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{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 "" 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Here comes the procedure to plot \+
a data point with error bar. Five arguments specify the point, and the
 sixth is for the colour." }}{PARA 0 "" 0 "" {TEXT -1 80 "The first tw
o arguments provide the location of the point, the next two the low-" 
}{TEXT 270 1 "y" }{TEXT -1 10 " and high-" }{TEXT 269 1 "y" }{TEXT -1 
112 " values, and the fifth argument specifies the width of the error \+
bar to either side, and the size of the symbol." }}}{EXCHG {PARA 0 "> \+
" 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 
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\"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$$\"1c'f`bp!*f\"F*$\"1I&QpK!R'=#F*7$F^]l$\"1U?3++++AF*F0Ff\\l
-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%!GFe\\m-%*AXESSTYLEG6#%$BO
XG-%%VIEWG6$;$!\"\"F7$Fi\\lF7;F7$\"\"%F7" 1 2 0 1 0 2 9 1 2 1 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
100 "If desired, one can fill in the circle with more than one, as don
e here, but it will slow plotting.." }}{PARA 0 "" 0 "" {TEXT -1 335 "W
e can now plot a circle that when reduced in size will appear  filled,
 and an 'error' bar that indicates the uncertainty of the  measurement
.  It is straightforward to write routines for other symbols (squares \+
and  diamonds at least)  We generate a colour table and have the luxur
y of distinguishing  subsequent data points by colour:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 61 "coltable:=[violet,magenta,blue,cyan,green,yell
ow,red,maroon];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)coltableG7*%'vio
letG%(magentaG%%blueG%%cyanG%&greenG%'yellowG%$redG%'maroonG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "imax:=7;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%imaxG\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "N
ow we set up a loop to prepare the data point structure:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "P3:=[]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
for i from 1 to imax do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dline:=
data[i]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "P3:=[Expoint(dline[1],d
line[2],dline[3],dline[4],0.01,coltable[i+1]),op(P3)]: od:" }}}{EXCHG 
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7F0$!1')eg?sUqSF*7$$\"1APRy,!z?\"F0$!1$H<bQ381%F*7$$\"15i^PVn37F0$!1)z
D(p_v\\SF*7$$\"1;K&4+D#47F0$!1sqw&o*fQSF*7$$\"1B#p))oa'47F0$!1^nRQ=0ES
F*7$$\"1mf`bp!*47F0$!1q91t'4O,%F*7$F^`v$!1fz\"*********RF*Fc]lFe]l-F$6
%7S7$$\"1++++++0>F0F87$$\"1!)p=\"=`\\!>F0$\"1@$)y74REo!#>7$$\"17%\\5!p
$[!>F0$\"1juzzWkm7Fa\\s7$$\"1I%RKHCY!>F0$\"1n\"GC;b:!>Fa\\s7$$\"1)Rfa`
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!>F0$\"1\"=Y/Nr(HNFa\\s7$$\"1gJ(yAeI!>F0$\"131wC&fc&RFa\\s7$$\"1a\"ydQ
