{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A:=randmatrix(4,4): # generating a random 4 by 4 matr ix (not needed by you)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "V:=evalm (transpose(A)&*A*1.); # creating a 4 by 4 symmetric, positive definite matrix,\n#which can serve as a variance-covariance matrix of 4 random variables (this is normally given)." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"VG-%'matrixG6#7&7&$\"&g5#\"\"!$\"&_A\"F,$\"%;%*F,$\"%c'*F,7&F-$ \"%e&*F,$\"&?.\"F,$\"$s#F,7&F/F6$\"&3&>F,$!%+iF,7&F1F8F=$\"&1j\"F," }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Converting a given var-cov matrix into a corresponding correlation matrix:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "C:=evalm(V):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "for i to 4 do for j to 4 do C[i,j]: =C[i,j]/sqrt(V[i,i]*V[j,j])end do end do;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "evalm(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr ixG6#7&7&$\"+++++5!\"*$\"+v_jN')!#5$\"+#>$[XYF-$\"+?%z1@&F-7&F+F($\"+> GqdvF-$\"+I9xy@!#67&F.F3F($!+jZDwMF-7&F0F5F9F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Program for computing a partial correlation coeffici ent (between Xi and Xj, given the value of Xk):" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "par1:=proc(i,j,k);(C[i,j]-C [i,k]*C[j,k])/sqrt((1-C[i,k]^2)*(1-C[j,k]^2)) end:" }}{PARA 0 "" 0 "" {TEXT -1 51 "Computing one such partial correlation coefficient:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "par1(2, 3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LMLTz!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Partial correlation between Xi and Xj gi ven the values of Xk and Xm:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "par2:=proc(i,j,k,m);\n(par1(i,j,m)-par1(i,k, m)*par1(j,k,m))/sqrt((1-par1(i,k,m)^2)*(1-par1(j,k,m)^2)) end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Computing one such partial correla tion coefficient (note the symmetry):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "par2(2,3,1,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +->BVG!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "par2(3,2,4,1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W?BVG!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "with(stats):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Finding a solution to A . A(trans ) = V, in form of a lower triangular matrix:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A:=matrix(4,4,[a1,0,0,0,a2,a 3,0,0,a4,a5,a6,0,a7,a8,a9,a10]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AG-%'matrixG6#7&7&%#a1G\"\"!F+F+7&%#a2G%#a3GF+F+7&%#a4G%#a5G%#a6GF+7 &%#a7G%#a8G%#a9G%$a10G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "e q:=evalm(A&*transpose(A)-V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG -%'matrixG6#7&7&,&*$)%#a1G\"\"#\"\"\"F/$\"&g5#\"\"!!\"\",&*&F-F/%#a2GF /F/$\"&_A\"F2F3,&*&F-F/%#a4GF/F/$\"%;%*F2F3,&*&F-F/%#a7GF/F/$\"%c'*F2F 37&F4,(*$)F6F.F/F/*$)%#a3GF.F/F/$\"%e&*F2F3,(*&F6F/F;F/F/*&FIF/%#a5GF/ F/$\"&?.\"F2F3,(*&F6F/F@F/F/*&FIF/%#a8GF/F/$\"$s#F2F37&F9FL,**$)F;F.F/ F/*$)FOF.F/F/*$)%#a6GF.F/F/$\"&3&>F2F3,**&F;F/F@F/F/*&FOF/FUF/F/*&FjnF /%#a9GF/F/$\"%+iF2F/7&F>FRF]o,,*$)F@F.F/F/*$)FUF.F/F/*$)FaoF.F/F/*$)%$ a10GF.F/F/$\"&1j\"F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e q:=convert(eq,set);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#eqG<,,**&%#a 4G\"\"\"%#a7GF)F)*&%#a5GF)%#a8GF)F)*&%#a6GF)%#a9GF)F)$\"%+i\"\"!F),&*$ )%#a1G\"\"#F)F)$\"&g5#F3!