{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "He lvetica" 1 18 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 11 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 16 0 0 0 0 0 0 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 258 "" 0 "" {TEXT 256 13 "Data analysis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 424 "In this worksh eet we provide some means to analyze experimental data. It represents an exercise in making use of drawing tools (to plot error bars), in \+ reading in data from an ASCII file (e.g., from a computer-interfaced \+ experiment, spreadsheet/graphing program, etc.), and finally in makin g use of the statistical analysis package. Ther is also a hand-writte n least-squares fitting procedure is attached at the end. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "First we make \+ use of the data interface. The file draw1 has to exist in the current directory. We assume four columns of data, containing x_i, y_i, and \+ low and high values for y_i (uncertainties)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "We follow the conventi on on PC's for the directory structure. Note, that instead of the forw ard-slash (borrowed from unix), one could use the double-backward slas h." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "readlib(readdata):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "#data:=readdata(`C:\\\\ComPh ys\\\\DataAnalysis\\\\draw1`,float,4): # this works as well" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "data:=readdata(`C:/ComPhys/D ataAnalysis/draw1`,float,4):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "A fter reading the data they are available as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "data[1],data[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7&$\"#5!\"\"$!\"&F&$!#b!\"#$!\"%F&7&$\"#7F&F,F'$!\"$F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dline:=map(evalf,data[1 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dlineG7&$\"#5!\"\"$!\"&F($!# b!\"#$!\"%F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Unfortunately th e Maple development team is not listening to user's complaints about t he meagre selection of plotting symbols." }}{PARA 0 "" 0 "" {TEXT -1 220 "Experimental physicists with data plus uncertainties are the last thing on their minds... So we write our own graphiscs primitives here (they are not very fast when it comes to plotting many points, but on e can survive)." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 55 "Graphics pri mitives to draw data points with error bars" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To plot a decent-sized open symbol we define our own circ le with the centre location [" }{TEXT 259 1 "x" }{TEXT -1 2 ", " } {TEXT 258 1 "y" }{TEXT -1 60 "] passed as the first argument, and the \+ radius as the third." }}{PARA 0 "" 0 "" {TEXT -1 48 "A fourth argument can be used to set the colour." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Circle:=proc() local col;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "if nargs<4 then col:=black else col:=args[4] fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot([args[3]*cos(t)+args[1],args[3]*sin(t)+args [2],t=0..2*Pi],scaling" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "=CONSTRAI NED,colour=col,thickness=2); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P1:=Circle(2,3,0.3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 " A line connecting points [" }{TEXT 260 1 "x" }{TEXT -1 3 "1, " }{TEXT 261 1 "y" }{TEXT -1 5 "1], [" }{TEXT 263 1 "x" }{TEXT -1 3 "2, " } {TEXT 262 1 "y" }{TEXT -1 8 "2] with " }{TEXT 264 1 "x" }{TEXT -1 3 "1 , " }{TEXT 265 1 "y" }{TEXT -1 2 "1," }{TEXT 266 2 " x" }{TEXT -1 3 "2 , " }{TEXT 268 1 "y" }{TEXT -1 99 "2 passed as the first four argument s to the procedure. The fifth argument allows to set the colour." