Fitting Data

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apter 5Fitting Data to Nonlinear Models

One of the more difficult topics in all of data analysis in the physical sciences is fitting data to nonlinear models. Often

such fits require large computational resources and great skill, patience, and intuition on the part of the analyst. These

difficulties are one of the reasons that, as we shall see, the whole topic of spectral line shapes is still a very active subject

of research spanning the fields of chemistry, physics, astronomy, and more. In addition, computational methods of

nonlinear fitting is still a current research topic in computer science.

However, since sometimes nature really is nonlinear, such fits are often unavoidable, and the principles and some tools

for nonlinear fitting are the topics of this chapter. The main EDA program introduced here is FindFit, which

accomplishes fits to arbitrary models.

FindFit is similar to the NonlinearFit function in the Statistics‘NonlinearFit‘ package which is

standard with Mathematica. The primary differences are that FindFit: (1) recognises EDA’s data format, including

errors in both coordinates; (2) estimates errors in fit parameters; (3) by default displays graphical information about the

fit; and (4) uses an algorithm that has been optimised for speed and stability for the types of nonlinear fits commonly

performed in the physical sciences and engineering.

The following command loads the initialization of EDA.

Needs "EDA‘Master‘"

5.1 Introduction

5.1.1 Overview of FindFit

The previous chapter, "Fitting Data to Linear Models by Least Square Techniques", introduced the distinction between

linear and nonlinear models. To briefly review, the terms refer to the way in which the parameters to which we are

fitting enter into the model.

In this chapter we discuss nonlinear models and the EDA program FindFit that can often find a reasonable fit to them.

Recall that if sos is the sum of the squares of the residuals, then we are seeking the minimum in its value. If we are

fitting to parameters a[0], a[1], ... , a[m], the answer is found by solving a set of simultaneous equations.

D[ sos, a[0] ] == 0D[ sos, a[1] ] == 0...D[ sos, a[m] ] == 0

This in general can be done analytically provided the model to which we are fitting is linear in the parameters. Similarly,

when there are explicit errors in the data, we form the chi−squared, say chisq, and we solve the corresponding equations.

D[ chisq, a[0] ] == 0D[ chisq, a[1] ] == 0...D[ chisq, a[m] ] == 0

This again will be analytic for a linear fit.

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For a nonlinear fit, no such analytic solutions are possible, so iteration is required to find the minimum in the sum of the

squares or the chi−squared.

If we imagine a plot of the value of the sum of the squares or the chi−squared as a function of the parameters to which

we are fitting, in general for a nonlinear fit there may be many local minima instead of one big one, as is the case for

linear fitting. For example, if we are fitting to two parameters, param1 and param2, the chi−squared as a function of the

values of the parameters might have two or more local minima.

Plot3D 1 1

param12 param22 35

1

param1 25 2 param2 10 2 25,

param1, 20, 50 , param2, 25, 25 ,PlotRange All, PlotPoints 40, Ticks Automatic, Automatic, None ,

AxesLabel "param1", "param2", "chi squared" , ViewPoint 1.3, 2.4, .75 ;

-200

2040 param1

-20-100 10 20

param2

chi-squared

-200

2040

- - 0 10 20

Thus, a nonlinear fitter must usually start off with initial values close to the real minimum.

The general technique for iteration, "steepest descent", is analogous to the

following situation. It was "a dark and stormy night". Foggy too. You are on the

side of a hill and want to find the valley. So you step in the direction in which the

slope goes down, and continue moving in the direction of the local definition of"down" until you are in the valley. Of course, if you take giant steps you might

step over the next hill and end up in the wrong valley. And when you get close to

the bottom of the valley you will want to start taking baby steps. The

Levenberg−Marquardt algorithm used by many nonlinear fitters, including

FindFit, is essentially some clever heuristics to define giant steps and baby

steps.

If the data has noise, which is almost a certainty for real experimental data, then there is a further difficulty. We can take

two sets of data from the same apparatus using the same sample, fit each dataset to a nonlinear model using identical

initial values for the fit parameters, and get very different final fits. This situation leads to ambiguity about which fit

results are "correct".

5.1.2 Providing Initial Parameter Values to FindFit

As already mentioned, unless we are fitting to a very simple model, FindFit must be provided with initial estimates of

the parameters.

We will first fit to a simple model where no such parameter estimates are necessary.

mydata Table n, 2 Cos 3 n , n, 0., 1.5, .5 ;

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FindFit mydata, b Cos a x , x, a, b

0 0.25 0.5 0.75 1 1.25 1.5

-2

-1

0

1

2

-0.0020

0.0020.0040.0060.008

Residuals

a

3., 0.34 , b

2., 0.71 , SumOfSquares0.0000952225702044433,

DegreesOfFreedom2

Actually, when not given initial values FindFit starts with initial values for the parameters, here a and b, equal to 1.

Now the syntax of the call to FindFit is given.

FindFit[data, model, ind,{param1, ... , paramM} ]

Here ind is the name of the independent variable in model. Also note that this model is nonlinear in the parameters to

which we are fitting, param1 ... paramM .

The EDA‘FindFit‘ package includes a Gaussian function.

Information "Gaussian", LongForm False

Gaussian[x,ampl,x0,sigma] is a Gaussian function of the independent variable x with amplitude

ampl, center value x0, and standard deviation sigma. The function is normalized so that its integral from −Infinity to +Infinity is equal to Sqrt[2Pi]*ampl*sigma.

We will use it to generate some data.

SeedRandom 1234 ; mydata Table x, Gaussian x, 10, 8, 2.5 Random Real, 1, 1 , x, 1, 16, .1 ;

We examine the data with EDAListPlot.

EDAListPlot mydata ;

2.5 5 7.5 10 12.5 15

2

4

6

8

10

We fit the data using FindFit; the fit takes a few seconds to complete.

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FindFit mydata, Gaussian x, ampl, x0, sigma , x, ampl, x0, sigma

0 2.5 5 7.5 10 12.5 15

0

2

4

6

8

10

-4

-2

0

2

4

6

Res ua s

ampl

3.9, 1.9 , x0

100., 3600. , sigma

1000., 23000. ,SumOfSquares

1812.095726856373, DegreesOfFreedom

148

This result is ridiculous. What has happened is that FindFit thinks it has found a local minimum in the sum of the

squares at these silly values.