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7$$\"1@[#*3pn)*=F0$\"1h4fuiw@[Fa\\s7$$\"1oW`XM6)*=F0$\"1!R6zEL/j%Fa\\s
7$$\"1*=jB%)yu*=F0$\"1@$Rl`UyJ%Fa\\s7$$\"1A=Eb-)p*=F0$\"1#3*>>]6&)RFa
\\s7$$\"1=m#=Mbk*=F0$\"1!y$=IgQENFa\\s7$$\"1Cs*y*y/'*=F0$\"1ae:#pBG1$F
a\\s7$$\"1&4ImMqc*=F0$\"1Zmu/-#3]#Fa\\s7$$\"1KM'=`%Q&*=F0$\"1MV(*fk(G#
>Fa\\s7$$\"1)HX#)=o^*=F0$\"1ZtI8>#fG\"Fa\\s7$$\"1\"H!*\\[Y]*=F0$\"1;-E
D]7-oFi_w7$$\"1e`/C++&*=F0$\"18hC=Zl]:Fbhu7$$\"1Q1*4yX]*=F0$!1ka*4Gf1v
'Fi_w7$$\"1tjEnH;&*=F0$!1ox89:9m7Fa\\s7$$\"1$[$*)o$o`*=F0$!1b#f\"o<h$)
=Fa\\s7$$\"1^R$p!Hm&*=F0$!1jnAVa*y[#Fa\\s7$$\"1gV6mp-'*=F0$!1s[^M@iNIF
a\\s7$$\"1a9.<ZW'*=F0$!15T.1Rn:NFa\\s7$$\"1T\\y1d(p*=F0$!1;Q!RHk;)RFa
\\s7$$\"1t3zbY](*=F0$!1En!eq6GL%Fa\\s7$$\"1S\"f#4O6)*=F0$!1`w1q**\\IYF
a\\s7$$\"1a$4sm&p)*=F0$!1sf)zMuo#[Fa\\s7$$\"1zBh;UN**=F0$!1(>Ml*47e\\F
a\\s7$$\"1`G38Z)***=F0$!1.='4jw***\\Fa\\s7$$\"1;oihQk+>F0$!1tBaA4Pe\\F
a\\s7$$\"1_Gn?vF,>F0$!1UUt9)RS$[Fa\\s7$$\"1zF>**y\">!>F0$!1^`3'fQvh%Fa
\\s7$$\"1\\!)G];]->F0$!1:!y_!Q<HVFa\\s7$$\"1W(*>vY0.>F0$!1xW#fk.%eRFa
\\s7$$\"1MK>]'\\N!>F0$!1'G%HI5O@NFa\\s7$$\"1ho>*3]R!>F0$!1`Y'eF>a1$Fa
\\s7$$\"10\"e(orL/>F0$!15**G'[jx[#Fa\\s7$$\"13mZ+Dh/>F0$!1)f`$)G%)*H>F
a\\s7$$\"16YVWt#[!>F0$!1Iv$)>>f-8Fa\\s7$$\"1$)zwxM&\\!>F0$!1r[tIl$[!oF
i_w7$Fb_w$\"1BmIq&o?5%Fi_vF[^mFe]l-F$6$7$7$Fd`oFbas7$Fi`oFbasF[ao-F$6%
7S7$$\"1++++++0<F0Ff]m7$$\"1!)p=\"=`\\q\"F0$!1=@(34O<$**FA7$$\"17%\\5!
p$[q\"F0$!1a--_bLt)*FA7$$\"1I%RKHCYq\"F0$!1%=dP[W)4)*FA7$$\"1)Rfa`EVq
\"F0$!1Dt-`*y$\\(*FA7$$\"1Pk,E@&Rq\"F0$!1`))\\;2s$p*FA7$$\"1?V3(GTNq\"
F0$!1#Qb\\'G-Z'*FA7$$\"1gJ(yAeIq\"F0$!1SR_ZSV/'*FA7$$\"1a\"ydQ0Dq\"F0$
!1T:1$p)Hn&*FA7$$\"1NB#)p+\">q\"F0$!1**pWC<#z`*FA7$$\"16\"QJpi7q\"F0$!
1KfF$f1i^*FA7$$\"138ToDn+<F0$!1@G\"=8WX]*FA7$$\"18^Y9q***p\"F0$!1&>M\"
*3++]*FA7$$\"1z.#)e(=$*p\"F0$!1F!>cjiY]*FA7$$\"1@[#*3pn)p\"F0$!1/4asL#
y^*FA7$$\"1oW`XM6)p\"F0$!1h)3Kncp`*FA7$$\"1*=jB%)yup\"F0$!1pgMYd@o&*FA
7$$\"1A=Eb-)pp\"F0$!1#4!3)\\)[,'*FA7$$\"1=m#=Mbkp\"F0$!1B;)pRhtk*FA7$$
\"1Cs*y*y/'p\"F0$!1:WyIwr$p*FA7$$\"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$!1a<GX$\\%)***FA7$$\"1
R1*4yX]p\"F0$!1&*4Gf1v15F*7$$\"1tjEnH;&p\"F0$!1y89:9m75F*7$$\"1$[$*)o$
o`p\"F0$!1$f\"o<h$)=5F*7$$\"1^R$p!Hm&p\"F0$!1oAVa*y[-\"F*7$$\"1gV6mp-'
p\"F0$!1\\^M@iNI5F*7$$\"1a9.<ZW'p\"F0$!1T.1Rn:N5F*7$$\"1T\\y1d(pp\"F0$
!1Q!RHk;)R5F*7$$\"1t3zbY](p\"F0$!1n!eq6GL/\"F*7$$\"1S\"f#4O6)p\"F0$!1x
1q**\\IY5F*7$$\"1a$4sm&p)p\"F0$!1g)zMuo#[5F*7$$\"1zBh;UN*p\"F0$!1U`'*4
7e\\5F*7$$\"1`G38Z)**p\"F0$!1='4jw***\\5F*7$$\"1;oihQk+<F0$!1CaA4Pe\\5
F*7$$\"1_Gn?vF,<F0$!1Ut9)RS$[5F*7$$\"1zF>**y\">q\"F0$!1a3'fQvh/\"F*7$$
\"1\\!)G];]-<F0$!1!y_!Q<HV5F*7$$\"1W(*>vY0.<F0$!1X#fk.%eR5F*7$$\"1MK>]
'\\Nq\"F0$!1VHI5O@N5F*7$$\"1ho>*3]Rq\"F0$!1Z'eF>a1.\"F*7$$\"10\"e(orL/
<F0$!1**G'[jx[-\"F*7$$\"13mZ+Dh/<F0$!1ON)G%)*H>5F*7$$\"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$$\"1N[$*)o$o`**F*$!1#f\"o<h$)=]F*7$$\"14&R$p!Hm&**F*$!