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Line:=proc() local col;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "if nargs<5 then col:=black else col :=args[5] fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot([[args[1],arg s[2]],[args[3],args[4]]],scaling=CONSTRAINED,colour=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "col); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P2:=Line(-1,-1,2,3,red):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "display(\{P1,P2\},axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT2D 679 213 213 {PLOTDATA 2 "6(-%'CURVESG6$7$7$$!\"\"\"\"!F(7$$ \"\"#F*$\"\"$F*-%'COLOURG6&%$RGBG$\"*++++\"!\")F*F*-F$6%7S7$$\"1++++++ +B!#:F.7$$\"1L)=7(3>(H#F=$\"1InZX$e4/$F=7$$\"1$pkH19-H#F=$\"1[yyo')*f2 $F=7$$\"1JeO%fduF#F=$\"1!pX(4L49JF=7$$\"11RcF@ffAF=$\"11O=GEP]JF=7$$\" 1_i)4cFrB#F=$\"1)o+,dnP=$F=7$$\"1!=f]AxC@#F=$\"1rn-\"G'y6KF=7$$\"1!e*Q sO\\$=#F=$\"1Pc[r&RtB$F=7$$\"1B#*oYJK]@F=$\"1wI;%y?'fKF=7$$\"1D,M*=/Y6 #F=$\"1,=LlpCxKF=7$$\"1WmG)ehd2#F=$\"1TW.WgF!H$F=7$$\"1a%yY5a./#F=$\"1 3B\"4_tsH$F=7$$\"1zn!z'3#)**>F=$\"1%[>l%*****H$F=7$$\"1bFAHb7f>F=$\"1% eG'=C?(H$F=7$$\"1&G*[NXh?>F=$\"1eaZwfI*G$F=7$$\"1p!o?t1o)=F=$\"1$oug*f #yF$F=7$$\"1D8zT0t[=F=$\"1fB>_02fKF=7$$\"1=$4dJ:)==F=$\"1X>:,p5RKF=7$$ \"1tqf40K(y\"F=$\"1F5\"=;$e6KF=7$$\"1BM$Q(Q(Gw\"F=$\"1^$H:UpP=$F=7$$\" 1#p0yz?-u\"F=$\"1*z%G7#\\+:$F=7$$\"11f!=\">2Bniy\"GF=7$$\"1gs)=-$o'y\"F=$ \"1`zjl&f!*y#F=7$$\"1lk42Ca==F=$\"1rdOU,5hFF=7$$\"1tBXZ$z-&=F=$\"1'f^w HJ+u#F=7$$\"1#Q[bb;o)=F=$\"1Tfz,+F=$\"1U37RvQ5FF=7$ $\"1kFu'*HDh>F=$\"1[z?SF^-FF=7$$\"1!>r\\y#3**>F=$\"1#HU@S,+q#F=7$$\"1n *3wpJ'Q?F=$\"1eukWx\\-FF=7$$\"1=6PS7lw?F=$\"1Yf66w&*4FF=7$$\"1Pnc^R2:@ F=$\"1z[B%oZHs#F=7$$\"1GHGLor&\\-u#F=7$$\"1ZY)>^!G$=#F=$ \"1JXC\"y&\\iFF=7$$\"1GSf6!zH@#F=$\"1VB=Q$=()y#F=7$$\"1c;\"=N0qB#F=$\" 1@\"[M%[2;GF=7$$\"1/j[D,BgAF=$\"10E#3>M2&GF=7$$\"1vkfG+vwAF=$\"1%y)pU4 ?%)GF=7$$\"1'owgmS'*G#F=$\"1[(4[[W=#HF=7$$\"1n*ygm3sH#F=$\"1\"f:3)4 \+ " 0 "" {MPLTEXT 1 0 48 "Expoint:=proc() local size,col; size:=2*args[5 ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "if nargs<6 then col:=black el se col:=args[6] fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Circle(args[ 1],args[2],0.5*args[5],col),Circle(args[1],args[2],args[5]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 ",col),Line(args[1],args[3],args[1],args[4], col),Line(args[1]-size,args" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "[3], args[1]+size,args[3],col),Line(args[1]-size,args[4],args[1]+size,args[ 4],col);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Pex:=Expoint(1.5,2.2,1.9,2.4,.1,magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "display(\{Pex\},axes=boxe d,view=[-1..2,0..4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 396 396 {PLOTDATA 2 "6+-%'CURVESG6$7$7$$\"1+++++++8!#:$\"1+++++++>F*7$$\"1++++ +++AF*7$$\"1%)Rfa`EV:F*$\"1osp/@1DAF*7$ $\"1vV;g7_R:F*$\"1:,NGziIAF*7$$\"1(>V3(GTN:F*$\"1iW]8xHNAF*7$$\"1(fJ(y AeI:F*$\"11wC&fc&RAF*7$$\"1P:ydQ0D:F*$\"1YQpI,FVAF*7$$\"1aLA)p+\">:F*$ \"1+`bFy?YAF*7$$\"126QJpi7:F*$\"12CnS$z$[AF*7$$\"1wI6%oDn]\"F*$\"1=(=o eX&\\AF*7$$\"1I6lW,(**\\\"F*$\"1\"e'3\"*****\\AF*7$$\"1$z.