\"\",**$)F(F8F)F)*$)F,F8F)F)*$)F/F8F)F)$\"&3& >F3F;,(*&%#a2GF)F*F)F)*&%#a3GF)F-F)F)$\"$s#F3F;,(*&FGF)F(F)F)*&FIF)F,F )F)$\"&?.\"F3F;,(*$)FGF8F)F)*$)FIF8F)F)$\"%e&*F3F;,,*$)F*F8F)F)*$)F-F8 F)F)*$)F0F8F)F)*$)%$a10GF8F)F)$\"&1j\"F3F;,&*&F7F)FGF)F)$\"&_A\"F3F;,& *&F7F)F(F)F)$\"%;%*F3F;,&*&F7F)F*F)F)$\"%c'*F3F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "solve(eq,\{a1,a2,a3,a4,a5,a6,a7,a8,a9,a10\})[ 1];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<,/%$a10G$!+U2Q$3\"!\")/%#a1G$! +&R17X\"!\"(/%#a4G$!+$y%R)['F(/%#a6G$!+B$**o^(F(/%#a9G$!+``!px\"!\"*/% #a2G$!+M4jU%)F(/%#a7G$!+HWx`mF(/%#a3G$!+STqH\\F(/%#a5G$!+h%oA#)*F(/%#a 8G$\"+KBN%3\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(% );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Verifying:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(A&*transpose(A ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&$\"+,++1@!\"&$ \"+++?D7F*$\"+-++;%*!\"'$\"+-++c'*F/7&F+$\"++++e&*F/$\"++++K5F*$\"*-++ s#F/7&F-F5$\"+++!3&>F*$!+%******>'F/7&F0F7F<$\"+****fI;F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Generating a RIS of size 20 from a 4-var iate Normal distribution with a given V and mu = [250, -400, 720, -110 0]:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Z :=matrix(4,20,[random[normald](4*20)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "X:=evalm(A&*Z+matrix(4,20,[250$20,-400$20,720$20,-110 0$20])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "transpose(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#767&$\"+\")GOKA!\"($!+M \">PF$F*$\"+xb;(\\)F*$!+&4Q,I\"!\"'7&$\"+ul&[R$F*$!+tL$\\y#F*$\"+)QaJ, \"F1$!+K@6*=\"F17&$\"+HG:^JF*$!+/!4bu#F*$\"+%*fKr*)F*$!+:P0#G\"F17&$\" +n7Y/8F*$!+*3%))GUF*$\"+M\"4!>rF*$!+0N@V7F17&$\"+LH5M@F*$!+q$46;%F*$\" +4_Y&Q(F*$!+XbR=6F17&$\"+0\\h@AF*$!+*eoh9&F*$\"+G6oEeF*$!+M'e@#*)F*7&$ \"+,XU)H\"F*$!+s%Q*ySF*$\"+2)Ga9)F*$!+9n]t7F17&$\"+/#*QG]F*$!+F*e7]#F* $\"+\\s&zk)F*$!+pdeJ)*F*7&$\"*\\L=q)F*$!+7r4(\\&F*$\"+2WBHaF*$!+7m?b5F 17&$\"+-&3It#F*$!+l`TYSF*$\"+6;UesF*$!+:EHo5F17&$\"+kT(zv%F*$!+&4S3n#F *$\"+\\G8^$)F*$!+Xd(y#**F*7&$\"+VY)Hj$F*$!+V\"=gk$F*$\"+\"R@<:'F*$!+GS Iz&*F*7&$\"+aPp_KF*$!+*=X\"F17&$\"+_(*HHZF*$!+@/mgCF*$\"+*)4=_'*F*$!+ e?+p5F17&$\"+!*=F*$!+^GjDTF*$\"+ya<:jF*$!+,^6-7F17&$\"+iZT#H$F*$!+h 8^iPF*$\"+7>2kmF*$!+KS<@5F17&$\"+72hfKF*$!+wN(QC$F*$\"+@He`rF*$!+-\\oW 6F17&$\"+1Vi=CF*$!+*)*y/e$F*$\"+O+\"**y)F*$!+5K>/7F17&$\"*hp(=HF*$!+Mv 0sgF*$\"+69*z-'F*$!+$enB.\"F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pointplot([seq([X[1,i],X[2,i]],i=1..20)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%'POINTSG667$$\"+\")GOKA!\"($!+M \">PF$F)7$$\"+ul&[R$F)$!+tL$\\y#F)7$$\"+HG:^JF)$!+/!4bu#F)7$$\"+n7Y/8F )$!+*3%))GUF)7$$\"+LH5M@F)$!+q$46;%F)7$$\"+0\\h@AF)$!+*eoh9&F)7$$\"+,X U)H\"F)$!+s%Q*ySF)7$$\"+/#*QG]F)$!+F*e7]#F)7$$\"*\\L=q)F)$!+7r4(\\&F)7 $$\"+-&3It#F)$!+l`TYSF)7$$\"+kT(zv%F)$!+&4S3n#F)7$$\"+VY)Hj$F)$!+V\"=g k$F)7$$\"+aPp_KF)$!+*=X!*=F)$!+^GjDTF)7$$\"+iZT#H$F)$!+h8^iPF)7$$\"+72hfKF)$!+ wN(QC$F)7$$\"+1Vi=CF)$!+*)*y/e$F)7$$\"*hp(=HF)$!+Mv0sgF)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "26 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }