx

|=

3

k=1 |XT

k

+T

k

|2

=

XT

1 2

1 2

+T

1 2

1 2

2

+

XT

1 2

1 3

+T

1 2

1 3

2

+

XT

1 2

2 3

+T

1 2

2 3

2

=

1 21 5

+

1 1

0 0

2

+

1 21 3

+

1 1

0 0

2

+ 2 25 3 + 1 10 0 2

XT k +T k= {i1,...,ir}

C{i1,...,ir}p k

,

{1, 2} {}

{1}

,{

2}{1, 2}.

C{i1,...,ir}p 2 2 i1, . . . , ir i1, . . . , ir

T k

XT k

1 21 5

+

1 1

0 0

=1 21 5

+

1 1

1 5

+ 1 20 0 +

1 1

0 0

1 21 5

+

1 1

0 0

=1 21 5

+1

1 1

1 5

1 21 3

+

1 1

0 0

=

1 21 3

+1

1 1

1 3


70/163

2 2

5 3 + 1 1

0 0 = 2 2

5 3+1 1 1

5 3

|Mx| =1 21 5

+1

1 1

1 5

2

+

1 21 3

+1

1 1

1 3

2

+

2 2

5 3

+1

1 1

5 3

2

.

X[1]

XTX[1] =1 2 2

1 5 3

1 11 51 3

X[1]TX[1] =

1 1 1

1 5 3

1 11 51 3

.

|XTX| = 1 21 52

+ 1 21 32

+ 2 25 32

,

|XTX[1]| =1 21 5

1 1

1 5

+1 21 3

1 1

1 3

+

2 2

5 3

1 1

5 3

,|X[1]TX[1]| =

1 1

1 5

2

+

1 1

1 3

2

+

1 1

5 3

2

|Mx

|=221

|X[1]

TX[1]|

+ 21|XTX[1]

|+

|XTX

|.

Mx i= 0, 1 i q

1 qp

i= 0, i > q

TAx=

qs=1

qr=1

rs|X[r]TX[s]| =Tq Aqxq,

bTx =

q

r=1 r|XTX[r]

|=bqx

Tq


71/163

Tq :=1, . . . , q ,

bqxT :=

|XTX[1]|, . . . , |XTX[q]|

,

Aqx:=

|X[1]TX[1]| . . . |X[1]TX[q]|

|X[q]TX[1]| . . . |X[q]TX[q]|

.

|Mx|

2Tq Aqxq+ 2b

qx

T + |XTX| = 0.

1, 2

1/2=2bqxTq

4

(bqxTq)

2 Tq Aqxq|XTX|

2Tq Aqxq

D:= 4

(bqxTq)

2 Tq Aqxq|XTX|

.

|Mx|

Aqx

Ru, u= 1, . . . , m n p n p

R=

|RT1 R1| |RT1 R2| . . . |RT1 Rm||RT2 R1| |RT2 R2| . . . |RT2 Rm|

|RTmR1| |RTmR2| . . . |RTmRm|

R= RT

R,

RT

:=

RT11 . . . RT1s

RTm 1 . . . RTm s Rms

s=

n

p R


72/163

|RTu Rv| =|RTv Ru|, u , v = 1, . . . , m

|RTu Ru| =

1j1...jpn

RTu

1 . . . p

j1 . . . jp

2

=s

k=1

RTuk2 , u= 1, . . . , m

|RTu Rv| =

1j1...jpn

RTu

1 . . . p

j1 . . . jp

RTv

1 . . . p

j1 . . . jp

=

s

k=1 RTuk

RTvk

u, v= 1, . . . , m , u =v.

s =

n

p

{j1,k, . . . , jp,k}, k = 1, . . . , s {1, . . . , n}

R= RT

R R

R

Aq

x= A

TA

AT

=

X[1]T 1 . . . X[1]T s

X[q]T 1 . . . X[q]T s Rqs,

Aqx

Tq A

qxq 0 Tq Aqxq > 0 |Mx|

Aqx bqx |XTX| = 0

B:= Aqx bqxb

qx

T

|XTX|

B 0


73/163

G:=

Aqx b

qx

bqxT |XTX|

=

|X[1]TX[1]| . . . |X[1]TX[q]|

|X[q]TX[1]| . . . |X[q]TX[q]|

|XTX[1]|

|XTX[q]|

|XTX[1]| . . . |XTX[q]| |XTX|

,

R(q+1)(q+1).

|XTX

|=

1j1...jpn XT 1 . . . pj1 . . . jp

2

=s

k=1

XT k2

=XTX

,

XT :=

XT 1 , . . . , XT s R1s.

AT

X

=

X[1]T 1 . . . X[1]T s

X[q]T 1 . . . X[q]

T s

XT 1

XT s

=

|X[1]TX|

|X[q]TX|

=|XTX[1]|

|XTX[q]|

=bqx,

sk=1

X[i]T k XT k= |X[i]TX|, i= 1, . . . , q ,

G= AT

A AT

X

XTA X

TX= A

T

XT A X R(q+1)(q+1)

G

Aqx

bqxbqx

T

|XTX|

0.


74/163

0 Tq

Aqx bqxbqxT|XTX|

q =Tq A

qxq (Tq bqx)(bqxTq)|XTX|

=Tq Aqxq

(Tq bqx)

2

|XTX| bqxT2 TAqx|XTX| 0,

D

|Mx|

|Mx| = 0 |Mx|

s=

n

p

=

n!

(n p)!p! =n(n 1) (n 2) . . . (p+ 1)

p (p 1) . . . 2 1 n, p < n

G

A X

Rs(q+1), qp

rank

A X

= q+ 1

G

q= p = n

s = 1

rank

A X

= 1

X

X

= 0

rank A X< q+ 1

j

X

c R

X

j

A

rank

A X

< q+ 1

G

G |Mx|

|Mx| |Mx| >0

Mx

|Mx| 0 Mx (X+ )

p


75/163

X np np

n p

j, j = 1, . . . , p

Mx=

X+T

X+

M1x =Mx

|Mx| = 2 M

quadx + M

linx + M

constx

2TAx + 2bTx + |XTX|

,

Mquadx :=

mquadx u,v

1u,vp

=

(1)u+v{u}TA

(uv)x {v}

1u,vp

,

M

lin

x := mlinx u,v1u,vp = (1)u+v b(uv)x T{v}+ b(vu)x T{u}1u,vp ,

Mconstx :=

mconstx u,v

1u,vp=

(1)u+v X{u}TX{v}

1u,vp

{u}:=

r

1rpr=u

R(p1)1,

b(uv)x :=

X{u}TX{v}[r]

1rpr=v R(p1)1,

A(uv)x :=

X{u}[r]TX{v}[s] 1r,spu=r;v=s

R(p1)(p1).