#)e(=$\\\"F* $\"1)4QktL&\\AF*7$$\"19#[#*3pn[\"F*$\"15fuiw@[AF*7$$\"1yYMbW8\"[\"F*$ \"19\"zEL/jC#F*7$$\"1))=jB%)yu9F*$\"1$Rl`UyJC#F*7$$\"1?#=Eb-)p9F*$\"1 \"*>>]6&)RAF*7$$\"1zhE=Mbk9F*$\"1Q=IgQENAF*7$$\"1PA(*y*y/Y\"F*$\"1f:#p BG1B#F*7$$\"1\\4ImMqc9F*$\"1nu/-#3]A#F*7$$\"1=Vj=`%QX\"F*$\"1W(*fk(G#> AF*7$$\"1!)HX#)=o^9F*$\"1uI8>#fG@#F*7$$\"1:H!*\\[Y]9F*$\"1h_-D@!o?#F*7 $$\"1%e`/C++X\"F*$\"1D=Zl]:+AF*7$$\"1&Q1*4yX]9F*$\"10!>2M\\K>#F*7$$\"1 KPms'H;X\"F*$\"1A'e[eQt=#F*7$$\"1M[$*)o$o`9F*$\"12%=B)Q;\"=#F*7$$\"14& R$p!HmX\"F*$\"1LxcX57v@F*7$$\"1+O9h'p-Y\"F*$\"1^[lyPkp@F*7$$\"1VXJqrWk 9F*$\"1f'R4EV[;#F*7$$\"16%\\y1d(p9F*$\"1i41dL=g@F*7$$\"1H(3zbY]Z\"F*$ \"1L>%H)=nc@F*7$$\"1(R\"f#4O6[\"F*$\"1C$*H+]p`@F*7$$\"1QN4sm&p[\"F*$\" 1T,_c7t^@F*7$$\"1%zBh;UN\\\"F*$\"1eY.!z=/:#F*7$$\"1K&G38Z)*\\\"F*$\"1# Q!pL-+]@F*7$$\"1h\"oihQk]\"F*$\"1wXx!H;/:#F*7$$\"1?&Gn?vF^\"F*$\"1eE&= gf;:#F*7$$\"1*yF>**y\">:F*$\"1Z\"RShCQ:#F*7$$\"1)[!)G];]_\"F*$\"1?s%>E 3n:#F*7$$\"1Tu*>vY0`\"F*$\"1c2ajfTg@F*7$$\"1QB$>]'\\N:F*$\"1dqp*Q'yk@F *7$$\"14'o>*3]R:F*$\"1a8C2eMp@F*7$$\"1^5e(orLa\"F*$\"1,r8lB7v@F*7$$\"1 zgw/]7Y:F*$\"1kk6d,q!=#F*7$$\"19hMWMF[:F*$\"1D;!33up=#F*7$$\"1G)zwxM& \\:F*$\"1l#pM;&>$>#F*7$FJ$\"1@5/++++AF*F0-%*THICKNESSG6#\"\"#-F$6%7S7$ $\"1+++++++;F*FL7$$\"16'RPij!*f\"F*$\"1xb#=y_O@#F*7$$\"1J#))4-Qnf\"F*$ \"1\\ff*)GLDAF*7$$\"15')yke[#f\"F*$\"1j&[K5J!QAF*7$$\"1pz=42`'e\"F*$\" 1OXR4U7]AF*7$$\"1^(G._U!z:F*$\"1I-qceDhAF*7$$\"1%R'oTd#3d\"F*$\"1C*3qU &fqAF*7$$\"1$>jubk6c\"F*$\"17_\\!>8\"zAF*7$$\"1uIc:x5]:F*$\"1#p(Qh-a'G #F*7$$\"13nW'R,#Q:F*$\"1,16bcT#H#F*7$$\"1:AwiQDD:F*$\"19[M\"oenH#F*7$$ \"1^hAo8X8:F*$\"1Oujt64*H#F*7$$\"1gAI*GS**\\\"F*$\"1iJ<#)*****H#F*7$$ \"1&e2kwGZn!*H#F*7$$\"1Hk\\y\"QNZ\"F*$\"1?=\\D`V'H#F*7$$ \"1c$*o5*oAY\"F*$\"1G#e`m3EH#F*7$$\"1vPEZod\\9F*$\"1(yI2&oN'G#F*7$$\"1 RkB0^gR9F*$\"1#)RQ+BqzAF*7$$\"1eB`Oo5H9F*$\"1wOg?x_qAF*7$$\"1uW%z&z&4U \"F*$\"1HvXQAF*7$$\"1hf!\\wjLS\"F*$\"1ZhEQ%=dA#F*7$$\"1Ie!)*pH4S\"F*$\" 1?00]Ug8AF*7$$\"1nr!4[++S\"F*$\"1\\O%485.?#F*7$$\"1pF\")>c\"4S\"F*$\"1 4!Q9o)\\'=#F*7$$\"1kuKX$fKS\"F*$\"1Xsrprnu@F*7$$\"1o'pyPntS\"F*$\"1:oj kxKi@F*7$$\"179nO?@F* 7$$\"1eu\"e6$4]9F*$\"1mQ)ewVL6#F*7$$\"1%z#=&=sAY\"F*$\"1Z')f++R2@F*7$$ \"1vq=WL\"RZ\"F*$\"1\"GSI^iM5#F*7$$\"1)eZAL%3([\"F*$\"1;$p+eP35#F*7$$ \"1jqlhUp*\\\"F*$\"1k2Qn/++@F*7$$\"1Aj`Ks(G^\"F*$\"1`\"\\:eK35#F*7$$\" 1RqX8/bD:F*$\"1:`q.#>L5#F*7$$\"1zb&Q)zNQ:F*$\"1$Hy!G#\\w5#F*7$$\"1w4w0 I.]:F*$\"1SW*Q_;M6#F*7$$\"1$)[*R]$4h:F*$\"16:3F>$37#F*7$$\"1wY'Q+$*4d \"F*$\"1:TRzFdH@F*7$$\"1>s$Ry,!z:F*$\"12F[9;pQ@F*7$$\"1,@;vLu'e\"F*$\" 1-UFIZC]@F*7$$\"1e@`4+D#f\"F*$\"1GHB9.Sh@F*7$$\"1HAp))oa'f\"F*$\"1]Kgh \"[R<#F*7$$\"1c'f`bp!*f\"F*$\"1I&QpK!R'=#F*7$F^]l$\"1U?3++++AF*F0Ff\\l -%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%!GFe\\m-%*AXESSTYLEG6#%$BO XG-%%VIEWG6$;$!\"\"F7$Fi\\lF7;F7$\"\"%F7" 1 2 0 1 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "If desired, one can fill in the circle with more than one, as don e here, but it will slow plotting.." }}{PARA 0 "" 0 "" {TEXT -1 335 "W e can now plot a circle that when reduced in size will appear filled, and an 'error' bar that indicates the uncertainty of the measurement . It is straightforward to write routines for other symbols (squares \+ and diamonds at least) We generate a colour table and have the luxur y of distinguishing subsequent data points by colour:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "coltable:=[violet,magenta,blue,cyan,green,yell ow,red,maroon];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)coltableG7*%'vio