{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 } {CSTYLE "" -1 256 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 " " 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Ti mes" 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 Plot" -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 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 21 "Mth 351 Least Squares" }}{PARA 257 "" 0 "" {TEXT 259 19 "Mth 351 Summer 2002" }}{PARA 0 "" 0 "" {TEXT 260 26 "July 28 2002 Maple 5 and 6" }}{PARA 0 "" 0 "" {TEXT 257 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 258 36 "Filename: 35 1u2002_least_squares.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "Maple has support for le ast squares fitting in the stats[fit] package and also in the linal g package. The stats[fit] package is the more convenient one to use for fitting data to a given equation." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(stats[fit]):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "It will also be useful to have some nice plot commands available:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name c hangecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We will need some random data to i llustrate the commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Xdata:=[seq(k/2,k=0..20)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&XdataG77\"\"!#\"\"\"\"\"#F(#\"\"$F) F)#\"\"&F)F+#\"\"(F)\"\"%#\"\"*F)F-#\"#6F)\"\"'#\"#8F)F/#\"#:F)\"\")# \"#F)\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Ydat a:=map(x->x+sin(x),evalf(Xdata));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %&YdataG77$\"\"!F'$\"+'QbUz*!#5$\"+&)4ZT=!\"*$\"+()\\\\(\\#F-$\"+FuH4H F-$\"+W@Z)4$F-$\"+3+7TJF-$\"+sn@\\JF-$\"+0v>VKF-$\"+#))pC_$F-$\"+Dd2TS F-$\"+u'fWz%F-$\"+-Xe?dF-$\"+))*>^r'F-$\"+*f')pl(F-$\"+x***zV)F-$\"+Z# e$*)*)F-$\"+8r[)H*F-$\"+&[=@T*F-$\"+!))[[U*F-$\"+*))yfX*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "These da ta points all lie on the graph of x + sin(x). Here's a plot of the g raph for comparison with the least squares fits that we obtain below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot(x+sin(x),x=0..10,title=\"x+sin(x)\",thickness=3, color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6( -%'CURVESG6#7S7$$\"\"!F)F(7$$\"3emmm;arz@!#=$\"37L8B)\\6AM%F-7$$\"3[LL $e9ui2%F-$\"3#z_??v'fS!)F-7$$\"3nmmm\"z_\"4iF-$\"3DS&=A7&p-7!#<7$$\"39 ommT&phN)F-$\"3cW]l5uKx:F:7$$\"3KLLe*=)H\\5F:$\"3_#>U5+sj\">F:7$$\"3sm m\"z/3uC\"F:$\"3#4S$**z@d&>#F:7$$\"3!****\\7LRDX\"F:$\"3ay3!y6bbW#F:7$ $\"3%om;zR'ok;F:$\"3Uav;i>GgEF:7$$\"33++D1J:w=F:$\"3hNinfG*)HGF:7$$\"3 oLLL3En$4#F:$\"3+lY3L!f+'HF:7$$\"3#pmmT!RE&G#F:$\"3VX_)Q512/$F:7$$\"3D +++D.&4]#F:$\"3!\\wm@&3m)4$F:7$$\"3;+++vB__2rJ*RJF:7$$\"3&om;z*ev:JF:$\"3`uK=z(*eTJF:7$$\"3_LL L347TLF:$\"3ZFlQTR\"H9$F:7$$\"3nLLLLY.KNF:$\"33o&)GkuV^JF:7$$\"33++D\" o7Tv$F:$\"3s_om1:=zJF:7$$\"3?LLL$Q*o]RF:$\"3Lknx2`-FKF:7$$\"3m++D\"=lj ;%F:$\"3@$GEX4pq4q%RVF:7$$\"3:++v=> Y2aF:$\"3dU!ow#\\XRYF:7$$\"3Znm;zXu9cF:$\"3iJ=rP:)\\*\\F:7$$\"34+++]y) )GeF:$\"3C`.V6n0!R&F:7$$\"3H++]i_QQgF:$\"3%[vym#H-'z&F:7$$\"3b++D\"y%3 TiF:$\"3)[_%4\"p'**)>'F:7$$\"3+++]P![hY'F:$\"3-V4Y%o\"4[mF:7$$\"3iKLL$ Qx$omF:$\"3?