1oAVa*y[-&F*7$$\"1+O9h'p-'**F*$!1\\^M@iNI]F*7$$\"1VXJqrWk**F*$!1T.1Rn:
N]F*7$$\"16%\\y1d(p**F*$!1Q!RHk;)R]F*7$$\"1H(3zbY](**F*$!1n!eq6GL/&F*7
$$\"1(R\"f#4O6)**F*$!1x1q**\\IY]F*7$$\"1QN4sm&p)**F*$!1g)zMuo#[]F*7$$
\"1%zBh;UN***F*$!1T`'*47e\\]F*7$$\"1K&G38Z)****F*$!1>'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_!Q<HV]F*7$$\"1W(*>vY0.5F0$!
1X#fk.%eR]F*7$$\"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<F.*&-F'6$*$)F*F@F<F1F.F4F<F6!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 154 "Everything goes as per textbook, except that the \+
sum of 1 from 1 to N  is not simplified to N. This is due to the fact \+
that the unevaluated  Sum was used." }}{PARA 0 "> " 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<F8F9F:*&F2F9F1F0F0!\"\"F:" }}}
{EXCHG {PARA 0 "> " 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#$\"#I\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 68 "We are ready to program the standard line
ar least squares algorithm:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i
':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "LLSQ:=proc(xv,yv) loc
al _i,N,a,b,c,d,Delta,A,B,sq,sigsq,dA,dB;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "N:=nops(xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "if
 nops(yv)<N then RETURN(`Mismatch in sizes of xv and yv`); fi;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c:=addpr(xv,xv);" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "d:=addme(xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "a:=addpr(xv,yv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b:=addme(yv
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Delta:=evalf(N*c-d^2);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=(c*b-d*a)/Delta;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "B:=(N*a-d*b)/Delta;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "sq:=[]; for _i from 1 to N do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "sq:=[op(sq),evalf(yv[i]-A-B*xv[i])]: od:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 26 "sigsq:=addpr(sq,sq)/(N-2);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "dA:=sigsq*c/Delta;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "dB:=evalf(N*sigsq)/Delta;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "print(`[A,B,sigma,dA,dB]`);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "[A,B,evalf(sqrt(sigsq),4),evalf(sqrt(dA),4),evalf(sqr
t(dB),4)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "LLSQ([1,2,3,4],[2.1,4.,5.9,8.2]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%2[A,B,sigma,dA,dB]G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#7'\"\"!$\"++++??!\"*$\"%\\:!\"%$\"%(*=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<!\"($\"+++++5F($\"+++++k!\")$\"+0R'
oO#F($\"+++]i:F(F+$\"+AhIlKF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 63 "addtr:=(a1,a2,a3)->evalf(Sum(a1[i]*a2[i]*a3[i],i=1..nops(a1)));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&addtrGR6%%#a1G%#a2G%#a3G6\"6$%)
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)<N then RETURN(`Mismatch in sizes of xv and yv`); fi;
" }}{PARA 0 "> " 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 "display(\{Pfit,Pwfit,op(P3)\},axes=boxed);" }}
{PARA 13 "" 1 "" {GLPLOT2D 709 303 303 {PLOTDATA 2 "6K-%'CURVESG6$7$7$
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