letG%(magentaG%%blueG%%cyanG%&greenG%'yellowG%$redG%'maroonG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "imax:=7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%imaxG\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "N ow we set up a loop to prepare the data point structure:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "P3:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " for i from 1 to imax do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dline:= data[i]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "P3:=[Expoint(dline[1],d line[2],dline[3],dline[4],0.01,coltable[i+1]),op(P3)]: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(P3,axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT2D 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\"1Cs*y*y/'p\"F0$!1:WyIwr$p*FA7$$\"1&4ImMqcp\"F0$!1N`_zz\"*\\(*FA7$$\" 1KM'=`%Q&p\"F0$!1nD+aBr2)*FA7$$\"1)HX#)=o^p\"F0$!1m#p'3ySr)*FA7$$\"1#H !*\\[Y]p\"F0$!1*RZ(\\(y>$**FA7$$\"1e`/C++&p\"F0$!1avY0.] '\\Nq\"F0$!1VHI5O@N5F*7$$\"1ho>*3]Rq\"F0$!1Z'eF>a1.\"F*7$$\"10\"e(orL/ 5F*7$$\"16YVWt#[q\"F0$! 1v$)>>f-85F*7$$\"1$)zwxM&\\q\"F0$!1N2`O[!o+\"F*7$F\\_x$!0z*e********** F*FEFe]l-F$6%7S7$$\"1++++++05F0Fj^m7$$\"1\")p=\"=`\\+\"F0$!16s34O<$*\\ F*7$$\"17%\\5!p$[+\"F0$!1D??bNL()\\F*7$$\"1I%RKHCY+\"F0$!1=dP[W)4)\\F* 7$$\"1)Rfa`EV+\"F0$!1KFI&*y$\\(\\F*7$$\"1Pk,E@&R+\"F0$!1&))\\;2s$p\\F* 7$$\"1?V3(GTN+\"F0$!1Qb\\'G-Z'\\F*7$$\"1gJ(yAeI+\"F0$!1%R_ZSV/'\\F*7$$ \"1a\"ydQ0D+\"F0$!1ahIp)Hn&\\F*7$$\"1NB#)p+\">+\"F0$!1+ZWs@z`\\F*7$$\" 16\"QJpi7+\"F0$!1$fF$f1i^\\F*7$$\"138ToDn+5F0$!1#G\"=8WX]\\F*7$$\"1I6l W,(*****F*$!1>M\"*3++]\\F*7$$\"1#z.#)e(=$***F*$!1.>cjiY]\\F*7$$\"19#[# *3pn)**F*$!1!4asL#y^\\F*7$$\"1yYMbW8\")**F*$!1')3Kncp`\\F*7$$\"1()=jB% )yu**F*$!12Yju:#o&\\F*7$$\"1?#=Eb-)p**F*$!14!3)\\)[,'\\F*7$$\"1zhE=Mbk **F*$!1i\")pRhtk\\F*7$$\"1PA(*y*y/'**F*$!1T%yIwr$p\\F*7$$\"1\\4ImMqc** F*$!1MD&zz\"*\\(\\F*7$$\"1=Vj=`%Q&**F*$!1c-SN7x!)\\F*7$$\"1\")HX#)=o^* *F*$!1Fp'3ySr)\\F*7$$\"1:H!*\\[Y]**F*$!1SZ(\\(y>$*\\F*7$$\"1%e`/C++&** F*$!1v\"GX$\\%)**\\F*7$$\"1&Q1*4yX]**F*$!1&*4Gf1v1]F*7$$\"1KPms'H;&**F *$!1x89:9m7]F*7$$\"1N[$*)o$o`**F*$!1#f\"o'4jw***\\]F*7$$\" 1;oihQk+5F0$!1CaA4Pe\\]F*7$$\"1_Gn?vF,5F0$!1Vt9)RS$[]F*7$$\"1zF>**y\"> +\"F0$!1a3'fQvh/&F*7$$\"1\\!)G];]-5F0$!1!y_!QvY0.5F0$! 1X#fk.%eR]F*7$$\"1MK>]'\\N+\"F0$!1VHI5O@N]F*7$$\"1ho>*3]R+\"F0$!1Z'eF> a1.&F*7$$\"10\"e(orL/5F0$!1**G'[jx[-&F*7$$\"13mZ+Dh/5F0$!1ON)G%)*H>]F* 7$$\"16YVWt#[+\"F0$!1v$)>>f-8]F*7$$\"1$)zwxM&\\+\"F0$!1N2`O[!o+&F*7$F` ^y$!1z*e*********\\F*F1Fe]l-F$6$7$7$$\"\"\"F8F+7$Fd]zFKF1-%(SCALINGG6# %,CONSTRAINEDG-%+AXESLABELSG6$%!GF^^z-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;$ \"\"*!\"\"$\"#AFi^z%(DEFAULTG" 1 2 0 1 0 2 9 1 2 1 1.000000 47.000000 43.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "A good question: g iven the uncertainty in the data, what is the uncertainty in the slop e and the " }{TEXT 274 1 "y" }{TEXT -1 63 " intercept? How well are th e data fitted by the straight line?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "describe[linearcorrelation](lxval,lyval): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+f!*Gr**!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "The linear correlation is strong (the number is close to 1; 0 would mean no linear relationship; -1 that an anticorrelation t o a linear relationship exists." }}{PARA 0 "" 0 "" {TEXT -1 459 "This \+ illustrates nicely that the question of fitting a straight line to mea sured data points is non-trivial. Note that the uncertainties have no t been used in the calculation. One can weigh the data points, if one \+ trusts some of them more than others. For example: the points on the r ight apparently follow a straight line rather well. The fit pays atten tion to this fact, but it does not 'know' that the two rightmost poin ts have the biggest error bars. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "What assum ptions enter the standard linear least squares fit? (J.R. Taylor, " } {TEXT 278 30 "Introduction to Error Analysis" }{TEXT -1 16 "; chapter \+ 8.2) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " 1) the " }{TEXT 279 1 "x" }{TEXT -1 69 " values are assumed to be very precise (negligible uncertainties); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "2) the " }{TEXT 280 1 "y" }{TEXT -1 157 " values are assumed to have equal uncertainties (if not then wei ghting should be applied) described by a Gaussian with a certain widt h common to all y_i; " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 421 "3) if the data were exact, they would satisfy y_i = A + B x_i , where A,B are physical constants. If the uncertainties \+ for each measurement y_i are governed by a Gaussian width sigma , the probability of obataining the entire data set for the physical const ants A,B is given as the product of probabilities for each point. The product of exponentials can be expressed as a single exponential with argument -chi^2/2 ." }{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 281 59 "Derivation expressions for the LLSQ fit with uncertainti es:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "y:='y': x:='x': i:='i ': chi:='chi': A:='A': B:='B':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "The expression for the total error is defined according to the da ta model (linear relationship, equal Gaussian errors in y_i) described above as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "E1:=chi^2=Sum ((y[i]-A-B*x[i])^2/sigma^2,i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#E1G/*$)%$chiG\"\"#\"\"\"-%$SumG6$*&*$),(&%\"yG6#%\"iG\"\"\"%\"AG! \"\"*&%\"BGF6&%\"xGF4F6F8F)F*F**$)%&sigmaG\"\"#F*!\"\"/F5;F6%\"NG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "The best values for A and B are t hose for which the probability P_\{A,B\}(y_1,...,y_N) is a maximum, o r chi-squared a minimum. They are found from the necessary condition: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E2a:=diff(rhs(E1),A)=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E2aG/-%$SumG6$,$*&,(&%\"yG6#%\"iG\" \"\"%\"AG!\"\"*&%\"BGF0&%\"xGF.F0F2\"\"\"*$)%&sigmaG\"\"#F7!\"\"!\"#/F /;F0%\"NG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E2b:=diff (rhs(E1),B)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E2bG/-%$SumG6$,$* &*&,(&%\"yG6#%\"iG\"\"\"%\"AG!\"\"*&%\"BGF1&%\"xGF/F1F3F1F6\"\"\"F8*$) %&sigmaG\"\"#F8!\"\"!\"#/F0;F1%\"NG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=solve(\{E2a,E2b\},\{A,B\}):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&* &-%$SumG6$&%\"yG6#%\"iG/F,;\"\"\"%\"NGF/-F'6$*$)&%\"xGF+\"\"#\"\"\"F-F /!\"\"*&-F'6$F5F-F/-F'6$*&F5F/F)F/F-F/F/F8,&*$)F;F7F8F/*&F1F8-F'6$F/F- F/F9!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&-%$SumG6$*&&%\"xG6#%\"iG\"\"\" &%\"yGF,F./F-;F.%\"NGF.-F'6$F.F1F.!\"\"*&-F'6$F*F1F.-F'6$F/F1F.F.\"\" \",&*$)F8\"\"#F " 0 "" {MPLTEXT 1 0 9 "value(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&-%$sumG6$&%\"y G6#%\"iG/F-;\"\"\"%\"NGF0-F(6$*$)&%\"xGF,\"\"#\"\"\"F.F0!\"\"*&-F(6$F6 F.F0-F(6$*&F6F0F*F0F.F0F0F9,&*$)F " 0 "" {MPLTEXT 1 0 9 "value(B);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,&*&-%$sumG6$*&&%\"xG6#%\"iG\"\"\"&%\"yGF,F./F-;F.