REGRESI RIDGE
PROGRAM STUDI STATISTIKA PROGRAM PASCA SARJANA INSTITUT PERTANIAN BOGOR
1
Regresi Ridge
2006
REGRESI RIDGE Pendahuluan Dalam pendugaan parameer p!pula"# $me!de %la"#%&' #n(eren"# mengena# p!pula"# d#da"ar%an "epenu)n*a pada #n(!rma"# *ang d#per!le) dar# "ample a+a% *ang d#am,#l dar# p!pula"#- Penduga *ang ,a#% adala) penduga *ang a% ,#a"' dan d#anara penduga.penduga *ang a% ,#a"' penduga *ang mem,er#%an ragam m#n#mumla) *ang merupa%an penduga *ang e(#"#enDalam penel##an *ang mengguna%an regre"# l#near ,erganda' er%adang penel## lang"ung mela%u%an pendugaan er)adap %!e(#"#en regre"# unu% menemu%an m!del regre"#n*a- Semenara "ala) "au a"um"# *ang )aru" d#penu)# dalam regre"# l#near ,erganda adala) #da% adan*a %!rela"# anar /ar#a,le pred#%!r- J#%a erad# %!rela"# d#anara /ar#e,el pred#%!r $erad# mul#%!l#near&' ma%a m!del regre"# menad# #da%
epa
lag#'
%arena dengan adan*a
mul#%!l#near
#n#
a%an
menga%#,a%an ragamn*a menad# ,e"ar' dan n#la# "a#"#% %e+#l' "e)#ngga +enderung mener#ma 1 0Menuru M!ng!mer* Pe+%' ,e,erapa pen*e,a, mul#%!l#near anara la#n 3 4- Dalam pengumpulan daa' n#la# /ar#a,el pred#%!r *ang d#guna%an d#,aa"#2- Penenuan ,an*a%n*a /ar#a,el pred#+!r le,#) ,an*a% dar# pada ,an*a%n*a !,"er/a"#5- Daa #me "er#e"' d#mana n#la# rend *ang er+a%up dalam /ar#a,el regre"!r mempun*a#
#ng%a penurunan aau pen#ng%aan *ang
"ama' "ealan dengan a%u-
2
Regresi Ridge
7- Spe"#(#%a"# m!del' m#"aln*a penamm,a)an ,enu% p!l*n!m#al er)adap m!del regre"#' %)u"un*a %e#%a n#la# ara% anar /ar#a,el pred#%!r "anga %e+#l-
Beberapa indikasi adanya multikolinear 4- N#la# %!e(#"#en deerm#na"# R 2 #ngg#' eap# "#gn#(#%an"# "a#"#% u# dar# %!e(#"#en penduga parameer renda)2- N#la# %!e(#"#en deerm#na"# R 2 #ngg#' eap# %!e(#"#en %!rela"# par"#al renda)5- Unu% m!del regre"# l#near ,erganda 2 /ar#a,el pred#%!r 3 n#la# %!e(#"#en %!rela"# anara 2 /ar#a,el pred#%!rn*a #ngg#7- N#la# R 2 #ngg#' #nd#%a"# /ar#a,el pred#%!r %e. ,er%!rela"# #ngg# dengan "#"a /ar#a,el ,e,a" la#nn*a8- Tanda dar# %!e(#"#en %!rela"# $anara /ar#a,el re"p!n dengan /ar#a,el pred#%!r& ,erlaanan anda dengan anda dar# %!e(#"#en parameer regre"#6- N#la# 9I: *ang ,e"ar- M*er" ;4<<0= n#la# 9I: > 40 #nd#%a"# adan*a mul#%!l#nearBe"arn*a %!l#near#a" dapa d#u%ur dengan 9ar#an+e In(la#!n :a+!r $9I:&- 9I: a%an mengu%ur "e,erapa ,e"ar %ena#%an ragam dar# %!e(#"#en penduga regre"# d#,and#ng%an dengan /ar#a,el pred#+!r *ang !r)!g!nal #%a d#)u,ung%an "e+ara l#near $:!? dan M!nee' 4<<2&Sema%#n ,e"ar n#la# 9I: menunu%%an %!rela"# d#anara /ar#a,el pred#+!r #ngg#- N#la# 9I: > 40 menunu%%an adan*a adan*a %!l#near#a" $ Neer' @a"errman and Kuner' 4<<0&-