Note that because all four quadrants of the plot of the data and the fit contain significant amounts of data, FindFit is

displaying the plot of the residuals separately.

Repeating the fit with reasonable initial guesses of the parameters gives a much better result.

FindFit mydata, Gaussian x, ampl, x0, sigma , x, ampl, 9 , x0, 9 , sigma, 3

0

8

6

4

2

0

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1

5

0

5

1

ampl 9.91, 0.18 , x0 8.075, 0.05400000000000001 ,sigma

2.488, 0.05400000000000001 , SumOfSquares

49.5618230640989,

DegreesOfFreedom148

Finding good initial values is sometimes subtle. As an example, we generate some made−up data for three peaks with a

Lorentzian shape using the Lorentzian function supplied with the EDA‘FindFit‘ package.

Information "Lorentzian", LongForm False

mydata Table x, Lorentzian x, 100, 18, 8

Lorentzian x, 10, 23, 4 Lorentzian x, 80, 35, 8 , x, 1, 60, .1 ;EDAListPlot mydata ;

Lorentzian[x,a,x0,gamma] is a nonrelativistic Lorentzian function of the independent variable x, center value x0, and full width at half−maximum gamma. The function is normalized so that its integral from −Infinity to +Infinity is equal to ’a’. The maximum amplitude of the peak is 2*a/(Pi*gamma). Lorentzian is a synonym for BreitWigner.

10 20 30 40 50 60

2

4

6

8

It is difficult to see the small peak on the shoulder of the leftmost peak, or to provide initial estimates for its values.

However, we can see the peak and probably make some sensible guesses of its parameters.

Despite the claims sometimes seen in glossy advertisements, there is no known software that can find and estimate peaksfor data such as this as well as a human expert. This is in part because of the great ability of the human visual system to

be an intuitive integrator.

Almost all versions of the notebook front end for Mathematica include a provision to use our visual ability. We can

display a graph of the data and using the mouse, point at a desired location on the graph. The coordinates will be

displayed in the notebook window, and can also be copied out using the cut−and−paste facility; consult the manual for

your version of the notebook to discover how to do this. For example, here we load and examine a part of a nuclear

spectrum.

LoadData Cobalt60

LoadData::name: Loading: Cobalt60Data

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Information "Cobalt60Data", LongForm False

Cobalt60Data is part of a Cobalt−60 nuclear spectrum. The format is: {channel, counts}.

The data were taken in the II Year Lab of the Dept. of Physics, Univ. of Toronto with a NaI scintillator and a PC−based multichannel analyzer running PC−MCA software by David Harrison (1990, unpublished).

EDAListPlot Cobalt60Data ;

1750 1800 1850 1900 1950

50

100

150

200

Using the mouse we can pick out the coordinates of the maximum, the points on the peak corresponding to roughly

one−half the maximum, and so on. So, the position of the peak is found.

1831.88, 181.251

In some earlier versions of the Windows front end, if the notebook is closed and then reopened, the coordinates will be

returned in PostScript units instead of the actual units of the plot; rerendering the plot works around this bug.

Finally, for simple spectra EDA supplies a function, FindPeaks, which can sometimes provide initial guesses of the

number of peaks and their parameter values close enough for FindFit. The function is described in Section 8.3.

5.1.3 Comparing LinearFit and FindFit

Although FindFit is intended for fitting to nonlinear models, it can also find the minimum in the sum of the squares

or the chi−squared for linear models, albeit more slowly than LinearFit.

For example, here we repeat a fit from Chapter 4.

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LoadData ThermocoupleLinearFit ThermocoupleData, 0, 1, 2 , a

LoadData::name: Loading: ThermocoupleData

0 20 40 60 80 100

-1

0

1

2

3

-0.06-0.04-0.02

00.020.040.06

Residuals

a 0 0.886, 0.03 , a 1 0.0352, 0.0014 , a 2 0.00006, 0.000013 ,ChiSquared


18

Using FindFit yields the same result.

FindFit ThermocoupleData, a 0 a 1 x a 2 x2, x, a 0 , a 1 , a 2

0 20 40 60 80 100

-1

0

1

2

3

-0.06-0.04-0.02

00.020.040.06

Residuals

a 0 0.886, 0.03 , a 1 0.0352, 0.0014 , a 2 0.00006, 0.000013 ,ChiSquared


18

For this example, FindFit took about four times as long as LinearFit.

Recall from the chapter on linear fitting that if the data have explicit errors in both coordinates, the effective variance

technique makes the fit essentially nonlinear unless the model is a straight line. Thus, in this case LinearFit iterates

until it finds a minimum in the chi−squared. When there are errors in both coordinates, FindFit also calculates the

error in the dependent variable based on the effective variance. However, although there is a fairly comprehensive

literature on using this technique in linear fits, the main justification of using effective variances in nonlinear fits is

based only on a series of experiments in which it was found that most often the algorithm of FindFit would produce

reasonable results.

We repeat another fit from the chapter on linear models.

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LoadData ReactanceLinearFit ReactanceData, 1, 1 , a

LoadData::name: Loading: ReactanceData

220002300024000250002600027000

-4

-2

0

2

4

6

8

-10123

4

Residuals

a 1 590000.0000000001, 110000. , a 1 0.00102, 0.0002 ,ChiSquared


3

FindFit gives a similar result if started sufficiently close to the "final" values.

FindFit ReactanceData,a 1

x

a 1 x, x, a 1 ,

590000 , a 1 , 0.001

220002300024000250002600027000

-4

-2

0

2

4

6

8

-101234

Res ua s

a 1 590000.0000000001,3.800000000000001

1014, a 1 0.001009, 0.000021 ,

ChiSquared2.203721816446138, DegreesOfFreedom

3

For the above results, FindFit appears to have done a poorer job of estimating the errors in the fit parameters than

LinearFit.

Another difference between the above two fits is that by default the graphs produced by LinearFit display the errors

in the fit parameters, while the graphs produced by FindFit do not. This is because for many nonlinear fits, sorting

out how to combine the various error terms is problematic. To display the fit errors the UseFitErrors option,

discussed in Section 5.3.2.2, may be set to True.

