統計基礎 19 10 - keio universityweb.sfc.keio.ac.jp/~maunz/dsa19/dsa19_10.pdf · u
TRANSCRIPT
•• X•••• !" #$% &$%• X• ' ("
•••
• f ), + l+ = -. + -0)
lv X
• vvl -. -0
l -. -0r X•
2 ..= - (- /
3A :(2 -(I 1 ( ( L
r-0 -.
12
)• n
n l• r lpS• n l l
nn
• r t
• Xl v
l
•)0 )"• v r+2 = -. + -0)02 + -")"2 + 12
v r
• + 3 )0, )", … , )5 6r 7 = 1,… , 6
+0 = -. + -0)00 +⋯+ -5)05 + 10+" = -. + -0)"0 +⋯+ -5)"5 + 1"
⋮+2 = -. + -0)20 +⋯+ -5)25 + 12
⋮+; = -. + -0);0 +⋯+ -5);5 + 1;
• l
< =
+0+"⋮+2⋮+;
, = =
1 )00 ⋯1⋮1
)"0⋮)20
⋯⋱⋯
⋮1
⋮);0
⋱⋯
)05)"5⋮)25⋮);5
, ? =
-.-0-"⋮-5
, @ =
101"⋮12⋮1;
• r
< = =? + @
-• R• R• RL• R• R 375 475• R X• R• R6• R
• @ = < − =? BB = < − =? C < − =? = <C< − 2?C=C=< + ?C=C=?
•EBE?
= −2=C< + 2=C=? = 0
• vvp
=C=? = =C<
• p =C= G0 p H?
H? = =C= G0=C<
)•
v x l
• = I Σ =~L I, Σ r=? + @~L =? + @, ?Σ?C
• @ 0 M" @~L 0, M"$x l
• v v p < =? M"$
< = =? + @~L =?, M"$
• ?Q @Q < pH?Q N<Q O
•H? = =C= G0=C<~L ?, M" =C= G0
•N< = =H? = = =C= G0=C< = P<~L =?, M"P
P = = =C= G0=C
•O = < − N< = $ − P <~L 0, M" $ − P
P
•
H?~L ?, M" =C= G0
• v
H? − ?
M" =C= G0~L 0, 1
• = ? M" n< r l
• 7 )2 = 1, )20, … , )25 +2r
Q +2|)2; ?, M" =1
2TM"U)Q −
+2 − )2? "
2M"
• o
•V +2|)2; ?, M" = )2?
•W +2|)2; ?, M" = M"
E yi | xi ;β,σ2( ) = xiβV yi | xi ;β,σ
2( ) =σ 2
• 7 l
Q +|=; ?, M" =X2Y0
;
Q +2|)2; ?, M"
=X2Y0
;1
2TM"U)Q −
+2 − )2? "
2M"
=1
2TM"
;
U)Q −∑2Y0; +2 − )2? "
2M"
•
ln Q +|=; ?, M" = ln X2Y0
;
Q +2|)2; ?, M"
= ln1
2TM"
;
U)Q −∑2Y0; +2 − )2? "
2M"
= −62ln2T −
62lnM" −
12M"
]2Y0
;+2 − )2? "
• v c
• vv
]2Y0
;+2 − )2? "
yi − xiβ( )2
i=1
n∑
-4 -2 0 2 4
-12
-10
-8-6
-4-2
Values of parameter
Log
likel
ihoo
d
山登り
対数尤度関数が最大となる点を点推定
• v
]2Y0
;+2 − )2? "
•H? =
∑2Y0; )2+2∑2Y0; )2
"
M" =1
6 − 3 + 1]2Y0
;
+2 − )2H?"
0 6 − (3 + 1)
• = ? M" n+ r
Q +|=; ?, M" =1
2TM"U)Q −
+ − =? "
2M"
• o
•V +|=; ?, M" = =?
