axjack
April 03, 2022
1.2k

# 統計学実践ワークブック 第16章 重回帰分析 pp.125-127

April 03, 2022

## Transcript

1. ౷ܭֶ࣮ફϫʔΫϒοΫ
ୈষॏճؼ෼ੳ
QQ

2. ه๏

1

0
X′

͸ཁૉ͕͢΂ͯͰ͋ΔॎϕΫτϧ ྻϕΫτϧ

͸ཁૉ͕͢΂ͯͰ͋ΔॎϕΫτϧ ྻϕΫτϧ

͸ߦྻ9ͷసஔߦྻ
X
n×m
͸OߦNྻͷߦྻ

3. ճؼ෼ੳͷزԿֶతͳೝࣝ
Im(X)
Y
e = Y− ̂
Y
̂
Y = PX
Y = X ̂
β
ℝn
Im(X) = {Xv ∣ v ∈ ℝ1+d} ⊂ ℝn
X
n×(1+d)
̂
β
(1+d)×1
̂
Y
n×1
=
, e ⊥ ̂
Y
:Λ*N 9
ʹࣹӨʹͨ͠Β
ͱͳΔɻ
̂
Y
࢒ࠩϕΫτϧ e

4. 9ͷ૾ɺࣹӨɺ࢒ࠩ
Y − ̂
Y
͸
Im(X) ʹࣹӨ͢Δߦྻɺ͢ͳΘࣹͪӨߦྻΛ PX ͱ͢Δͱɺ
̂
Y = PX
Y Y Λ Im(X) ΁ࣹӨͨ͠ϕΫτϧͰ͋Δɻ
ͳͷͰɺ ̂
Y
ͱ ͸௚ަ͢Δɻ
̂
Y
ͱ͜ΖͰɺ ͸ Im(X) ʹଐ͢ΔϕΫτϧͰ΋͋Δ͔Βɺ
̂
Y = X ̂
β Λຬͨ͢Α͏ͳ ̂
β ∈ ℝ1+d ͕ଘࡏ͢Δɻ
X′

e = 0
࢒ࠩϕΫτϧ e ʹ͍ͭͯɺ
͕੒Γཱͭɻ F͸*N 9
ͷ௚ަิۭؒʹؚ·ΕΔͱ͍͏͜ͱ

e = Y − ̂
Y = Y − PX
Y = Y − X ̂
β
࢒ࠩϕΫτϧ e Λ
ͱͨ͠ͱ͖ɺ
Im(X) = {Xv ∣ v ∈ ℝ1+d} ⊂ ℝn
9ͷ૾Λ*N 9
ͱද͢ɻ*N 9
͸ू߹Ͱ͋ͬͯҎԼͷΑ͏ʹࣔ͞ΕΔɻ
Y − PX
Y ͱ
͕௚ަ͢ΔΑ͏ͳม׵ ʁ
͕
PX ͳͷͰ͋Δɻ
PX
Y
ͱ X

5. 9F
∣ ∣ ∣ ⋯ ∣

1 x1
x2
⋯ xd
∣ ∣ ∣ ⋯ ∣
X =
− ⃗
1 ′

−x′

1−
−x′

2−

−x′

d

X′

=
X′

e =
− ⃗
1 ′

−x′

1−
−x′

2−

−x′

d

e =

1 ′

e
x1

e

xd

e
=
0
0

0
= ⃗
0
n × (1 + d)
(1 + d) × n
(1 + d ) × n n × 1 (1 + d) × 1
9͸Eݸͷجఈ ॎϕΫτϧ
Λ
ԣʹฒ΂ͨ΋ͷͩͱߟ͑Δɻ
సஔ͢Δͱɺ
ɾ֤ॎϕΫτϧΛసஔ
ɾԣฒͼˠॎฒͼ
ͱͳΔɻ
جఈ͸*N 9
ʹؚ·Ε͍ͯͯɺ
*N 9
ͷ೚ҙͷཁૉ͸࢒ࠩϕΫτϧFͱ௚ަ͢
Δɻ ͦͷΑ͏ʹߏ੒ͨ͠΋ͷ͕࢒ࠩϕΫτϧ
Ͱ͋Δɺͱ΋ݴ͑Δɻ

