contoh makalah regresi ridge

7/22/2019 Contoh Makalah Regresi Ridge

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REGRESI RIDGE

PROGRAM STUDI STATISTIKA

PROGRAM PASCA SARJANA

INSTITUT PERTANIAN BOGOR

Regresi Ridge1


2/16

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(#"#en-

Dalam 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 10-

Menuru 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-

Regresi Ridge2


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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"# R2#ngg#' eap# "#gn#(#%an"# "a#"#% u#

dar# %!e(#"#en penduga parameer renda)-

2- N#la# %!e(#"#en deerm#na"# R2 #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# R2 #ngg#' #nd#%a"# /ar#a,el pred#%!r %e. ,er%!rela"# #ngg#

dengan "#"a /ar#a,el ,e,a" la#nn*a-

8- 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


4/16

+= Xy

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 ,eru%uran p?4 adala) /e+!r parameer

p!pula"# dan /e%!r ,eru%uran n?4 merupa%an /e+!r gala

pengamaan *ang mempun*a# "#(a ( ) 0=E dan ( )n

E2

' = -

Per"amaan regre"# er"e,u mempun*a# pen*ele"a#an

( ) YXXX '' 1

=

J#%a %!l#near#a" d#anara /ar#a,el pred#%!r %ua' ma%a elemen.

elemen d#ag!nal mar#%" ( )XX' ,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#%" ( )XX' "#mer#" dengan a%ar +#r#

k ,....,, 21 ma%a erdapa mar#%" !r)!g!nal P "e)#ngga

),....,,(')'()'( 21'

kdiagPXXPPXXP ==

Karena mar#%" P !r)!g!nal' ma%a per"amaan regre"# ,ergandadapa d#ul#"%an dalam ,enu% %an!n#%'

+= PXPy '

aau

+= *Xy

Regresi Ridge4


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Dengan '* XPX = dan

==

k

P

.

.

.

1

0

Penduga dar# adala) **

.bP=

"e)#ngga d#per!le) penduga regre"#

r#dge *a#u 3

= ** '.Pb

Unu% mem#n#mum%an umla) %uadra gala m!del %an!n#%

eXy +=

** . ' d#am,a)%an $%4& pengal# lagrange *a#u ( )kddd ,....,, 10 '

d#mana 0>jd ' unu% kj ,....,1,0= Dengan me!de %uadra er%e+#l

d#per!le) *** ).( gDA =+ ' *ang mem,er#%an pen*ele"a#an 3

*1** .)( gDA

+=

dengan XXA '** = ' dan yXg '*

= -

1al er"e,u "ama ar#n*a dengan menam,a)%an %!n"ana

erenu pada elemen.elemen d#ag!nal ( )XX' ' 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 1* )( +DA menad# le,#)

%e+#l' "e)#ngga penduga %!e(#"#en regre"#n*a menad# le,#) "a,#l-

M!del regre"# a%an !p#mum #%a

k

j

jj

E

0

*

)( m#n#mum- 1al

#n# a%an d#penu)# #%a

2

2

j

jd = ' unu% kj ,....,1,0= - D#mana 2 d#duga

dengan s ' dan 2

jd#duga dengan

*

j

-

Regresi Ridge5


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Algoritma regresi ridge

4- menenu%an ( ) *1** ' gXX

=

2- Menenu%an 22 s=

5- Menenu%an =

2

2

j

j

sd

7- "!lu"# per"amaan adala) ( ) *1*** .' gDXX

+=

8- =

=k

j

j

0

2***

'

6- Ulang# #era"# dar# lang%a) 5 "ampa# 8' dengan

*

j pada lang%a) 7

dan enu%an** '

- Iera"# d#la%u%an "ampa# d#per!le) %e"a,#lan**

'

- D#per!le) %!e(#"#en regre"# r#dge

= ** ' P

Regre"# r#dge 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

eXy +=

*.

dengan = 2

*

k

'


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( ) yXcIXX '' 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 ZZrxx '= *ang merupa%an mar#%" %!rela"# dar# /ar#a,el

pred#%!r-

5- Meng)#ung yZrxy '= ' *ang merupa%an mar#%" %!rela"# /ar#a,el

pred#+!r er)adap re"p!n-

7- meng)#ung penduga parameer *unu% ,er,aga# eapan +'

$d#mula# dar# + H0&-

8- Meng)#ung n#la#

k

VIF

dan2

s dar# ,er,aga# eapan +-

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

Regresi Ridge7


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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) Appl#+a#!n !( R#dge Regre""#!n !

