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"" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 18 "Least Squares Line" }} {PARA 19 "" 0 "" {TEXT -1 14 "Brian E. Blank" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 " Scatter Pl ots - Hubble Data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "If N observations " }{XPPEDIT 18 0 "[xi[j], eta[j]] ,` `*j = 1 .. N;" "6$7$&%#xiG6#%\"jG&%$etaG6#F'/*&%&~~~~~G\"\"\"F' F.;F.%\"NG" }{TEXT -1 93 " are made of two variables then the resul ting plot of the observations is often called a " }{TEXT 256 12 "scat ter plot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 7 "Example" }{TEXT 258 1 ":" }{TEXT -1 2 " " }{TEXT 259 19 "Random scatter plot" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 348 "Here is \+ a list of fifteen pairs of observations based on data associated with \+ fifteen randomly selected students in a differential equations class. \+ The first number in each pair is the two digit number formed from the \+ last two digits of a student's ID number. The second number in the pai r is that student's final score, scaled to a maximum of 100." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "randomList := [ [35,53.8], [24,93. 8], [36,91.1], [05,55.0], [83,28.5], [91,69.4], [48,86.3], [86,60.0], \+ [27,57.5], [46,72.5], [75,66.4], [14,72.5], [49,88.4], [55,71.3], [46, 75.0]]; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+randomListG717$\"#N$\"$ Q&!\"\"7$\"#C$\"$Q*F*7$\"#O$\"$6*F*7$\"\"&$\"$]&F*7$\"#$)$\"$&GF*7$\"# \"*$\"$%pF*7$\"#[$\"$j)F*7$\"#')$\"$+'F*7$\"#F$\"$v&F*7$\"#Y$\"$D(F*7$ \"#v$\"$k'F*7$\"#9FM7$\"#\\$\"$%))F*7$\"#b$\"$8(F*7$FL$\"$](F*" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 25 "Here is the scatter plot:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot( randomList, style=POINT, symbol=circle, color=NAVY );" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6#717$$\"#N\" \"!$\"1++++++!Q&!#97$$\"#CF*$\"1++++++!Q*F-7$$\"#OF*$\"1************4 \"*F-7$$\"\"&F*$\"#bF*7$$\"#$)F*$\"1++++++]GF-7$$\"#\"*F*$\"1,+++++SpF -7$$\"#[F*$\"1++++++I')F-7$$\"#')F*$\"#gF*7$$\"#FF*$\"1++++++]dF-7$$\" #YF*$\"1++++++]sF-7$$\"#vF*$\"1,+++++SmF-7$$\"#9F*FY7$$\"#\\F*$\"1,+++ ++S))F-7$F;$\"1++++++IrF-7$FWFfn-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGB G$\")!\\DP\"!\")F^p$\")viobF`p-%&STYLEG6#%&POINTG" 2 375 375 375 5 4 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "The " } {TEXT 260 11 "style=POINT" }{TEXT -1 190 " option is used when you wan t to see only the observed values (and not have then connected by line segments). Without specifying that plot option we get the data conne cted by line segments:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot( r andomList, symbol=circle, color=NAVY );" }}{PARA 13 "" 1 "" {INLPLOT " 6%-%'CURVESG6#717$$\"#N\"\"!$\"1++++++!Q&!#97$$\"#CF*$\"1++++++!Q*F-7$ $\"#OF*$\"1************4\"*F-7$$\"\"&F*$\"#bF*7$$\"#$)F*$\"1++++++]GF- 7$$\"#\"*F*$\"1,+++++SpF-7$$\"#[F*$\"1++++++I')F-7$$\"#')F*$\"#gF*7$$ \"#FF*$\"1++++++]dF-7$$\"#YF*$\"1++++++]sF-7$$\"#vF*$\"1,+++++SmF-7$$ \"#9F*FY7$$\"#\\F*$\"1,+++++S))F-7$F;$\"1++++++IrF-7$FWFfn-%'SYMBOLG6# %'CIRCLEG-%'COLOURG6&%$RGBG$\")!\\DP\"!\")F^p$\")viobF`p" 2 375 375 375 2 4 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " The other options are..optional. See " }{HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 40 " for details of the available opti ons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 8 "E xample:" }{TEXT -1 4 " " }{TEXT 288 20 "(Edwin Hubble Data) " } {TEXT -1 2 "\n " }{TEXT 287 340 " Virgo Pegasus Perseus Ur sa Major 1 Leo Gemini Bootes Hydra Coma Berenices \n d \+ 22 68 108 255 315 \+ 405 685 1100 137 \n v 0.75 2.4 3.2 9.3 12.0 14.4 24 .5 38.0 \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Hub bleData := [[22,0.75],[68,2.4],[108,3.2],[255,9.3],[315,12.0],[405,14. 4],[685,24.5],[1100,38.0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+Hubb leDataG7*7$\"#A$\"#v!\"#7$\"#o$\"#C!