X{u} Rn(p1) u X X{u}[r]

X

r

X

u

Mx =:

mxu,v

1u,vp Mx

X{u}, {u} X{v}, {v} n (p 1)

X

u

v

mxv,u = mxu,v = (1)u+v

X{u}+{u}TX{v}+{v}= (1)u+v

sk=1

X{u}T k +{u}T k X{v}T k +{v}T k ,

s :=

n

p 1

(p 1)

{1, . . . , n}

k

k= 1, . . . , s


76/163

X{u}T k +{u}T k

:=

X{u}T 1 . . . p 1j1,k . . . jp1,k +{u}T 1 . . . p 1j1,k . . . jp1,k .

X{u}T k +{u}T k= |X{u}T k | + pr=1

r=u

r|X{u}[r]T k |

X{v}T

k

+

{v

}

T

k

= |X{v}T

k

|+

p

s=1s=v

s

|X

{v

}[s]

T

k

|,

X{u}[r], r = 1, . . . , p, r=u X{u}

r X X{u}

mxu,v = (1)u+vs

k=1

|X{u}T k ||X{v}T k |

+

p

s=1s=vs|X{u}T k ||X{v}[s]T k | +

p

r=1r=ur|X{v}T k ||X{u}[r]T k |

+2ps=1

s=v

pr=1

r=u

rs|X{u}[r]T k ||X{v}[s]T k |

= (1)u+v2 p

s=1s=v

pr=1r=u

rs|X{u}[r]TX{v}[s]| +ps=1s=v

s|X{u}TX{v}[s]|

+

p

r=1r=ur|X{v}TX{u}[r]| + |X{u}TX{v}|

=2mquadx u,v+mlinx u,v+ m

constx u,v

Mquadx

Mlinx

Mconstx p p

Mconstx X

TX

= 0

XTX


77/163

|Mx|

Mx

Z Rnp, n > p

M =

Z+T

Z+

= ZTZ+2T,

Z

ZTZ[v] =

Z12 . . . Z 1TZv1 0 Z1TZv+1 . . . Z 1TZp

ZpTZ1 . . . Z pTZv1 0 ZpTZv+1 . . . Zp2

|ZTZ[v]| = 0, v= 1, . . . , p

bx= 0

Z[u]TZ[v] =

Z12 . . . Z 1TZv1 0 Z1TZv+1 . . . Z 1TZp

Zu1TZ1. . . Z u1TZv1 0Zu1TZv+1. . . Z u1TZp0 . . . 0 n 0 . . . 0

Zu+1TZ1. . . Z u+1

TZv1 0Zu+1TZv+1. . . Z u+1TZp

ZpTZ1 . . . Z p

TZv1 0 ZpTZv+1 . . . Zp2

|Z[u]TZ[v]| = (1)u+vn|Z{u}TZ{v}|, p 2, u , v= 1, . . . , p .

Z

Ax

n

|ZTZ|

Ax = n

(1)u+v Z{u}TZ{v}

1u,vp

r =u

s =v|Z{u}TZ{v}[s]| = 0

|Z{u}[r]T

Z{v}[s]| = (1)r+s

n|Z{ur}T

Z{vs}|, p 3,


78/163

Z{ur}, r = 1, . . . , p u r Z

Z{1}[2]= Z{2}[1] = 1n

p= 2

|Z{u}[r]TZ{v}[s]| =n, u, v , r, s = 1, 2; r =u; s =v.

b(uv)x = 0

A(uv)x = n , p= 2

(1)r+sn|Z{ur}TZ{vs}|1r,spu=r;v=s

, p 3.

(5) mu,v = (1)u+vn2 p

r=1r=u

ps=1s=v

(1)r+srs|Z{ur}TZ{vs}| + |Z{u}TZ{v}|

,

1 u, vp, p 3

p= 2

mu,v = (1)u+vn2 2

r=1

r=u

2s=1

s=v

rs+ |Z{u}TZ{v}|

, u, v= 1, 2.

Z

Z

n p

n > p, p 2

np

j, j = 1, . . . , p M =

(Z+)T(Z+) =ZTZ+2T

M1 =M

|M| = n2 M

quad+ M

const

n2T Mconst

+ |ZTZ|,

Mquad

:=

mquadu,v

1u,vp

:=

(1)u+v{u}TA

(uv){v}

1u,vp

,

Mconst

:= mconstu,v 1u,vp:= (1)u+v Z{u}

TZ

{v

}1u,vp ,


79/163

{u}:= r 1rpr=u

R(p1)1,

A(uv)

:=

1 , p= 2(1)r+s|Z{ur}TZ{vs}|

1r,spu=r;v=s

, p 3 R(p1)(p1).

Mconst

ZTZ

Mconst

=

|ZTZ

| ZTZ

1

Rpp.

ZTZ

ZTZ |ZTZ| >0

Mconst

n2T M

const 0 |ZTZ| >0 Z

|M| >0

M

Z

ZT =

2 1 12 2 0

,

X

T =

1 1 1

0 0 0

.

ZTZ= 6 66 8

=

1 0

.

A(11)

= A(12)

= A(21)

= A(22)

= 1,

{1}= 0,

{2}= 1


80/163

Mquad=0 0

0 21

,

Mconst

=

8 66 6

M1 =

n2

0 0

0 21

+

8 66 6

n2218 + 6 66 8

= 1

24221+ 12

3221

0 0

0 1

+

8 66 6

.

|M|

M

M1

= (ZTZ+2T)1ZTy

= n2 M

quad+ M

const

n2T Mconst

+ |ZTZ|ZTy,

Mquad

=

mquadu,v

1u,vp

=

(1)u+v{u}TA

(uv){v}

1u,vp

,

Mconst =

mconstu,v

1u,vp =

(1)u+v Z{u}TZ{v}

1u,vp.