•W +|=; ?, M" = M"
E y | X ;β,σ 2( ) = Xβ
• l
Q +|=; ?, M" =1
2TM"U)Q −
+ − =? "
2M"
=1
2TM"U)Q −
+ − =? C + − =?2M"
• l
ln Q +|=; ?, M" = ln1
2TM"U)Q −
+ − =? C + − =?2M"
= −62ln2T −
62lnM" −
+ − =? C + − =?2M"
• d
H? = =C= G0=C+
M" =1a+ − =H?
C+ − =H?
a = 6 − 3 + 1 Vn
• b2 cde % cd ce• f ) = )0,… , ); + = +0,… , +; l
), g+ v r•
cde =∑2Y0; ) − )2 g+ − +2
6 − 1•
cd =∑2Y0; ) − )2 "
6 − 1
ce =∑2Y0; g+ − +2 "
6 − 1
b =cdecdce
=∑2Y0; ) − )2 g+ − +2
∑2Y0; ) − )2 " ∑2Y0
; g+ − +2 "
• v px
• −1 ≤ b ≤ 1• b = 0 r• b = 1 r
• + cee cdd• v
cee =]2Y2
;+2 − g+ "
cdd =]2Y2
;)2 − ) "
• t + ceex cC
• vv ci = ⁄cde" cdd ckci l v r
ck = cee − ci =]2Y2
;+2 − +2 "
ci = cee − ck =]2Y2
;+2 − g+ "
• ceeR +
• ckR r l• ciR r
• ci cee !"
!" =cicee
= 1 −ckcee
• X!" l t• X
b" =cdecdce
"
=⁄cde" cddcee
=cicee
= !"
• Mk" ck l Mk" =lm;G5
• x !"x ! l
! = 1 −∑2G0; +2 − +2 "
∑2G0; +2 − g+ " = 1 −
ckcee
• n )0 )" f 3 = 2X rl )0
)" )0 )" fl n v p Y
l• 3 r o
25 v• xY l p
p Adj. !"X#$% &$%
Adj. !"
• X!" e t• l x x
Adj. !" l• X !" ck cee
x
Adj. !" = 1 −⁄ck 6 − 3 − 1⁄cee 6 − 1
= 1 −
∑2G0; +2 − +2 "
6 − 3 − 1∑2G0; +2 − g+ "
6 − 1
• vv 6 3
#$%
• #$%lnrs 3 l
#$% = −2lnrs + 23
• vv
lnrs = −62ln 2TM" −
+ − =H?C+ − =H?
2M"• M" M"
M" =+ − =H?
C+ − =H?
6
#$%
• x lnrs r
lnrs = −62ln 2T
+ − =H?C+ − =H?
6−62
• #$%
#$% = −2lnrs + 23
= 6 ln 2T∑2Y2; +2 − +2 "
6+ 1 + 23
#$%
• #$% 62T 6X ll
6 t ln∑2Y2; +2 − +2 "
6+ 23
• #$% l l t #$% ll
L &$%
• n
&$% = −2lnrs + 23 t ln 6
• x
&$% = 6 ln 2T∑2Y2; +2 − +2 "
6+ 1 + 23 t ln 6
• #$% &$% l lln
• U2 = +2 − +2X p l
• U2 l 7 X M"U2~L 0, M"
• ckck =]
2Y0
;+2 − +2 " =]
2Y0
;+2 − u-. − u-0)2 − ⋯− -5)25
"
• v 6 − 3 + 1 − 1 = 6 − 3vk"
vk" =∑2Y0; +2 − +2 "
6 − 3
• U2 = +2 − +2 f
•
]2Y0
;U2 =]
2Y0
;+2 − +2 = 0
•
]2Y0
;U2)2 =]
2Y0
;+2 − +2 )2 = 0
• 1 1~L 0, M"$ U
U~L 0, M" $ − = =C= G0=C
= =
1 )00 ⋯1⋮1
)"0⋮)20
⋯⋱⋯
⋮1
⋮);0
⋱⋯
)05)"5⋮)25⋮);5
• vv P = = =C= G0=C
-
• P 7 X ℎ22 l•
16< ℎ22 < 1
• 3
]2Y0
;ℎ22 = 3 + 1
• X lv r