\"\"7$\"$3\"$\"#KF/7$\"$b#$\"#$*F/ 7$\"$:$$\"$?\"F/7$\"$0%$\"$W\"F/7$\"$&o$\"$X#F/7$\"%+6$\"$!QF/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "HubbleDataPlot := plot(Hubbl eData,style=POINT,symbol=CIRCLE,color=NAVY):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plots[display](HubbleDataPlot);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6&7*7$$\"#A\"\"!$\"1+++++++v!#;7$$\"#oF*$\" 1+++++++C!#:7$$\"$3\"F*$\"1+++++++KF37$$\"$b#F*$\"1,++++++$*F37$$\"$:$ F*$\"#7F*7$$\"$0%F*$\"1++++++S9!#97$$\"$&oF*$\"1++++++]CFH7$$\"%+6F*$ \"#QF*-%'COLOURG6&%$RGBG$\")!\\DP\"!\")FW$\")viobFY-%'SYMBOLG6#%'CIRCL EG-%&STYLEG6#%&POINTG" 2 375 375 375 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 14880 8253 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The plotted points clearly lie near a line " }{XPPEDIT 18 0 "y = m*x+b;" "/%\"yG,&*&%\"mG\"\"\"%\"xGF'F'%\"bGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 64 " The B est-Fitting Line (aka Least Squares Line, Regression Line)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 38 "We can measure the vertical distance " }{XPPEDIT 18 0 "abs(m*x[j]+b-y[j]);" "-%$absG6#,(*&%\"mG\"\"\"&%\"xG6#%\"jGF(F(%\"bGF (&%\"yG6#F,!\"\"" }{TEXT -1 27 " by which a data point " } {XPPEDIT 18 0 "[x[j], y[j]];" "7$&%\"xG6#%\"jG&%\"yG6#F&" }{TEXT -1 29 " misses the proposed line\n " }{XPPEDIT 18 0 "y = m*x+b;" "/%\"y G,&*&%\"mG\"\"\"%\"xGF'F'%\"bGF'" }{TEXT -1 92 ". We use absolute valu es so that errors on each side of the line do not tend to cancel out. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The sum of the absolute errors is\n \n " }{XPPEDIT 18 0 "SAE = Sum(abs(m*x[j ]+b-y[j]),j = 1 .. N);" "/%$SAEG-%$SumG6$-%$absG6#,(*&%\"mG\"\"\"&%\"x G6#%\"jGF-F-%\"bGF-&%\"yG6#F1!\"\"/F1;F-%\"NG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The idea is to choose " } {XPPEDIT 18 0 "m;" "I\"mG6\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "b; " "I\"bG6\"" }{TEXT -1 53 " so as to minimize this expression (or one like it)." }}{PARA 0 "" 0 "" {TEXT -1 115 "However, the absolute valu e function is not differentiable and so the tools of calculus fail. \n Instead, we use the " }{TEXT 309 25 "sum of the squared errors" } {TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 32 " \n \+ " }{XPPEDIT 18 0 "SSE = Sum((m*x[j]+b-y[j])^2,j = 1 .. N); " "/%$SSEG-%$SumG6$*$,(*&%\"mG\"\"\"&%\"xG6#%\"jGF+F+%\"bGF+&%\"yG6#F/ !\"\"\"\"#/F/;F+%\"NG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Let us apply the ideas of multivariable calculus to find the critical point:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "N := 'N': # Restores the \"literal\" (unassi gned) value of N" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "SSE := \+ sum((m*x[j]+b-y[j])^2,j = 1 .. N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$SSEG,&*&%\"NG\"\"\")%\"bG\"\"#F(F(-%$sumG6$,,*&)%\"mGF+F()&%\"xG6#% \"jGF+F(F(**F+F(F2F(F4F(F*F(F(**F+F(F2F(F4F(&%\"yGF6F(!\"\"*(F+F(F*F(F :F(F<*$)F:F+F(F(/F7;F(F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CriticalPointEquation1 := diff(SSE,m) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%7CriticalPointEquation1G/-%$sumG6$,(*(\"\"#\"\"\"%\"m GF,)&%\"xG6#%\"jGF+F,F,*(F+F,F/F,%\"bGF,F,*(F+F,F/F,&%\"yGF1F,!\"\"/F2 ;F,%\"NG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CriticalPo intEquation2 := diff(SSE,b) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 7CriticalPointEquation2G/,&*(\"\"#\"\"\"%\"NGF)%\"bGF)F)-%$sumG6$,&*(F (F)%\"mGF)&%\"xG6#%\"jGF)F)*&F(F)&%\"yGF4F)!\"\"/F5;F)F*F)\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve( \{CriticalPointEquati on1, CriticalPointEquation2\}, \{m,b\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"bG,$*&,&*&-%$sumG6$&%\"xG6#%\"jG/F0;\"\"\"%\"NGF3 -F+6$*&F-F3&%\"yGF/F3F1F3F3*&-F+6$F8F1F3-F+6$*$)F-\"\"#F3F1F3!\"\"F3,& *&F4F3F=F3F3*$)F*FAF3FBFBFB/%\"mG*&,&*&F4F3F5F3F3*&F*F3F;F3FBF3FCFB" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "The line " }{XPPEDIT 18 0 "y = m*x+b;" " /%\"yG,&*&%\"mG\"\"\"%\"xGF'F'%\"bGF'" }{TEXT -1 38 " that results fr om these values of " }{XPPEDIT 18 0 "m;" "I\"mG6\"" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "b;" "I\"bG6\"" }{TEXT -1 16 " is called the " } {TEXT 312 18 "least squares line" }{TEXT -1 24 ". It is also called th e " }{TEXT 311 17 "best-fitting line" }{TEXT -1 28 " and, in some cont exts, the " }{TEXT 310 15 "regression line" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "In the next se ction we use these formulas to write a command that outputs the equati on of the least squares line." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 " Code for " }{TEXT 302 16 "leastSquaresLine" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1855 "leastSquaresLine := proc()\nlocal jj, NN, xData, yD ata, xx, yy, mm, bb:\n\nif nargs < 2 or nargs > 3 then\nERROR(`leastSq uaresLine expects two or three arguments`):\nfi:\n\nif not type(args[n args], list) then\nERROR(`leastSquaresLine expects its last argument t o be a list`):\n\nelif not nops(args[nargs]) = 2 then\nERROR(`leastSqu aresLine expects its last argument to have the form [x,y]`):\n\nelif n ot type(args[nargs][1], name) or not type(args[nargs][2], name) then \+ \nERROR(`leastSquaresLine expects the entries of its last argument to \+ be names`):\n\nelse\nxx := args[nargs][1]:\nyy := args[nargs][2]:\nfi: \n\n\nif nargs = 3 then\n if not type(args[1],list) or not type(args[ 2],list) then\n ERROR(`leastSquaresLine expects a list for each of it s arguments`):\n elif nops(args[1]) <> nops(args[2]) then\n ERROR(`u nequal sizes for lists`):\n else\n NN := nops(args[1]):\n for jj fr om 1 to NN do\n xData[jj] := args[1][jj]:\n yData[jj] := args[2][jj] :\n od:\n fi:\n jj := 'jj':\nelse\n if not type(args[1],list) then \n ERROR(`leastSquaresLine expects a list for each of its arguments`) :\n fi:\n\n NN := nops(args[1]):\n \n for jj from 1 to NN do\n i f not type(args[1][jj],list) then\n ERROR(`each entry in the first a rgument should have the form [x,y]`):\n elif nops(args[1][jj])<> 2 t hen\n ERROR(`each entry in the first argument should have the form [ x,y]`):\n fi:\n\n xData[jj] := args[1][jj][1]:\n yData[jj] := ar gs[1][jj][2]:\n od:\njj := 'jj':\nfi:\n\nmm := (NN*sum(xData[jj]*yDat a[jj],jj = 1 .. NN)-sum(xData[jj],jj = 1 .. NN)*sum(yData[jj],jj = 1 . . NN))/(NN*sum(xData[jj]^2,jj = 1 .. NN)-sum(xData[jj],jj = 1 .. NN)^2 ):\njj := 'jj':\n\nbb := -(sum(xData[jj],jj = 1 .. NN)*sum(xData[jj]*y Data[jj],jj = 1 .. NN)-sum(yData[jj],jj = 1 .. NN)*sum(xData[jj]^2,jj \+ = 1 .. NN))/(NN*sum(xData[jj]^2,jj = 1 .. NN)-sum(xData[jj],jj = 1 .. \+ NN)^2): \n \nRETURN(yy = mm*xx + bb):\nend:\n\n\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 " Description of \+ " }{TEXT 303 16 "leastSquaresLine" }}{PARA 3 "" 0 "" {TEXT 304 16 "le astSquaresLine" }{TEXT 30 50 " - calculate the line that best-fits a s catterplot" }}{PARA 4 "" 0 "usage" {TEXT -1 16 "Calling Sequence" }} {PARA 0 "" 0 "" {TEXT -1 22 " leastSquaresLine(" }{TEXT 35 1 "L" } {TEXT -1 2 ", " }{TEXT 35 1 "v" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 22 " leastSquaresLine(" }{TEXT 35 2 "L1" }{TEXT -1 2 ", \+ " }{TEXT 35 2 "L2" }{TEXT -1 2 ", " }{TEXT 35 1 "v" }{TEXT -1 1 ")" }} {PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "L - " }{TEXT -1 20 "list of data points " }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "L1 - " }{TEXT -1 14 "l ist of values" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "L2 - \+ " }{TEXT -1 14 "list of values" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {TEXT 23 38 "v - an ordered pair of variable names" }}{SECT 0 {PARA 4 "" 0 "info" {TEXT -1 11 "Description" }}{PARA 15 "" 0 "" {TEXT -1 18 "A typical call to " }{TEXT 305 17 "leastSquaresLine " }{TEXT -1 4 " is " }{TEXT 35 54 "leastSquaresLine([[x1,y1],[x2,y2],...,[xN,yN]], [ x,y])" }{TEXT -1 41 ". The return is an equation of the form " } {XPPEDIT 18 0 "y = m*x+b;" "/%\"yG,&*&%\"mG\"\"\"%\"xGF'F'%\"bGF'" } {TEXT -1 9 " where " }{XPPEDIT 18 0 "m;" "I\"mG6\"" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "b;" "I\"bG6\"" }{TEXT -1 58 " are calculated from the data in the first argument and " }{XPPEDIT 18 0 "x;" "I\"xG6\"" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "y;" "I\"yG6\"" }{TEXT -1 38 " the \+ names given in the last argument." }}{PARA 15 "" 0 "" {TEXT -1 10 "The call " }{TEXT 35 55 "leastSquaresLine([x1, x2,...,xN],[y1,y2,...,yN] , [x,y])" }{TEXT -1 27 " is equivalent to the call" }{TEXT 35 54 "lea stSquaresLine([[x1,y1],[x2,y2],...,[xN,yN]], [x,y])" }{TEXT -1 3 ". \+ " }}{PARA 15 "" 0 "" {TEXT -1 88 "The return is in exact form if all t he entered data is. In such cases the composed call " }{TEXT 35 63 " ( evalf@leastSquaresLine)([[x1,y1],[x2,y2],...,[xN,yN]], [x,y])" }{TEXT -1 17 " may be desired." }}}{SECT 0 {PARA 4 "" 0 "examples" {TEXT -1 10 "Examples\n\n" }}{PARA 0 "" 0 "" {TEXT -1 41 "Recall the transforme d weight-pulse data " }{TEXT 306 1 "X" }{TEXT -1 5 " and " }{TEXT 307 1 "Y" }{TEXT -1 51 " from an earlier section. The paired list of data " }{TEXT 308 2 "XY" }{TEXT -1 19 " was also defined." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "(evalf@leastSquaresLine)(HubbleData , [d,v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"vG,&%\"dG$\"+\\KX&[$! #6$\"+zg'G\"=!#5\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 " leastSquaresLinePlot := \n plot(rhs((evalf@leastSquaresLine)(HubbleD ata, [d,v])), d = 22..1100,numpoints=600):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "plots[display](HubbleDataPlot,leastSquaresLinePlot) ;" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6&7*7$$\"#A\"\"!$\"1++++++ +v!#;7$$\"#oF*$\"1+++++++C!#:7$$\"$3\"F*$\"1+++++++KF37$$\"$b#F*$\"1,+ +++++$*F37$$\"$:$F*$\"#7F*7$$\"$0%F*$\"1++++++S9!#97$$\"$&oF*$\"1+++++ +]CFH7$$\"%+6F*$\"#QF*-%'COLOURG6&%$RGBG$\")!\\DP\"!\")FW$\")viobFY-%' SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-F$6$7d`m7$F($\"1++!oAj3[*F-7$$\" 1^0LyCH)Q#FH$\"1[fr&y9P,\"F37$$\"1`:$pvC@b#FH$\"1U+\\gx\"32\"F37$$\"1. :!*)Grjt#FH$\"11s$y-O]8\"F37$$\"1ST[E\"R=#HFH$\"1Zfav*z'*>\"F37$$\"1$o 4yYDk5$FH$\"1g`&=o;SE\"F37$$\"1e[\"\\jgvF$FH$\"1]eFW]mB8F37$$\"1Fl\"z= hZX$FH$\"1wUfouU&Q\"F37$$\"1k)=LDA!QOFH$\"1aksH^\"F37$$\"1'*yt'e(f3SFH$\"1;H;cXYy:F37$$\"1dUbg<5uTFH$\"1X !)>i-:O;F37$$\"1CXrg-UgVFH$\"1'GrVq!4,'3&FH$\"1q)\\rRbS&>F37$$\"1].@a=6^_FH$\"1))>@&4P:,#F37$$\" 10I!ep^HW&FH$\"1k'G^(>Sy?F37$$\"1/FihCw7cFH$\"14GnB()eP@F37$$\"1waG&yr !*z&FH$\"1\\2Unf_-AF37$$\"1\\&H(>B[wfFH$\"1U#*yz:OkAF37$$\"1DZ$3g\"fhh FH$\"1]SJa0))GBF37$$\"1AHv0(z:L'FH$\"1**=d8\"H\")Q#F37$$\"1.D]T[$\\^'F H$\"1?Q$*)oO?X#F37$$\"12U_'H*Q0nFH$\"17!Q))p=%=DF37$$\"1Fe\\f/=roFH$\" 1/:gLW?wDF37$$\"10D]d%R-0(FH$\"15!Q-h9'QEF37$$\"1l*yL#\\ANsFH$\"1*pV&H /4.FF37$$\"1APB7r>;uFH$\"1Wf!*Ru;mFF37$$\"1zsO5qH\"f(FH$\"1i3z!p(>FGF3 7$$\"1;#HNf:dy(FH$\"1Pc-z8'\\*GF37$$\"1DghP$4/'zFH$\"1rj\"o1]e&HF37$$ \"1q<1(>Kp9)FH$\"1mJYP<'3-$F37$$\"1XuYI'\\fJ)FH$\"15Lqt>xzIF37$$\"1BU` G.t+&)FH$\"1Wi9Ak\"oKF37$$\"1[%p;7:S.*FH$\"1\"zLOM]+L$F37$$\"1J'yr0K+A*F H$\"1Tx&on&)[R$F37$$\"1&=6ZZ)=*R*FH$\"1YXh3)HtX$F37$$\"1bHx(y,Ce*FH$\" 1l;M_z=@NF37$$\"1n,5-!)4k(*FH$\"1VuR=t^%e$F37$$\"1Z!GG&z0J**FH$\"1\"=p 'G/rUOF37$$\"1(Hy(*GTA,\"!#8$\"1<(H,I1%4PF37$$\"1_4gM n'>SF37$$\"1*QL5$zJ?6F^]l$\"1$GL0Q-h3%F37$$\"1m&R@l*oP6F^]l$\"1HNI-2lY TF37$$\"1\"fa\\e\"pc6F^]l$\"1Q4x]5)G@%F37$$\"15jQRq3t6F^]l$\"12))oSm-q UF37$$\"1l*y\"[&f<>\"F^]l$\"1Xr7(z3^L%F37$$\"19%[)piz47F^]l$\"1N7FZ\\( zR%F37$$\"1SRc37#yA\"F^]l$\"1\"H![G+!3Y%F37$$\"1q>yO)zdC\"F^]l$\"11/b( )RRBXF37$$\"13\\McC.j7F^]l$\"15)fk=FNe%F37$$\"1cO*\\*Ho\"G\"F^]l$\"1*o ))pwK&[YF37$$\"1O9m5$e%*H\"F^]l$\"1_\\t)z([5ZF37$$\"19$)Q$*f;=8F^]l$\" 16mF_DpvZF37$$\"1Wm=^45N8F^]l$\"1![cfb=Z$[F37$$\"1D]\")\\s\"QN\"F^]l$ \"1B+#)RL&***[F37$$\"1$ypC?W&*oVi\\F37$$\"1L+U?vi*Q\"F^ ]l$\"1=y%QOoZ-&F37$$\"1?AL:nI39F^]l$\"10KWCQ()*3&F37$$\"1\"fad$4^D9F^] l$\"1p3TI$Q)\\^F37$$\"1#Hv2TIJW\"F^]l$\"1Yu'o=]7@&F37$$\"1n*zZS%ei9F^] l$\"1lqZth0z_F37$$\"1&zwGS)>![\"F^]l$\"1N:PS*[/M&F37$$\"1*o8_,7#)\\\"F ^]l$\"1\"R'\\RJl^\"F^]l$\"1%e!o&)e3naF37$$\"1;$*yp/PL :F^]l$\"1Eyu$pxd_&F37$$\"13^Ey.K^:F^]l$\"1JScg7M)e&F37$$\"1&yq3lL\"p:F ^]l$\"12GW*eG/l&F37$$\"1QD_:$)=)e\"F^]l$\"1l=EKF%or&F37$$\"1`?jJK+0;F^ ]l$\"1SuCO.XvdF37$$\"10I?aTSC;F^]l$\"1gf&[MrI%eF37$$\"1$zv5*>\">k\"F^] l$\"1!