(=I!HX6WqF:7$$\"3Y+++v.I%)oF:$\"3km!eB!G')\\uF:7$$\"3?mm\" zpe*zqF:$\"3;/%)>eI1&z(F:7$$\"3;,++D\\'QH(F:$\"3!e\"p/O!e59)F:7$$\"3%H L$e9S8&\\(F:$\"3^y-uqhVJ%)F:7$$\"3s++D1#=bq(F:$\"3wZgJRx^%p)F:7$$\"3\" HLL$3s?6zF:$\"3@!\\T>Fq&4*)F:7$$\"3a***\\7`Wl7)F:$\"3Uwn&ylG'*3*F:7$$ \"3enmmm*RRL)F:$\"3Z75xHT&4A*F:7$$\"3%zmmTvJga)F:$\"3H-udr6(fJ*F:7$$\" 3]MLe9tOc()F:$\"3)p'4(yu1hP*F:7$$\"31,++]Qk\\*)F:$\"3GApzc:52%*F:7$$\" 3![LL3dg6<*F:$\"3rt`yPy1A%*F:7$$\"3%ymmmw(Gp$*F:$\"3Q&\\Sz[\\ZU*F:7$$ \"3C++D\"oK0e*F:$\"3/k*fM&pSD%*F:7$$\"35,+v=5s#y*F:$\"3j]v&[hsBV*F:7$$ \"#5F)$\"3WI16*))yfX*F:-%&TITLEG6#Q)x+sin(x)6\"-%*THICKNESSG6#\"\"$-%' COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\"xFjzQ!6\"-%%VIEWG6 $;F(Fbz%(DEFAULTG" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "Let's try a leastsquares line fit:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eqn1:=y=m*x+b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"yG ,&*&%\"mG\"\"\"%\"xGF*F*%\"bGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "lsq1:=leastsquare[[x,y],eqn1,\{m,b\}]([Xdata,Ydata]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%lsq1G/%\"yG,&%\"xG$\"+rNy*y*!#5$\"+ [vSOEF+\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 194 "Note [x,y] tells Maple the names of the variables i n eqn1 and \{m,b\} tells Maple which parameters to adjust. If you \+ want to leave b as an undetermined parameter you could do it as fool ws:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "lsq1b:=leastsquare[[x,y],eqn1,\{m\}]([Xdata,Ydata]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&lsq1bG/%\"yG,&*&,&%\"bG$!+MYTj9 !#5$\"+J*fv,\"!\"*\"\"\"F1%\"xGF1F1F*F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "To plot the solution ls q1 we note we just want to plot the right-hand side of the equation \+ lsq1. Let's plot it together with x + sin(x) so we can compare them. 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First we need the \+ Cartesian coordinates corresponding to our data:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "L:=[]: for \+ k from 1 to nops(Xdata) do L:=[op(L),[Xdata[k],Ydata[k]]]: od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(L,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#777$$\"\"!F&F%7$$\"%+]!\"%$\"%%z*F*7$$\"\"\"F&$\"% T=!\"$7$$\"%+:F2$\"%(\\#F27$$\"\"#F&$\"%4HF27$$\"%+DF2$\"%)4$F27$$\"\" $F&$\"%TJF27$$\"%+NF2$\"%\\JF27$$\"\"%F&$\"%VKF27$$\"%+XF2$\"%ANF27$$ \"\"&F&$\"%TSF27$$\"%+bF2$\"%%z%F27$$\"\"'F&$\"%@dF27$$\"%+lF2$\"%:nF2 7$$\"\"(F&$\"%dwF27$$\"%+vF2$\"%Q%)F27$$\"\")F&$\"%*)*)F27$$\"%+&)F2$ \"%)H*F27$$\"\"*F&$\"%7%*F27$$\"%+&*F2$\"%D%*F27$$\"#5F&$\"%c%*F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "img0:=PLOT(POINTS(op(L)),SYM BOL(CROSS,20),COLOR(RGB,0,0,1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "img1:=plot(rhs(lsq1),x=0..10,color=red,thickness=3,ti tle=\"Line fit\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "displ ay([img0,img1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'POINTSG697$\"\"!