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What is the proposed approach if one wants to simultaneously fit multiple functions to multiple datasets with shared parameters?

As an example consider the following case: We have to measurements of Gaussian line profiles and we would like to fit a Gaussian to each of them but we expect them to be at the same line center, i.e. the fitting should use the same line center for both Gaussians.

The solution I came up with looks a little clumsy. Any ideas on how to do this better, especially in cases where we have more than 2 datasets and more than one shared parameter?

Example:

f[x_, amplitude_, centroid_, sigma_] := amplitude Exp[-((x - centroid)^2/sigma^2)] data1 = Table[{x, RandomReal[{-.1, .1}] + f[x, 1, 1, 1]}, {x, -4, 6, 0.25}]; data2 = Table[{x, RandomReal[{-.1, .1}] + f[x, .5, 1, 2]}, {x, -8, 10, 0.5}]; gauss1 = NonlinearModelFit[data1, f[x, a1, x1, b1], {a1, x1, b1}, x, MaxIterations -> 1000, Method -> NMinimize]; gauss2 = NonlinearModelFit[data2, Evaluate[f[x, a2, x1, b2] /. gauss1["BestFitParameters"]], {a2, b2}, x, MaxIterations -> 1000, Method -> NMinimize]; Join[gauss1["BestFitParameters"],gauss2["BestFitParameters"]] datpl = ListPlot[{data1, data2}, Joined -> True, PlotRange -> {{-10, 10}, All}, Frame -> True, PlotStyle -> {Black, Red}, Axes -> False, InterpolationOrder -> 0]; Show[{datpl, Plot[{Evaluate[f[x, a1, x1, b1] /. gauss1["BestFitParameters"]], Evaluate[ f[x, a2, x1 /. gauss1["BestFitParameters"], b2] /. gauss2["BestFitParameters"]]}, {x, -10, 10}, PlotRange -> All, PlotStyle -> {Black, Red}, Frame -> True, Axes -> False]}]

enter image description here

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    $\begingroup$ Note at forums.wolfram.com/mathgroup/archive/2011/Sep/msg00555.html has URLs to several techniques for doing this sort of thing. $\endgroup$ Commented Jan 28, 2012 at 20:56
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    $\begingroup$ @Daniel Thanks for the links. They present a handy way to do this. $\endgroup$ Commented Jan 29, 2012 at 10:38

5 Answers 5

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This is an extension of Heike's answer to address the question of error estimates. I'll follow the book Data Analysis: A Bayesian Tutorial by D.S. Sivia and J. Skilling (Oxford University Press).

Basically, any error estimate depends on the basic assumptions you make. The previous answers implicitly assume uniform normally distributed noise: $\epsilon \sim N(0, \sigma)$. If you know $\sigma$ the error estimate is straightforward.

With the same definitions:

data1 = Table[{x, RandomReal[{-.1, .1}] + f[x, 1, 1, 1]}, {x, -4, 6, 0.25}]; data2 = Table[{x, RandomReal[{-.1, .1}] + f[x, .5, 1, 2]}, {x, -8, 10, 0.5}]; f[x_, amplitude_, centroid_, sigma_] := amplitude Exp[-((x - centroid)^2/sigma^2)]

Add the variables:

vars = {mu, au1, s1, au2, s2};

The variance of the error is (analytically, from the definition above):

noiseVariance = Integrate [x^2, {x, -0.1, 0.1}];

The log-likelihood of the model is: 

logModel = -Total[ (data1[[All, 2]] - (f[#, au1, mu, s1] & /@ data1[[All, 1]]) )^2 /noiseVariance]/2 - Total[ (data2[[All, 2]] - (f[#, au2, mu, s2] & /@ data2[[All, 1]]) )^2 /noiseVariance]/2;

Optimize the log-likelihood (note the change of sign leading to a maximization instead of minimization)

fit = FindMaximum[logModel, vars]

The fit will be the same, as the variance estimation doesn't affect the maximum, so I won't repeat it here.

For the error estimates, the covariance matrix is found as minus the inverse of the hessian of the log-likelihood function, so (DA p.50):

$$ \sigma_{ij}^2 = -[\nabla \nabla L]^{-1}_{ij} $$

hessianL = D[logModel {vars, 2}]; parameterStdDeviations = Sqrt[- Diagonal@Inverse@hessianL]; {vars, #1 \[PlusMinus] #2 & @@@ ({vars /. fit[[2]], parameterStdDeviations}\[Transpose]) }\[Transpose] // TableForm

If $\sigma$ is unknown the analysis is slightly trickier, but the results are easily implemented. If the error is additive guassian noise of unknown variance the correct estimator is (DA p. 67):

$$ s^2 = \frac{1}{N-1} \sum_{k=1}^N (data_k - f[x_k; model])^2 $$

estimatedVariance1 = Total[(data1[[All, 2]] - (f[#, au1, mu, s1] & /@ data1[[All, 1]]) )^2] / (Length@data2 - 1); estimatedVariance2 = Total[(data2[[All, 2]] - (f[#, au2, mu, s2] & /@ data2[[All, 1]]) )^2] / (Length@data2 - 1);

As stated above the magnitude of the variance won't affect our point estimates in the model, so we can use the same code above, and just inject the newly estimated variance into the log-likelihood function. This seems to be equivalent to the default behaviour of NonlinearModelFit.

As you seem to indicate that you are fitting spectra from a counting experiment, you might have better performance if you assume Poisson counting noise instead, then the variance for each channel is estimated as the number of counts in that channel: $$ \sigma^2_k \approx data_k $$ You might also want to consider adding a background model (a constant background is a simple extension of the above), depending on the noise level.

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I once had to do this to fit some spectroscopic data. This was my solution...

Here is a simple model of the intensity and phase of a laser after passing through a medium with a complex refractive index 

n[den_, det_] := 1 + den ((I - 2 det)/(1 + 4 det^2)); int[den_, det_] := Exp[-2 Arg[n[den, det]] den]; phase[den_, det_] := Re[n[den, det]] den;

some noisy data

d1 = Table[{x, int[1.34, x] + RandomReal[{-.1, .1}]}, {x, -20, 20}]; d2 = Table[{x, phase[1.34, x] + RandomReal[{-.1, .1}]}, {x, -20, 20}];

define a dummy variable labelling the data sets, and join the data into one list

dat = Join[d1 /. {x_, y_} -> {1, x, y}, d2 /. {x_, y_} -> {2, x, y}];

define a fit function that depends on the value of the "set" variable

fitmodel[set_, den_, det_] := Which[set == 1, Evaluate@int[den, det], set == 2, Evaluate@phase[den, det]]

now NonlinearModelFit fits both the datasets simultaneously

fit = NonlinearModelFit[dat, fitmodel[set, den, det], {{den, 2}}, {set, det}]; fitparams = fit["BestFitParameters"] dd = {d1, d2}; mm = {int[den, det], phase[den, det]}; Show[ ListPlot[dd, PlotRange -> All], Plot[Evaluate[mm /. fitparams], {det, -20, 20}] ]
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    $\begingroup$ Welcome to our site. This is a well presented answer, and I would like to see more of your expertise on this site. So, consider registering your account, as it will allow you to participate more fully. $\endgroup$ Commented May 8, 2012 at 0:48
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You could use NMinimize to fit the two models. For example with

data1 = Table[{x, RandomReal[{-.1, .1}] + f[x, 1, 1, 1]}, {x, -4, 6, 0.25}]; data2 = Table[{x, RandomReal[{-.1, .1}] + f[x, .5, 1, 2]}, {x, -8, 10, 0.5}]; f[x_, amplitude_, centroid_, sigma_] := amplitude Exp[-((x - centroid)^2/sigma^2)]

We could find a least squares solution by doing something like

min = NMinimize[Total[(#.#) & /@ {data1[[All, 2]] - (f[#, a1, mu, s1] & /@ data1[[All, 1]]), data2[[All, 2]] - (f[#, a2, mu, s2] & /@ data2[[All, 1]])} ], {a1, a2, s1, s2, mu}] (* ==> {0.253998, {a1 -> 0.984464, a2 -> 0.451312, s1 -> 0.980629, s2 -> -2.07535, mu -> 0.988739}} *)

To compare the found fit with the data

datpl = ListPlot[{data1, data2}, Joined -> True, PlotRange -> {{-10, 10}, All}, Frame -> True, PlotStyle -> {Black, Red}, Axes -> False, InterpolationOrder -> 0]; Show[datpl, Plot[Evaluate[{f[x, a1, mu, s1], f[x, a2, mu, s2]} /. min[[2]]], {x, -10, 10}, PlotRange -> All, PlotStyle -> {Black, Red}]]

Mathematica graphics

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    $\begingroup$ Thank you for the answer. I tried something similar, but was not satisfied with the fit. Yours looks much better. The advantage of the NonlinearModelFit is that it provides error estimates of the fit parameters. Do you know how to estimate the error using NMinimize? $\endgroup$ Commented Jan 28, 2012 at 17:29
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Here's a way that uses NonlinearModelFit[]:

