matricial model
TRANSCRIPT
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•
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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
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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
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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
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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
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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
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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
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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}
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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
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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
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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
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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
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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
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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)
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• 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
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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
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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
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