Regresi Ridge M!del regre"# l#near ,erganda
3
Regresi Ridge
y
=
X β + ε
Mar#%" ,eru%uran n?p' $"ela#n ,ar#" perama& ,ar#" %e.# men*aa%an n#la# pengamaan ? *ang men#m,ul%an re"p!n %e.#- /e+!r * men*aa%an re"p!n amaan %e.#- 9e%!r p!pula"#
dan
/e%!r ε
,eru%uran p?4 adala) /e+!r parameer
,eru%uran
n?4
merupa%an
pengamaan *ang mempun*a# "#(a E ( ε ) = 0 dan E ( ε ' ε ) =
/e+!r σ
gala
2
n
-
Per"amaan regre"# er"e,u mempun*a# pen*ele"a#an ∧
1 −
β = ( X ' X ) X ' Y
J#%a %!l#near#a" d#anara /ar#a,el pred#%!r %ua' ma%a elemen. elemen d#ag!nal mar#%" ( X ' X ) ,e"ar "e%al# dan mar#%"n*a menad# "#ngular-
Se)#ngga
pendugaan
dengan
me!de
%uadra
er%e+#l
meng)a"#l%an penduga %!e(#"#en regre"# *ang a% ,#a" eap# ragamn*a menad# ,e"ar- 1al #n# menga%#,a%an pendugaan %!e(#"#en regre"# menad# #da% a%ura lag#- Dalam penel##an "er#ng%al# "emua pred#+!r )aru" d##%u"era%an' d# "#"# la#n %!rela"# d#anara /ar#a,el pred#+!r "ul# d#)#ndar#Adan*a
mul#%!l#near
er"e,u
dapa
d#aa"#
dengan
menam,a)%an "eumla) ,#a" erenu "e)#ngga penduga ragamn*a dapa d#m#n#mum%an- Karena mar#%" ( X ' X ) "#mer#" dengan a%ar +#r# , 2 ,...., λ k ma%a erdapa mar#%" !r)!g!nal P "e)#ngga λ 1 λ
P ' ( X ' X ) P = P ( X ' X ) P ' = diag (λ 1 , λ 2 ,...., λ k )
Karena mar#%" P !r)!g!nal' ma%a per"amaan regre"# ,erganda dapa d#ul#"%an dalam ,enu% %an!n#%' y = XP ' P β + ε
aau y
=
*
X α + ε
4
Regresi Ridge
α α . = XP ' dan α = P β = . . α
0
1
Dengan X *
k
Penduga dar#
α
*
adala) α ∧
P .b * "e)#ngga
=
d#per!le) penduga regre"#
r#dge *a#u 3 ∧
b * = P '.α *
Unu% mem#n#mum%an umla) %uadra gala m!del %an!n#% ∧
y
=
X * .α * + e '
d#mana d j
d#am,a)%an $%4& pengal# lagrange *a#u ( d 0 , d 1 ,...., d k ) '
>0'
unu% j = 0,1,...., k Dengan me!de %uadra er%e+#l
d#per!le) ( A* + D).α *
*
= g
' *ang mem,er#%an pen*ele"a#an 3 ∧
α * = ( A* + D) −1 .g *
dengan A *
* X *' X ' dan g = X ' y -
=
1al er"e,u "ama ar#n*a dengan menam,a)%an %!n"ana erenu
pada
elemen.elemen
d#ag!nal
( X ' X ) '
dan
a%an
menga%#,a%an penduga %!e(#"#en regre"#n*a menad# ,#a"- D#"#"# la#n penam,a)an %!n"ana er"e,u a%an mem,ua mar#%" er"e,u "e!la). !la) !r)!g!nal- Elemen.elemen d#ag!nal ( A* + D ) −1 menad# le,#) %e+#l' "e)#ngga penduga %!e(#"#en regre"#n*a menad# le,#) "a,#lk
M!del regre"# a%an !p#mum #%a ∑ E ( j −0
#n# a%an d#penu)# #%a d j =
σ
*
β − β ) m#n#mum- 1al j
j
2 2
α j
' unu% j = 0,1,...., k - D#mana σ 2 d#duga
2
∧ *
dengan s ' dan α j d#duga dengan α j -
5
Regresi Ridge
Algoritma regresi ridge ∧
4- menenu%an α = ( X * ' X * ) 1 g * −
∧
2- Menenu%an σ = s 2
∧
5- Menenu%an d j
=
2
s
2
∧
α j2 ∧
7- "!lu"# per"amaan adala) α * ∧
8-
α
*
∧
'α
k
*
∧
=
( X
*
' X *
D ) .g * 1
−
+
2
= ∑ α j* j = 0
∧
6- Ulang# #era"# dar# lang%a) 5 "ampa# 8' dengan α j* pada lang%a) 7 ∧
∧
dan enu%an α * 'α * ∧
∧
- Iera"# d#la%u%an "ampa# d#per!le) %e"a,#lan α * 'α * ∧
- D#per!le) %!e(#"#en regre"# r#dge β *
Regre"# r#dge
∧
=
P ' α *
d#la%u%an dengan uuan memper%e+#l ragam dar#
penduga %!e(#"#en regre"#' alaupun penduga *ang d#per!le) ,er,#a"Penduga regre"# r#dge dapa d#per!le) dengan mem#n#mum%an umla) %uadra gala dar# m!del ∧
y k
dengan
∑
∧
β *
2
= ρ '
0
< ρ < ∞
X . β * +e
=
- Dengan me!de pengal# angrange' 2
∧ ∧ ∧ ∧ k L = ∑ y i − β − β x i − .... − β k xki c β ρ + − ∑ j = j 2
*
*
0
1
*
*
1
0
∧
∧
∧
d#urun%an er)adap β 0 , β 1 ,...., β k dan d#"ama%an dengan n!l ma%a d#per!le)
( X ' X
∧
+
cI ) β *
=
X ' y
Dan penduga %!e(#"#en regre"# r#dge adala)
6
Regresi Ridge
∧
β *
=
( X ' X
+
cI ) X ' y 1
−
Pendugaan %!e(#"#en regre"# r#dge d#mula# dar#
c = 0
' "ampa# d#per!le)
n#la# + *ang mem,er#%an "emua %!e(#"#en regre"# *ang "a,#l- Dalam menenu%an n#la# + *ang mana *ang mem,er#%an n#la# %!e(#"#en regre"# *ang "a,#l' dapa d#la%u%an dengan menggam,ar%an gra(#% n#la#.n#la# %!e(#"#en regre"# dengan eapan + padanann*a' $d#"e,u ea% r#dge&Ta)apan.a)apan dalam runu regre"# adala) "e,aga# ,er#%u 3 4- 9ar#a,el pred#%!r dan /ar#a,el re"p!n d#ran"(!rma"# pem,a%uan menad# /ar#a,el F dan *2- Meng)#ung r xx
=
Z ' Z *ang merupa%an mar#%" %!rela"# dar# /ar#a,el
pred#%!r5- Meng)#ung r xy
Z ' y
=
' *ang merupa%an mar#%" %!rela"# /ar#a,el
pred#+!r er)adap re"p!n7- meng)#ung penduga parameer
*
β unu%
,er,aga# eapan +'
$d#mula# dar# + H0&8- Meng)#ung n#la#
∧ β dan s 2 dar# ,er,aga# eapan +
VIF
k
6- Menggam,ar%an ea% r#dge dengan ,er,aga# eapan +- Meneap%an n#la# eapan ,#a" + dengan memper#m,ang%an n#la# 9I: "era pl! ea% r#dge- Menenu%an penduga %!e(#"#en regre"# r#dge dar# eapan + *ang mem,er#%an pendugaan *ang "a,#l-
Aplikasi regresi ridge Ber#%u adala) daa "ur/e* e%!n!m# d# Pa%#"an a)un 2000.2004 Y
X1
X2
X3
X4
X5
20.30
19.55
0.2671
3286
68.924
22.2
20.08
19.82
0.1166
3248
71.033
22.5
21.89
19.76
0.1178
3373
73.205
22.8
22.73
21.10
0.0779
3676
75.444
23.2
23.62
19.98
0.0663
3715
77.516
23.4
24.15
20.23
0.1072
3750
80.130
23.7
24.70
20.30
0.1237
3815
82.580
24.0
7
Regresi Ridge
25.27
20.42
0.1000
3882
84.254
26.2
25.85
20.31
0.0448
3931
87.758
26.5
26.40
20.33
0.0836
4047
90.480
26.9
26.96
20.61
0.0746
4423
93.286
27.2
27.93
20.67
0.0483
4349
96.180
27.5
28.70
21.92
0.0387
4544
99.162
27.9
28.99
20.66
0.3884
4573
102.230
28.0
29.99
20.73
0.3087
4595
105.409
28.1
30.82
20.73
0.3854
4543
108.678
28.3
31.78
20.77
0.3886
4589
111.938
28.6
31.78
20.96
0.2910
4656
111.938
34.9
31.94
21.06
0.4112
4849
113.610
36.0
32.45
21.40
0.2129
4809
116.470
37.2
33.29
21.51
0.6121
4852
119.390
38.4
33.60
21.55
0.4291
4998
122.361
39.6
34.42
21.68
0.1231
5072
125.387
40.9
36.84
21.98
0.5120
4992
128.421
42.2
37.73
21.96
0.4001
4924
131.510
43.6
38.59
21.93
0.4014
4992
134.511
45.0
40.40
21.99
0.4423
5081
137.512
47.1
41.20
21.99
0.4328
5128
140.473
52.0
Sum,er 3 G-R- Pa")a and Mu)ammad A%,ar Al# S)a) (2004) Mul#+!ll#near Daa '
Research'
Appl#+a#!n !( R#dge Regre""#!n !
48' <.406-
Keterangan : H umla) pe%era $ua& 4 H lua" ana) *ang d#d#r#%an ,angunan $ ua )e%ar& 2 H #ng%a #n(la"# $& 5 H umla) ,angunan 7 H umla) pendudu% $ua& 8 H #ng%a l#era"# $&
Dengan pr!gram m#n#a,' dapa d#per!le) 3 %!rela"# anar /ar#a,el pred#+!r' per"amaan regre"# dan an!/a dar# daa er"e,u "e,aga# ,er#%u 3
8
Regresi Ridge
Correlations: Y X1 X2 X3 X4 X5
Y
X1
X2
X3
X4
X1
0.892
0.000
X2
0.712
0.544
0.000
0.003
0.944
0.883
0.659
0.000
0.000
0.000
0.993
0.890
0.729
0.963
0.000
0.000
0.000
0.000
0.957
0.870
0.681
0.867
0.951
0.000
0.000
0.000
0.000
0.000
X3
X4
X5
Cell Contents: Pearson correlation
P-Value
Re!ression "nal#sis: Y $ers%s X1 X2 X3 X4 X5 The regression equation is
Y = - 1.22 0.353 X1 - 1.16 X2 - 0.00201 X3 0.312 X4 0.0152 X5
Pre!ictor
Coe"
#$ Coe"
%
P
Constant
-1.217
7.119
-0.17
0.866
X1
0.3532
0.4211
0.84
0.411
6.2
X2
-1.162
1.197
-0.97
0.342
2.5
X3
-0.002007
0.001108
-1.81
0.084
26.1
X4
0.31157
0.04818
6.47
0.000
67.2
X5
0.01524
0.06405
0.24
0.814
18.5
# = 0.6700
(-#) = 99.0*
V&'
(-#)+a!, = 98.7*
Analysis of Variance #ource
'
##
/#
'
P
5
945.01
189.00
420.98
0.000
(esi!ual $rror
22
9.88
0.45
%otal
27
954.89
(eression
Unu% men+ar# per"amaan regre"# r#dge dapa d#guna%an pr!gram SAS "e,aga# ,er#%uL
SAS PROGRAM data ri!e
inut 1 2 3 4 5 car!s
9
Regresi Ridge
20.30 19.55 0.2671
3286
68.924
22.2
20.08 19.82 0.1166
3248
71.033
22.5
21.89 19.76 0.1178
3373
73.205
22.8
22.73 21.10 0.0779
3676
75.444
23.2
23.62 19.98 0.0663
3715
77.516
23.4
24.15 20.23 0.1072
3750
80.130
23.7
24.70 20.30 0.1237
3815
82.580
24.0
25.27 20.42 0.1000
3882
84.254
26.2
25.85 20.31 0.0448
3931
87.758
26.5
26.40 20.33 0.0836
4047
90.480
26.9
26.96 20.61 0.0746
4423
93.286
27.2
27.93 20.67 0.0483
4349
96.180
27.5
28.70 21.92 0.0387
4544
99.162
27.9
28.99 20.66 0.3884
4573
102.230
28.0
29.99 20.73 0.3087
4595
105.409
28.1
30.82 20.73 0.3854
4543
108.678