• U2
Wyb U2 = M" 1 − ℎ22
• XxU2z l
U2z =
U2M
• U2z ℎ22 l
r
U2M 1 − ℎ22
• vv vk" p
M" = vk" =∑2Y0; U2
"
6 − 3 − 1
• L 0,1
• Wyb U2 = M" 1 − ℎ22 v p ℎ22 eU2 M" t
• v r 7 tvv n rl ln
• N< = P< v p
+2 = ℎ20+0 + ℎ2"+" +⋯+ ℎ22+2 ⋯+ ℎ2;+;
• ℎ22 rt 7 l +2 +2n rt
{•
H? = =C= G0=C<~L ?, M" =C= G0
• l
u-5 = ~L -5, M" =C= 55G0
• v
u-5 − -5M" =C= 55
G0 ~L 0, 1
{
• M" M" r n
u-5 − -5M" =C= 55
G0 ~{ 6 − 3 − 1
• vvp Y l v xP.: u-5 = 0 { n l
S• v Y
l l l
• P.: H? = 0 P.: u-0 = ⋯ = u-5 = 0 r l
• v ci vk"M"
• ci 3 ("
ci =]2Y2
;+2 − g+ "
S• +2 +2~L =2?, M"P22
• g+2 = g+ v p +2 x e}Gge~ �}}
+2 − g+
M P22~L 0, 1
• e}Gge~ �}}
"3 ("
• 3 + 1 n p v p3 + 1 − 1 = 3
("
• =2 I M" =2~L I, M" rÄ2 =
Å}GÇ~Ä2~L 0,1
• v rÄ2" (" (" 1 v l
Ä2" =
=2 − IM
"
~(" 1
• É = ∑2Y05 Ä2
" 3 (" (" 3
É =]2Y0
5Ä2" ~(" 3
0 5 10 15 20 25 30
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x
dchi
sq(x
, 1)
("
("
(" (" 1
(" 3
(" 5(" 10
(" 20
("
• 3 (" (" 3 Ü ); 3 0 ≤ )v r ) < 0 r
Ü ); 3 =1
25"Γ
32
UGd")
5"G0
• vv Γ à
• 3 (" l3o 23
'
• ci 3 vk"v '
' =⁄ci 3
vk"
• '
' =⁄ci 3vk"
= â∑2Y2; +2 − g+ "
3∑2Y0; U2
"
6 − 3 − 1
• v 3 (" 6 − 3 − 1 ("
' ("
• É0 30 (" (" 30 l É" 3" ("(" 3" l É0 É" l
• v r É0 É"⁄äã 5ã⁄äå 5å
'll 30, 3" ' ' 30, 3"
' =ڃ0 30ڃ" 3"
~' 30, 3"
'
• 30, 3" ' ' 30, 3" Ü ); 30, 3"v r
Ü ); 30, 3" =Γ30 + 3"
2 )5ãG""
Γ 302 Γ 3"
2 1 + 303")
5ãç5å"
303"
5ã"
' {
• L 0,1 Ä 3 (" (" 3É r p { 3 {
{ =Ä
⁄É 3• (
{" =Ä"
⁄É 3• Ä" (" (" 1 v p {" 1, 3' ' 1, 3
• cC ll6 − 1 (" cC cee X
cC =]2Y2
;+2 − g+ "
• ck 6 − 3 − 1 ("
ck =]2Y2
;+2 − +2 "
• v r cC ck ci
cC = ck + ci
• v r
]2Y2
;+2 − g+ " =]
2Y2
;+2 − +2 " +]
2Y2
;+2 − g+ "
6 − 1 6 − 3 − 1 3
• ci 3 vk" '3, 6 − 3 − 1 ' ' 3, 6 − 3 − 1
' =⁄ci 3
vk"= â∑2Y2; +2 − g+ "
3
∑2Y0; U2
"
6 − 3 − 1
• r P.: u-0 = ⋯ = u-5 = 0• + v 3, 6 − 3 −1 ' /+ 'é• ' 3, 6 − 3 − 1 > 'é r• ' 3, 6 − 3 − 1 ≤ 'é r l
• 'l
'
ci 3 ci ' =⁄ci 3vk"
ck 6 − 3 − 1 vk" =ck
6 − 3 − 1
cC 6 − 1
(• ( - / f j Y e
ep• B L Lp• Y• )02 Y )02 Y +2 Y l• Y .