ͷجఈ͸ɺ9ͷୈྻ ୈྻ ʜ ୈEྻ
Ͱ͋Δɻ
Im(X) = {Xv ∣ v ∈ ℝ1+d} ⊂ ℝn
ˠ

6. ࣹӨߦྻͷಋग़
X′

e = X′

(Y − ̂
Y) = X′

(Y − X ̂
β) = ⃗
0 ΑΓɺ
⟺ X′

Y = X′

X ̂
β ⟺ ̂
β = (X′

X)−1X′

Y
̂
Y = X ̂
β = X(X′

X)−1X′

Y
·ͨɺ
PX
= X(X′

X)−1X′

Ͱ͋Δ͔Β ̂
Y = PX
Y ͱൺֱͯ͠
X′

X͸Մٯͱ͢Δ
X′

e = ⃗
0 ΑΓɺ
ͱͳΔɻ

7. ࣹӨߦྻ1Yͷੑ࣭

PX
= X(X′

X)−1X′

ͱͯ͠ɺ
P2
X
= PX
PX
= (X(X′

X)−1X′

)(X(X′

X)−1X′

) = X(X′

X)−1(X′

X)(X′

X)−1X′

= X(X′

X)−1X′

= PX
P′

X
= (X(X′

X)−1X′

)′

= X′

((X′

X)−1)′

X′

= X(X′

X)−1X′

= PX
Im(X) = {Xv|v ∈ ℝ1+d}
Im(PX
) = {PX
w|w ∈ ℝn}
X ͸ n × (1 + d) ߦྻͳͷͰɺ
͸
PX n × n ߦྻ O࣍ਖ਼ํߦྻ
Ͱ͋Δɻ
ࣹӨߦྻͷೋ৐ͱసஔ͸࠶ͼࣹӨߦྻʹ໭Δ͜ͱΛ֬ೝ͢Δɻ
ͯ͞ɺ
ͱ
͸౳͍͠ͱͷ͜ͱͳͷͰɺ͜ΕΛࣔͯ͠ΈΔɻ
*N 1Y
΋*N 9
΋ͲͪΒ΋ू߹Ͱ͋Δ͜ͱΑΓɺ
l1
∈ Im(X) ⟹ l1
∈ Im(PX
)
l2
∈ Im(PX
) ⟹ l2
∈ Im(X)
Λࣔͤ͹ྑ͍ɻ

8. ࣹӨߦྻ1Yͷੑ࣭

[1]l1
∈ Im(X) ⟹ l1
∈ Im(PX
)
[2]l2
∈ Im(PX
) ⟹ l2
∈ Im(X)
l1
∈ Im(X)
Im(X) = {Xv|v ∈ ℝ1+d}
Im(PX
) = {PX
w|w ∈ ℝn}
v1
∈ ℝ1+d
l1
= Xv1
∈ ℝn
w1
= Xv1
∈ ℝn
l2
∈ Im(PX
) l2
= PX
w2
= X(X′

X)−1X′

w2
∈ ℝn
l1
= PX
w1
∈ Im(PX
)
l2
= X(X′

X)−1X′

w2
= Xv2
v2
= (X′

X)−1X′

w2
∈ ℝ1+d l2
= Xv2
∈ Im(X)
ͱͳΔl1
ΛऔΔͱɺ ͱͳΔ ͕औΕΔɻ
l1
= Xv1
= X(X′

X)−1(X′

X)v1
= PX
Xv1 ͱදͤ͹
ͱͯ͠
ͱͳΔl2ΛऔΔͱɺ
ͱͳΔw2
∈ ℝn ͕औΕΔɻ ͱදͤ͹
ͱͯ͠
Ҏ্ΑΓɺ
Im(PX
) = Im(X)