28.3
31.78 20.77 0.3886
4589
111.938
28.6
31.78 20.96 0.2910
4656
111.938
34.9
31.94 21.06 0.4112
4849
113.610
36.0
32.45 21.40 0.2129
4809
116.470
37.2
33.29 21.51 0.6121
4852
119.390
38.4
33.60 21.55 0.4291
4998
122.361
39.6
34.42 21.68 0.1231
5072
125.387
40.9
36.84 21.98 0.5120
4992
128.421
42.2
37.73 21.96 0.4001
4924
131.510
43.6
38.59 21.93 0.4014
4992
134.511
45.0
40.40 21.99 0.4423
5081
137.512
47.1
41.20 21.99 0.4328
5128
140.473
52.0
proc reg
o!el =1 2 3 4 5collin V&' run proc reg outest=te outst norint
o!el =1 2 3 4 5 ri!e=+0 to 0.8 0.01 outi" run
10
Regresi Ridge
title (i!e %race sol1 = =0.5 c=lac; sol2 =circle =0.5 c=re! sol3 =s)uare =0.5 c=reen sol4 =trianle =0.5 c=lue sol5 =lus =0.5 c=aenta leen!1 lael=none osition=+to center insi!e o!e=sare ais1 lael=+anle=90 (i!e Coe""icients proc gplot
<ere te=(&>$#%? lot +1 2 3 4 5@ri!e oerla leen!=leen!1 ais=ais1 re"=0 run proc print
<ere te=(&>$#%? <ere te=(&>$V&' ar ri!e 1 2 3 4 5 run quit
11
Regresi Ridge
1. 2
x1
x2
x3
x4
x5
1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0. 0 - 0 . 1 - 0 . 2 - 0 . 3 0. 0
0. 1
0. 2
0. 3 Ri d ge
0. 4 r egr es s i on
0. 5 c on t r o l
0. 6
0. 7
v al u e
1 The REG Procedure Model: MODEL1 Dependent Variable: y
u! of
Mean
D"
#uares
#uare
" Value
Pr $ "
Model
%
&'%()1*++
1+&())*%+
'*)(&+
,()))1
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**
&(+--).
)(''+&.
ource
12
Regresi Ridge
0. 8
/orrected Total
*-
Root ME
&%'(++&&'
)(.-))'
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)(&+&-
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*&(-*+%-
Ad R0#
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oeff Var
*(*%2+-
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tandard
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Variable
D"
Esti!ate
Error
t Value
Pr $ 3t3
4nflation
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Referensi Norman Draper dan Harry Smi!, Analisis Regresi Terapan, edi"i 2, #$ %ramedia #&"a'a ama a'ara, 1992.
#ro*. Dr. S&d+ana .-.,.S., Me!da "aisika, #ener/i $ar"io, and&n, 2002.
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Regresi Ridge
$!oma" #. yan, M!dern Regressi!n Me#!ds, o!n iey Son" n., anada, 1997. Draper N. ., Smi! H, 1981, Applied Regressi!n Analysis, Seond :diion, /y o!n iey Son", n. Neer, . a""erman, ., dan ;&ner, . H. 1997 M!del Linier Terapan (er+.) &r&"an Sai"i'a <#- #, oor. apoe yer", , m& Pel$ang dan "aisika $n$k Insiny$r dan Il%$&an, edi"i 4, $ and&n. N$- S-S =er"ion ei!
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Regresi Ridge