Mul#+!ll#near Daa ' Research' 48'


9/16

Correlations: Y X1 X2 X3 X4 X5

Y X1 X2 X3 X4X1 0.892

0.000

X2 0.712 0.544

0.000 0.003

X3 0.944 0.883 0.659

0.000 0.000 0.000

X4 0.993 0.890 0.729 0.963

0.000 0.000 0.000 0.000

X5 0.957 0.870 0.681 0.867 0.951

0.000 0.000 0.000 0.000 0.000

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 V&'

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* (-#)+a!, = 98.7*

Analysis of Variance

#ource ' ## /# ' P

(eression 5 945.01 189.00 420.98 0.000

(esi!ual $rror 22 9.88 0.45

%otal 27 954.89

Unu% men+ar# per"amaan regre"# r#dge dapa d#guna%an pr!gram SAS

"e,aga# ,er#%uL

SAS PROGRAM

datari!e

inut 1 2 3 4 5

car!s

Regresi Ridge9


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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 22.822.73 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

procreg

o!el=1 2 3 4 5collin V&'

run

procregoutest=te outst norint

o!el=1 2 3 4 5

ri!e=+0to 0.8 0.01

outi"

run

Regresi Ridge10


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title(i!e %race

sol1= =0.5c=lac;

sol2=circle =0.5c=re!sol3=s)uare =0.5c=reen

sol4=trianle =0.5c=lue

sol5=lus =0.5c=aenta

leen!1lael=none osition=+to center insi!e

o!e=sare

ais1lael=+anle=90(i!e Coe""icients

procgplot

$#%?

lot+1 2 3 4 5@ri!e oerlaleen!=leen!1

ais=ais1 re"=0

run

procprint

$#%?

$V&'

arri!e 1 2 3 4 5

run

quit

Regresi Ridge11


12/16

x 1 x 2 x 3 x 4 x 5

- 0 . 3

- 0 . 2

- 0 . 1

0 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0

1 . 1

1 . 2

R i d g e r e g r e s s i o n c o n t r o l v a l u e

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8

1

The REG Procedure

Model: MODEL1

Dependent Variable: y

Analysis of Varianceu! of Mean

ource D" #uares #uare " Value Pr $ "

Model % &'%()1*++ 1+&())*%+ '*)(&+ ,()))1

Error ** &(+--). )(''+&.

Regresi Ridge12


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14/16

1- )()% 2()11%+ 1(.*&'* *(+-%% *(22&2 2()1%2

*) )(). *(.&-11 1(%'+)2 *('&)* 1(+)+' *(.%1-

*2 )()- *('2**- 1('-%*) *(1&2% 1('%22 *(2.11

*. )()+ *(*).&- 1(')&2* 1(&%.1 1(*)** *(1***

*& )()& *()12%. 1(2'&*' 1(-.)& 1()1-) 1(&*1-

2* )(1) 1(+'.*) 1(*&')+ 1(%&-2 )(+-%+ 1(-%)+

2% )(11 1(-))21 1(*'21. 1('%+) )(-.%* 1(.)2%

2+ )(1* 1(%-*21 1(1&%&% 1(22+) )(.-.. 1('-%'

'1 )(12 1('%&21 1(1%1&& 1(*22. )(.)'' 1(2.*&

'' )(1' 1(2%&)* 1(11)&1 1(1'*) )(%''% 1(*.2-

'- )(1% 1(*.&%. 1()-*'2 1().11 )('&'2 1(1-%%

%) )(1. 1(1+&2- 1()2.*. )(&+&* )('%1- 1()&.-

%2 )(1- 1(11-1& 1())**) )(&*%) )('1%1 1()*.)

%. )(1+ 1()%1&- )(&-))% )(+.-* )(2+2% )(&.*2%& )(1& )(&&*+) )(&2&.% )(+1%* )(2%%& )(&)'-

.* )(*) )(&2+&' )(&1)+' )(-.+) )(221- )(+%*%

.% )(*1 )(++&-. )(++2%1 )(-*%1 )(21)2 )(+)'+

.+ )(** )(+''-* )(+%-%' )(.+.) )(*&12 )(-.12

-1 )(*2 )(+)22' )(+2*+2 )(.%)2 )(*-'2 )(-*1%

-' )(*' )(-.%*2 )(+)&*+ )(.1-% )(*%&) )(.+'+

-- )(*% )(-2))' )(-+.+* )(%+-2 )(*'%2 )(.%11

+) )(*. )(.&-'+ )(-.%2+ )(%%&' )(*2*+ )(.*))

+2 )(*- )(..-*- )(-''++ )(%22- )(**1% )(%&1*

+. )(*+ )(.2&1& )(-*%*. )(%)&& )(*11* )(%.'%

+& )(*& )(.12)2 )(-).'+ )('+-- )(*)1+ )(%2&-

&* )(2) )(%++.2 )(.++'+ )('.-* )(1&21 )(%1..

&% )(21 )(%.%+1 )(.-1*1 )(''+) )(1+%* )('&%1

&+ )(2* )(%'''' )(.%'.' )('2)1 )(1--+ )('-%)

1)1 )(22 )(%*'') )(.2+-1 )('122 )(1-11 )('%.*

1)' )(2' )(%)%%- )(.*2') )(2&-. )(1.'+ )('2+.

