Applications of Multiple Biological Signals
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1 Applcatos of Multple Bologcal Sgals I the Hosptal of Natoal Tawa Uversty, curatve gastrectomy could be performed o patets of gastrc cacers who are udergoe the curatve resecto to acqure sgal resposes from arteral pressure waves from radal artery based o three vessels from rght gastro-epploc, rght gastrc ad left gastrc arteres whch are dssected routely ad lgated sequetally Fgure 1: Illustrato of smulatg three sgal waves The purpose of ths research s to apply the techques of Statstcal Patter Recogto to extract the features based o the recorded waves collected from the patets of gastrc cacer or other dseases, ad to do clusterg ad classfcato o the extracted feature vectors. The goal s to mprove the qualty of dagoss ad provde better medcal servce. We ams to udergo the followg tasks. Sgal wave recordg (Medcal Doctors) Feature Extracto (Computer Scetsts & Egeers) Patter Recogto (Egeers & Data Aalysts) 0
2 Bayesa Classfer Cosder j trag vectors {v (j) 1, v (j) 2,, v (j) j } from class ω j for j =1, 2,,K. The maxmum lkelhood estmate of mea vector ad covarace matrx for patter class ω j ca be computed by m j = 1 j j =1 v (j) Ŝ j = 1 j j (v (j) =1 m j )(v (j) m j ) t, 1 j K (1) To determe the class of a ukow vector x, a quadratc classfer based o the multvarate ormal (Gaussa) model s defed as x belogs to class ω k f (x m j ) tŝj 1 (x mj ) t (x m k ) tŝk 1 (x mk ) t 2l p j det(ŝk), j k, (2) p k det(ŝj) where p j = j K s the estmated pror probablty for class ω j. If the uderlyg class-codtoal probablty destes are multvarate ormal wth kow parameters, the form of the above decso rule s Bayesa optmal. 1
3 Fsher Lear Classfer Cosder j trag vectors {v (j) 1, v (j) 2,, v (j) j } from class ω for j =1, 2,,K wth K j=1 j =. The maxmum lkelhood estmate of mea vector ad covarace matrx for patter class ω j ca be computed by m j = 1 j j =1 v (j) Ŝ j = 1 j j (v (j) =1 m j )(v (j) m j ) t, 1 j K (3) The pool mea vector ad wth-class scatter matrx ca be estmated (as troduced Bayesa classfer) by m = 1 K j m j = 1 j=1 Ŵ = 1 K jŝj = 1 j=1 K j [v (j) j=1 =1 K j j=1 =1 v (j) m j ][v (j) m j ] t (4) by The betwee-class scatter matrx B ad the total scatter matrx T ca be estmated B = 1 K j (m j m)(m j m) t (5) j=1 T = 1 K j (v (j) j=1 =1 m)(v (j) m) t (6) It ca be proved that Ŵ + B = T. The Mahalaobs dstace betwee two vectors x ad y wth respect to a vertble symmetrc matrx C s defed as δ(x, y) =(x y) t C 1 (x y) The Fsher lear classfer ca be defed as x belogs to class ω k f k = arg m j (x m j ) tŵ 1 (x m j ) t (7) 2
4 K Nearest Neghbor (K-NN) Classfer A. Nearest Neghbor (NN) Classfer The earest decso rule, τ NN assgs category k to the pot x the patter space f the trag patter closest to x s from patter class ω k.thats, τ NN (x) =ω k f δ(x, x k ) <δ(x, x ) 1 j k. where δ(x, y) s the dstace betwee patters x, y R d uder some sutable metrc δ, for example, Eucldea dstace. B. K Nearest Neghbor (K-NN) Classfer The atural exteso of the earest eghbor rule s to exame the K earest eghbors to x ad classfy x accordg to the patter class most heavly represeted amog the k eghbors. I mathematcal termology, let H (K, ) betheumberof patters from class ω amog the K earest eghbors of x. The earest eghbors are computed from the trag patters. The K-NN rule s defed as follows. τ K NN (x) =ω j f H j (K, ) >H (K, ) j Tes may be hadled by radomzato, so we try to avod them by choosg K