$F'F'7$#\"\"\"\"\"#$\"+'QbUz*!#57$F+$\"+&)4ZT= !\"*7$#\"\"$F,$\"+()\\\\(\\#F37$F,$\"+FuH4HF37$#\"\"&F,$\"+W@Z)4$F37$F 6$\"+3+7TJF37$#\"\"(F,$\"+sn@\\JF37$\"\"%$\"+0v>VKF37$#\"\"*F,$\"+#))p C_$F37$F>$\"+Dd2TSF37$#\"#6F,$\"+u'fWz%F37$\"\"'$\"+-Xe?dF37$#\"#8F,$ \"+))*>^r'F37$FF$\"+*f')pl(F37$#\"#:F,$\"+x***zV)F37$\"\")$\"+Z#e$*)*) F37$#\"#F,$\"+!))[[U*F37$\"#5$\" +*))yfX*F3-%&COLORG6&%$RGBGF'F'F+-%'SYMBOLG6$%&CROSSG\"#?-%'CURVESG6%7 S7$F($\"3))*****za2kj#!#=7$$\"3emmm;arz@Fjq$\"3'[(Qbl(4FL$*=F_s7$$\"33++D1J:w= F_s$\"3*poC//a.5#F_s7$$\"3oLLL3En$4#F_s$\"3$*Q9@D4I8BF_s7$$\"3#pmmT!RE &G#F_s$\"3P:5CdY'3]#F_s7$$\"3D+++D.&4]#F_s$\"3^6s:&*p,7FF_s7$$\"3;+++v B_e$)QRF_s7$$\"3 ?LLL$Q*o]RF_s$\"3!)y!3n:!GJTF_s7$$\"3m++D\"=lj;%F_s$\"3up(=]4ACM%F_s7$ $\"3S++vV&Rz%45:zMa%F_s7$$\"3CML$e9Ege%F_s$\"3%Gclm4hKv%F_s 7$$\"3]LLeR\"3Gy%F_s$\"3o*z8'4k!f%\\F_s7$$\"3emm;/T1&*\\F_s$\"3[mi50/q `^F_s7$$\"3=nm\"zRQb@&F_s$\"3gCW)p'*R&p`F_s7$$\"3:++v=>Y2aF_s$\"3[)o)4 S*Gub&F_s7$$\"3Znm;zXu9cF_s$\"3!\\r[%yTNgdF_s7$$\"34+++]y))GeF_s$\"3G^ J\"f!e**pfF_s7$$\"3H++]i_QQgF_s$\"3'H5>'Q#*3vhF_s7$$\"3b++D\"y%3TiF_s$ \"3MQ*p/oFNP'F_s7$$\"3+++]P![hY'F_s$\"3-RrJP(fQf'F_s7$$\"3iKLL$Qx$omF_ s$\"3gi%e+*y$=z'F_s7$$\"3Y+++v.I%)oF_s$\"3j9/qD=A.qF_s7$$\"3?mm\"zpe*z qF_s$\"3y7BA*3nZ>(F_s7$$\"3;,++D\\'QH(F_s$\"3U;e)flwTS(F_s7$$\"3%HL$e9 S8&\\(F_s$\"3j=6kQZ@,wF_s7$$\"3s++D1#=bq(F_s$\"3Zk(e*3j<2yF_s7$$\"3\"H LL$3s?6zF_s$\"3JW=H!RT&3!)F_s7$$\"3a***\\7`Wl7)F_s$\"3f:J!*o>N>#)F_s7$ $\"3enmmm*RRL)F_s$\"3[jt`6wQA%)F_s7$$\"3%zmmTvJga)F_s$\"3&o^=7)3-I')F_ s7$$\"3]MLe9tOc()F_s$\"3XK\\fU[$f$))F_s7$$\"31,++]Qk\\*)F_s$\"3*37.xQ[ ^-*F_s7$$\"3![LL3dg6<*F_s$\"3e;Z8j%3?C*F_s7$$\"3%ymmmw(Gp$*F_s$\"3#HY3 )*pqfV*F_s7$$\"3C++D\"oK0e*F_s$\"3D%)HB**[xU'*F_s7$$\"35,+v=5s#y*F_s$ \"3k:Nq0HrS)*F_s7$$FgpF'$\"3=++eKCM05!#;-%'COLOURG6&F]q$\"*++++\"!\")F (F(-%*THICKNESSG6#F6-%&TITLEG6#Q)Line~fit6\"-%+AXESLABELSG6$Q\"xFialQ! 6\"-%%VIEWG6$;F(Fh`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Looking at the data you may feel a polynomial of degree 4 ought to fit better. 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Extrapolat ion is definitely out. For example the degree 4 polynomial here decrea ses after x = 10 whereas x + sin(x) continues to increase! If you we re dealing with experimental data the good fit might mislead you! You \+ really need to have an underlying model which implies a certain functi onal relation before a least squares fit with that function allows you to reasonably extrapolate or draw any conclusions other than statisti cal ones concerning properties of the actual data." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 21 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }