2
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So I'm trying to fit experimental data to a nonlinear model using the NonLinearModelFit command, but it isn't quite working.

The plotted data set looks like this:

The data is to be fitted with

1+β*exp⁡(-2γ*t),

so I tried using the NonLinearModelFit command as follows:

nlm = NonlinearModelFit[data, 1 + beta*Exp[-2 gamma*t], {beta, gamma},
    t];

LogLinearPlot[nlm[t], {t, 0, data[[Length[data], 1]]}, 
 PlotRange -> Automatic]

Show[ListLogLinearPlot[data], 
 LogLinearPlot[nlm[t], {t, 0, data[[Length[data], 1]]}, 
  PlotStyle -> Red]]

which looks like

The proposed function obviously doesn't fit the data set and is a scaled down version shifted to the right, without the last almost constant part of the data.

What's happening here? How can I accurately fit the data and maybe use some initial values for the parameters β and γ?

The data used to plot and fit:

082222  1251 PM
Experiment Type Sizing
Pseudo Cross Correlation
Scattering angle    40.0
Duration (s)    20
Wavelength (nm) 660.0
Refractive index    1.330
Viscosity (mPas)    1.000
Temperature (K) 293.2
Laser intensity (mW)    0.147463
Average Count rate  A (kHz) 191.3
Average Count rate  B (kHz) 192.1
Intercept   0.9941
Cumulant 1st    667.97
Cumulant 2nd    667.97  0.00
Cumulant 3rd    641.63  0.40