= 0

0=M

const

|ZTZ| ZTy= .

Z

y=

10

1

.


81/163

= 1

24221+ 12

3221

0 0

0 1

+

8 66 6

2 1 12 2 0

101

= 1

2221+ 1

1

0.5(221 1)

.

= 0

= 1

0.5 .

1= 1

0 1 2 3 4 5

0.5

0

0.5

1

1

2


82/163

=:

1 . . .

p

T

() = (ZTZ+2T)1ZT(y)Z(ZTZ+2T)1

= (ZTZ+2T)1ZT()Z(ZTZ+2T)1

= (ZTZ+2T)1ZTP()PTZ(ZTZ+2T)1

=2(ZTZ+2T)1ZT

In 1n

1n1Tn

Z(ZTZ+2T)1

=2(ZTZ+2T)1ZTZ(ZTZ+2T)1,

ZT1n Z

() =2(ZTZ+2T)1(ZTZ+2T2T)(ZTZ+2T)1

=2(ZTZ+2T)1 22(ZTZ+2T)1T(ZTZ+2T)1

=2 n2 M

quad+ M

const

n2T Mconst

+ |ZTZ|

22 n2 M

quad+ M

const

n2

T

M

const

+ |ZT

Z|

T

n2 Mquad

+ Mconst

n2

T

M

const

+ |ZT

Z|.

A Rp(p+1)A{r}[l]= (1)(lr+1) A{l}[r] , 1 r, l p, l =r, A{r}[l] A l

r

A{r}[l] A{l}[r]

l > r

A{r}[l] A{l}[r] r

(r+ 1)

(r+ 1)

(r + 2)

(l 1)

A{l}[r]

= l 2 l 1 . . .r+ 1 r+ 2r r+ 1 .


83/163

sign() = (1)(l1)r = (1)lr+1.

sign()

A{r}[l]= (1)(lr+1) A{l}[r] , 1 r < l p.

l

r

l < r A{l}[r]= (1)(rl+1) A{r}[l]

A{r}[l]= (1)(lr+1) A{l}[r] .

Mquadx

Mquadx = 0.

(u, v)

M

quadx Rnp

u

Mquadx (u, v) =

pl=1

(1)l+vl{l}TA(lv)x {v},

A(lv)x =|X{l}[r]TX{v}[s]|

1r,spl=r;v=s

Mquadx (u, v) =

pl=1

ps=1

s=v

pr=1

r=l

(1)l+vrslX{l}[r]TX{v}[s]

=

ps=1s=v

pl=2

l1r=1

(1)l+vrlX{l}[r]TX{v}[s]

+

p

r=2

r1

l=1

(

1)l+vr

l X{l}[r]TX{v}[s]s


84/163

=

p

s=1s=v

p

l=2l1

r=1(1)l+vrl X{l}[r]TX{v}[s]

+

pl=2

l1r=1

(1)r+vrlX{r}[l]TX{v}[s]

s.

s=

n

p 1

X{r}[l]TX{v}[s]

=

sk=1

X{r}[l]T k

X{v}[s]T k

= (1)lr+1

sk=1

X{l}[r]T k X{v}[s]T k= (1)lr+1 X{l}[r]TX{v}[s] .

M

quadx n p

Z

M

quad

Mquad

= 0

Rnp

() =2 n

2 Mquad

+ Mconst

n2T Mconst

+ |ZTZ|

22

(n2T Mconst

+ |ZTZ|)2M

const

T M

const.

Mconst

T

Mconst

=

n

ps=1

pr=1

(1)u+v+r+srs|Z{u}TZ{r}||Z{v}TZ{s}|1u,vp

M

const

T M

const

tr

Mconst

T

Mconst

= n

pj=1

pr=1

(1)rr|Z{j}TZ{r}|2

.


85/163

cov(u , v) =

2 n2mquadu,v + mconstu,v

n2T Mconst

+ |ZTZ|

22n

ps=1

pr=1(1)u+v+r+srs|Z{u}TZ{r}||Z{v}TZ{s}|(n2T M

const + |ZTZ|)2

, u, v= 1, . . . , p ,

mquadu,v mconstu,v

tr(()) =

p

j=1

var(

j)

=2p

j=1

n2{j}TA(jj){j}+ |Z{j}TZ{j}|

n2T Mconst

+ |ZTZ|

n2p

r=1(1)rr|Z{j}TZ{r}|2

(n2T Mconst

+ |ZTZ|)2

.

2 1.5 1 0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

var(1

)/

2

var(2

)/

2


86/163

() =2 n

2 Mquad + Mconst

n2T Mconst

+ |ZTZ|

22

(n2T Mconst

+ |ZTZ|)2M

const

T M

const

= 2

24221+ 12

8 66 6 + 3221

2221(24221+ 12)

2

192 144144 108

=2

23

(2221+1)2

12

(2221+1)2

12

(2221

+1)2

12(441+

221+1)

(2221

+1)2

.

var(1) 1

1 :=

var(1)

var(1)

=

163(22 + 1)3

var(1) = 0 = 0

1

1

1/

2

2/2

1= 1

Bias() = E() =

ZTZ+2T1

ZTE(y) = ZTZ+2T

1ZT (Z+ E())

=

ZTZ+2T1

ZT (Z+ PE()) =

ZTZ+2T1

ZTZ =

ZTZ+2T1

ZTZ+2T 2T = 2(ZTZ+2T)1T

= 2 MT

n2T Mconst

+ |ZTZ|

=

2 M

const

T

n2T Mconst

+ |ZTZ| Rp1,


87/163

BiasT()Bias() = 4TT( M

const)2T

(n2T Mconst

+ |ZTZ|)2 R.

Bias() = 2 M

const

T

n2T Mconst

+

|ZTZ

|

=

2

8 66 6

1 1 1

0 0 0

1 01 01 0

24221+ 12

.

=

1 1

T

Bias() =

2

2221+12(2+7221)2(2221+1)

.

2 1.5 1 0.5 0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9

10

Bias()

TBias(

)


88/163

BiasT()Bias() =144(8 + 28221+ 49441)

(2221+ 1)2

.