9. F
0 = ⃗
1 ′

e
Im(X)
Y
̂
Y = PX
Y = X ̂
β
ℝn
Im(X) = {Xv ∣ v ∈ ℝ1+d} ⊂ ℝn
e
e

1
ˠ
ͱ͸ɺ

1 ͱe͕௚ަ͢Δɺͱݴ͍ͬͯΔɻ

10. 0 = ⃗
1 ′

e = ⃗
1 ′

(Y − X ̂
β) = n(¯
y − (1,¯
x′

) ̂
β)ʹ͍ͭͯ෼ղͯ͠ߟ͑Δɻ

1 ′

Y =
(
1

1
)

y1

yn
= ∑ yi
= n¯
y

1 ′

X =
(
1

1
)

1 , −x′

1

⋮ , ⋮
1 , −x′

n

= (n, ∑ x′

i) = (n, n ¯
x′

)
0 = ⃗
1 ′

e = ⃗
1 ′

(Y − X ̂
β) = (n¯
y − (n, n ¯
x′

) ̂
β) = n(¯
y − (1, ¯
x′

) ̂
β)
͜ͷΧϯϚ͸ྻΛ۠෼͚͢ΔͨΊͷه߸ɻ
಺ੵͰ͸ͳ͍ɻ
−x′

i
− = (xi,1
⋯ xi,d
)
͸ԣϕΫτϧ
Ҏ্ΑΓɺ
ͱͳͬͯɺ ¯
y = (1, ¯
x′

) ̂
β ΛಘΔɻ
¯
y
ԣϕΫτϧͨͪͷ࿨͸
ରԠ͢Δ੒෼͝ͱʹ࿨ΛऔΔ

11. QԼஈ
Y = X ̂
β + e Y − ⃗
1 ¯
y = X ̂
β − ⃗
1 ¯
y + e
ΑΓ
Y − ⃗
1 ¯
y = X ̂
β − ⃗
1 (1,¯
x′

) ̂
β + e
¯
y = (1, ¯
x′

) ̂
β
͞Βʹ ΑΓ
= (X − ⃗
1 (1,¯
x′

))
̂
β + e
= (
1 x1

⋮ ⋮
1 xn

− ⃗
1 (1,¯
x′

))
̂
β + e
=
1 − 1 x1

− x′

⋮ ⋮
1 − 1 xn

− x′

̂
β + e
=
0 (x1
− ¯
x)′

⋮ ⋮
0 (xn
− ¯
x)′

̂
β + e

12. Qதஈ
Im(X)
Y− ⃗
1 ¯
y
X ̂
β − ⃗
1 ¯
y
ℝn
Im(X) = {Xv ∣ v ∈ ℝ1+d} ⊂ ℝn
e
Y
X ̂
β
e

1 ¯
y͚ͩͣΒ͢
͜ͷɹɹ্ͷࡾ֯ܗͰ
ࡾฏํͷఆཧΛ͢Ε͹ྑ͍ɻ
ℝn

13. Qதஈ
Y − ⃗
1 ¯
y = X ̂
β − ⃗
1 ¯
y + e
⟺ Y − ⃗
1 ¯
y = X ̂
β − ⃗
1 (1,¯
x′

) ̂
β + e
ΑΓɺ ∥Y − ⃗
1 ¯
y∥2 = ∥X ̂
β − ⃗
1 (1,¯
x′

) ̂
β∥2 + ∥e∥2
⟺ ∑ (yi
− ¯
y)2 = ∑ ((xi
− ¯
x)′

̂
β1:d)
2
+ ∑ e2
i
ͱͳΔɻ
̂
β0
܎਺
஫໨ϙΠϯτ
͸ɺ
͕ফ͑Δ
ͷฏۉ͔Βͷภࠩ
xi
xi
− ¯
x ͕ग़ݱ͢Δ

1 ¯
y Λ࢖ͬͨ͜ͱͰɺ
͓ೃછΈ
ͷมಈͷ෼ղͷ͕ࣜग़ݱ͢Δ
͜ͱͰ͋Δɻ