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FindFit ReactanceData,a 1

x

a 1 x, x, a 1 , 590000 , a 1 , 0.001 ,

UseFitErrors True

220002300024000250002600027000

-4

-2

0

2

4

6

8

-10123

4

Residuals

a1

590000.0000000001,

3.800000000000001

1014, a 1

0.001009, 0.000021 ,

ChiSquared


3

Note that as opposed to the plot produced by LinearFit, here the lines representing the errors in the fit parameters are

parallel. This is an artifact of a heuristic used by the FindFit package to combine the two terms; the heuristic is often

reasonable for true nonlinear fits.

As a final comparison between LinearFit and FindFit, we load and examine some data on oscillations in

interneuron networks.

LoadData Interneuron

LoadData::name: Loading: InterneuronData

Information "InterneuronData", LongForm

False

InterneuronData are data of oscillations of interneuron networks driven by metabotopic glutamate receptor activation in slices of rat hippocampus and neocortex. The data are from Miles A. Whittington, Roger D. Traub and John G.R. Jefferys, Nature 373, (1995) pg 612. The format of the data is {ipscTau, frequency}, where ’ipscTau’ is the decay constant of the inhibitory postsynapic currents in ms and ’frequency’ is the oscillation frequency of the network in Hz.

EDAListPlot InterneuronData ;

10 15 20 25 30

25

30

35

40

45

The data appears to be modeled by an exponential relationship.

frequency = a*Exp[−b*ipscTau]

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This is nonlinear in the parameters a and b. However, as mentioned in the previous chapter, we can linearize the

relationship by taking the logarithms of both sides.

Log[frequency] = Log[a] − b*ipscTau

Thus, we form a data set of {ipscTau, Log[ frequency]}.

mydata First #1 , Log Last #1 &

InterneuronData;

We fit this transformed data to a straight line.

linresult LinearFit mydata, 0, 1 , a

5 10 15 20 25 302.8

3

3.2

3.4

3.6

3.8

-0.3-0.2-0.1

00.10.2

Residuals

a 0

4.027000000000001, 0.05700000000000001 ,a 1 0.04240000000000001, 0.0033 , PseudoErrorY 0.1075978101620072,SumOfSquares 0.2547003525365055, DegreesOfFreedom 22

Thus b = − 0.0424 +/− 0.0033 in (ms)^(−1). Although there are problems with the above fit, which are discussed below,

we will calculate a and its errors using the Datum construct discussed in Chapter 3.

Exp Datum a 0 . linresult . Datum Identity

56.10000000000001, 3.2

Using FindFit, we can fit directly to the exponential.

FindFit InterneuronData, a Exp b x , x, a, 56 , b, 0.04

5 10 15 20 25 30

20

25

30

35

40

45

-6

-4-2024

Res ua s

a

59.90000000000001, 1. , b

0.04680000000000001, 0.0012 ,SumOfSquares


22

The two fits are similar, and both show some problems in the residuals. Also, FindFit seems more confident about the

uncertainties in the values of the fit parameters, probably without justification. The difference in the SumOfSquares is

because of the different values being used for the dependent variable of the data.

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One likely reason for the problems here is that the data does not seem to asymptotically approach zero, as assumed by

our model, but some other value c.

frequncy = a*Exp[−b*ipscTau] + c

Fitting to this model seems to confirm our hypothesis.

FindFit InterneuronData, a Exp b x c, x, a, 60 , b, 0.05 , c, 15

5 10 15 20 25 30

20

25

30

35

40

45

-4-2024

Res ua s

a

59.00000000000001, 2.3 , b

0.0925, 0.0094 , c

13.8, 1.5 ,SumOfSquares


21

This essentially duplicates a fit presented by the experimenters in their original paper.

The model can be linearized as follows:

frequncy = a*Exp[−b*ipscTau] + cfrequency − c = a*Exp[−b*ipscTau]Log[frequency − c] = Log[a] − b*ipscTau

Thus, we can form another data set in which we subtract 13.8 from each value of frequency before taking the logarithms.

mydata2 First #1 , Log Last #1

13.8 &

InterneuronData;

We fit this to a straight line.

linresult2 LinearFit mydata2, 0, 1 , a

5 10 15 20 25 30

1.5

2

2.5

3

3.5

-0.6-0.4-0.2

00.20.40.6

Residuals

a 0

4.030000000000001, 0.11 , a 1

0.0901, 0.006600000000000001 ,PseudoErrorY

0.2152138256526544, SumOfSquares

1.018973796545124,

DegreesOfFreedom22

Thus, a can be calculated.

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Exp Datum a 0 . linresult2 . Datum Identity

56.3, 6.200000000000001

This number is within errors of the result found by FindFit.

Of course, without FindFit or some other nonlinear fitter available, it would be difficult to get an objective estimateof the value that should be subtracted from each value of the dependent variable frequency.

Also, although not an issue here, this sort of linearization procedure can introduce biases in the values of the estimates of

the fit parameters; this is discussed further in Chapter 8.

5.1.4 References

Philip R. Bevington, Data Reduction and Error Analysis (McGraw−Hill, 1969), Chapter 11. A classic introduction to

nonlinear fitting techniques.

Xiang Ouyang and Philip L. Varghese, Appl. Optics 28 (1989), p. 1538. A discussion of a popular Fouriertransform−based algorithm for fitting spectra to Galatry and Voigt profiles.

William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, Numerical Recipes: The Art of

Scientific Computing or Numerical Recipes in C: The Art of Scientific Computing (Cambridge Univ. Press), Section

14.4. A good brief introduction to nonlinear fitting and the Levenberg−Marquardt algorithm used by FindFit.

A.P. De Weljer, C.B. Lucasius, L. Buydens, G. Kateman, H.M. Heuvel, and H. Mannee, Anal. Chem. 66 (1994), p 23.

An illuminating discussion of the research into fitting to curves using neural networks, this article also has a good

review of some of the problems with conventional techniques.

5.2 Examples

5.2.1 Fitting to a Single Peak with a Background

We begin by loading and examining some data that was briefly examined in Section 5.1.2.

LoadData Cobalt60

LoadData::name: Loading: Cobalt60Data

Information "Cobalt60Data", LongForm False

Cobalt60Data is part of a Cobalt−60 nuclear spectrum. The format is: {channel, counts}.

