{VERSION 1 0 "X11/Motif" "1.0"}{GLOBALS 3 1}{FONT 0 "-adobe-helve tica-bold-r-normal--*-140-*-*-*-*-*-*" "helvetica" "Helvetica-Bol d" 8 14 0 "Helvetica-Bold" 12}{FONT 1 "-adobe-times-bold-r-normal --*-140-*-*-*-*-*-*" "times" "Times-Bold" 8 14 64 "Times-Bold" 12 }{FONT 2 "-adobe-courier-medium-r-normal--*-140-*-*-*-*-*-*" "cou rier" "Courier" 4 14 192 "Courier" 12}{FONT 3 "-adobe-times-mediu m-i-normal--*-120-*-*-*-*-*-*" "times" "Times-Italic" 4 12 96 "Ti mes-Italic" 12}{SCP_R 1 0 51{COM_R 2 0{TEXT 1 94 "Here we discuss least-squares fitting of some given data.\015\015First, lets typ e in our data to fit"}}{INP_R 3 0 "> "{TEXT 0 51 "pts := [ [0,0.2 5], [1/2,2], [1,4], [5,8], [6,10] ];"}}{OUT_R 4 0 3{DAG :3n3\`pts `[2,6[2,3j2x0000e3j2x0025i2x0002[2,3/3j2x0001j2x0002p13[2,3p11j2x 0004[2,3j2x0005j2x0008[2,3j2x0006j2x0010}}{INP_R 5 0 "> "{TEXT 0 34 "plot(pts,style=point, axes=boxed);"}}{GRH_R 6 0{RASTER 535 228 1 67 000000000000E0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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0000000000000000078000000000000000006000000000000000C000000000000 000033000000000000000003000000000000006600000000000000C00C0000000 0000000000000000000003000000000000000C03F00000000000000FC03000000 000000001E000000000000000003000000000000003C000000000000008007000 00000000000}}{COM_R 7 0{TEXT 1 279 "We could use maple's built-in regression package to find the answer, but that wouldn't help mu ch in explaining what's going on. Let's do it anyway, though. \ 015\015Note that the least-squares package wants the x values and y values in separate lists, so we have to adjust them a bit."}} {INP_R 8 0 "> "{TEXT 0 84 "with(stats):\012fit[leastsquare[[x,y]] ]([[ pts[i][1] $ i=1..5 ], [pts[i][2] $i=1..5]]);"}}{OUT_R 9 0 8 {DAG =3n3\`y`+5e3j4x001271370968i2x0009j2x0001n3\`x`e3j4x00143145 1613p9}}{COM_R 10 0{TEXT 1 175 "\015Let's do the same thing ourse lves, \"by hand\". \015\015First, lets define the \"error functio nal\" which gives us the distance from a line (actually, its coef ficients) to the data. "}}{INP_R 11 0 "> "{TEXT 0 105 "F := (m, b) -> sum( ( (m*pts[i][1] + b) - pts[i][2] ) ^2, \012 \+ i=1..5);"}}{OUT_R 12 0 11{DAG :3n3\`F` @8,3n3\`m`n3\`b`,1,3n5\`operator`n4\`arrow`p9(3n3\`sum`,3*3+7*5a2 x0001j2x0001b3b3n3\`pts`,2n3\`i`,2p1Cp1Cp1Ca2x0002p1Cb3p1F,2j2x00 02i2x0001p2F=3p23~3p1Cj2x0005p9p9}}{COM_R 13 0{TEXT 1 0 ""}} {COM_R 14 0{TEXT 1 87 "For example, if we guessed that the line w as y= 2x + 1, we could compute its distance:"}}{INP_R 15 0 "> " {TEXT 0 7 "F(2,1);"}}{OUT_R 16 0 15{DAG e3j3x00195625i2x0004}} {COM_R 17 0{TEXT 1 85 "but we see that decreasing the slope and i ncreasing the intercept gives a better fit:"}}{INP_R 18 0 "> " {TEXT 0 11 "F(1.5,1.2);"}}{OUT_R 19 0 18{DAG e3j4x003125000000i2x 0009}}{COM_R 20 0{TEXT 1 108 "Now we want to minimize this quanti ty. Note that maple will happily compute the partial derivatives for us."}}{INP_R 21 0 "> "{TEXT 0 18 "diff ( F(m,b), m);"}} {OUT_R 22 0 21{DAG +7n3\`m`/3j2x0249j2x0002n3\`b`j2x0025i2x0210j2 x0001}}{COM_R 23 0{TEXT 1 66 "\015So, we can ask maple to solve, \+ setting each of the partials to 0."}}{INP_R 24 0 "> "{TEXT 0 79 " solve( \{ diff ( F(m,b), m)=0 , diff ( F(m,b), b) = 0\},\012 \+ \{m, b\} );"}}{OUT_R 25 0 24{DAG %2,3=3n3\`b`e3j4x00127 1370968i2x0009=3n3\`m`e3j4x001431451613pA}}{COM_R 26 0{TEXT 1 211 "Not surprisingly, this was the same solution we got before. I f we now want to use THIS value of m and b without retyping it, w e can use maple's assign command to let b and m be constants with the values given."}}{INP_R 27 0 "> "{TEXT 0 11 " assign(\");"}} {COM_R 28 0{TEXT 1 60 "Maple just does the assignment, but doesn' t say anything... "}}{COM_R 29 0{TEXT 3 93 " \"When I have no thing to say, my lips are sealed. Say something once, why say it again?\""}}{COM_R 30 0{TEXT 1 85 "\015But, m and b are now cons tants, with values of about 1.43 and 1.27, respectively. "}} {INP_R 31 0 "> "{TEXT 0 2 "m;"}}{OUT_R 32 0 31{DAG e3j4x001431451 613i2x0009}}{COM_R 33 0{TEXT 1 69 "If we want, we can compute the distance of this line to the points..."