ensembled multivariate adaptive regression ......mars. ohd dibldi blour method ppp proposed is...
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CO S 2010COMPSTAT2010 in Paris
Ensembled Multivariate Adaptive Regression SplinesEnsembled Multivariate Adaptive Regression Splines ith N ti G t E ti twith Nonnegative Garrote Estimator
Hiroki MotogaitogOsaka UniversityOsaka University
M hi G tMasashi GotoBiostatistical Research Association NPOBiostatistical Research Association, NPO.
JAPANJAPAN
AgendaAgendag
• Introduction and motivation• Introduction and motivationTree methods• Tree methods M lti i t Ad ti R iMultivariate Adaptive RegressionMultivariate Adaptive Regression
Splines(MARS)Splines(MARS)p ( )Bagging MARSBagging MARSgg g
O th d d• Our method proposedOur method proposedEnsembled MARS with nonnegative garroteEnsembled MARS with nonnegative garrote
• Example and simulation• Example and simulation• Concluding remarks• Concluding remarksg
22
AgendaAgendag
• Introduction and motivation• Introduction and motivationTree methods• Tree methods M lti i t Ad ti R iMultivariate Adaptive RegressionMultivariate Adaptive Regression
Splines(MARS)Splines(MARS)p ( )Bagging MARSBagging MARSgg g
O th d d• Our method proposedOur method proposedEnsembled MARS with nonnegative garroteEnsembled MARS with nonnegative garrote
• Example and simulation• Example and simulation• Concluding remarks• Concluding remarksg
33
Introduction and motivationIntroduction and motivation
Unstable Less interpretableUnstable Less interpretable
)(f )(xf
ˆ)(ˆ xf )(ˆ xfSt bili i)(ˆ xf)(xf )(xfStabilizing
BaggingMARS gg g(Breiman,1996)(Friedman,1991) (Breiman,1996)(Friedman,1991)
M ti tiMotivation
i MARS th t h b th t bilit d i t t bilita new version MARS that has both stability and interpretability
44
AgendaAgendag
• Introduction and motivation• Introduction and motivationTree methods• Tree methodsM lti i t Ad ti R iMultivariate Adaptive RegressionMultivariate Adaptive Regression
Splines(MARS)Splines(MARS)p ( )Bagging MARSBagging MARSgg g
O th d d• Our method proposedOur method proposedEnsembled MARS with nonnegative garroteEnsembled MARS with nonnegative garrote
• Example and simulation• Example and simulation• Concluding remarks• Concluding remarksg
55
M lti i t Ad ti R i S li (F i d 1991)Multivariate Adaptive Regression Splines(Friedman,1991)
• Model form• Model formRegression model Basis function
M KRegression model Basis function
M
Bf )(ˆˆˆ mK
qiB )]([)( mmBf 0MARS )(x qmkmkpmkm txiB ),(),(),( )]([)(x
m 1
k
mkmkpmkm1
),(),(),(
• Algorithms• AlgorithmsForward stepwise
0.5
Forward stepwise 0.45
Increase basis functions0.4
Increase basis functions 0 3
0.35
Backward stepwise 0 25
0.3
数の値Backward stepwise
P ff 0 2
0.25
基底関
数
Prune off 0.15
0.2基
Select the best tree 0.1
0.15
)]5.0([ px )]5.0([ px Select the best tree0.05
)]([ p)]([ p
0 0 2 0 4 0 6 0 8 10
0 2 0 4 0 5 0 6 0 8 1 x0 0.2 0.4 0.6 0.8 1
px0.2 0.4 0.5 0.6 0.8 1 px
1 d k t t 0 5q=1 and knot t=0.5
66
Bagging (Breiman 1996)Bagging (Breiman,1996)gg g ( , )
• Model form(Bagging MARS)• Model form(Bagging MARS)Regression model Each treeRegression model
1Each tree
)(ˆ1ˆ E ff MARS d l)(f)(1MARS Bagging x
e efE
f : MARS model)(xef1gg g eE
• Algorithms• AlgorithmsSamplep
Bootstrap sample Bootstrap sample Bootstrap sampleBootstrap sample Bootstrap sample
+ +・・・+ +・・・+
)(ˆ xf )(f )(f )(f)(1 xf )(2 xf )( xef )( xEf
ˆ7
averaging )(xf7
AgendaAgendag
• Introduction and motivation• Introduction and motivationPre io s research• Previous research M lti i t Ad ti R iMultivariate Adaptive RegressionMultivariate Adaptive Regression
Splines(MARS)Splines(MARS)p ( )Bagging MARSBagging MARSgg g