The data were taken in the II Year Lab of the Dept. of Physics, Univ. of Toronto with a NaI scintillator and a PC−based multichannel analyzer running PC−MCA software by David Harrison (1990, unpublished).

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EDAListPlot Cobalt60Data ;

1750 1800 1850 1900 1950

50

100

150

200

The theoretical prediction for the peak is that it should be a Gaussian, so part of the model for the fit will be the

Gaussian function included in the EDA‘FindFit‘ package.

Information "Gaussian", LongForm False

Gaussian[x,ampl,x0,sigma] is a Gaussian function of the independent variable x with amplitude ampl, center value x0, and standard deviation sigma. The function is normalized so that its integral from −Infinity to +Infinity is equal to Sqrt[2Pi]*ampl*sigma.

We also note that there is a background under the peak, i.e. ,counts in addition to just the Gaussian peak. We will

approximate the background as a straight line with the following slope.

a 1 N16 44

1950 1700

0.112

Here the numbers in the calculation are based on values obtained by pointing and clicking at the left− and right−hand

sides of the plot of the data with the mouse. We calculate the intercept.

a 0 16 a 1 1950

234.4

From the plot of the data we estimate the center of the peak to be at channel 1830, and the amplitude above the

background is about 140 counts. The full width at half−maximum is about 90 channels, so we will try an initial value for

sigma of 45 channels.

Since we are going to use a[0] and a[1] as names for fit parameters, we must clear the definitions we have just made for

them.

Remove "a "

Now we fit to the data.

FindFit Cobalt60Data, a 0 a 1 chan Gaussian chan, ampl, x0, sigma ,chan, a 0 , 234 , a 1 , 0.11 , ampl, 140 , x0, 1830 , sigma, 45

0

0

0

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0

0

0

0

0

0

0

a 0 201.6, 1.6 , a 1 0.09559, 0.00087 , ampl 154.03, 0.17 ,x0 1833.514, 0.05400000000000001 , sigma 43.498, 0.06900000000000001 ,SumOfSquares


281

Note that the Reweight option, which is True by default for LinearFit, is False by default for FindFit.

Section 5.3.1.9 discusses this further.

Here is the syntax of the call to FindFit.

FindFit[ data, model, ind, params ]

The model is the model to which we are fitting and ind is the independent variable. The params can be a list of the

parameter names.

{param1, param2, ... , paramM}

More often params is a list of { parameter name, initial value} pairs.

{ {param1, value1}, {param2, value2}, ... ,{paramM, valueM} }

The world will end if model contains undefined arguments in addition to ind and those named in params.

Other forms of parameter specification include {name, min, start, max}, which causes FindFit to begin parameter

name at start and not allow it to go below min or above max. Or the parameter can be specified as {name, min, max}, in

which case the value starts at (min + max)/2.

Note that in comparison to fits you may have done using LinearFit, FindFit is very slow. On a fairly fast UNIX

workstation, fitting to Cobalt60Data took over 30 seconds.

From the theory of nuclear physics, we expect the error in the number of counts in each channel to be Sqrt[counts].

Thus, we form a new data set with these errors included.

mydata N #1 1 , #1 2 , #1 2 & Cobalt60Data ;

We fit to this new data set.

FindFit mydata, a 0 a 1 chan Gaussian chan, ampl, x0, sigma , chan,a 0 , 234 , a 1 , 0.11 , ampl, 140 , x0, 1830 , sigma, 45

50

100

150

200

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-40

-20

0

20

40

Residuals

a 0 194., 10. , a 1 0.0919, 0.005200000000000001 , ampl 154.8, 1.7 ,x0 1833., 0.51 , sigma 42.89, 0.55 , ChiSquared 299.4179776308529,DegreesOfFreedom

281

Although the error bars in the data have obscured the curve representing the fit, the residuals and the ChiSquared per

DegreesOfFreedom show that the fit is reasonable, except perhaps on the far right−hand side.

There is another sort of bell−shaped curve that high−energy physicists usually call a "Breit Wigner". The same curve is

usually called a "Lorentzian" by spectroscopists. The EDA‘FindFit‘ package includes a BreitWigner function.

Information "BreitWigner", LongForm False

BreitWigner[x,a,x0,gamma] is a nonrelativistic Breit−Wigner function of the independent variable x, center value x0, and full width at half−maximum gamma. The function is normalized so that its integral from −Infinity to +Infinity is equal to ’a’. The maximum amplitude of the peak is 2*a/(Pi*gamma). BreitWigner is a synonym for Lorentzian.

We can compare the shape of a Breit−Wigner to a Gaussian.

Needs "Graphics‘Legend‘"Plot Gaussian x, 10, 10, 2.5 , BreitWigner x, 80, 10, 5 ,x, 0, 20 , PlotStyle GrayLevel 0 , Dashing 0.03 ,PlotLegend "Gaussian", "BreitWigner" , LegendSize 1, .4 ;

5 10 15 20

2

4

6

8

10

BreitWigne

Gaussian

Note that, consistent with a common convention, BreitWigner does not take the maximum amplitude as an

argument, but instead the total area under the curve. If the peak in the Cobalt−60 data is a Breit−Wigner, this

corresponds to the total number of counts in the peak. The total number of counts in the data, peak plus background, can

be calculated.

Last Plus Cobalt60Data

24057

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Also, the width is specified by the full width at the half−maximum, not the standard deviation. We fit the Cobalt−60 data

with errors to a Breit−Wigner plus a linear background.

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18/35

-50

0

50

100

150

Residuals

a 0

272.1666687,4.7

106, a 1

0.3632, 0.004400000000000001 ,

a 2 0.00009,2.3

106, ampl 215.2443773,

6.800000000000001

106,

x0

1212.69729812,6.1

107, sigma

704.3369377000001,

5.600000000000001

107,

ChiSquared 9042.73001502903, DegreesOfFreedom 280

This is a ridiculous result because FindFit fell into the wrong minimum in the ChiSquared . Try again with the

quadratic term set initially to zero, storing the answer in result.

result

FindFit mydata, a 0 a 1 chan a 2 chan2 Gaussian chan, ampl, x0, sigma , chan,a 0 , 234 , a 1 ,

0.11 , a 2 , 0.0 , ampl, 140 , x0, 1830 , sigma, 45

1700 1750 1800 1850 1900 1950 2000

0

50

100

150

200

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-20

0

20

40

60Residuals

a 0

233.8485, 0.001 , a 1

0.1236, 0.0058 ,

a 2

6.