1)- )(2% )('+-+% )(.)+.+ )(2+*& )(1%+& )('**)

11) )(2. )('-11. )(%&'%1 )(2.&1 )(1%2% )(').%

112 )(2- )('%%'1 )(%+)+. )(2%.1 )(1'+' )(2&1&

11. )(2+ )('')%2 )(%.--) )(2'2+ )(1'2- )(2-+*

11& )(2& )('*.'. )(%%%)* )(22*2 )(12&* )(2.%*

1** )(') )('1212 )(%'*-+ )(2*12 )(12%) )(2%*&

1*% )('1 )('))%) )(%2)&- )(211) )(1211 )(2'12

1*+ )('* )(2++%1 )(%1&%- )(2)12 )(1*-% )(22)'

121 )('2 )(2--1* )(%)+%% )(*&*) )(1*') )(2*))

12' )('' )(2..*+ )('&-&) )(*+2* )(1*)- )(21)1

12- )('% )(2%%&- )('+-%& )(*-'& )(11-. )(2))+

1') )('. )(2'.1' )('--.2 )(*..& )(11'- )(*&1&

1'2 )('- )(22.-. )('.-&+ )(*%&' )(111& )(*+2%

1'. )('+ )(2*-+1 )('%+.' )(*%** )(1)&2 )(*-%'

1'& )('& )(21&*- )(''&.) )(*'%' )(1).+ )(*.--

Regresi Ridge14


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1%* )(%) )(211)& )('')+2 )(*2+& )(1)'' )(*.)'

Rid6e race'

Obs 9R4DGE9 51 5* 52 5' 5%

1%% )(%1 )(2)2*- )('2*22 )(*2*.2 )(1)*1. )(*%2'.

1%+ )(%* )(*&%-+ )('*')& )(**..- )(1)))1 )(*'.+)

1.1 )(%2 )(*++.) )('1.1) )(**)&+ )()&-&% )(*')'2

1.' )(%' )(*+1-* )(')+2' )(*1%%' )()&%&& )(*2'2'

1.- )(%% )(*-%11 )('))+1 )(*1)2* )()&'1* )(**+%1

1-) )(%. )(*.+-- )(2&2%) )(*)%2* )()&*22 )(***&*

1-2 )(%- )(*.*.- )(2+.2& )(*))%2 )()&).* )(*1-%-

1-. )(%+ )(*%.+1 )(2-&'& )(1&%&2 )()++&- )(*1*''

1-& )(%& )(*%11- )(2-*-+ )(1&1%* )()+-') )(*)-%1

1+* )(.) )(*'%-' )(2..*. )(1+-*+ )()+%+& )(*)*-+

1+% )(.1 )(*')%* )(2%&&1 )(1+2*1 )()+''' )(1&+*2

1++ )(.* )(*2%'+ )(2%2-' )(1-&*+ )()+2)' )(1&2+.

1&1 )(.2 )(*2).* )(2'--' )(1-%%1 )()+1-) )(1+&..

1&' )(.' )(**%&2 )(2'1+& )(1-1++ )()+)'1 )(1+%.1

1&- )(.% )(**1'1 )(22.*) )(1.+2+ )()-&1- )(1+1-1

*)) )(.. )(*1-)' )(22).. )(1.%)) )()--&- )(1--&%

*)2 )(.- )(*1*+* )(2*%*. )(1.1-% )()-.+1 )(1-'22

*). )(.+ )(*)+-' )(2*))) )(1%+.1 )()-%-) )(1-)+'

*)& )(.& )(*)'-& )(21'+- )(1%%%+ )()-'.* )(1.-'-

*1* )(-) )(*))&- )(2)&+- )(1%*.% )()-2%+ )(1.'**

*1% )(-1 )(1&-*+ )(2)%)) )(1'&+* )()-*%- )(1.1)+

*1+ )(-* )(1&2-) )(2))*% )(1'-)& )()-1.) )(1%+)'

**1 )(-2 )(1&)*2 )(*&%.1 )(1'''' )()-).% )(1%%1)

**' )(-' )(1+.+- )(*&1)+ )(1'1++ )().&-' )(1%**.

**- )(-% )(1+2.1 )(*+..- )(12&') )().++. )(1'&%*

*2) )(-. )(1+)'% )(*+*2. )(12-)) )().+)) )(1'.+.

*22 )(-- )(1--2& )(*-+1% )(12'.+ )().-1- )(1''*+

*2. )(-+ )(1-''1 )(*-')' )(12*'* )()..2. )(1'1-+

*2& )(-& )(1-1%2 )(*-))2 )(12)*' )().%%- )(12&2-

*'* )(+) )(1.+-* )(*..11 )(1*+1* )().'+1 )(12-)*

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.

Regresi Ridge15


16/16

$!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

contoh makalah regresi ridge

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