% \"NGF.F3F.F.*&-F'6$F*F1F.-F'6$F/F1F.!\"\"\"\"\",&*$)F5\"\"#F:F9*&-F'6$ *$)F*F>F:F1F.F3F:F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A1:=subs(N=imax,value(A)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "i:='i': for j from 1 to imax do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A1:=subs(x[j]=xval[j],y[j]=yval[j],A1); od:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "simplify(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+qOSM5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B1:=s ubs(N=imax,value(B)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "i: ='i': for j from 1 to imax do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "B 1:=subs(x[j]=xval[j],y[j]=yval[j],B1); od:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "simplify(B1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+(f:(ea!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 330 "Thus, we know ho w the simple linear least squares result was obtained. Now we are int erested in understanding the estimate of uncertainties. The best esti mate for sigma is obtained in analogy with the best estimate for A an d B: We find the maximum probability to obtain the measured values \{ y_i\} from the necessary condition. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "First we indicate schematically how t he probability depends on sigma: psi is the same as chi except for the sigma^2 that has been removed:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " Prob:=exp(-psi^2/(2*sigma^2))/sigma^N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ProbG*&-%$expG6#,$*&*$)%$psiG\"\"#\"\"\"F/*$)%&sigmaG\"\"#F/! \"\"#!\"\"F.F/)F2%\"NGF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E3:=diff(Prob,sigma)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G/,&* &*&)%$psiG\"\"#\"\"\"-%$expG6#,$*&*$F)F,F,*$)%&sigmaG\"\"#F,!\"\"#!\" \"F+\"\"\"F,*&)F5\"\"$F,)F5%\"NG\"\"\"F7F:*&*&F-F,F?F:F,*&F>\"\"\"F5\" \"\"F7F9\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solsig:=so lve(E3,sigma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'solsigG*$-%%sqrtG 6#*&*$)%$psiG\"\"#\"\"\"F.%\"NG!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "E4:=sigma^2=solsig[1]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E4G/*$)%&sigmaG\"\"#\"\"\"*$)&*$-%%sqrtG6#*&*$)%$psi GF)F*F*%\"NG!\"\"F*6#\"\"\"F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "Thus, the sum of squares (psi^2) per degree of freedom (N) provid es an estimate for the uncertainty of the data. In fact, we need to c orrect for the fact that A and B are not exactly known, but rather de duced from the data to divide by (N-2). " }}{PARA 0 "" 0 "" {TEXT -1 24 "For our example we have:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:=' i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sigsq:=evalf(Sum((yv al[i]-A1-B1*xval[i])^2,i=1..imax)/(imax-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigsqG$\"+w\"45@$!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 327 "This number represents a mean square deviation between t he data points and the line fit. That this is a realistic estimate fo llows from taking a typical deviation from the graph and squaring it \+ (e.g., 0.02 for the green point #4). We are, however, more interested in the uncertainty in the extracted coefficients A and B. " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(yval[4]-A1-B1*xval[4])^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+>_XKC!