106,

3.

106, ampl

139.87, 0.05300000000000001 ,

x0

1833.34, 0.57 , sigma

44.02000000000001, 0.6200000000000001 ,


280

Although not ridiculous, this fit is not nearly as good as the one without the quadratic background term. In fact, thechi−square probability is pretty small.

ChiSquareProbability ChiSquared, DegreesOfFreedom . result

0.0128891929874729

These sorts of difficulties are common when the number of fit parameters begins to get large. They are often particularly

acute when we are trying to fit to one or more peaks with a significant background under them. The "noise" in the data

further compounds the difficulty.

One frequently useful technique is to "sneak up" on the values. For example, we will fix all values of the fit at the values

of our best fit so far and allow the quadratic term to vary.

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FindFit mydata, 194 0.0919 chan a 2 chan2 Gaussian chan, 155, 1833, 42.9 ,chan, a 2 , 0.0

1700 1750 1800 1850 1900 1950 2000

0

50

100

150

200

-40

-20

0

20

40

Residuals

a 2

7.000000000000001

108,

1.1

107, ChiSquared

299.4365730959574,

DegreesOfFreedom285

Then we can try the full fit again, starting a[2] with the value from this fit, and storing the answer in result.

result

FindFit mydata, a 0 a 1 chan a 2 chan2 Gaussian chan, ampl, x0, sigma , chan,

a 0 , 194 , a 1 , 0.092 , a 2 ,7

108, ampl, 154 , x0, 1833 , sigma, 42.9

1700 1750 1800 1850 1900 1950 2000

0

50

100

150

200

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-40

-20

0

20

40

Residuals

a 0

194.00251, 0.00081 ,

a 1

0.0918, 0.005700000000000001 , a 2

0,2.9

106, ampl

154.009, 0.045 ,

x0

1832.98, 0.51 , sigma

42.99000000000001, 0.5400000000000001 ,


280

The point to all these fits is to demonstrate just how sensitive the results can be to apparently small changes in the initialvalues. Here we have a reasonable ChiSquared per DegreesOfFreedom and the quadratic term is now zero within

errors. The fact that we have not managed to account for the small upturn in the data on the right may be because it is

not an artifact of the background, but because there is another peak beyond the range of our data that is bringing up

these values. The fact that each of these fits takes a minute or more, depending on the hardware, means that nonlinear

fitting is often somewhat laborious and time consuming.

Finally, we can compare the total number of counts in the data, 24057, to the total predicted from the fit.

N NIntegrate Gaussian x, 154, 1833, 43 194 0.0918 x, x, 1700, 1985

23663.73555975553

Whether or not this number is close to the experimental number cannot be answered until we do some error analysis.We can find the contributions to this number from the background and the Gaussian, including errors, using some tools

discussed in Chapter 3.

We ignore the quadratic term in the background. Using the Datum construct, we can find the counts and error in the

counts due to the background.

bkgd 1700

1985

a b x x . a Datum a 0 , b Datum a 1 . result

Datum 7100., 3000.

We use the fact that the Gaussian is essentially zero at both the left and right of the data , so we can use the fullnormalization of the Gaussian to calculate the counts under the peak.

peak N 2 Datum ampl Datum sigma . result

Datum 16600., 220.

Thus, the total number of counts predicted by the fit is calculated.

bkgd peak . Datum Identity

23700., 3000.

This compares well with the experimental value of 24057.

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However, you should be aware that this fit pushed FindFit pretty hard. Not only is the fit very sensitive to the initial

values, but on a fairly fast UNIX workstation each took over five minutes of cpu to perform. This is why specialized

software to solve these sorts of problems has been written. See Ouyang and Varghese, listed in the references, Section

5.1.4, for an example of a technique for fitting this particular type of spectrum.

5.3 Options, Utilities, and Details

There are many options to FindFit that control both how it does the fit and what it returns. These are discussed in

Section 5.3.1.

The package also includes programs that are used by FindFit, but may also be used directly. These are the topic of

Section 5.3.2, which also discusses some convenience functions of various peak shapes.

5.3.1 Options to FindFit

The options and default values used directly by FindFit are given by using Options.

TableForm Options FindFit

AbsoluteChiSquaredTolerance −> 0.1

MaximumIterations −> 30

RelativeChiSquaredTolerance −> 0.005

ReturnCovariance −> False

ReturnEffectiveVariance −> False

ReturnErrors −> True

ReturnFunction −> False

ReturnResiduals −> False

Reweight −> False

ShowFit −> TrueShowProgress −> False UseSignificantFigures −> True

ValueTolerance −> 0.002

These options will be discussed in order.

In addition, if ShowFit is True, the default, FindFit, uses the function ShowFitResult. This function is

discussed in Section 5.3.2. Options to ShowFitResult given to FindFit are passed to that function.

If FindFit is called with ReturnFunction set to True, or if the ShowFit option is set to True, the default, the

function ToFitFunction is called. This function is discussed in Section 5.4.2. Options to ToFitFunction given

to FindFit are passed to that function.

5.3.1.1 The AbsoluteChiSquaredTolerance Option

FindFit uses three separate tests to determine if a fit has converged to a final value. The

AbsoluteChiSquaredTolerance option is used by one of those tests.

If the chi−squared (or the sum of the squares if there are no explicit errors in the data) is decreasing and, in the current

iteration, its value is less than the value in the previous iteration by AbsoluteChiSquaredTolerance, the fit is

udged to have converged.

If there are declared errors in the data being fit, so that the test is comparing chi−squared statistics, the default value of

0.1 for this option is usually reasonable. If there are no declared errors, so the test is comparing the sum of the squares of

the residuals, then the actual value of the sum of the squares depends on the magnitude of the values of the dependentvariable. In this case some adjustment of the value of AbsoluteChiSquaredTolerance may be appropriate.

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For information on the other two tests used by FindFit to determine convergence, see the discussion of the

RelativeChiSquaredTolerance and ValueTolerance options in 5.3.1.3 and in 5.3.1.13, respectively.