properly, for example, a dchotomous problem wth two patter classes, a odd umber of K such as K =3orK =5sused. Refereces 1. P.A. Devjver ad J. Kttler, Patter Recogto: A Statstcal Approach, P.A. Devjver, A multclass K-NN approach to Bayes rsk estmato, Patter Recogto Letters, vol. 3, pp.1-6, R.O. Duda, P.E. Hart, ad D.G. Stork, Patter Classfcato, Joh Wley, A.K. Ja ad R.C. Dubes, Algorthms for Clusterg Data, T. Haste, R. Tbshra, J. Fredma, The Elemets of Statstcal Learg: Data Mg, Iferece, ad Predcto, D. Haselma ad B. Lttlefeld, Masterg MATLAB 7 (2005) 7. S. Theodords ad K. Koutroumbas, Patter Recogto, cche/isa5305/rs.txt 3
5 Examples for Testg Classfers The data set rs.txt cossted of 150 patter vectors equally from 3 patter classes: Setosa, Vergca, ad Verscolor. For each flower, four measuremets: petal legth, petal wdth, sepal legth, ad sepal wdth are recorded. We used 75 patters selected from the frst 25 patters of each category as trag ad the remag 75 patters for testg the above classfers. The results are summarzed as follows. Classfer Quadratc Fsher 3-NN Error Rate 3/75 0/75 0/75 A vsualzed result s show Fgure 1. PCP for rs Data wth 75 Trag Patters PCP for rs Data wth 75 Testg Patters Fgure 2: Prcpal Compoet Projecto of rs trag ad testg data A. Quadratc Classfer 1 = 2 = 3 =25, 1 = 2 = 3 = =75, p 1 = p 2 = p 3 =25/75 = 1/3. B. Fsher Lear Classfer 1 = 2 = 3 =25, 1 = 2 = 3 = =75, p 1 = p 2 = p 3 =25/75 = 1/3. 4
6 The mea vectors of each category, the class-codtoal covarace matrces S 1,S 2,S 3, ad the wt-class scatter matrx W are obtaed as follows. m 1 =[5.0280, , , ] t m 2 s =[6.0120, , , ] t m 3 =[6.5760, , , ] t S S S 3 W
7 Scrpt fle: rslda.m Trag Lear Dscrmat Classfer from datars.txt (75 patters) d=4; N=150; 1=25; 2=25; 3=25; =1+2+3; f=fope( datars.txt ); fgetl(f); fgetl(f); fgetl(f); skp three header les A=fscaf(f, f,[d+1 N]); X=A(1:d,:); X1=X(:, 1:25); X2=X(:,51:75); X3=X(:,101:125); Y1=X(:,25:50); Y2=X(:,76:100); Y3=X(:,126:150); Compute pror probabltes: p1, p2, p3 mea vectors: u1\, u2\, u3\ class-codtoal covarace matrces S1,S2,S3 of each category wth-class scatter matrx W p1=1/3; p2=1/3; p3=1/3; u1=mea(x1,2); u2=mea(x2,2); u3=mea(x3,2); S1=cov(X1 ); S2=cov(X2 ); S3=cov(X3 ); usg a ubased estmator W=((1-1)*S1+(2-1)*S2+(3-1)*S3)/3; wth-class scatter matrx fout=fope( rslda.txt, w ); fprtf(fout, probabltes, mea vectors, ); fprtf(fout, class-codtoal S ad wth-class scatter W matrces\ ); fprtf(fout, 8.4f\t8.4f\t8.4f\,p1,p2,p3); for =1:d-1 fprtf(fout, 8.4f\t,u1()); fprtf(fout, 8.4f\,u1(d)); for =1:d-1 fprtf(fout, 8.4f\t,u2()); fprtf(fout, 8.4f\,u2(d)); for =1:d-1 fprtf(fout, 8.4f\t,u3()); fprtf(fout, 8.4f\,u3(d)); Prt class-codtoal matrces: S1, S2, S3 for =1:d for j=1:d 6
8 fprtf(fout, 8.4f,S1(,j)); fprtf(fout, \ ); Le 49: Ed-of-Page for =1:d for j=1:d fprtf(fout, 8.4f,S2(,j)); fprtf(fout, \ ); for =1:d for j=1:d fprtf(fout, 8.4f,S3(,j)); fprtf(fout, \ ); Prt wth-class scatter matrx W for =1:d for j=1:d fprtf(fout, 8.4f,W(,j)); fprtf(fout, \ ); For Testg err1=0; err2=0; err3=0; f (<=1) x=y1(:,); R1=mhdst(x,u1,W); R2=mhdst(x,u2,W); R3=mhdst(x,u3,W); f (R1>R2 R1>R3) err1=err1+1; f (<=2) x=y2(:,); R1=mhdst(x,u1,W); R2=mhdst(x,u2,W); R3=mhdst(x,u3,W); f (R2>R1 R2>R3) 7
9 err2=err2+1; f (<=3) x=y3(:,); R1=mhdst(x,u1,W); R2=mhdst(x,u2,W); R3=mhdst(x,u3,W); f (R3>R1 R3>R2) err3=err3+1; [err1,err2,err3,] fprtf(fout, \ ); fprtf(fout, (err1,err2,err3,)=(2d,2d,2d,3d)\,err1,err2,err3,); 8
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