1 = 1

= 0

MSE() =

pj=1

var(j) + BiasT()Bias(),

pj=1

var(j) =2

pj=1

n2{j}TA(jj){j}+ |Z{j}TZ{j}|

n2T Mconst

+ |ZTZ|

n2p


(n2T Mconst

+ |ZTZ|)2

BiasT()Bias() = 4TT( M

const)2T

(n2T Mconst

+ |ZTZ|)2,

MSE()

MSE() MSE()

MSE() =

pj=1

var

j

+

BiasT()Bias()

,

BiasT()Bias()

=

4TT( M

const)2T

(n2T Mconst

+ |ZTZ|)2

=43TT( M

const)2T(n2T M

const + |ZTZ|)

(n2T Mconst

+ |ZTZ|)3

4n5T M

constTT( M

const)2T

(n2T Mconst

+ |ZTZ|)3

= 43|ZTZ|TT( Mconst)2T

(n2T Mconst

+ |ZTZ|)3,


89/163

pj=1

var(j) =2

pj=1

n2{j}TA(jj){j}+ |Z{j}TZ{j}|n2T M

const + |ZTZ|

n2p


(n2T Mconst

+ |ZTZ|)2

=2p

j=1

2n{j}TA(jj){j}(n2T Mconst + |ZTZ|)

(n2T Mconst

+ |ZTZ|)2

2nT M

const

n2{j}TA

(jj){j}+ |Z{j}TZ{j}|(n2T Mconst + |ZTZ|)2

2np


(n2T Mconst

+ |ZTZ|)(n2T M

const + |ZTZ|)3

+4n23T M

constp


(n2T Mconst

+ |ZTZ|)3

=2p

j=1

2n|ZTZ|{j}TA(jj){j} 2nT Mconst|Z{j}TZ{j}|

(n2T Mconst

+ |ZTZ|)2

2n|ZTZ| 2n23T Mconst

pr=1(1)rr|Z{j}TZ{r}|2(n2T M

const + |ZTZ|)3

= 2n2p

j=1

{j}TA(jj){j}|ZTZ| T Mconst|Z{j}TZ{j}|

(n2T Mconst

+ |ZTZ|)2

|ZTZ| n2T Mconstpr=1(1)rr|Z{j}TZ{r}|2

(n2T Mconst

+ |ZTZ|)3

= 2n2p

j=1nT M

consts1(j)

3 + |ZTZ|s2(j)(n2T M

const

+ |ZTZ|)3,

s1(j) :={j}TA

(jj){j}|ZTZ| T M

const|Z{j}TZ{j}|

+

pr=1

(1)rr|Z{j}TZ{r}|2

,

s2(j) :={j}TA

(jj){j}|ZTZ| T M

const|Z{j}TZ{j}|

p

r=1

(

1)rr|Z{j}

TZ{r}|

2

.


90/163

MSE()

= 2n2

pj=1

nT Mconsts1(j)3 + |ZTZ|s2(j)(n2T M

const + |ZTZ|)3

+43|ZTZ|TT( Mconst)2T

(n2T Mconst

+ |ZTZ|)3.

|ZTZ| >0

M

const

n2T Mconst

> 0.

MSE()

=0

= 0,

= 0

[, ]

= 0

[, 0)

(0, ]

= 0 >0 [, ]\{0} : MSE()< MSE().

= 0

= 0

= 0

= 0

= 0


91/163

A= [ar,s]1r,sp

|A{i,j}||A{k,l}| |A{i,l}||A{j,k}| =(1)

i+j|A| p= 2|A{ik,jl}||A| p 3

, i =k, j=l,

A{q,v} R(p1)(p1), q , v = i ,j,k,l A

q

v

A{ik,jl} R(p2)(p2) i

k

j

l

p= 2

i ,j,k,l= 1, 2

i =k

j=l

i= j

k= l

i =j

k=l

(1) i = 1, j = 1, k= 2, l= 2

(2) i = 2, j = 1, k= 1, l= 2

(3) i = 1, j = 2, k= 2, l= 1

(4) i = 2, j = 2, k= 1, l= 1.

|A{1,1}||A{2,2}| |A{1,2}|2 =a1,1a2,2 a21,2= |A| = (1)i+j|A|

|A{1,2}|2 |A{1,1}||A{2,2}| =a21,2 a1,1a2,2= |A| = (1)i+j |A|.

p3

i A

(1)i1

(i 1)

i

A A

(i 1)

j

k

l

A

j

k

l

|A| =

ai,i ai,j ai,k ai,l

ai,j aj,j aj,k aj,l

ai,k aj,k ak,k ak,l

ai,l aj,l ak,l al,l

aiajakal

aTi aTj a

Tk a

Tl A{ijkl,ijkl}

,


92/163

aq = aq,1, . . . , aq,i1, aq,i+1, . . . , aq,j1 , aq,j+1 , . . .

. . . , aq,k1, aq,k+1, . . . , aq,l1 , aq,l+1, . . . , aq,p

R1(p4), q= i,j,k,l.A{ijkl,ijkl} A i j k l

i

j

k

l

A{ijkl,ijkl}

p Rp1

i

j

k

l

pT :=p1, . . . , pi1, 0, pi+1, . . . , pj1, 0, pj+1, . . .

. . . , pk1, 0, pk+1, . . . , pl1, 0, pl+1, . . . , pp i =k, j=l.

pTAp= p{ijkl}TA{ijkl,ijkl}p{ijkl},

p{ijkl}T :=

p1, . . . , pi1, pi+1, . . . , pj1, pj+1, . . .

. . . , pk1, pk+1, . . . , pl1, pl+1, . . . , pp R1(p4).