Same function:

f[x_, amplitude_, centroid_, sigma_] := amplitude Exp[-((x - centroid)^2/sigma^2)]

Same form for the data (but numbers are different due to different random seeds):

data1 = Table[{x, RandomReal[{-.1, .1}] + f[x, 1, 1, 1]}, {x, -4, 6, 0.25}]; data2 = Table[{x, RandomReal[{-.1, .1}] + f[x, .5, 1, 2]}, {x, -8, 10, 0.5}];

But lets make this a single dataset by shifting data2 so its maximum x value is slightly less than the minimum x value from data1:

min1 = Min[data1[[All, 1]]]; max2 = Max[data2[[All, 1]]] + 1; data = Join[data2 /. {x_Real, y_Real} :> {x + min1 - max2, y}, data1]; ListLinePlot[data, InterpolationOrder -> 0, AxesLabel -> {"x", "y"}]

Mathematica graphics

Here is a two-Gaussian model fit:

gauss12 = NonlinearModelFit[data, f[x, a1, x1, b1] + f[x, a2, x1 + min1 - max2, b2], {a1, x1, b1, a2, b2}, x]; gauss12["BestFitParameters"] (*{a1 -> 1.00363, x1 -> 0.982859, b1 -> 0.979613, a2 -> 0.56506, b2 -> 1.65061}*)

(Note: you could compare the a1 and a2 parameters with the maximum of each dataset to automate associating the fit parameters with their datasets. You would do this by inspection in the way presented here.)

And,

datpl = ListPlot[{data1, data2}, Joined -> True, PlotRange -> {{-10, 10}, All}, Frame -> True, PlotStyle -> {Black, Red}, Axes -> False, InterpolationOrder -> 0]; Show[{datpl, Plot[{Evaluate[f[x, a1, x1, b1] /. gauss12["BestFitParameters"]], Evaluate[f[x, a2, x1, b2] /. gauss12["BestFitParameters"]]}, {x, -10, 10}, PlotRange -> All, PlotStyle -> {Black, Red}, Frame -> True, Axes -> False]}]

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Taking advantage of NonlinearModelFit functionality:

gauss12["ParameterTable"]

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EDIT

To address the comment about multiple peaks, use multiple-peak model. For example, two peaks that have the same spacing:

f[x_, amplitudes_, centroids_, sigmas_] := Total[amplitudes MapThread[ Exp[-((x - #1)^2/#2^2)] &, {centroids, sigmas}]]; data1 = Table[{x, RandomReal[{-.1, .1}] + f[x, {1, 0.9}, {1, 3}, {.5, .7}]}, {x, -4, 6, 0.25}]; data2 = Table[{x, RandomReal[{-.1, .1}] + f[x, {.5, .6}, {1, 3}, {0.7, 1.1}]}, {x, -8, 10, 0.5}]; min1 = Min[data1[[All, 1]]]; max2 = Max[data2[[All, 1]]] + 1; data = Join[data2 /. {x_Real, y_Real} :> {x + min1 - max2, y}, data1]; ListLinePlot[data, InterpolationOrder -> 0, AxesLabel -> {"x", "y"}]

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Here is the fit using NonlinearModelFit[]:

gauss12 = NonlinearModelFit[data, {f[x, {a11, a12, a21, a22}, {x11, x12, x11 + min1 - max2, x12 + min1 - max2}, {b11, b12, b21, b22}]}, {a11, a12, a21, a22, x11, x12, b11, b12, b21, b22}, x]; datpl = ListPlot[{data1, data2}, Joined -> True, PlotRange -> {{-10, 10}, All}, Frame -> True, PlotStyle -> {Black, Red}, Axes -> False, InterpolationOrder -> 0]; Show[{datpl, Plot[{Evaluate[f[x, {a11, a12}, {x11, x12}, {b11, b12}] /. gauss12["BestFitParameters"]], Evaluate[f[x, {a21, a22}, {x11, x12}, {b21, b22}] /. gauss12["BestFitParameters"]]}, {x, -10, 10}, PlotRange -> All, PlotStyle ->{Directive[Dashed, Black], Directive[Dashed, Red]}, Frame -> True, Axes -> False]}, FrameLabel -> {"x", "y"}]

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gauss12["ParameterTable"]

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The approach can be extended to three or more datasets and multiple peaks (programmatically, using parameters of the form a[idataset,jpeak], etc.).

