Curve fitting tool does not respect the upper and lower limits of parameters of custom equation

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Hi. I hope you can help me. I have the data from measurements in vectors vel_3_Hor and tau_fr_3_Hor:
vel_3_Hor = [206.5139956
179.4945199
162.7919326
151.195134
138.3501347
130.3942364
122.4986401
106.4425864
98.45750292
89.01579654
82.19190831
70.47075771
60.70021288
49.84117539
42.22794379
32.74655746
];
tau_fr_3_Hor = [0.137735032
0.139278888
0.125791442
0.125100852
0.125945036
0.125767804
0.122046211
0.116725322
0.121744852
0.119606
0.117339514
0.115280309
0.111458901
0.111580077
0.113360071
0.11463187
];
The elements with same index in the previous vectors correspond to each other. I want to fit these points to this custom equation:
y = tau_C + (T_S - tau_C)*exp(-(x/vel_S)^delta_S) + c_vel*x
I tried to fit, manually, this function to my data with this page and I got these results:
(There is a value of tau_fr_3_Hor at vel_3_Hor = 0 here, but please ignore it. tau_fr_3_Hor = 0.1208 is an average result that I got from several experiments at vel_3_Hor where I measured tau_fr_3_Hor. I cannot rely on this average for specific reasons, but still it works as an estimate).
I set this values in Curve Fitting Tool:
Then I got these results:
It just seems to be something wrong with the confidence bounds in each parameter. For instance, vel_S cannot be negative, and also should be less than 80 at first sight. Also, tau_C should be between 0.10 - 0.12.
Thanks in advance.

Accepted Answer

John D'Errico
John D'Errico on 12 Aug 2022
Sorry, but the confidence limits are not impacted by the bounds. To get better confidence limits they would need to do considerably more work. Those confidence limits were computed using a fairly simplistic scheme.
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