A

pTAp > 0

p

p{ijkl}TA{ijkl,ijkl}p{ijkl}> 0 p{ijkl}

A{ijkl,ijkl}

W = [wr,s]1r,sp4 := (A{ijkl,ijkl})1.

|A| = A{ijkl,ijkl}

ai,i ai,j ai,k ai,l

ai,j aj,j aj,k aj,l

ai,k aj,k ak,k ak,l

ai,l aj,l ak,l al,l

aiajakal

W

aTi aTj aTk aTl

=: |W|1

ai,i ai,j ai,k ai,l

ai,j aj,j aj,k aj,l

ai,k aj,k ak,k ak,l

ai,l aj,l ak,l al,l

m1,1 m1,2 m1,3 m1,4

m1,2 m2,2 m2,3 m2,4

m1,3 m2,3 m3,3 m3,4

m1,4 m2,4 m3,4 m4,4

.

|A{i,j}| = (1)i+j|W|1

ai,j aj,k aj,lai,k ak,k ak,l

ai,l ak,l al,l

a

j

akal

W

aTi aTk aTl


93/163

= (1)i+j

|W|1

ai,j aj,k aj,l

ai,k ak,k ak,lai,l ak,l al,l

m1,2 m2,3 m2,4

m1,3 m3,3 m3,4m1,4 m3,4 m4,4

,

j

A{i,j} (1)j1 (j 1)

i

A{i,j} A{i,j} (i 1)

k

l

A{i,j}

k

l

(1)i+j2 = (1)i+j

|A{i,j}||A{k,l}| |A{i,l}||A{j,k}| = (1)i+j+k+l|W|2

ai,j aj,k aj,lai,k ak,k ak,l

ai,l ak,l al,l

m1,2 m2,3 m2,4m1,3 m3,3 m3,4

m1,4 m3,4 m4,4

ai,i ai,j ai,kai,j aj,j aj,k

ai,l aj,l ak,l

m1,1 m1,2 m1,3m1,2 m2,2 m2,3

m1,4 m2,4 m3,4

ai,j aj,j aj,kai,k aj,k ak,k

ai,l aj,l ak,l

m1,2 m2,2 m2,3m1,3 m2,3 m3,3m1,4 m2,4 m3,4

ai,i ai,j ai,lai,k aj,k ak,l

ai,l aj,l al,l

m1,1 m1,2 m1,4m1,3 m2,3 m3,4

m1,4 m2,4 m4,4

.

3 3

|A{i,j}||

A{k,l}| |

A{i,l}||

A{j,k}|

= (1)i+j+k+l|W|2

ai,j aj,k

ai,l ak,l

m1,2 m2,3

m1,4 m3,4

ai,i ai,j ai,k ai,l

ai,j aj,j aj,k aj,l

ai,k aj,k ak,k ak,l

ai,l aj,l ak,l al,l

m1,1 m1,2 m1,3 m1,4

m1,2 m2,2 m2,3 m2,4

m1,3 m2,3 m3,3 m3,4

m1,4 m2,4 m3,4 m4,4

= |A{ik,jl}||A|,


94/163

|A{ik,jl}| = (1)i+j+k+l

aj,i aj,kal,i al,k

ajal

aTi aTk A{ijkl,ijkl}

= (1)i+j+k+l|W|1

ai,j aj,k

ai,l ak,l

ajal

W

aTi aTk

= (

1)i+j+k+l

|W

|1 ai,j aj,kai,l ak,l

m1,2 m2,3

m1,4 m3,4 .

s1(j) s2(j)

s:=p

j=1

{j}

TA(jj)

{j}|ZTZ| T Mconst

|Z{j}TZ{j}|

.

p= 2

s= 22|ZTZ| |Z{1}TZ{1}|2

s=1

2r=1

(1)r+srs|Z{r}TZ{s}| +21|ZTZ|

|Z{2}TZ{2}|2

s=1

2r=1

(1)r+srs|Z{r}TZ{s}|.

2

s=12

r=1(1)r+srs|Z{r}TZ{s}| =21|Z{1}TZ{1}| 212|Z{1}TZ{2}|

+22|Z{2}TZ{2}|

s= 22|ZTZ| |Z{1}TZ{1}||Z{2}TZ{2}|

+21|ZTZ| |Z{1}TZ{1}||Z{2}TZ{2}| + 212|Z{1}TZ{1}||Z{1}TZ{2}|

+ 212|Z{2}TZ{2}||Z{1}TZ{2}| 21|Z{1}TZ{1}|2 22|Z{2}TZ{2}|2.

|ZT

Z| |Z{1}T

Z{1}||Z{2}T

Z{2}| = |Z{1}T

Z{2}|2


95/163

s= 21|Z{1}TZ{1}|2 212|Z{1}TZ{1}||Z{1}TZ{2}| +22|Z{1}TZ{2}|2 21|Z{1}TZ{2}|2 212|Z{2}TZ{2}||Z{1}TZ{2}| +22|Z{2}TZ{2}|2

= 2

j=1

2

r=1

(1)rr|Z{j}TZ{r}|2

.

p= 3

s

s=p

j=1

|ZTZ|

p

s=1s=jp

r=1r=j(1)r+srs|Z{jr}TZ{js}|

|Z{j}TZ{j}|p

s=1

pr=1

(1)r+srs|Z{r}TZ{s}|

=

pj=1

|ZTZ| p

s=1s=j

pr=1r=j

(1)r+srs|Z{jr}TZ{js}| |Z{j}TZ{j}|

2j |Z{j}TZ{j}| + 2pr=1

r=j

(1)r+jrj|Z{j}TZ{r}|

+

ps=1s=j

pr=1r=j

(1)r+srs|Z{r}TZ{s}|

=

pj=1

2j |Z{j}TZ{j}|2 2|Z{j}TZ{j}| p

r=1r=j

(1)r+j rj|Z{j}TZ{r}|

+

ps=1

s=j

pr=1

r=j

(1)r+srs|ZTZ||Z{jr}TZ{js}| |Z{j}TZ{j}||Z{r}TZ{s}|

.

i= j

k= r

l= s

|ZTZ||Z{jr}TZ{js}| |Z{j}TZ{j}||Z{r}TZ{s}| = |Z{j}TZ{r}||Z{j}TZ{s}|


96/163

s=

p

j=1

2j |Z{j}TZ{j}|2 2|Z{j}TZ{j}|p

r=1r=j

(1)r+j

rj |Z{j}T

Z{r}|

ps=1s=j

pr=1r=j

(1)r+srs|Z{j}TZ{r}||Z{j}TZ{s}|

= p

j=1

ps=1

pr=1

(1)r+srs|Z{j}TZ{r}||Z{j}TZ{s}|

= p

j=1 p

r=1(1)rr|Z{j}TZ{r}|

2

,

Z p 2p

j=1

s1(j) =

pj=1

{j}

TA(jj)

{j}|ZTZ| T Mconst

|Z{j}TZ{j}|

+

pr=1

(1)rr|Z{j}TZ{r}|2= 0

pj=1

s2(j) =

pj=1

{j}TA(jj){j}|ZTZ| T Mconst|Z{j}TZ{j}|

pr=1

(1)rr|Z{j}TZ{r}|2

= 2p

j=1

pr=1

(1)rr|Z{j}TZ{r}|2

.