5.3.1.2 The MaximumIterations Option

As discussed above, FindFit uses an iterative technique to try to determine the minimum in the chi−squared or sum ofthe squares. The MaximumIterations option controls the number of iterations FindFit will attempt before giving

up.

When FindFit gives up, it issues a warning message, but also presents the result of the fit just as if it had converged.

For example, here we repeat a fit we performed in Section 5.1.3, but we restrict the number of iterations to less than that

required by FindFit to achieve convergence.

FindFit GanglionData, a 0 a 1 x, x, a 0 , a 1 , MaximumIterations 2

FindFit::maxiters:FindFit failed to converge after 2

iterations. The maximum number of iterations can be increased with the

MaximumIterations option.

0 20 40 60 80 100 120 140

2.5

5

7.5

10

12.5

15

17.5

-2-10123

Residuals

a 0

0.16, 0.5 , a 1

0.1098, 0.006700000000000001 ,SumOfSquares


12

5.3.1.3 The RelativeChiSquaredTolerance Option

FindFit uses three separate tests to determine if a fit has converged to a final value. The

RelativeChiSquaredTolerance option is used by one of those tests.

If the chi−squared (or the sum of the squares if there are no explicit errors in the data) is decreasing, and the value of the

previous iteration minus the current value divided by the previous value is less than the value of

RelativeChiSquaredTolerance , then the fit is judged to have converged.

If there are declared errors, so that the test is comparing chi−squared statistics, the default value of 0.005 for this option

is usually reasonable. If there are no declared errors, so the test is comparing the sum of the squares of the residuals, then

the actual value of the sum of the squares depends on the magnitude of the values of the dependent variable; in this case

some adjustment of the value of RelativeChiSquaredTolerance may be appropriate.

For information on the other two tests used by FindFit to determinate convergence, see the discussion of the

AbsoluteChiSquaredTolerance option and of the ValueTolerance option in 5.3.1.1 and in 5.3.1.13,

respectively.

5.3.1.4 The ReturnCovariance Option

If the ReturnCovariance option is set to True, then FindFit returns the full covariance matrix of the fit inaddition to other rules about the result.

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A discussion of the meaning of the covariance matrix is in Section 4.4.18.

5.3.1.5 The ReturnEffectiveVariance Option

When both coordinates have errors, FindFit uses an "effective variance" technique. This method is discussed in

Section 4.1.3.

If the ReturnEffectiveVariance option to FindFit is set to True, then the program returns the values of the

effective variance in addition to other rules about the result.

This option is identical to the option of the same name used by LinearFit.

5.3.1.6 The ReturnErrors Option

By default, FindFit calculates and returns the errors in the fit parameters. It also uses these errors to adjust the

significant figures in the values of the parameters. If ReturnErrors is set to False, no errors are returned and no

significant figure adjustment is performed.

For example, here we repeat a fit performed before in this chapter, one with and the other without ReturnErrors set

to True. We also set ShowFit to False to suppress the graphs of the fit.

FindFit GanglionData, a 0 a 1 x, x, a 0 , a 1 , ShowFit False

a 0

0.03, 0.5 , a 1

0.1068, 0.006700000000000001 ,SumOfSquares


12

FindFit GanglionData, a 0 a 1 x, x, a 0 , a 1 , ShowFit False,ReturnErrors False

a 0 0.03098373593165319, a 1 0.1067875703568358,SumOfSquares


12


5.3.1.7 The ReturnFunction Option

Be default, FindFit returns a set of rules for the result of the fit. By setting ReturnFunction to True, FindFit

instead returns a function of the independent variable.

For example, here we repeat a fit performed a few times already in this section, but with ReturnFunction set to

True. We also set ShowFit to False to suppress the graphs of the fit.

FindFit GanglionData, a 0 a 1 x, x, a 0 , a 1 , ShowFit False,

ReturnFunction True

0.03 0.1068 x


5.3.1.8 The ReturnResiduals Option

If set to True, the ReturnResiduals option causes FindFit to return the residuals of the fit along with the other

results of the fit.


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5.3.1.9 The Reweight Option

When the data has no explicit errors, FindFit finds the minimum in the sum of the squares of the residuals. However,

if the scatter in the data points can be considered to be random and statistical then it is often reasonable to assume that

the effective error in the dependent variable is given by PseudoErrorY.

PseudoErrorY =Sqrt[ SumOfSquares / DegreesOfFreedom ]

The Reweight option is identical to the option of the same name used by LinearFit except that by default it is

True for LinearFit, but False for FindFit. This is because in a nonlinear fit, reweighting the data can change

the values of the parameters to which we are fitting. This is not true for a linear fit, where reweighting only affects the

calculated errors in those parameters.

5.3.1.10 The ShowFit Option

Setting ShowFit to False suppresses the display of the graphical information about the fit. As discussed in Section

5.3.2.1, the graphs can be created later from the result returned by FindFit.


5.3.1.11 The ShowProgress Option

Setting ShowProgress to True causes FindFit to print information about its progress in performing the fit. This

option is identical to the one of the same name used by LinearFit, although for FindFit the information is much

more verbose and more often of use in finding the "best" fit of the data to a model.

Here we repeat a fit done a few times already in this Section, setting ShowProgress to True and setting ShowFit

to suppress the graphs of the fit.

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FindFit GanglionData, a 0 a 1 x, x, a 0 , a 1 , ShowFit False,ShowProgress True

14 data points with 2 variables.Fitting to 2 parametersBeginning iterations.Iteration 1 Sum of squares = 62556.7 308 Previous sum of squares: 1.79769 10 Current answer: {a[0] −> 1, a[1] −> 1} Evaluating matrices for the current\

answer. Scale for step size (lambda) = 0.001 Calling SingularValues to find\

corrections. Corrections to parameters:

{−1.15781, −0.890167}Iteration 2 Sum of squares = 25.5589 Previous sum of squares: 62556.7 Current answer: {a[0] −> −0.15781,

a[1] −> 0.109833} Evaluating matrices for the current\

answer. Scale for step size (lambda) = 0.0001

Calling SingularValues to find\ corrections.

