Matricial Model

8/21/2019 Matricial Model

1/19

•

•

•

•


2/19

A =

2.1 1.74.0 −2.35.0 0.34.7 −1.5

B =

1.0 .32 .55.32 1.0 .61

.55 .61 1.0

D =

−1 0 00 2 0

0 0 4

C =

c11 c12c21 c22

c31 c32

A 4 × 2 B D 3 × 3 C 3 × 2 C ci j

B D B D

D

I

I =

1 0 00 1 0

0 0 1

0 =

0 0 00 0 0

A

A

A

A A = [aij ] A = [aji]

A A

A =

2.1 4.0 5.0 4.71.7 −2.3 0.3 −1.5

A AT 4 × 2 2 × 4

A

= A

a


3/19

b

c λ

a =

1.23

1.7

b = 1.1 .5 2 c = 22.6 α = .05

A B

A =

0 42 5

7 1

B =

−1 33 −2

5 0

A+ B =

−1 75 3

12 1

A− B =

1 1−1 7

2 1

a

b

ab =

2 4 1 3

1352

=

2 × 1 = 24 × 3 = 121 × 5 = 53 × 2 = 6

ab = 2 + 12 + 5 + 6 = 25

A B

A A

A

B AB AB


4/19

A

B

A

B

A

B

3 × 2 2 × 5 3 × 5 1 × 5 5 × 1 1 × 1 5 × 1 1 × 5 5 × 5

A = 0 42 57 1

B = 3 42 5

AB11 = (0 × 3) + (4 × 2) = 8AB12 = (0 × 4) + (4 × 5) = 20AB21 = (2 × 3) + (5 × 2) = 16

AB22 = (2 × 4) + (5 × 5) = 33AB31 = (7 × 3) + (1 × 2) = 23AB32 = (7 × 4) + (1 × 5) = 33

AB =

8 2016 33

23 33

W

X

Y

Z 4 × 2 2 × 3 3 × 7 7 × 5

WXYZ 4 × 5

n × 1 1 × n 1 × 4 4 × 1 1 × 1 1 × 4 4 × 1 4 × 4 AB A B

B A

A 0

A I A

A D

A

D

A

D

A

D

X XX XX


5/19

k

AkIA = kAIA = AIAk

b =

b1 b2

c =

20 10

A =

2 43 1

2b1 + 4b2 = 20

3b1 + b2 = 10

2 43 1

b1b2

=

2010

Ab = c

Ab = c

x 1/x x−1 a(1/x) = a/x

A A−1

A−1A

AA−1

I (1/a)a = 1

A =

2 43 1

A−1 =

−.1 .4.3 −.2


6/19

AA−1 = I

Ab = c A b A−1

Ab = c

A−1Ab = A−1c

Ib = A−1c

b = A−1c

Ab = c A−1c

−.1 .4.3 −.2

2010

=

24

b1 = 2 b2 = 4

AA−1

I

(A)−1 = (A−1)

(A−1)−1 = A

I−1 = I

a bc d


7/19

1/(ad − bc)

d −b−c a

a d b c 1/(ad − bc)