=n<4%4/fF37$$\"1TX_!3N#f;F^]l$\"1P!p,\"HZkfF37$$\"1gfxmH'yn\"F^] l$\"1Zg#o`*RHgF37$$\"17rm1ic'p\"F^]l$\"1!\\r]!))e%4'F37$$\"1^5j&Q$e86`'F37$$\"13^m!f(\\R=F^]l$\"1u8\"=YpFf'F37$$\"1m% z7N*>d=F^]l$\"11x&37oWl'F37$$\"1(*zfESDv=F^]l$\"1>jgZoR\">F^]l$\"1,MF^]l$\"1IZrvRj0pF37$$\"1%fc`\"pxZ>F^]l$\"1Ro&e)>F^]l$\"1Z>R-&pX.(F37$$\"1(RQiPOG)>F^]l$\"13bp(4qB4(F37$$\"1aEf&e![-?F ^]l$\"1RL,r!R3;(F37$$\"1[!H!\\z\"*=?F^]l$\"17x!GtI\"=sF37$$\"1#4b=HV\" Q?F^]l$\"1%)QUm(R^G(F37$$\"1J&=\">?\\c?F^]l$\"1`#zuR$4\\tF37$$\"1J'ypC vG2#F^]l$\"1YLDsj>1uF37$$\"1Ec<+**H\"4#F^]l$\"170gRYTquF37$$\"1!*Q$RoY )4@F^]l$\"1^#4tee]`(F37$$\"1Wk1=`IG@F^]l$\"1m'=OH&R*f(F37$$\"1hpxM)=a9 #F^]l$\"1`\"RglV!fwF37$$\"1Gr2!*)QJ;#F^]l$\"1\"ed.313s(F37$$\"1ith'*\\ Y\"=#F^]l$\"1e(Rz'3o%y(F37$$\"1-:5JBt*>#F^]l$\"1v!o#33N[yF37$$\"1l#f*H D_=AF^]l$\"1?i#z;VQ\"zF37$$\"1,4MZH2NAF^]l$\"1]8'R()G:(zF37$$\"1GpN(z/ PD#F^]l$\"1$fMhJpk.)F37$$\"1K#R::8CF#F^]l$\"1YF+5kn,\")F37$$\"18wO<@W! 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\"1r$e]r)4ePFH7$$\"1#>&p$*[vu5Fciu$\"1c&)*ofOTw$FH7$$\"1M.K#o3\"Fciu$\"1n:Fx\"y$*y$FH7$$\"1RJkt/w$3\" Fciu$\"1csyC^_&z$FH7$$\"1rBKNAn&3\"Fciu$\"1\\Mu![)=-QFH7$$\"1.J/H,3#QFH7$$\"1\\#4Z9\"y#4\"Fciu$\"1!zi?Dmp#QFH7$$\"1%oq2 K^X4\"Fciu$\"1Bu'z6OJ$QFH7$$\"1QDI)ycj4\"Fciu$\"1&GU1**G%RQFH7$$\"1L+k `0F)4\"Fciu$\"1$ob-L*4YQFH7$FO$\"1++pMs7_QFH-FT6&FV$\"#5!\"\"F*F*" 2 365 365 365 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 8233 8253 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 " Code for " }{TEXT 313 20 "leastSquaresLinePlot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2754 " leastSquaresLinePlot := proc()\nlocal jj, NN, xData, yData, xx, yy, mm , bb, xyData, \n xMin, xMax, yMin, yMax, scatterPlot, bestLinePlot: \n\nif nargs < 2 or nargs > 3 then\nERROR(`leastSquaresLinePlot expect s two or three arguments`):\nfi:\n\nif not type(args[nargs], list) the n\nERROR(`leastSquaresLine expects its last argument to be a list`):\n elif not nops(args[nargs]) = 2 then\nERROR(`leastSquaresLinePlot expec ts its last argument to have the form [x,y]`):\nelif not type(args[nar gs][1], name) or not type(args[nargs][2], name) then \nERROR(`leastSqu aresLinePlot expects the entries of its last argument to be names`):\n else\nxx := args[nargs][1]:\nyy := args[nargs][2]:\nfi:\n\n\nif nargs \+ = 3 then\n if not type(args[1],list) or not type(args[2],list) then\n ERROR(`leastSquaresLinePlot expects a list for each of its arguments `):\n elif nops(args[1]) <> nops(args[1]) then\n ERROR(`unequal size s for lists`):\n else\n NN := nops(args[1]):\n xyData := []:\n xMi n := args[1][1]:\n xMax := args[1][1]:\n yMin := args[2][1]:\n yMax := args[2][1]:\n for jj from 1 to NN do\n xData[jj] := args[1][jj]: \n xMin := min(xMin,xData[jj]):\n xMax := max(xMax,xData[jj]): \n \+ yData[jj] := args[2][jj]:\n yMin := min(yMin,yData[jj]):\n yMax := m ax(yMin,yData[jj]): \n xyData := [op(xyData),[xData[jj],yData[jj]]]: \n od:\n fi:\n jj := 'jj':\nelse\n if not type(args[1],list) then \n ERROR(`leastSquaresLine expects a list for each of its arguments`) :\n fi:\n\n NN := nops(args[1]):\n xMin := args[1][1][1]:\n xMax : = args[1][1][1]:\n yMin := args[1][1][2]:\n yMax := args[1][1][2]:\n \n for jj from 1 to NN do\n if not type(args[1][jj],list) then\n \+ ERROR(`each entry in the first argument should have the form [x,y]`): \n elif nops(args[1][jj])<> 2 then\n ERROR(`each entry in the firs t argument should have the form [x,y]`):\n fi:\n\n xData[jj] := ar gs[1][jj][1]:\n xMin := min(xMin,xData[jj]):\n xMax := max(xMax,xD ata[jj]):\n yData[jj] := args[1][jj][2]:\n yMin := min(yMin,yData[ jj]):\n yMax := max(yMin,yData[jj]): \n od:\n\nxyData := args[1] :\njj := 'jj':\nfi:\n\nmm := (NN*sum(xData[jj]*yData[jj],jj = 1 .. NN) -sum(xData[jj],jj = 1 .. NN)*sum(yData[jj],jj = 1 .. NN))/(NN*sum(xDat a[jj]^2,jj = 1 .. NN)-sum(xData[jj],jj = 1 .. NN)^2):\njj := 'jj':\n\n bb := -(sum(xData[jj],jj = 1 .. NN)*sum(xData[jj]*yData[jj],jj = 1 .. \+ NN)-sum(yData[jj],jj = 1 .. NN)*sum(xData[jj]^2,jj = 1 .. NN))/(NN*sum (xData[jj]^2,jj = 1 .. NN)-sum(xData[jj],jj = 1 .. NN)^2): \n\nyMin := min(yMin,mm*xMin+bb,mm*xMax+bb);\nyMax := max(yMax,mm*xMin+bb,mm*xMax +bb);\n\nscatterPlot := plot(xyData, style=POINT, symbol=circle, color =NAVY ):\nbestLinePlot := plot(mm*xx + bb,xx = xMin..xMax, yy = yMin.. yMax,numpoints=600,color=COLOR(RGB,.01,.6,.999)): \nRETURN( plots[dis play](scatterPlot,bestLinePlot) );\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 " Description of " }{TEXT 314 20 "leastSquaresLinePlot" }} {PARA 3 "" 0 "" {TEXT 315 21 "leastSquaresLinePlot " }{TEXT 30 61 " - \+ plots a scatterplot and superimposes the best-fitting line" }}{PARA 4 "" 0 "usage" {TEXT -1 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT -1 26 " leastSquaresLinePlot(" }{TEXT 35 1 "L" }{TEXT -1 2 ", " } {TEXT 35 1 "v" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 26 " lea stSquaresLinePlot(" }{TEXT 35 2 "L1" }{TEXT -1 2 ", " }{TEXT 35 2 "L2 " }{TEXT -1 2 ", " }{TEXT 35 1 "v" }{TEXT -1 1 ")" }}{PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "L - " }{TEXT -1 20 "list of data points " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "L1 - " }{TEXT -1 14 "list of values" } }{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "L2 - " }{TEXT -1 14 " list of values" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 38 "v \+ - an ordered pair of variable names" }}{SECT 0 {PARA 4 "" 0 "info" {TEXT -1 11 "Description" }}{PARA 15 "" 0 "" {TEXT -1 18 "A typical ca ll to " }{TEXT 316 21 "leastSquaresLinePlot " }{TEXT -1 4 " is " } {TEXT 35 58 "leastSquaresLinePlot([[x1,y1],[x2,y2],...,[xN,yN]], [x,y] )" }{TEXT -1 37 ". The return is the plot of the data " }{TEXT 35 29 " [[x1,y1],[x2,y2],...,[xN,yN]]" }{TEXT -1 51 " together with a plot of an equation of the form " }{XPPEDIT 18 0 "y = m*x+b;" "/%\"yG,&*&%\" mG\"\"\"%\"xGF'F'%\"bGF'" }{TEXT -1 9 " where " }{XPPEDIT 18 0 "m;" "I\"mG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b;" "I\"bG6\"" }{TEXT -1 58 " are calculated from the data in the first argument and " } {XPPEDIT 18 0 "x;" "I\"xG6\"" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "y;" "I\"yG6\"" }{TEXT -1 38 " the names given in the last argument." }} {PARA 15 "" 0 "" {TEXT -1 10 "The call " }{TEXT 35 59 "leastSquaresLi nePlot([x1, x2,...,xN],[y1,y2,...,yN], [x,y])" }{TEXT -1 28 " is equi valent to the call " }{TEXT 35 58 "leastSquaresLinePlot([[x1,y1],[x2,y 2],...,[xN,yN]], [x,y])" }{TEXT -1 3 ". " }}}{SECT 0 {PARA 4 "" 0 "ex amples" {TEXT -1 9 "Examples\n" }}{PARA 0 "" 0 "" {TEXT -1 41 "Recall \+ the transformed weight-pulse data " }{TEXT 317 1 "X" }{TEXT -1 5 " and " }{TEXT 318 1 "Y" }{TEXT -1 51 " from an earlier section. 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K#o3\"Fciu$\"1n:Fx\"y$*y$FH7$$ \"1RJkt/w$3\"Fciu$\"1csyC^_&z$FH7$$\"1rBKNAn&3\"Fciu$\"1\\Mu![)=-QFH7$ $\"1.J/H,3#QFH7$$\"1\\#4Z9\"y#4\"Fciu$\"1!zi?Dmp#Q FH7$$\"1%oq2K^X4\"Fciu$\"1Bu'z6OJ$QFH7$$\"1QDI)ycj4\"Fciu$\"1&GU1**G%R QFH7$$\"1L+k`0F)4\"Fciu$\"1$ob-L*4YQFH7$FO$\"1++pMs7_QFH-%&COLORG6&FV$ \"\"\"!\"#$\"\"'!\"\"$\"$***!\"$-%+AXESLABELSG6$%!G%\"vG-%%VIEWG6$%(DE FAULTG;$\"#vF\\[w$\"+Ns7_QFY" 2 365 365 365 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 489 8714 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " leastSquaresLinePlot(XY, [x,y]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The data \+ points can also be entered as separate lists of x-values and y-values ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 150 "C ountry Cigarette Consumption CHD Mortality per 100,00 0\n Per Adult per year: (a ges 35-64)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 95 "United States 3900 \+ 256.9" }}{PARA 259 "" 0 "" {TEXT -1 98 "Can ada 3350 \+ 211.6" }}{PARA 260 "" 0 "" {TEXT -1 99 "Australia \+ 3220 \+ 238.1" }}{PARA 261 "" 0 "" {TEXT -1 94 "New Zealand \+ 3220 \+ 211.8" }}{PARA 262 "" 0 "" {TEXT -1 92 "United Kingdom \+ 2790 194.1" }} {PARA 263 "" 0 "" {TEXT -1 97 "Switzerland 2780 \+ 124.3" }}{PARA 264 "" 0 "" {TEXT -1 100 "Ireland 2770 \+ 187.3" }}{PARA 265 "" 0 "" {TEXT -1 100 "Iceland 2290 \+ 110.5" }}{PARA 266 "" 0 "" {TEXT -1 100 "Finland 2160 \+ 233.1" }}{PARA 267 "" 0 " " {TEXT -1 92 "West Germany 1890 \+ 150.3" }}{PARA 268 "" 0 "" {TEXT -1 96 "Netherlands 1810 \+ 124.7" }}{PARA 269 "" 0 "" {TEXT -1 100 "Gree ce 1800 \+ 41.2" }}{PARA 270 "" 0 "" {TEXT -1 100 "Austria \+ 1770 \+ 182.1" }}{PARA 271 "" 0 "" {TEXT -1 98 "Belgium \+ 1700 \+ 118.1" }}{PARA 272 "" 0 "" {TEXT -1 99 "Mexico \+ 1680 \+ 31.9" }}{PARA 273 "" 0 "" {TEXT -1 102 "Italy \+ 1510 \+ 114.3" }}{PARA 274 "" 0 "" {TEXT -1 96 "Denmark \+ 1500 144.9" }}{PARA 275 "" 0 "" {TEXT -1 100 "France 14 10 59.7" }} {PARA 276 "" 0 "" {TEXT -1 98 "Sweden 1270 \+ 126.9" }}{PARA 277 "" 0 "" {TEXT -1 102 "Spain 1200 \+ 43.9" }}{PARA 278 "" 0 "" {TEXT -1 99 "Norway 1090 \+ 136.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "X := [3900,3350,3220,3220,2790,2780,2770,2290,216 0,1890,1810,1800,1770,1700,1680,1510,1500,1310,1270,1200,1090];" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"XG77\"%+R\"%]L\"%?KF(\"%!z#\"%!y# \"%qF\"%!H#\"%g@\"%!*=\"%5=\"%+=\"%q<\"%+<\"%!o\"\"%5:\"%+:\"%58\"%q7 \"%+7\"%!4\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Y := [256. 9,211.6,238.1,211.8,194.1,124.5,187.3,110.5,233.1,150.3,124,7,41.2,182 .1,118.1,31.9,114.3,144.9,59.7,126.9,43.9,136.3];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"YG78$\"%pD!\"\"$\"%;@F($\"%\"Q#F($\"%=@F($\"%T>F($ \"%X7F($\"%t=F($\"%06F($\"%JBF($\"%.:F(\"$C\"\"\"($\"$7%F($\"%@=F($\"% 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J. Clark, Comparative Physiology of the Heart, Macmi llan, 1972" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Here are lists, " }{TEXT 292 1 "W" }{TEXT -1 7 " and " } {TEXT 293 1 "P" }{TEXT -1 31 ", of the weights and pulses :\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "W := [25,200,300,2000,5000,3 0000,50000,70000,450000,3000000];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"WG7,\"#D\"$+#\"$+$\"%+?\"%+]\"&++$\"&++&\"&++(\"'++X\"(+++$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P := [670, 420, 300, 205, 12 0, 85, 70, 72, 38,48];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG7,\"$q '\"$?%\"$+$\"$0#\"$?\"\"#&)\"#q\"#s\"#Q\"#[" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Next we create a list, " } {TEXT 294 2 "WP" }{TEXT -1 10 ", of the " }{TEXT 291 5 "pairs" } {TEXT -1 36 " of observations using a standard " }{TEXT 290 5 "Maple " }{TEXT -1 20 " idiom. There are " }{TEXT 295 2 " N" }{TEXT -1 33 " observations in each list where" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "N := nops(W);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "testeq( N = nops(P) ); #Make sure that we haven't omitted a value." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "We start by assigning the name " }{TEXT 296 2 "WP" } {TEXT -1 71 " to the empty list. One at a time we augment the current contents of " }{TEXT 297 2 "WP" }{TEXT -1 26 " with the next data pa ir. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "WP := []:\nfor i from 1 to N do\nWP := [op(WP),[W[i], P[i]]]:\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "To obtain a managable scale, we will omit the extreme tw o data points at each end, plotting pairs 3 through 8. Here is the sca tter plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot( [seq(WP[j],j =3..8)], style=POINT, symbol=circle, color=NAVY );" }}{PARA 13 "" 1 " " {INLPLOT "6&-%'CURVESG6#7(7$$\"$+$\"\"!F(7$$\"%+?F*$\"$0#F*7$$\"%+]F *$\"$?\"F*7$$\"&++$F*$\"#&)F*7$$\"&++&F*$\"#qF*7$$\"&++(F*$\"#sF*-%'CO LOURG6&%$RGBG$\")!\\DP\"!