Robust and Accurate Cancer Classification with Gene Expression Profiling

Transcription

1 Robus and Accurae Cancer Classfcaon wh Gene Expresson Proflng (Compuaonal ysems Bology, 2005) Auhor: Hafeng L, Keshu Zhang, ao Jang

2 Oulne Background LDA (lnear dscrmnan analyss) and small sample sze problem Prevous mehods and Generalzed Lnear Dscrmnan Analyss (GLDA) Expermenal resuls Concluson

3 Background Accurae dagnoss of human cancer s essenal n cancer reamen. Gene expresson proflng could be a precse and sysemac mehod for cancer classfcaon. Mcroarray echnology enables us o smulaneously observe he expresson levels of many housands of genes on he ranscrpon level. Mcroarray: chp wh a marx of housands of spos prned on o Each spo bnds o a specfc gene

4 Background Mcroarray chps Images scanned by laser Gene Value D26528_a 93 D2656_cds_a -70 D2656_cds2_a 44 D2656_cds3_a 33 D26579_a 38 D26598_a 764 D26599_a 537 D26600_a 204 D284_a 707 New sample Predcon: Cancer or Normal Classfers D26528 D63874 D63880 ample ample ample ample ample ample ample Daases

5 Background wo Major Challenges : hgh dmenson of daa (oo many columns (genes), usually >,000) he number of samples s small. (oo few samples, usually < 00) Peakng Phenomenon : a large number of feaures may degrade he performance of classfers f he number of ranng samples s small relave o he number of feaures. I needs a leas 5 0 mes as many ranng samples per class as he number of feaures o oban well-raned (robus) classfers (A.K.Jan e al. 982) Consequenly, dmensonaly reducon s essenal o cancer classfcaon.

6 Dmenson Reducon Feaure elecon: choose a bes subse of feaures from a large nal se. Advanage: he seleced feaures rean her orgnal bologcal nerpreaon. Mehod: ngle-gene-rankng, gene-par-rankng, GA/KNN, VM wh RFE (recursve feaure elmnaon) Dsadvanage: A leas 50 feaures need be chosen n general. hs number s far from he 5-0 mes rao of samples o feaures.

7 Dmenson Reducon Feaure Exracon: ransform he orgnal feaures space no a reduced feaure space. Advanage: provdes a beer dscrmnave ably han feaure selecon; mees he recommended 5-0 mes rao of samples o feaures per class. Dsadvanage: he new feaures generaed by feaure exracon may no have a clear bologcal meanng.

8 LDA (Lnear Dscrmnan Analyss) ha knd of projecon s desrable? (wo dmensonal and wo classes example) m p m n m p m n Inuon: Afer projecon, he pons from same classes are clusered whle pons from dfferen classes are as far away as possble. (he dsance beween he projeced sample and he correspondng projeced mean s as shor as possble and he dsance beween he projeced means s as long as possble)

9 Lnear Dscrmna Analysswo classes Projecon: Beween-Class caer Marx hn-class caer Marx oal caer Marx w b N P x m x m x = = ) )( ( n p n p b m m m m ) )( ( = = N x n j n j P x p p w j m x m x m x m x ) )( ( ) )( ( x y = b y b = ) ( w y w = ) (

10 Lnear Dscrmnan Analyss In he projeced low dmensonal space we ry o maxmze: oluon for he projecon: s he egenvecors wh he larges egenvalues for ) ( ) ( w b y w y b = max arg w b op = b w

11 Lnear Dscrmnan Analyssmulple classes Exenson of wo-class LDA (Fsher Dscrmnan Analyss) o Mul-Class Case wh assumpon of C classes. Formulas have o be rewren ) Beween-Class caer Marx 2) hn-class caer Marx 3) arge funcon op = C b = P ( m m )( m m ) = = C P arg = x Class _ max ( x m s he egenvecors wh he larges egenvalues for )( x B m ) w b

12 mall ample ze Problem wh LDA: mall ample ze Problem: he number of samples (n) s smaller han he dmensonaly of samples (D). Rank( w )<D. o w s sngular and w - doesn exs. e can no ge he opmal, whch s he egenvecors wh he larges egenvalue for w b

13 Prevous mehods ) Frs reduce he dmensonaly wh some oher feaure selecon/exracon mehod and hen apply LDA on he dmensonaly-reduced daa. 2) Pseudo-nverse w : replace w - wh w. s he larges egenvecors of w b. 3) Regularzaon: add small quanes o he dagonal elemens of w o make nonsngular. For example: PDA (Penalzed dscrmnan analyss). w s replaced wh w λω and hen LDA proceeds as usual, where Ω s a symmerc and nonnegave defne penaly marx. he choce of Ω depends on he problem.

14 Generalzed Lnear Dscrmnan Analyss (GLDA) heorem: he column vecors of are he egenvecors of b Defnon (Moore-Penrose Inverse) : A marx A sasfyng he followng condons s unque and s called he Moore-Penrose nverse of A: AA A =A A AA = A (A A) = A A (AA ) = AA Lemma 2: Lemma 3: Lemma 4: J ( ) = r ( ( x m) = b w = ) w = x m w w = r (( Dagonalze symmerc marces b o Λ and o I P ( b )P = Λ, P ( )P = I where P s a d d nonsngular marx. J() = J(P) ) ( b ))

15 Generalzed Lnear Dscrmnan Analyss (GLDA) Denong, In oher words, he opmal ransformaon s he column vecors of are he egenvecors of

16 Generalzed Lnear Dscrmnan Analyss (GLDA) Boh and b s D D. nce D s usually many housands, s very me and memory consumng o calculae and he egenvecors of b usng common compuaonal mehods. Devse a fas algorhm o effcenly calculae and he egenvecors of b va sngular value decomposon (VD).

17 A fas Algorhm = K = 2 2 K 2 = UΛU = UΛ U 2 = UΛ U = XX wh X = [( x m ),..., ( x n m )] n hus, we can oban he egenvalues and correspondng egenvecors hrough he VD of X. Noe ha dmensonaly of X s D n, where n s he number of samples <<D 2 he column vecors of K are he egenvecors of 2 b Recall b = MM wh M = [ P, we ( m m),..., Pc ( mc m) ] can oban 2 b 2 = ( 2 M )( 2 M ) hus, we can oban 2 K by he VD of 2 M nce 2 M has he dmensonaly D c and he number of classes c s <<D. 2

18 A fas Algorhm

19 Expermens es GLDA on seven publc daases and compare wh PDA, Random Foress, VM, DLDA, KNN wh feaure selecon of Dudo and GA/KNN. For each daase, randomly dvde no hree par, of whch wo pars are used for ranng (boh feaure selecon/exracon mehods and classfers) and he las par s kep for es. hs procedure s repeaed for 200 mes and he averages and sandard devaons of error raes are lsed n able.

20 Resuls

21 Resuls

22 Resuls

23 Concluson Gene expresson proflng has grea poenal for accurae cancer dagnoss. I also brngs machne learnng researchers wo challenges, he curse of dmensonaly and he small sample sze problem. In hs paper, he auhor has presened a novel mehod GLDA o solve hese wo problems. ) hey gve us a mahemacal proof o show ha he column vecors of are he egenvecors of b 2) hey provde a fas algorhm o compue he. 3) Exensve expermens on seven publc daases demonsrae ha he mehod s able o classfy umors robusly wh a hgh accuracy. Besdes cancer classfcaon, GLDA may also fnd applcaons n oher areas where he small sample sze problem and he curse of dmensonaly arse.