}}{INP_R 34 0 "> "{TEXT 0 8 "F(m, b);"}}{OUT_R 35 0 34{DAG e3j4x002929334677i2x0009}} {COM_R 36 0{TEXT 1 179 "\015To get an idea of how good our fit is , we can make a plot with the data points and the line on the sam e graph.\015We first need to load the 'plots' package, so we can \+ use 'display'."}}{INP_R 37 0 "> "{TEXT 0 12 "with(plots):"}} {COM_R 38 0{TEXT 1 133 "\015We can now compute both plots, and us e display to show them together. Note that we use a : to supress the output of the plot calls."}}{INP_R 39 0 "> "{TEXT 0 102 "lpl ot := plot(m*x+ b, x=-1..6.5, axes=boxed):\012pplot:= plot(pts, s tyle=point):\012display(\{lplot, pplot\});"}}{GRH_R 40 0{RASTER 530 222 1 67 000000000000E0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF F1F00000000000000000000000000F0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFF3F0000000000000000000000000030000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 00000000030000000000000000000000000003000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 00000000000003000000000000000000000000000300000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 00000000000000000300000000000000000000000000030000000000000000000 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0000000000000000000030000000003000000000000060000000000000F000000 0000000E001000000000000020000000000F00F0000000000C003000000000000 00000000000000000000000038000000007800000000000070000000000000980 10000000000300300000000000003000000000030000000000000600600000000 00000000000000000000000000003C00000000CC0000000000007800000000000 00C03000000000018060000000000800300000000003000000000000030040000 000000000000000000000000000000003600000000CC0000000000006C0000000 000000C03000000000018060000000000C0030000000000300000000000003000 00000000000000000000000000000000000030000000008601000000000060000 00000000000030000000000000600000000006003000000000030000000000000 30000000000000000000000000000000000000003000000000860100000000006 0000000000000000300000000000003000000000030030000000000B003000000 0000B00300000000000000000000000000000000807F300000000086010000000 0006000000000000080010000000000C001000000000018030000000000700600 00000000700600000000000000000000000000000000000030000000008601000 000000060000000000000E0000000000000000300000000001803000000000000 0C0000000000300C0000000000000000000000000000000000003000000000860 10000000000600000000000003000000000000000060000000000F80700000000 00000C0000000000300C000000000000000000000000000000000000300000000 0CC00000000000060000000000000180000000000001806000000000000030000 000000000C0000000000300C00000000000000000000000000000000000030000 00000CC000000000000600000000000000C000000000000180600000000000003 0000000000300C0000000000300C0000000000000000000000000000000000003 00000000078000000000000600000000000000C00000000000030030000000000 00030000000000600600000000006006000000000000000000000000000000000 000FE0100000030000000000000FC030000000000FC030000000000E001000000 000000030000000000C0030000000000C00300000000000000000000}}{COM_R 41 0{TEXT 1 350 "\015\015One minor point:\015 Since we defined \+ 'm' and 'b' to be constants, we can no longer treat them as vari ables. For example, the statement we used before to compute the derivative of F with respect to m no longer works, since it is a sking maple to compute the derivative of 2.929334677 with respec t to 1.431451613. This, of course, makes no sense."}}{INP_R 42 0 "> "{TEXT 0 15 "diff(F(m,b),m);"}}{OUT_R 43 0 42{TEXT 2 61 "Erro r, wrong number (or type) of parameters in function diff\012"}} {COM_R 44 0{TEXT 1 60 "Of course, we can do this by substituting \+ in other variables"}}{INP_R 45 0 "> "{TEXT 0 15 "diff(F(n,c),n);" }}{OUT_R 46 0 45{DAG +7n3\`n`/3j2x0249j2x0002n3\`c`j2x0025i2x0210 j2x0001}}{COM_R 47 0{TEXT 1 81 "Alternatively, we can tell maple \+ to \"forget\" the previous definitions of m and b:"}}{INP_R 48 0 "> "{TEXT 0 15 "m:='m';\012b:='b';"}}{OUT_R 49 0 48{DAG :3n3\`m`p 1}}{OUT_R 50 0 48{DAG :3n3\`b`p1}}{INP_R 51 0 "> "{TEXT 0 15 "dif f(F(m,b),m);"}}{OUT_R 52 0 51{DAG +7n3\`m`/3j2x0249j2x0002n3\`b`j 2x0025i2x0210j2x0001}}}{END}