MSE()

MSE()

= 4n2

pj=1

|ZTZ| pr=1(1)rr|Z{j}TZ{r}|2 (n2T M

const + |ZTZ|)3

+43|ZTZ|TT( Mconst)2T

(n2T Mconst

+ |ZTZ|)3.

pj=1

var(j) = 4n2p

j=1


const + |ZTZ|)3


97/163

BiasT()Bias()

= 4

3|ZTZ|TT( Mconst)2T(n2T M

const + |ZTZ|)3

.

Z Rnp, n > p, p 2

= 0 > 0 [, ]\{0} : MSE()< MSE().

Z

[, ]

[, ]

MSE()

= 4n2

pj=1


const + |ZTZ|)3

+ O(3)

(n2T Mconst

+ |ZTZ|)3.

|ZTZ| > 0 pj=1 pr=1(1)rr|Z{j}TZ{r}|2 > 0 MSE()

= 0

MSE()

>0, [, 0)0, = 0


98/163

Z

T

Z

Z

T

Z

1= . . .= p= 0

= 0 p

j=1

pr=1

(1)rr|Z{j}TZ{r}|2

>0

= 0

1= 0

2,3= npj=1pr=1(1)rr|Z{j}TZ{r}|2

TT( Mconst

)2T.

Z

limMSE() =

2

pj=1

T{j} A

(jj){j}

T Mconst

+

TT( Mconst

)2T

n2 T Mconst

2 .

1= 0 2= . . .= p= 0

pj=1

s2(j) = 221|ZTZ|p

j=1

|Z{1}TZ{j}|2

TT( Mconst

)2T= n24

1

2

1

p

j=1 |

Z{1}

TZ{j}|2 >0

2=

2

n2121

3=

2

n2121

.


99/163

MSE() =2

23

(22

2

1+ 1)2

+12(

441+221+ 1)

(22

2

1+ 1)2

+14

4(8 + 28221+ 49441)

(2221+ 1)2

= 1

12

142 + 62221+ 6

2441+ 24441+ 84

661+ 147881

(2221+ 1)

2 .

2 =21 = 1

2,3=

1

3= 0.5774.

MSE() = () + Bias()BiasT()


100/163

MtxMSE() =2(ZTZ+2T)1ZTZ(ZTZ+2T)1

+4(ZTZ+2T)1TTT(ZTZ+2T)1

= (ZTZ+2T)1

2ZTZ+4TTT

(ZTZ+2T)1.

:= MtxMSE() MtxMSE() 0

MtxMSE() =2(ZTZ)1

2

ZTZ1 (ZTZ+2T)1 2ZTZ

+4TTT

(ZTZ+2T)1 0.

2 ZTZ+2T Z

TZ1

ZTZ+2T 2ZTZ 4TTT 0

2

ZTZ+ 22T +4T

ZTZ1

T

2ZTZ 4TTT 0

2 22Ip 2TTT +24T ZTZ1 T 0.

0

22 n2

ps=1

pr=1

rsrs

0.

T

T =n

p

r=11r1r . . .p

r=11rpr

pr=1pr1r . . . pr=1prpr


101/163

T

TT= n2

pr=1ps=121rsrs . . .pr=1ps=11prsrs

pr=1

ps=11prsrs . . .

pr=1

ps=1

2prsrs

=n

pr=1

ps=1

rsrsT

.

22T 2TTT=

22 n2p

r=1p

s=1rsrs

T.

c:=

22 n2ps=1pr=1rsrs 0

pTTp 0,

pTcTp= cpTTp 0

p

Rp1

ZTZ

1 = 12 12

ZTZ

ZTZ

1=

VVT1

=V1VT =

1

2 VTT

1

2 VT

pT

cT +24T

ZTZ1

T

p

=cpTTp +24pT

12 VTT

T 1

2 VTT

p 0,

c 0

c 0

c


102/163

j = j

1

Sjj

12

, j= 1, . . . , p

0 = y p

j=1

j Xj .

0

0 = y 1 X1 . . . p Xp

j =

j

1

Sjj

12

, j= 1, . . . , p .

=+ QD1 R(p+1)1,

=

y, 0, . . . , 0T R(p+1)1,

Q=

X1 X2 . . . Xp1 0 . . . 0

0 1 . . . 0

0 0 . . . 1

R

(p+1)p

D=

S11

Spp

Rpp.

Bias() = E(QD1+) = QD1E() + E() .


103/163

E() =

0+1 X1+. . .+pXp

0

0

01

p

=

1 X1+. . .+p Xp

1

p

= Q{0}

T{0} = 1, . . . , p {0} = D1

Bias() =QD1E() QD1

=QD1Bias().

() = (QD1+) =QD

1()D1QT

MSE() = tr

QD1()D1QT

+ BiasT()D

1QTQD1Bias()

= tr

D1QTQD1()

+ BiasT()D1QTQD1Bias().

QTQ=

X21 + 1 X1X2 . . . X1 Xp

X1 X2 X22 + 1 . . .

X2 Xp

X1Xp X2Xp . . . X2

p + 1

=

X21 X1 X2 . . . X1 Xp

X1X2 X22 . . .

X2 Xp

X1Xp X2 Xp . . . X2

p

+ Ip

=

X1

Xp

X1, . . . , Xp + Ip=: XXT + Ip


104/163

D1 XXTD1 =

X21S11

X1

X2S11S22 . . .

X1

XpS11Spp

X1 X2S11

S22

X22S22

. . . X2 Xp

S22

Spp

X1 XpS11

Spp

X2 XpS22

Spp

. . .X2pSpp

,

MSE() = tr

D1( XXT + Ip)D1()

+ Bias()TD1( XXT + Ip)D1Bias()

= tr

D1 XXTD1()

+ tr

D2()

+ Bias()TD1 XXTD1Bias() + Bias()

TD2Bias().