Corrections to parameters:

{0.188662, −0.00304403}Iteration 3 Sum of squares = 25.3546 Previous sum of squares: 25.5589 Current answer: {a[0] −> 0.0308523,

a[1] −> 0.106789} Evaluating matrices for the current\

answer. Scale for step size (lambda) = 0.00001 Calling SingularValues to find\

corrections. Corrections to parameters:

−6 {0.000131454, −1.80534 10 }Iteration 4 Sum of squares = 25.3546 Previous sum of squares: 25.3546 Current answer: {a[0] −> 0.0309837,

a[1] −> 0.106788} Convergence: the previous sumofsquares\

minus the current sumofsquares is less than: 0.1 Convergence: the relative decrease in the sumofsquares is less than: 0.005Calculating covariance matrix using\

SingularValues.

a 0

0.03, 0.5 , a 1

0.1068, 0.006700000000000001 ,SumOfSquares


12

Note that the final iteration satisfied two of the three possible criteria for convergence. This is common when FindFit has found a true minimum in the sum of the squares.

5.3.1.12 The UseSignificantFigures Option

As discussed in Chapter 3, in the physical sciences, a specification of an error associated with a quantity essentially

defines what the significant figures of that quantity are. By default, FindFit uses this definition of significant figures

in returning the value of fit parameters. The UseSignificantFigures option allows this default behavior to be

turned off. For example, here we repeat a fit we have already done a few times.

FindFit GanglionData, a 0 a 1 x, x, a 0 , a 1 , ShowFit False

a 0 0.03, 0.5 , a 1 0.1068, 0.006700000000000001 ,SumOfSquares


12

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Note that the syntax and functionality of ShowFitResult are very similar to that of ShowLinearFit in the

EDA‘LinearFit‘ package. ShowFitResult is somewhat more general.

By default, ShowFitResult tries to find a quadrant in the data−fit graph in which to place the residual plot. If no

such quadrant can be found, the residual plot is displayed separately. Setting ResidualPlacement to Separate

causes the residual plot to always be displayed separately. ResidualPlacement can also be set to an integer

between 1 and 4, which causes the residual plot to be placed in that quadrant of the data−fit plot.

ShowFitResult GanglionData, a 0 a 1 x a 2 x2, x, a 0 , a 1 , a 2 ,result, ResidualPlacement 4

0 20 40 60 80 100 120 140

2.5

5

7.5

10

12.5

15

17.5

-1-0.5

00.5

1Residuals

Graphics

If ResidualPlacement is set to None, no residual plot is displayed.

Internally, ShowFitResult uses EDAListPlot, Plot, and ToFitFunction. Options given to

ShowFitResult for these are passed to the appropriate program. In addition, ShowFitResult itself uses only the

two options ResidualPlacement and UseSignificantFigures.

There can be some minor differences between the graphs displayed by FindFit and those displayed byShowFitResult. For example, if the data set has no explicit errors and the Reweight option is set to True in the

call to FindFit, then the PseudoErrorY is used in the calculation of the errors in the residuals by FindFit.

ShowFitResult is unaware of this, and will then have slightly different errors in the residual graph. By specifying

ReturnResiduals to True in the call to FindFit, the "better" numbers will be used by ShowFitResult.

We give an example by fitting GanglionData with the Reweight option.

result FindFit GanglionData, a 0 a 1 x, x, a 0 , a 1 , Reweight

True, ShowFit

True

0 20 40 60 80 100 120 140

2.5

5

7.5

10

12.5

15

17.5

-2

02

4

Residuals

a 0

0.03, 0.72 , a 1

0.107, 0.01 , PseudoErrorY1.453577429308659,

SumOfSquares25.35464811594684, DegreesOfFreedom

12

Next we use ShowFitResult on result.

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result FindFit ThermocoupleData, a 0 a 1 x, x, a 0 , a 1 , ShowFit False

LoadData::name: Loading: ThermocoupleData

a 0

0.981, 0.021 , a 1

0.04122, 0.00036 , ChiSquared21.05345454545474,

DegreesOfFreedom 19

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We pass the result to ToFitFunction.

ToFitFunction result, a 0 a 1 x, x, a 0 , a 1

0.981 0.04122 x

The ToFitFunction is similar to the ToLinearFunction function supplied in the EDA‘LinearFit‘ package,but somewhat more general. However, one difference is that when the fit parameters have errors, by default,

ToFitFunction does not return two functions. In contrast, ToLinearFunction will return two functions, the first

being the result of the fit and the second the estimated errors in the function. ToFitFunction can return this second

"error function" if the UseFitErrors option is set to True.

ToFitFunction result, a 0 a 1 x, x, a 0 , a 1 , UseFitErrors True

0.981

0.04122 x, 0.0004410000000000001

1.296 x2

107

5.3.3 Peak Shape Routines

The EDA‘FindFit‘ package includes some convenience functions to define peak shapes. They are BreitWigner,

Galatry, Gaussian, Lorentzian, PearsonVII, RelavitivisticBreitWigner, and Voigt.

5.4 Summary of the FindFit Package

Information "FindFit", LongForm False

FindFit[data, model, ind, parameters] finds a fit of ’data’ to ’model’ that minimises the sum of the squares of the residuals or the chi−squared.

The model is expected to be a function of the independent variable ’ind’ and the variables named in ’parameters’. The parameters can be of the form {name1, name2, ... , nameM}, but will more usually be of the form { {name1,start1}, {name2, start2}, ... , {nameM, startM}}, where ’starti’ are initial values of the parameter.

In addition, each parameter can be specified as {name, min, start, max}, where ’name’ is the name, ’start’ is the starting value, and ’min’ and ’max’ specify the minimum and maximum values of the parameter that can be returned. Finally, the parameter can be specified as {name, min, max} in which case the starting value is (min + max)/2.

Information "AbsoluteChiSquaredTolerance", LongForm False

AbsoluteChiSquaredTolerance is used in the default convergence test of FindFit. When the

chi−squared of the current iteration is less than the chi−squared of the previous iteration and their difference is less than AbsoluteChiSquaredTolerance, the fit is judged to have converged and no more iterations are performed.

Information "MaximumIterations", LongForm False

MaximumIterations is an option to various routines that use iterative techniques, and specifies the maximum number of iterations to perform before quitting.

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Information "RelativeChiSquaredTolerance", LongForm False

RelativeChiSquaredTolerance is used in the default convergence test of FindFit. When the chi−squared of the current iteration is less than the chi−squared of the previous iteration and their relative difference is less than RelativeChiSquaredTolerance, the fit is judged to have converged and no more iterations are performed.

