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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 21 "Fitting of noisy data" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "We used \+
a linear least squares fit to analyze the linear relationship between \+
given data. Linear data fitting is applicable for many problems, and i
n fact can also be used for exponential laws (simply take the logarith
m of the " }{TEXT 259 2 "y-" }{TEXT -1 69 "values and then fit), and e
ven power laws (take the logarithm of the " }{TEXT 257 1 "x" }{TEXT 
-1 5 " and " }{TEXT 258 1 "y" }{TEXT -1 190 " values and fit). In this
 worksheet we continue along the same lines, but extend the fit to all
ow for a quadratic function. The fit is still linear in the parameters
 to be determined, i.e., " }{XPPEDIT 18 0 "y = a+bx+cx^2;" "6#/%\"yG,(
%\"aG\"\"\"%#bxGF'*$)%#cxG\"\"#F'F'" }{TEXT -1 404 ". Note that the pr
oblem where nonlinear parameters are to be determined, and where the p
roblem cannot be reduced to a linear fit of a related problem (as ment
ioned above), is considerably more difficult. No built-in procedures a
re available in Maple for nonlinear fitting at the present time. There
 exist implementations of some algorithms for this case described in t
he literature (cf. W. Press et al., " }{TEXT 270 17 "Numerical Recipes
" }{TEXT -1 30 ", Cambridge University Press)." }}{PARA 12 "" 1 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 408 "We follow another approa
ch in the following respect: we will use a random number generator to \+
produce data with uncertainties of a known variance. Then we will test
 the fitting techniques and the error estimates. Thus, we will use Map
le to perform a computer experiment. We will rely solely on Maple's bu
ilt-in fit procedure, even though it does not provide us with uncertai
nties for the fitted coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 136 "We start with the drawing tools that we \+
developed for plotting symbols with error bars. They work reasonably w
ell if the scaling on the " }{TEXT 261 1 "x" }{TEXT -1 5 " and " }
{TEXT 260 1 "y" }{TEXT -1 143 " axes is not very different. You may ne
ed to scale the data if they don't perform well for you (or replace th
e Circle procedure by an Ellipse)." }}}{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 0 112 "plot([args[3]*cos(t)+args[1],args[
3]*sin(t)+args[2],t=0..2*Pi],scaling=CONSTRAINED,colour=col,thickness=
2); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Line:=proc() lo
cal col;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "if nargs<5 then col:=bl
ack else col:=args[5] fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "plot([
[args[1],args[2]],[args[3],args[4]]],scaling=CONSTRAINED,colour=col); \+
end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Expoint:=proc() loc
al size,col; size:=2*args[5];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "if
 nargs<6 then col:=black else col:=args[6] fi;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 222 "Circle(args[1],args[2],0.5*args[5],col),Circle(args[
1],args[2],args[5],col),Line(args[1],args[3],args[1],args[4],col),Line
(args[1]-size,args[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 "en
d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 236 "We need a random number generator
 that produces random numbers on the interval [-1,1], which are then e
asily transformed to some other interval. Maple has by default an inte
ger random number generator. The stats package is of some help." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7*%&anovaG%)describeG%$fitG%+importdataG%'randomG%
*statevalfG%*statplotsG%*transformG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "with(stats[statplots]):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 366 "We learn that there is a random number generator for nor
mally or uniformly distributed data, and a histogram function that wor
ks either with tallied (adjusted) intervals so that each bin has equal
 area, or for equidistant bins. The bin size is calculated by the prog
ram, and the area under the histogram equals unity to allow for a dire
ct probability interpretation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "histogram([random[normald](250)]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 303 303 303 {PLOTDATA 2 "6%-%)POLYGONSG6-7&7$$!+k$y_e#!\"*\"
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2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "data:=random[normald](250):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "histogram[1]([data]);" }}{PARA 13 "" 1 "
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o$\"+#GLIe#F87$$\"+])3&*z*F8Fio7$F\\pF+7&F^p7$F\\p$\"+QcgT>F87$$\"+Gp;
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6&%$RGBG$\"\"(!\"\"Fcq$\"#5Feq-%%VIEWG6$;$!+)\\l#GPF*$\"+0\")=!z$F*%(D
EFAULTG" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "xrand:=stats[random, uniform
](20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&xrandG66$\"+.LU0F!#5$\"+W
eTW)*F($\"+a\">Gm&!#6$\"+$*zoimF($\"+B&*RCFF($\"+q@0e(*F($\"+9d_*z&F($
\"+Y'3a%GF-$\"+`Ek%f\"F($\"+oP$=^$F($\"+\\'H,B#F($\"+6m*=z$F($\"+X[l!