#8" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 226 "Now that we have an error estimate for the data \{y_i \} we can use standard error propagation to estimate the errors in A \+ and B from their respective expressions. Straightforward calculus res ults in the following expressions:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "sigBsq:=evalf (sigsq*imax/(imax*Sum(xval[i]^2,i=1..imax)-Sum(xval[i],i=1..imax)^2)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigBsqG$\"+v*foV$!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "sigAsq:=evalf(sigsq*Sum(xval[i]^2,i=1..imax) /(imax*Sum(xval[i]^2,i=1..imax)-Sum(xval[i],i=1..imax)^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigAsqG$\"+PT/*[)!#8" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 120 "These expressions are not easily derived from A a nd B. The following example shows Maple's difficulty with the problem :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diff(sum(z[i]*eta[i],i=1..6),z [2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%$etaG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diff(sum(z[i]*eta[i],i=1..N),z[2]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT 282 51 "Hand-written LLSQ fit procedure with uncertainties:" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "We define procedures which implem ent the derived formulae for the LLSQ fit:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "addme:=arg->evalf(Sum(arg[i],i=1..nops(arg)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&addmeGR6#%$argG6\"6$%)operatorG%&ar rowGF(-%&evalfG6#-%$SumG6$&9$6#%\"iG/F5;\"\"\"-%%nopsG6#F3F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "addme([1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#5\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "addpr:=(arg1,arg2)->evalf(Sum(arg1[i]*arg2[i],i=1..no ps(arg1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&addprGR6$%%arg1G%%ar g2G6\"6$%)operatorG%&arrowGF)-%&evalfG6#-%$SumG6$*&&9$6#%\"iG\"\"\"&9% F6F8/F7;F8-%%nopsG6#F5F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "addpr([1,2,3,4],[1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"#I\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(sum(i^2, i=1..4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#I\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "We are ready to program the standard line ar least squares algorithm:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i ':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "LLSQ:=proc(xv,yv) loc al _i,N,a,b,c,d,Delta,A,B,sq,sigsq,dA,dB;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "N:=nops(xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "if nops(yv) " 0 "" {MPLTEXT 1 0 16 "c:=addpr(xv,xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d:=addme(xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a:=addpr(xv,yv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b:=addme(yv );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Delta:=evalf(N*c-d^2);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=(c*b-d*a)/Delta;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "B:=(N*a-d*b)/Delta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sq:=[]; for _i from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sq:=[op(sq),evalf(yv[i]-A-B*xv[i])]: od:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "sigsq:=addpr(sq,sq)/(N-2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dA:=sigsq*c/Delta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dB:=evalf(N*sigsq)/Delta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "print(`[A,B,sigma,dA,dB]`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "[A,B,evalf(sqrt(sigsq),4),evalf(sqrt(dA),4),evalf(sqr t(dB),4)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "LLSQ([1,2,3,4],[2.1,4.,5.9,8.2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%2[A,B,sigma,dA,dB]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!