O th d d• Our method proposedOur method proposedEnsembled MARS with nonnegative garroteEnsembled MARS with nonnegative garrote
• Example and simulation• Example and simulation• Concluding remarks• Concluding remarksg
88
Proposed methodProposed methodp
Motivation
a new version MARS that has both stability and interpretabilitya new version MARS that has both stability and interpretability
Stable, but less interpretable Stable and interpretable
233
1Selection 14
1
&RankingRanking
45Typical tree
45Typical tree
nonnegativeBagging Proposed methodnonnegativetBagging Proposed methodgarrote
(B i 1995)(Breiman,1995)
99
Ensembled MARS ith non negati e garrote(1/2)Ensembled MARS with non-negative garrote(1/2)g g
• Model form• Model formRegression model Each treeg
E )(ˆˆˆ x E fcf : MARS model :non-negative garrote estimator)(ˆ xf c)(1
x
e ee fcf : MARS model , :non negative garrote estimator )(xef ec
• Algorithms• Algorithms
Generate Bagging trees Generate Bagging trees. Att h h t d ti t i tiˆ Attach on each tree and estimate using nonnegative ec ec
garrote(Breiman,1995). e e
g ( , )Select candidate trees(If the tree is removed)0ˆ c― Select candidate trees(If , the tree is removed).0ec
)(ˆˆˆ E ff Get .)(1
x
e ee fcf• Interpretable structure through typical tree(max )
1e
c• Interpretable structure through typical tree(max )ec
1010
Ensembled MARS ith non negati e garrote(2/2)Ensembled MARS with non-negative garrote(2/2)g g
ti t (B i 1995)non-negative garrote (Breiman,1995)
N P
P 2)(ˆP
pnppn
Pp xcYc
P
2)(1 )ˆ(min arg}ˆ{ scc pp ,0 , subject to ,
n pnppn
cp P
p 1 1}{1 )(g}{
1
p
pp 1
,, j ,
h i th l t ti t d P1where is the least square estimator and .p Ps 1
Ensembled MARS with non-negative garrote g g
N E E
N E
E fcYc 2))(ˆ(minarg}ˆ{ x 10 E
ccbj t t neenc
e fcYcE
1 1}{1 ))((min arg}{
1
x 1,01
ee cc , subject to , n ece 1 1}{ 1 1e
ˆwhere is MARS model.)(ˆnef x )( nef
h t i ticharacteristics
• All indicates BaggingEc /1All indicates Bagging.Ece /1
• Selection of optimal is unnecessary( ). s 1sp y( )s s
1111
AgendaAgendag
• Introduction and motivation• Introduction and motivationPre io s research• Previous research M lti i t Ad ti R iMultivariate Adaptive RegressionMultivariate Adaptive Regression
Splines(MARS)Splines(MARS)p ( )Bagging MARSBagging MARSgg g
O th d d• Our method proposedOur method proposedEnsembled MARS with non-negative garroteEnsembled MARS with non negative garrote
• Example and simulation• Example and simulation• Concluding remarks• Concluding remarksg
1212
Literature exampleLiterature examplep
Prostate cancer data (Stamney et al 1989: Tibshirani 1996)Prostate cancer data (Stamney et al.,1989: Tibshirani,1996)
L l f t t ifi tiy• : Level of prostate-specific antigenyT• : Clinical measuresT
81 ),...,( xxx: Log of tumor size
81 ), ,(x : Log of tumor size1x
: Weight of prostate2x : Weight of prostate P ti t’
2
: Patient’s age 3x: Log of benign prostatic hyperplasia amount4x : Log of benign prostatic hyperplasia amount4x: Dummy variables of whether it is metastasizing to seminal vesicle5x y g: Log of capsular penetration
5
x : Log of capsular penetration 6x: Gleason score7x : Gleason scoreGl ’ ti f 4 5
7
x : Gleason score’s ratio of 4 or 58x
• Sample size : 97Np
1313
Literature exampleLiterature examplep
0 50.5
0.45
0 40.4
V
0 35CV
0.35
GC
G
0.3
0 250.25
0 20.2
Ensembled BaggingMARS NNG
gg gMARSMARS-NNG MARS
1414
Literature exampleLiterature examplep
• Number of trees• Number of treesBagging Ensembled MARS-NNG
97 997 9
• Structure• StructureBagging Ensembled MARS-NNG
1x 2x 2x2x
x 1x4x
Typical treecandidates
Typical treecandidates
1515
Small simulationSmall simulation
• Design• Design Model(Friedman 1991) Model(Friedman,1991)
510)50(20)sin(10 2 xxxxxy where is )10(N,510)5.0(20)sin(10 54321 xxxxxy where is . )1,0(N
T i i l i 100 Training sample size: 100 Testing sample size: 1 000 Testing sample size: 1,000 Number of simulation: 100
M th d• MethodMethod MARS B i MARS E bl d MARS NNG MARS, Bagging MARS, Ensembled MARS-NNG
E l ti• EvaluationMSE (St d di d )MSESTD(Standardized mean square error)( q )
1616
Small simulationSmall simulation
0.07
0.065
DE
ST
0 06MS
E
0.06M
0 0550.055
0.05
Ensembledse b edMARS-NNG Bagging MARS MARSMARS NNG
Number 11 6Numberof trees
11.6( d)
100 1of trees (averaged)
1717
AgendaAgendag
• Introduction and motivation• Introduction and motivationPre io s research• Previous research M lti i t Ad ti R iMultivariate Adaptive RegressionMultivariate Adaptive Regression
Splines(MARS)Splines(MARS)p ( )Bagging MARSBagging MARSgg g
O th d d• Our method proposedOur method proposedEnsembled MARS with non-negative garroteEnsembled MARS with non negative garrote
• Example and simulation• Example and simulation• Concluding remarks• Concluding remarksg
1818
Concluding remarksConcluding remarksg
• We proposed a new ensembled method of• We proposed a new ensembled method of p pMARSMARS.O h d d i bl d i blOur method proposed is stable and interpretable.p p p
• Ensembled MARS NNG provided superior• Ensembled MARS-NNG provided superior p por comparable results to MARS andor comparable results to MARS and pBagging MARSBagging MARS.gg g
1919
ReferencesReferences
• Breiman, L. (1995) Better subset regression using the nonnegative , ( ) g g ggarrote. Technometrics, 37,373-384.g , ,
• Breiman L (1996) Bagging predictors Machine Learning 24 123-• Breiman, L. (1996). Bagging predictors. Machine Learning, 24, 123-140140. B i L F i d J H Ol h R A & St C J (1984)• Breiman, L., Friedman, J. H., Olshen, R. A. & Stone, C. J. (1984). Cl ifi ti A d R i T W d thClassification And Regression Trees. Wadsworth.
• Friedman, J. H. (1991) Multivariate Adaptive Regression SplinesFriedman, J. H. (1991) Multivariate Adaptive Regression Splines (with discussion) Annals of Statistics 19 1-141(with discussion). Annals of Statistics,19, 1 141.
• Friedman J H (2001) Greedy function approximation: a gradient• Friedman, J. H. (2001). Greedy function approximation: a gradient boosting machine Ann Statist 29(5) 1189 1232boosting machine. Ann. Statist., 29(5), 1189-1232.
• Meinshausen N (2009): Node Harvest: simple and interpretableMeinshausen, N. (2009): Node Harvest: simple and interpretable regression and classification Arxiv preprint arXiv:0910 2145regression and classification. Arxiv preprint arXiv:0910.2145.
• Motogaito, H., Sugimoto, T. & Goto, M. (2007): Multivariate AdaptiveMotogaito, H., Sugimoto, T. & Goto, M. (2007): Multivariate Adaptive Regression Splines with Non negative Garrote Estimator JapaneseRegression Splines with Non-negative Garrote Estimator. Japanese J A l St ti t 36 99 118 (i J )J. Appl. Statist., 36, 99-118 (in Japanese).
• Yuan M & Lin Y (2007) On the non negative garrote estimator J• Yuan, M. & Lin, Y. (2007) On the non-negative garrote estimator. J. R St ti t S B 69(2) 143 161R. Statist. Soc., B 69(2), 143-161.
2020
Thank you very much for your attentionThank you very much for your attentiony y y
h-motogt@sigmath es osaka-u ac [email protected]
2121
Back pBack upBack upp
2222
Small simulationSmall simulation
0 06620 07
0.0597 0.06620.07
0 05950 065
0.05950.065
0 05450,0545
0 060.0567
0.06
0 0 440.0544
0 0550.0550 06090.0609
0.05 0.05730.0570
0.05730.0570
EnsembledEnsembledMARS-NNG Bagging MARS MARSMARS-NNG gg g
2323
Literature exampleLiterature examplep
1x x x1x 1x 8x 8x 1x1
x x x2x 3x 6x
MARSMARSMARSMARSMARSMARS24242424
Literature exampleLiterature examplep
x1x 2x 2x2x1
xx 1x4x
EnsembledEnsembledEnsembledEnsembledMARS NNGMARS NNGMARS-NNGMARS-NNG
25252525