{0}:=

1, . . . ,

p

{0}

pj=1

var(j) = tr

(D1)

= tr

D1()D1= tr D2() .

Q {0}

Bias({0}) =D

1Bias().

MSE({0}) = tr

D2()

+ Bias()

TD2Bias()

=

pj=1

var(j)

Sjj+ Bias()

TD2Bias()

MSE(0) = tr

D1 XXTD1()

+ Bias()TD1 XXTD1Bias()

=

pj=1

pi=1

Xi XjSii

Sjjcov(i ,

j)

+4TT M

constD1 XXTD1 M

const

T

(n2T Mconst

+ |ZTZ|)2.


105/163

X

= 0 >0 [, ]\{0} : MSE()< MSE().

{0}

MSE({0}) =

pj=1

var(

j)Sjj

+

BiasT()D

2Bias()

.

pj=1

var(j)

Sjj= 4n2

pj=1

|ZTZ| pr=1(1)rr|Z{j}TZ{r}|2 Sjj (n2

T Mconst

+ |ZTZ|)3

BiasT()D

2Bias()= 43|ZTZ|TT MconstD2 MconstT

(n2

T

M

const

+ |ZT

Z|)3

.

MSE(

{0})

=0

= 0.

[1, 1]\{0} 1

MSE({0}) = 4n2|ZTZ|

pj=1

pr=1(1)rr|Z{j}TZ{r}|

2Sjj (n2

T Mconst

+ |ZTZ|)3

+ O(3

)(n2T M

const + |ZTZ|)3

.

pr=1

(1)rr|Z{j}TZ{r}|2

>0

(n2T Mconst

+ |ZTZ|)3 >0 Z Sjj

Sjj >0


106/163


107/163

var(

0

) = 2p

j=1

p

i=1

(

1)i+j Xi Xj

SiiSjj

n2{i}TA

(ij){j}+ |Z{i}TZ{j}|

n2T Mconst + |ZTZ|

n2p

s=1

pr=1(1)r+srs|Z{i}TZ{r}||Z{j}TZ{s}|n2T M

const + |ZTZ|

2

=2p

j=1

pi=1

(1)i+j XiXjSii

Sjj

2n{i}TA

(ij){j}

n2T M

const + |ZTZ|

n2T Mconst

+ |ZTZ|2

2nT Mconst

n2{i}TA

(ij){j}+ |Z{i}TZ{j}|n2T Mconst + |ZTZ|2

2np

s=1

pr=1(1)r+srs|Z{i}TZ{r}||Z{j}TZ{s}|n2T M

const + |ZTZ|

2

+4n23T M

constp

s=1

pr=1(1)r+srs|Z{i}TZ{r}||Z{j}TZ{s}|

n2T Mconst

+ |ZTZ|3

= 2n2p

j=1p

i=1(1)i+j

XiXjSiiSjj

{i}TA(ij){j}|ZTZ| |Z{i}TZ{j}|T Mconst

n2T Mconst

+ |ZTZ|2

|ZTZ| n2T Mconst

n2T Mconst

+ |ZTZ|3

p

s=1p

r=1(1)r+srs|Z{i}TZ{r}||Z{j}TZ{s}|

.

[2, 2]\{0} 2


108/163

var(0) = 2n2

|ZTZ

|

p

j=1p

i=1(1)i+j

Xi Xj

SiiSjj{i}TA(ij){j}|ZTZ| |ZT{i}Z{j}|T Mconst

n2T Mconst

+ |ZTZ|3

p

s=1

pr=1(1)r+srs|Z{i}TZ{r}||Z{j}TZ{s}|

n2T Mconst

+ |ZTZ|3

+ O(3)

n2T Mconst + |ZTZ|3

Bias(0)

TBias(0) =43|ZTZ|TT MconstD1 XXTD1 MconstT

(n2T Mconst

+ |ZTZ|)3

= O(3)

n2T Mconst

+ |ZTZ|3 .

MSE(0)=0 = 0.

[2, 0)

(0, 2] MSE({0})

MSE() = 0

s+ :=

pj=1

pi=1

(1)i+j Xi Xj

Sii

Sjj

|ZTZ|{i}TA

(ij){j}

|Z{i}TZ{j}|T Mconst

.

p 3

s+ =

pj=1

pi=1

(1)i+j Xi Xj

Sii

Sjj

|ZTZ| p

s=1

s=j

pr=1

r=i

(1)r+srs|Z{ir}TZ{js}|

|Z{i}TZ{j}|p

s=1

pr=1

(1)r+srs|Z{r}TZ{s}|

.


109/163

p

s=1p

r=1(1)r+srs

|Z

{r

}

TZ

{s

}|= (

1)i+jij

|Z

{i

}

TZ

{j

}|+

ps=1s=j

(1)i+sis|Z{i}TZ{s}|

+

pr=1r=i

(1)r+jrj|Z{r}TZ{j}| +ps=1s=j

pr=1r=i

(1)r+srs|Z{r}TZ{s}|

s+

=

pj=1

pi=1

(1)i+j

XiXjSiiSjj (1)i+jij|Z{i}TZ{j}|2

|Z{i}TZ{j}|ps=1s=j

(1)i+sis|Z{i}TZ{s}|

|Z{i}TZ{j}|pr=1r=i

(1)r+jrj|Z{r}TZ{j}|

+

p

s=1s=j

p

r=1r=i

(

1)r+srs |Z

TZ

||Z{

ir

}

TZ{js

}| |Z{

i

}

TZ{j

}||Z{

r

}

TZ{

s

}| .

s+ =

pj=1

pi=1

(1)i+j XiXj

Sii

Sjj

(1)i+jij|Z{i}TZ{j}|2

|Z{i}TZ{j}|p

s=1s=j(1)i+sis|Z{i}TZ{s}|

|Z{i}TZ{j}|pr=1r=i

(1)r+jrj|Z{r}TZ{j}|

ps=1s=j

pr=1r=i

(1)r+srs|Z{i}TZ{s}||Z{j}TZ{r}|

= p

j=1

pi=1

(1)i+j XiXj

Sii

Sjj

ps=1

pr=1

(1)r+srs|Z{i}TZ{s}||Z{j}TZ{r}|


110/163

=

p

i=1p

s=1(1)i+s

Xi

Siis

|Z

{i

}

TZ

{s

}|

p

j=1p

r=1(1)j+r

XjSjj r|Z{j}TZ

{r

}|

=

ps=1

pi=1

(1)i+sXi

Siis|Z{i}TZ{s}|

2.