\")FH$\")viobFJ-%'SYMBOLG6#%'CIRCLEG-%&STYLEG 6#%&POINTG" 2 375 375 375 5 4 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 " Transforming Variables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 12 "Weight-Pulse" }{TEXT -1 130 " scatter plot in the preceding se ction would suggest to the experienced observer a relationship of the \+ form \n\n " }{XPPEDIT 18 0 "P = k*W^p;" "/%\"PG*&% \"kG\"\"\")%\"WG%\"pGF&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "for some positive constant " }{XPPEDIT 18 0 "k;" "I \"kG6\"" }{TEXT -1 31 " and some negative constant " }{XPPEDIT 18 0 "p;" "I\"pG6\"" }{TEXT -1 96 ". These are our unknowns. There are \+ a number of standard transformations that convert certain " }{TEXT 262 9 "nonlinear" }{TEXT -1 25 " relationships (such as " }{XPPEDIT 18 0 "P = k*W^p" "/%\"PG*&%\"kG\"\"\")%\"WG%\"pGF&" }{TEXT -1 29 ") t o linear relationships " }{XPPEDIT 18 0 "y = m*x+b;" "/%\"yG,&*&%\"m G\"\"\"%\"xGF'F'%\"bGF'" }{TEXT -1 58 ". In this case, if we take the logarithm of each side of " }{XPPEDIT 18 0 "P = k*W^p" "/%\"PG*&%\"kG \"\"\")%\"WG%\"pGF&" }{TEXT -1 18 " then we obtain " }{XPPEDIT 18 0 "ln(P) = ln(k)+p*ln(W);" "/-%#lnG6#%\"PG,&-F$6#%\"kG\"\"\"*&%\"pGF+-F$ 6#%\"WGF+F+" }{TEXT -1 62 ". This is a linear relationship in the tran sformed variables " }{XPPEDIT 18 0 "x = ln(W);" "/%\"xG-%#lnG6#%\"WG " }{TEXT -1 8 " and " }{XPPEDIT 18 0 "y = ln(P);" "/%\"yG-%#lnG6#% \"PG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Let us test whether our conjectured relationship " }{XPPEDIT 18 0 "P = k*W^p" "/%\"PG*&%\"kG \"\"\")%\"WG%\"pGF&" }{TEXT -1 132 " is reasonable by plotting the tr ansformed data. If we get a scatter plot that is like a straight line \+ then we are on to something." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "X \+ := map(ln, W); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG7,-%#lnG6#\"# D-F'6#\"$+#-F'6#\"$+$-F'6#\"%+?-F'6#\"%+]-F'6#\"&++$-F'6#\"&++&-F'6#\" &++(-F'6#\"'++X-F'6#\"(+++$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Y := map(ln,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG7,-%#lnG6 #\"$q'-F'6#\"$?%-F'6#\"$+$-F'6#\"$0#-F'6#\"$?\"-F'6#\"#&)-F'6#\"#q-F'6 #\"#s-F'6#\"#Q-F'6#\"#[" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " XY := []:\nfor i from 1 to N do\nXY := [op(XY),[X[i],Y[i]]]:\nod:" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "Here is \+ the transformed scatter plot:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "scatterPlot := plot(XY, styl e=POINT, symbol=circle, color=NAVY ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 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RS%GMF*7$$\"1e.%)f!oeZ\"FB$\"1*G)y+#GJU$F*7$$\"1:l46Fmx9FB$\"1:T&*y;`= MF*7$$\"1:\"4L$Rsz9FB$\"1+,_1CD8MF*7$$\"1')4b!fh;[\"FB$\"19;*Gh*G3MF*7 $$\"12q8K0e$[\"FB$\"1BuVaZP.MF*7$$\"1'Rb-+,b[\"FB$\"17m<')fX)R$F*7$$\" 1n> " 0 "" {MPLTEXT 1 0 51 "(evalf@leastSquaresLine)(XY, [ln_weight,ln_Pulse]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)ln_PulseG,&%*ln_weightG$!+0SBhD!#5$\"+l;< .s!\"*\"\"\"" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 281 1 " " }{TEXT 283 32 "Copyright and Author Information" }{TEXT 284 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 48 "leastSquaresLineR4.mws A Maple R4 worksheet." }} {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 80 "Autho r: Brian E. Blank \nWritten: 23 October 2002\nLast Revised: 16 Mar ch 2005" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, with out" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the au thor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " " }{TEXT 285 25 "Department of Mathematics" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington University in St. Louis" }}{PARA 0 "" 0 " " {TEXT -1 26 " St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-67 63" }}{PARA 256 "" 0 "" {TEXT -1 44 " e-mail: brian@mat h.wustl.edu" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 282 61 "Copyright: \251 2002-2005 Brian E. Blank, All Rights R eserved." }}}}{MARK "7" 0 }{VIEWOPTS 1 1 0 1 1 1803 }