$\"++++??!\"*$\"%\\:!\"%$\"%(*=F*$\"%Gp!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol1:=LLSQ(lxval,lyval); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2[A,B,sigma,dA,dB]G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%sol1G7'$!+qOSM5!\"*$\"+'f:(ea!#5$\"%#z\"!\"&$ \"%9HF.$\"%a=F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "yfit:=so l1[1]+sol1[2]*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yfitG,&$!+qOSM5 !\"*\"\"\"%\"xG$\"+'f:(ea!#5" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 283 28 "Weighted LLSQ fit procedure:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 431 "We know that our procedure works and we have estimates for the u ncertainties in A and B now. We can modify the results to take into a ccount individual uncertainties for the datapoints y_i by means of a \+ weighted least squares fit (J.R. Taylor problem 8.4). The individual \+ uncertainties have to be known for this method to be meaningful. One \+ uses for the inverse of the individual sigma_i-squared as a weight fo r data point i." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "i:='i': for i f rom 1 to imax do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "wval[i]:=1./(0 .5*(data[i][3]-data[i][4]))^2; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "lwval:=convert(wval,list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&lwvalG7)$\"+yxxx " 0 "" {MPLTEXT 1 0 63 "addtr:=(a1,a2,a3)->evalf(Sum(a1[i]*a2[i]*a3[i],i=1..nops(a1))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&addtrGR6%%#a1G%#a2G%#a3G6\"6$%) operatorG%&arrowGF*-%&evalfG6#-%$SumG6$*(&9$6#%\"iG\"\"\"&9%F7F9&9&F7F 9/F8;F9-%%nopsG6#F6F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "WLLSQ:=proc(xv,yv,wv) local Delta,N,a,b,c,d,e,A,B;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "N:=nops(xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "if nops(yv) " 0 "" {MPLTEXT 1 0 19 "a:=addtr(wv,xv,xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "b:=addpr(wv,yv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c:=addpr(wv,xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d:=addtr(wv,xv,yv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e:=addme (wv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Delta:=evalf(e*a-c^2);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=(a*b-c*d)/Delta;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "B:=(e*d-c*b)/Delta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "print(`[A,B]`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " [A,B];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "WLLSQ(lxval,lyval,lwval);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% &[A,B]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+Q?/c5!\"*$\"++8=Bc!#5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ywlsq:=%[1]+%[2]*x;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ywlsqG,&$!+Q?/c5!\"*\"\"\"%\"xG$\"+ +8=Bc!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Pwfit:=plot(ywl sq,x=xval[1]-0.1..xval[imax]+0.1,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 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