p= 2

s+ =2

j=1

2i=1

(1)i+j Xi Xj

Sii

Sjj

|ZTZ| 2

s=1s=j

2r=1r=i

rs

|Z{i}TZ{j}|2

s=1

2r=1

(1)r+srs|Z{r}TZ{s}| .

p= 2

2s=1

s=j

2r=1

r=i

rs|ZTZ| (1)r+s|Z{i}TZ{j}||Z{r}TZ{s}|

=2

s=1s=j

2r=1r=i

rs

(1)i+j+r+s|ZTZ| (1)r+s|Z{i}TZ{j}||Z{r}TZ{s}|

= 2

s=1s=j

2r=1r=i

(1)r+srs|Z{i}TZ{s}||Z{j}TZ{r}|.

s+

p = 2

pj=1

pi=1

ps=1

pr=1

(1)i+j+r+s XiXjSii

Sjjrs|Z{i}TZ{r}||Z{j}TZ{s}|

=

p

j=1p

i=1p

s=1p

r=1(1)i+j+r+s

Xi Xj

SiiSjj rs|Z{i}TZ

{s

}||Z{j

}TZ

{r

}|

=

pi=1

ps=1

(1)s+iXi

Siis|Z{i}TZ{s}|

pj=1

pr=1

(1)r+j Xj

Sjjr|Z{j}TZ{r}|

=

pi=1

ps=1

(1)i+sXi

Siis|Z{i}TZ{s}|

2


111/163

var(

0) = 4n2

|ZT

Z|ps=1

pi=1(1)i+s

XiSii

s|Z{i}TZ{s}|2

n2T M

const + |ZTZ|3

+ O(3)

n2T Mconst

+ |ZTZ|3

Z

MSE(0) = 4n2|ZTZ|

ps=1

pi=1(1)i+s

XiSii

s|Z{i}TZ{s}|2

n2T M

const + |ZTZ|

3

+ O(3)n2T M

const + |ZTZ|

3 .

t:=

ps=1

pi=1

(1)i+sXi

Siis|Z{i}TZ{s}|

2 0

t= 0

t > 0

2

MSE(

0)

[2, 0) (0, 2]

:=

min(1, 2) , t >01 , t= 0

Z


112/163

X

X=

1 x1,1 . . . x1,p

1 xn,1 . . . xn,p

Rn(p+1).

=

(X+)T (X+)1

(X+)T y.

= 2

M

quad

x +

M

lin

x +

M

const

x2TAx + 2b

Tx + |XTX| (X+

)T y

=2( M

quadx X

Ty + Mlinx

Ty) +( Mlinx X

Ty + Mconstx

Ty) + Mconstx X

Ty

2TAx + 2bTx + |XTX|

.

X

|X[j]TX[j]| = |XTX[j]| = 0, j = 2, . . . , p

|X[1]TX[1]| = |XTX[1]| = |XTX|.

bTx =|XTX| 0 . . . 0

R1p,

Ax=

|XTX| 0 . . . 00 0 . . . 0

0 0 . . . 0

Rpp.

|Mx| =2|XTX|21+ 2|XTX|1+ |XTX|= |XTX|(221+ 21+ 1).

|X{u}TX{v}[r]| = 0, v=r = 1

|X{u}[r]TX{v}[s]

|= 0, v

=s

= 1

u

=r

= 1.


113/163

b(uv)x T{v} = |X{u}TX{v}[1]| 0 . . . 0 {v}=1|X{u}TX{v}[1]|, v= 1,

b(u1)x

T{1} =|X{u}TX{1}[2]| . . . |X{u}TX{1}[p]|

{1}

=

pr=2

r|X{u}TX{1}[r]|,

T{1} A(11)x {1}= {1}

T|X{1}[2]

TX{1}[2]

| . . .

|X{1}[2]

TX{1}[p]

|

|X{1}[p]TX{1}[2]| . . . |X{1}[p]TX{1}[p]|

{1}

=

pr=2

2r |X{1}[r]TX{1}[r]| + 2p

s=3

s1r=2

rs|X{1}[r]TX{1}[s]|,

{u}TA

(uv)x {v}= {u}

T

|X{u}[1]TX{v}[1]| 0 . . . 00 0 . . . 0

0 0 . . . 0

{v}

=21|X{u}[1]TX{v}[1]|, u, v= 1,

{1}TA

(1v)x {v}= {1}

T

|X{1}[2]TX{v}[1]| 0 . . . 0|X{1}[3]TX{v}[1]| 0 . . . 0

|X{1}[p]TX{v}[1]| 0 . . . 0

{v}

=1

pr=2

r|X{1}[r]TX{v}[1]|, v= 1,

{u}TA

(u1)x {1}= {u}

T

|X{

u

}[1]

TX{1

}[2]

| . . .

|X{

u

}[1]

TX{1

}[p]

|0 . . . 0

0 . . . 0

{1}

=1

pr=2

r|X{u}[1]TX{1}[r]|, u = 1.


114/163

X

u = 1M

quadx X

T + Mlinx

T1up1vn

=21

M

constx X

T1up1vn

,

Mlinx X

T + Mconstx

T1up1vn

= 21

M

constx X

T1up1vn

.

(u, v)

Mquad

x XT

Rpn

u = 1

Mquadx X

T(u, v) =

pr=1

(1)u+rxv,r{u}TA(ur)x {r}

= (1)u+1xv,1{u}TA(u1)x {1}+

pr=2

(1)u+rxv,r{u}TA(ur)x {r}

= (1)u+11xv,1p

r=2

r|X{u}[1]TX{1}[r]|

+21p

r=2

(1)u+rxv,r|X{u}[1]TX{r}[1]|.

Mquadx X

T(u, v) = (1)u+11p

r=2

r|X{u}TX{1}[r]|

+21

pr=2

(1)u+rxv,r|X{u}TX{r}|

=1

pr=2

(1)u+r+1r|X{u}TX{r}| +21p

r=2

Date post:	04-Jun-2018
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