Information "ReturnCovariance", LongForm False

ReturnCovariance is an option to various fitting routines. If True, then the routine returns the full covariance matrix; otherwise it does not.

Information "ReturnEffectiveVariance", LongForm False

ReturnEffectiveVariance is an option to various fitting routines. If True the routine returns the effective variance as part of the result of the fit.

Information "ReturnErrors", LongForm False

ReturnErrors is an option to various fitting routines. If True, then the routine returns errors in the fitted paramters; otherwise it does not.

Information "ReturnFunction", LongForm False

ReturnFunction is an option to various fitting routines. If set to False, then the fit will be returned as a set of Rules involving the ’parameter’ given in the call to the routine. If set to True, then the fit will be returned as a function and the independent variable is taken to be ’parameter’.

Information "ReturnResiduals", LongForm False

ReturnResiduals is an option to various fitting

routines. If True, the Residuals of the fit are returned along with other information about the fit. The Residuals returned always include a value for the independent variable; if none is in the data the values are {1,2, ... , N}.

Information "Reweight", LongForm False

Reweight is an option to LinearFit and FindFit, which controls whether or not to re−weight the data if it contains no explicit errors. The default is True for LinearFit and False for FindFit. If set to True, the data is weighted using a "statistical assumption", where the error in the dependent variables is the square root of the sum of the squares divided by the number of degrees of freedom; see Taylor, "An Introduction to Error Analysis," Eqn 8.14 on pg. 158 for further information. When set to True,

all subsequent processing assumes that the generated errors in the dependent variable are real.

Information "ShowFit", LongForm False

ShowFit is an option to various fitting routines.If True, then a graphical display of the results

of the fit is displayed.

Information "ShowProgress", LongForm False

ShowProgress is an option to various routines.When True, the routine will Print messages

showing its progress in performing its tasks.If set to False no such information is printed.

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Information "UseSignificantFigures", LongForm False

UseSignificantFigures is an option to various routines. When True, the routine will use AdjustSignificantFigures on numbers with associated errors so that the error determined the number of significant figures in the number itself. If set to False no such adjustment is performed.

Information "ValueTolerance", LongForm False

ValueTolerance is used in the default convergence test of FindFit. When the chi−squared of the current iteration is less than the chi−squared of the previous iteration and the maximum change in the values of the parameters being fit divided by the value of that parameter is less than ValueTolerance, the fit is judged to have converged and no more iterations are performed.

Information "ShowFitResult", LongForm False

ShowFitResult[data, model, ind, parameters,result] takes result from FindFit and displays graphic information about the fit. The meaning of the first four arguments are identical to arguments

of the same name for FindFit. Note that if parameters contains {param,value} pairs, the values are ignored by ShowFitResult.

ShowFitResult[data, model, ind, parameters, result, residuals] is the same as the first form except the residuals of the fit are given instead of calculated.

Information "ResidualPlacement", LongForm False

ResidualPlacement is an option to ShowLinearFit and ShowFitResult. If set to Automatic, they try to place the plot of residuals as a small graph inside the graph of the data and results of the fit; if a quadrant cannot be found then the residual plot is displayed separately. If the option is set to Separate, then the residual plot is always displayed separately. If the option is set to an integer between 1 and 4,

that quadrant is used to display the residuals.If the option is set to None, no residuals are displayed.

Information "ToFitFunction", LongForm False

ToFitFunction[result,model,ind,parameters] takes result which is assumed to be a set of Rules such as is returned by default by FindFit and returns a function of the independent variable ’ind’ for ’model’. The model is assumed to be a function of ’ind’ and the ’parameters’. The ’parameters’ may be a list of Symbols, or {Symbol, value} pairs, although the values are in all cases ignored. By default a single function is returned, which is evaluated at the result of the fit. If UseFitErrors is set to True, the routine returns a list of two functions; the first is as before and the second

is the estimated error in the function due to the errors in the fit parameters if any.

Information "UseFitErrors", LongForm False

UseFitErrors is an option to ToLinearFunction and ToFitFunction. If set to True, the default for ToLinearFunction, and the result contains errors in the fitted parameters then two functions are returned. The first is the function for the values of the parameters, the second is the function for the values of the errors in the parameters. A heuristic method is used to choose the sign of each term in the function of the errors in the parameters. If UseFitErrors is set to False, the default for ToFitFunction, only a single function evaluated at the values of the parameters is used.

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Information "BreitWigner", LongForm False

BreitWigner[x,a,x0,gamma] is a nonrelativistic Breit−Wigner function of the independent variable x, center value x0, and full width at half−maximum gamma. The function is normalized so that its integral from −Infinity to +Infinity is equal to ’a’. The maximum amplitude of the peak is 2*a/(Pi*gamma). BreitWigner is a synonym for Lorentzian.

Information "Galatry", LongForm False

Galatry[y,z,tau] is a Galatry profile with collision broadening parameter ’y’, collison narrowing parameter ’z’, and dimensionless time ’tau’. The profile is normalized so that its value is one when tau = 0.

Information "Lorentzian", LongForm False

Lorentzian[x,a,x0,gamma] is a nonrelativistic Lorentzian function of the independent variable x, center value x0, and full width at half−maximum gamma. The function is normalized so that its integral from −Infinity to +Infinity is equal to ’a’. The maximum amplitude of the

peak is 2*a/(Pi*gamma). Lorentzian is a synonym for BreitWigner.

Information "PearsonVII", LongForm False

PearsonVII[x,a,x0,gamma,m] is a Pearson VII function of the independent variable x, absorbance a, center value x0, full width at half maximum gamma, and tailing factor m.

Information "RelativisticBreitWigner", LongForm False

RelativisticBreitWigner[m_, mr_, gamma_, m1_, m2_] generates a relativistic Breit−Wigner for a resonance of nominal mass mr and total width gamma decaying into two particles of rest mass m1 and m2. It returns the fraction of such decays whose invariant mass is m. The function

is normalized so that when m equals mr the function returns 1.

Information "Voigt", LongForm False

Voigt[y,tau] is a Voigt profile with collison broadening parameter ’y’ and dimensionless time ’tau’. The profile is the Limit[ Galatry[ y,z,tau], z −> 0].

Chapter5.nb 35

Fitting Data

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