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  • $\begingroup$ Thanks for the idea. Shifting and joining works for many of my applications but not for all of them. Consider two data sets containing not one but multiple peaks. If they, e.g. represent fine-structure emission their individual components have a fixed distance. Joining the data sets might lead to confusion problems. Still a good idea. $\endgroup$ Commented Jan 29, 2012 at 10:30
  • $\begingroup$ @Markus Can you add to the question a more realistic example where my approach would fail? $\endgroup$ Commented Jan 29, 2012 at 13:36
  • $\begingroup$ actually I can't come up with a good counter example. All answers were really helpful and I wish I could accept all of them. $\endgroup$ Commented Feb 8, 2012 at 12:20
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I just wrote a nice wrapper function to handle this problem in a systematic way. My basic approach is to add an extra index variable to the datasets that can be used as an extra independent parameter. Here's the definition:

MultiNonlinearModelFit[ datasets_, expressions_, params_, constraints : _ : True, independents_Symbol, opts : OptionsPattern[] ] := MultiNonlinearModelFit[datasets, expressions, constraints, params, {independents}, opts]; MultiNonlinearModelFit[ datasets : {__?(MatrixQ[#, NumericQ] &)}, expressions_List, constraints : _ : True, {fitParams__Symbol}, {independents__Symbol}, opts : OptionsPattern[] ] /; Length[expressions] === Length[datasets] := Module[{ fitfun, numSets = Length[expressions], augmentedData = Catenate @ MapIndexed[ Join[ (* Attach indices to the data *) ConstantArray[N[#2], Length[#1]], #1, 2 ] &, datasets ] }, fitfun = With[{ conditions = Map[ {\[FormalN] == #, expressions[[#]]} &, N @ Range[numSets] ] }, Which @@ Catenate[conditions] ]; NonlinearModelFit[ augmentedData, If[TrueQ[constraints], fitfun, {fitfun, constraints} ], {fitParams}, {\[FormalN], independents}, (* use dataset index as extra independent variable *) opts ] ];

Example usage: fitting two Gaussian peaks with a shared location parameter. First define some data for testing:

xvals = Range[-5, 5, 0.1]; gauss[x_] := Evaluate@PDF[NormalDistribution[], x]; With[{amp1 = 1.2, amp2 = 0.5, width1 = 1, width2 = 2, sharedOffset = 0.5, eps = 0.05}, dat1 = Table[{x, amp1 gauss[(x - sharedOffset)/width1] + eps RandomVariate[NormalDistribution[]]}, {x, xvals}]; dat2 = Table[{x, amp2 gauss[(x - sharedOffset)/width2] + eps RandomVariate[NormalDistribution[]]}, {x, xvals}] ]; plot = ListPlot[{dat1, dat2}]

enter image description here

Fit the two Gaussians with a shared location parameter:

fit = MultiNonlinearModelFit[ {dat1, dat2}, { amp1 gauss[(x - sharedOffset)/width1], amp2 gauss[(x - sharedOffset)/width2] }, {amp1, amp2, width1, width2, sharedOffset}, {x} ]; fit["BestFitParameters"]

{amp1 -> 1.21652, amp2 -> 0.466029, width1 -> 0.988112, width2 -> 2.07783, sharedOffset -> 0.512484}

Extract the fits as a list of expressions and plot the fits:

fits = Table[Normal[fit], {\[FormalN], {1, 2}}] Show[ plot, Plot[fits, {x, -5, 5}] ]

enter image description here

This function is on the Wolfram function repository: https://resources.wolframcloud.com/FunctionRepository/resources/MultiNonlinearModelFit

Edit

Upon request, here is an example of how to use my resource function with parameter constraints. Please note that the resource function has changed a little since I wrote this answer and I recommend looking at the notebook for the most up-to-date definition of MultiNonlinearModelFit.

Define some test data (two parallel lines, basically):

data = RandomVariate[BinormalDistribution[0.7], {2, 100}]; data[[1, All, 2]] += 1.; data[[2, All, 2]] += -1.;

As a slight variation of the first example of the documentation page, we'll fit both datasets with 1st order polynomials where the slope parameters cannot differ by more than 0.1 (note the use of a quadratic constraint expression (a1 - a2)^2, which is differentiable and therefore easier for Mathematica to work with):

fit = ResourceFunction["MultiNonlinearModelFit"][ data, <| "Expressions" -> {a1 x + b1, a2 x + b2}, "Constraints" -> (a1 - a2)^2 < 0.1^2 && b1 > 0 && b2 < 0 |>, {a1, a2, b1, b2}, {x} ]

Show the fits:

Show[ ListPlot[data], Plot[Evaluate[Table[Normal[fit], {\[FormalN], Length[data]}]], {x, -5, 5}] ]

enter image description here

I submitted an update to the resource function that includes this example.

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