Lag time (s)         g2-1
1.250000E-8 0.981694
2.500000E-8 0.991310
3.750000E-8 0.987267
5.000000E-8 0.986830
6.250000E-8 0.995900
7.500000E-8 0.988141
8.750000E-8 0.985300
1.000000E-7 1.028684
1.125000E-7 0.977760
1.250000E-7 1.000162
1.375000E-7 0.988360
1.500000E-7 0.991529
1.625000E-7 0.992731
1.750000E-7 1.001255
1.875000E-7 1.003659
2.000000E-7 0.992731
2.125000E-7 0.996173
2.375000E-7 1.001583
2.625000E-7 0.976339
2.875000E-7 0.992731
3.125000E-7 1.007101
3.375000E-7 0.977377
3.625000E-7 0.994370
3.875000E-7 1.004151
4.125000E-7 0.997539
4.625000E-7 0.989972
5.125000E-7 1.003577
5.625000E-7 1.000408
6.125000E-7 0.990054
6.625000E-7 0.985109
7.125000E-7 0.992977
7.625000E-7 0.994452
8.125000E-7 0.987171
9.125000E-7 1.003509
1.012500E-6 1.004055
1.112500E-6 0.994712
1.212500E-6 0.990600
1.312500E-6 0.985559
1.412500E-6 0.988237
1.512500E-6 1.000039
1.612500E-6 1.001002
1.812500E-6 0.983599
2.012500E-6 0.993154
2.212500E-6 0.994131
2.412500E-6 0.994090
2.612500E-6 0.997109
2.812500E-6 0.998311
3.012500E-6 0.997553
3.212500E-6 0.993281
3.612500E-6 0.994408
4.012500E-6 0.989310
4.412500E-6 0.996071
4.812500E-6 0.993520
5.212500E-6 0.990286
5.612500E-6 0.992212
6.012500E-6 0.992093
6.412500E-6 0.992017
7.212500E-6 0.994756
8.012500E-6 0.994418
8.812500E-6 0.991991
9.612500E-6 0.990677
1.041250E-5 0.991290
1.121250E-5 0.995478
1.201250E-5 0.993559
1.281250E-5 0.996065
1.441250E-5 0.996347
1.601250E-5 0.991484
1.761250E-5 0.990913
1.921250E-5 0.993117
2.081250E-5 0.991552
2.241250E-5 0.990403
2.401250E-5 0.991326
2.561250E-5 0.991138
2.881250E-5 0.991134
3.201250E-5 0.991837
3.521250E-5 0.990236
3.841250E-5 0.990808
4.161250E-5 0.989931
4.481250E-5 0.990017
4.801250E-5 0.990026
5.121250E-5 0.988744
5.761250E-5 0.987654
6.401250E-5 0.987021
7.041250E-5 0.986750
7.681250E-5 0.985235
8.321250E-5 0.985939
8.961250E-5 0.985592
9.601250E-5 0.985029
0.000102    0.984424
0.000115    0.982993
0.000128    0.982606
0.000141    0.980761
0.000154    0.978720
0.000166    0.978311
0.000179    0.977051
0.000192    0.975604
0.000205    0.974399
0.000230    0.972358
0.000256    0.969710
0.000282    0.966684
0.000307    0.965229
0.000333    0.961571
0.000358    0.960360
0.000384    0.957262
0.000410    0.955180
0.000461    0.950771
0.000512    0.945581
0.000563    0.941056
0.000614    0.936345
0.000666    0.931493
0.000717    0.927869
0.000768    0.922947
0.000819    0.918401
0.000922    0.908841
0.001024    0.899845
0.001126    0.890876
0.001229    0.881773
0.001331    0.873013
0.001434    0.864277
0.001536    0.855824
0.001638    0.847578
0.001843    0.830522
0.002048    0.814198
0.002253    0.798193
0.002458    0.782513
0.002662    0.767277
0.002867    0.752026
0.003072    0.736805
0.003277    0.721951
0.003686    0.693111
0.004096    0.665850
0.004506    0.640100
0.004915    0.615177
0.005325    0.591402
0.005734    0.569494
0.006144    0.548935
0.006554    0.529993
0.007373    0.493404
0.008192    0.458930
0.009011    0.425248
0.009830    0.394020
0.010650    0.364432
0.011469    0.337440
0.012288    0.311853
0.013107    0.287955
0.014746    0.243217
0.016384    0.205101
0.018022    0.173352
0.019661    0.144798
0.021299    0.120097
0.022938    0.098263
0.024576    0.079198
0.026214    0.064734
0.029491    0.041283
0.032768    0.024546
0.036045    0.009757
0.039322    -0.007328
0.042598    -0.023800
0.045875    -0.027768
0.049152    -0.025468
0.052429    -0.020721
0.058982    0.002437
0.065536    0.022033
0.072090    0.038237
0.078643    0.044000
0.085197    0.025870
0.091750    0.002250
0.098304    -0.003367
0.104858    -0.018009
0.117965    -0.031016
0.131072    -0.029388
0.144179    -0.026414
0.157286    -0.028317
0.170394    -0.016462
0.183501    0.007269
0.196608    0.010597
0.209715    0.015417
0.235930    0.019428
0.262144    0.020318
0.288358    0.032608
0.314573    0.035763
0.340787    0.022082
0.367002    -0.006260
0.393216    0.004914
0.419430    0.007707
0.471859    -0.005449
0.524288    0.006633
0.576717    -0.004173
0.629146    0.007739
0.681574    -0.001696
0.734003    0.000670
0.786432    0.001787
0.838861    -0.009630
0.943718    -0.003229
1.048576    -0.003674
1.153434    -0.011465
1.258291    0.010513
1.363149    -0.031880
1.468006    -0.017126
1.572864    0.004514
1.677722    -0.003074
1.887437    0.000161
2.097152    -0.005843
2.306867    -0.007362
2.516582    0.012358
2.726298    0.011785
2.936013    0.028037
3.145728    0.024937
3.355443    0.019830
3.774873    0.004959
4.194304    0.009447
4.613734    0.028008
5.033165    0.031347
5.452595    0.021929
5.872025    0.046492
6.291456    0.031570
6.710886    0.027511
7.549747    0.018476
8.388608    0.043572
9.227468    0.068573
10.066330   0.027640
10.905190   0.042270
11.744051   0.065827
12.582912   0.064546
13.421773   0.053715
15.099494   0.105235
16.777216   -0.051853
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  • $\begingroup$ I haven't tried fitting your data. Have you drop parts of the data apparently not fitting into 1+β*exp⁡(-2γ*t), say the tails. $\endgroup$
    – Lacia
    Sep 11, 2022 at 19:18
  • $\begingroup$ I tried excluding the plotpoints with t<10^-6, but it just shifted the fitted function a tiny bit in negative y direction $\endgroup$ Sep 11, 2022 at 19:27

1 Answer 1

6
$\begingroup$

I think your proposed model may be inappropriate for your data. The main trend in your data is a simple exponential decay, a model that fits remarkably well:

nlm = NonlinearModelFit[data, beta Exp[-gamma t], {{beta, 1}, {gamma, 100}}, t]

Show[
  LogLinearPlot[
    Legended[nlm[t], "fit"], {t, data[[1, 1]], data[[-1, 1]]},
    Frame -> True, Axes -> False,
    PlotRange -> All,
    PlotStyle -> Directive[Opacity[0.3, Blue], Thickness[0.02]]
  ],
  ListLogLinearPlot[Legended[data, "data"], PlotStyle -> Black]
]

fit and original data on log-linear plot


data comes from data = ImportString[" your data table minus the headers ", "Table"].

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3
  • $\begingroup$ Thank very much, I've been sitting on this the whole day :D $\endgroup$ Sep 11, 2022 at 20:51
  • $\begingroup$ may I ask how you picked the initial values for gamma and beta? $\endgroup$ Sep 11, 2022 at 21:31
  • 1
    $\begingroup$ @xPigeonDestroyer2000 Beta is easy: for $t=0 $ the model reduces to $\beta\ e^0 = \beta$ and this value should correspond to the initial value of the data, which is around 1, so $\beta = 1$ is a good starting point. For $\gamma$, it was just trial and error. $\endgroup$
    – MarcoB
    Sep 11, 2022 at 23:24

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