\\#F-$\"+><b6rF($\"+W*y4)=F($\"+S#3jj%F($\"++wNMaF($\"+)>IwO&F($\"+,do
)G#F($\"+iAs$>(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "histog
ram[1]([xrand]);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 
2 "6%-%)POLYGONSG6(7&7$$\"+X[l!\\#!#6\"\"!7$F($\"+_eiQ7!\"*7$$\"+`Ek%f
\"!#5F-7$F1F+7&F47$F1$\"+D!)Q,CF/7$$\"+-do)G#F3F77$F:F+7&F<7$F:$\"+c=j
A7F/7$$\"+!>l=l$F3F?7$FBF+7&FD7$FB$\"+WM?]$*F37$$\"++wNMaF3FG7$FJF+7&F
L7$FJ$\"+c<BP**F37$$\"+?<b6rF3FO7$FRF+7&FT7$FR$\"+wyg)4'F37$$\"+WeTW)*
F3FW7$FZF+-%&COLORG6&%$RGBG$\"\"(!\"\"F[o$\"#5F]o-%%VIEWG6$;$!+:bp/rF*
$\"+)3&R!3\"F/%(DEFAULTG" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "These are (suppose
dly) uniformly distributed over [0,1]. " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 262 16 "Linear fit case:" }}{PARA 0 "" 0 "
" {TEXT -1 58 "We generate 5 data points with random errors according \+
to " }{TEXT 265 2 "y " }{TEXT -1 10 "= 1 + 0.5 " }{TEXT 264 1 "x" }
{TEXT -1 13 ". We use the " }{TEXT 266 1 "x" }{TEXT -1 15 " interval 0
..2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Y:=x->1+0.5*x;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YGR6#%\"xG6\"6$%)operatorG%&arrowG
F(,&\"\"\"F-9$$\"\"&!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "errsize:=0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(errsizeG$
\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "imax:=5: for
 i from 1 to imax do: xval[i]:=2/imax*(i-0.5):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "yval[i]:=Y(xval[i])+errsize*(2.*xrand[i]-1.); od:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "P1:=plot(Y(x),x=0..2,color=
red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "P2:=\{\}: i:='i': \+
symbolsize:=0.02: for i from 1 to imax do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 102 "P2:=\{op(P2),Expoint(xval[i],yval[i],yval[i]+errsize
,yval[i]-errsize,symbolsize,blue)\}: od: P2:=op(P2):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{P1,P2\});" }}{PARA 13 "" 1 "" 
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or bounds are not standard errors, but brackets on the error as we hav
e a uniform distribution of the random numbers used to produce the dat
a points from exact values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 214 "We are ready to perform a least squares fit. W
e can get help by requesting ?fit. We need to do some experimenting to
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Thus we convert in the call:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG$\"+>7\"G%[!#5$\"+7<t05!\"*\"\"
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The difference in the slope and " 
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{PARA 0 "" 0 "" {TEXT 263 19 "Quadratic fit case:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We generate 6 data points
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bX=F>$\"1-jT<j@V=F>7$$\"1NHxL>`Z=F>$\"1B)yy[k\\%=F>7$$\"1Wuy*oL\"\\=F>
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "For comparison:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sort(y); sort(Y(x));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"$\"+<?ipc!#5F&$!+CaR'*fF+$
\"+jT^=5!\"*\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#
\"\"\"$\"\"&!\"\"F&$!\"&F+\"\"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 210 "We learn that a least squares fit for a quadratic function can
 be a trickier problem: the added flexibility puts a greater demand on
 the problem. The coefficients deviate more from the 'true' model (red
 curve)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 
10 "Exercises:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 142 "1) Imp
rove the situation by adding more data points in the interval. Also tr
y fewer data points. Repeat the fits with increased error margins." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "2) Obser
ve the deviation between the 'true' and the model result based on rand
omized data fitting outside the interval in which the fitting was perf
ormed (extrapolation problem)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 127 "3) Repeat the data fitting problems with
 normally distributed errors (the error size plays the role of the sta
ndard deviation)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 50 "4) Test the data fit problem for a cubic function." }}}
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