A

(ABCD) = DCBA

(ABCD)−1 = D−1C−1B−1A−1

5 × 4

X =

1 2 1 01 4 1 01 3 1 01 7 0 11 4 0 1

X c1 c2 · · · c4

r × c c λ1, λ2, · · · , λc

λ1c1 + λ2c2 + · · · + λccc = 0

λ X

λ


8/19

1, 0, −1, −1

X

n × 1 =

1 2 · · · n

E{} =

E{1} E{2} · · · E {n}

E{} = 0

σ2

{}

n × 1 n × n

σ2{} =

σ2{1} σ{1, 2} · · · σ{1, n}σ{2, 1} σ

2{2} · · · σ{2, n}

σ{n, 1} σ{n, 2} · · · σ2{n}


9/19

σ2

σ2{} =

σ2

0 · · · 00 σ2 · · · 0

0 0 · · · σ2

σ2{} = E {} = σ2I

yi = β 0 + β 1xi + i i = 1,...,n

i = 1,...,n

y1 = β 0 + β 1x1 + 1

y2 = β 0 + β 1x2 + 2

· · ·

yn = β 0 + β 1xn + n

y,X, β

y =

y1y2

yn

X =

1 x11 x2

1 xn

β =

β 0β 1

=

12

n

y,X n n

n

y = Xβ +

β 0


10/19

yi = β 0 + β 1xi,1 + β 2xi,2 + · · · + β p−1xi,p−1 + i i = 1,...,n

p − 1 x1 x p−1 x0 p

p − 1 p − 1 + 1 = p y

X =

1 x1,1 x1,2 · · · x1,p−11 x2,1 X 2,2 · · · x2,p−1

1 xn,1 xn,2 · · · xn,p−1

β =

β 0β 1β 2

β p−1

y = Xβ +

X

n × p β p × 1

• E{} = 0

σ2{} = E {} = σ2I

•

y

E{y} = Xb

y

σ2{y} = E {(y −Xb)(y − Xb)} = E {} = σ2I

β

Q =

ni=1

(Y i − b0 − b1X i,1 − b2X i,2 − ·· · − b p−1X i,p−1)2


11/19

b β b = b0 b1 · · · b p−1

XXb = Xy

XX Xy

p

p

b

XX

XXb = Xy

(XX)−1XXb = (XX)−1Xy

Ib = (XX)−1Xy

b = (XX)−1Xy

b = (XX)−1Xy

b (XX)−1X

b

ŷi ei H

Ŷ i Y i

ŷi = b0 + b1X i,1 + ... + b p−1X i,p−1 i = 1, · · · , n

n × 1 Ŷ Ŷ =

Ŷ 1 Ŷ 2 · · · Ŷ n

ŷ = Xb

b Xb

ŷ = X(XX)−1Xy

ŷ = Hy


12/19

H = X(XX)−1X

H H

• H n × n X

• ŷ y H

hii H i yi i ŷi

• H HH = H

H HH H

X

ei i

ei = yi − ŷi

n × 1 e

e

=

e1 e2 · · · en

e = y − ŷ

e = y − ŷ

e = y −Hy

e = (I −H)y

H

I− H n × n H(I− H) = 0

y

n − 2 n − p


13/19

SS R =

(ŷi − ȳ)2 p − 1 MSR = SSR/( p − 1)

SS E =

(yi − ŷi)2 n − p MSE = SSE/(n − p)

SSTO =

(yi − ȳ)2 n − 1 V ar(Y ) = SSTO/(n − 1)

(yi − ȳ) = (ŷi − ȳ) + (yi − ŷi)

yi ȳ

yi

ŷi −ŷi

SSTO = SSR + SS E

df

• SST O df = (n − 1) 1 df

ȳ

• SS R df = ( p − 1)

• SSE df = (n − p) p p df ŷi ei

S 2 y df,

SS R SSTO n × 1 u yuuy (

yi)

2 yuuy =

(yu)(uy) = (

yi)(

yi) = (

yi)2

yAy A


14/19

SS R [Hy − (1/n)uuy][Hy − (1/n)uuy] y[H− (1/n)uu]y p − 1SS E ee = (y −Hy)(y −Hy) y(I− H)y n − pSSTO [y − (1/n)uuy][y − (1/n)uuy] y[I− (1/n)uu]y n − 1

yAy A

yAy =

ni=1

aijyiyj

aij = aji yAy 1 × 1 yAy

yi 5y12 + 6y1y2 + 4y22 yAy

y =

y1 y2

A =

5 33 4

A n × n

n

χ2

• A y ∼ yAy A z2 χ2(1) 1 df A A

• A A A A

A n

i=1 aii

• SSR,SSE, SSTO p − 1 n − p n − 1 df

• k χ2(1)

1 df

χ

2

(k)

k df

• y ∼ SS R SS E SSTO χ2( p −1) χ2(n −

p) χ2(n − 1)


15/19

• df F (df 1, df 2) df 1 df 2 df F ∗ = MSR/MSE = (SSR/( p −1))/(SSE/(n − p)) F ( p − 1, n − p)

ŷi ei y

b ŷh y Xh e

•

• b ŷh e

y

Ay

A

y

•

Ay

σ2{Ay} = Aσ2{y}A y

b

σ2{b} b

b = (XX)−1Xy = Ay

σ2{b} = σ2{Ay}

= Aσ2{y}A

= (XX)−1Xσ2IX(XX)

−1

= σ2(XX)−1XX(XX)

−1

σ2{b} = σ2(XX)−1


16/19

b

σ2 MSE

s2{b} = M SE (XX)−1

p × p b

2{b0} 2{b1}

b

ŷh

x

h =

1 xh,1 xh,2 · · · xh,p−1

x

h

X xh

E{yh}

ŷh = x

hb

ŷh = x

hb b ŷh

σ2{ŷh} ŷh

ŷh = x

hb

σ2

{ŷh} = x

hσ2

{b}xh

= xhσ2(XX)

−1xh

= σ2xh(XX)

−1xh

ŷh

s2{ŷh} = M SE (x

h(XX)

−1xh)

ŷh

s{ŷh} =

MSE (xh(X

X)−1xh)

x

h(X

X)−1

xh


17/19

e

σ2

{e}

e

e = (I −H)y

σ2{e} = (I −H)σ2{y}(I−H)

= σ2(I− H)I(I− H)

= σ2(I− H)(I− H)

σ2{e} = σ2(I− H)

e

s2{e} = M SE (I −H)

n × n e

F ∗ = MSR/MSE

F ∗ = MSR/MSE F ( p − 1, n − p) ( p − 1) (n − p) df


18/19


19/19

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Matricial Model