(•
S• lS
•S
• v l
INC1
20 30 40 50
50100
150
200
2030
4050
CONS1
50 100 150 200 0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
WORK
•
/.+ ./.+ . /. . /
• {+ = 12.04 + 0.187)0 + 2.98)"
• u-0 = 0.19 > 0p u-0 { 2 ) . +• u-" = 2.98 > 0p u-" { 2( + +• !" = 0.977 l
• u-0 u-" x r nrl pS
.( ) .!" = 0.977
( +
•+ = 12.04 + 0.187)0 + 2.98)"
• u-0 u-" x r nrl pS
• )0 e n .- 2 .-☓ X n• )" e e n (/. 2( /.☓ X n• x lS
.( ) .!" = 0.977
( + {
• eR + = -. + -0)0• fR + = -. + -0)0 + -")"• R + = -. + -")"• #$% &$% x
Adj. !" #$% &$%
e / / -+ -. )f /-- - ) - /
-() / -
• l v lp p r
l
•• 9 9• 8• vX
Q• +2 U2
l• 2M n
tl
ra
• 7 = 1, 4, 13 20 30 40 50
−3−2
−10
12
Fitted valuesRe
sidua
lslm(CONS1 ~ INC1 + WORK)
Residuals vs Fitted
13
1
4
Q•
9 9• 9 9
r +t
+• f n
l
• 7 = 1, 13, 18−2 −1 0 1 2
−2−1
01
Theoretical Quantiles
Stan
dard
ized
resid
uals
lm(CONS1 ~ INC1 + WORK)
Normal Q−Q
1318
1
Q•
xl
• 2 nv
• 7 = 1, 13, 1820 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fitted valuesSt
anda
rdize
d re
sidua
ls
lm(CONS1 ~ INC1 + WORK)
Scale−Location13
181
• ℎ22 ⁄2 3 + 1 6
• î2 0.2 <î2 ≤ 0.5 Y 0.5 < î2
Y pxl
• 7 = 18 xt 7 = 1, 13
r 0.0 0.2 0.4 0.6
−2−1
01
Leverage
Stan
dard
ized
resid
uals
lm(CONS1 ~ INC1 + WORK)
Cook's distance
1
0.5
0.5
1
Residuals vs Leverage
181
13
• e x l
î2 =∑ïY0; +ï − +ï 2
"
Q t vk"=∑ïY0; +ï − +ï 2
"
3 + 1 t vk"
• vv +ï l +ï 27 x Q
rQ = 3 + 1X vk"
• l lRi X l
• Q
• ) ñdñd =
) − )vó )
• x l xl
• n +2
• (/ ++2 − 29.1110.05
ñe
• ñe Q
. / (( () ( ( (+ (- ) ) ) (/ )( ) + ++
+ - + + ( / / / (/ / . +. ( +.
•
• lx l v
p
−1 0 1 2 3
−10
12
z.kakei[, 4]
z.ka
kei[,
5]
X
(• ñd0 ñd"o ñe l
òñe = 0.00 + 0.853ñd0 + 0.162ñd"
• l+• rlp ln rl l• x x l s
x ot
) .!" = 0.977
( + {
•l
p x l•
INC1
20 30 40 50
50100
150
200
2030
4050
CONS1
50 100 150 200 0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
WORK/.+. . /
• X ll Xn
•
x x l pS• p x
rpS
/.+
.
. /
• v
• { t X• X rt X• rt• r
• B CF L C L 0 <76 l
• )0 )" Md0 Md"o Md0d"pb r
b =Md0d"Md0Md"
• b x b" l <76
W$' =1
1 − b"
• <76 r l( X <76 r p x
• )0 )" W$' =0
0G..ö0õå≈ 3.04
• v f ln l