Derive equation from dataset

Hello!

I have the following four sets of data:

x11=[0  8  32  35  38  46  50  62];
y11=[288  288  224  183  155  29  23  0];

x12=[0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  26  35  36  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  71];
y12=[1830  1830  1830  1779  1779  1779  1779  1481  1481  1225  1225  1128  1128  995  995  707  707  682  682  661  661  607  515  511  344  344  344  339  339  339  332  332  332  319  319  319  305  305  305  261  261  261  249];

x21=[0  12  18  20  22  23  24  26  31  32  35  37  38  39  48  50  54  57];
y21=[715  715  670  643  639  616  612  581  521  466  421  369  318  326  109  55  7  0];

x22=[0  5  6  9  16  21  22  23  24  35  37  38  40  44  48  58  72  73  77];
y22=[6833  6833  6776  5704  2497  1893  1779  1706  1741  1444  1335  1256  1217  967  947  812  649  593  551];

All x-y pairs have quite the similar shape, with the curve decaying rapidly in the first half and then slowly in the second one.

The goal is to interpolate each x-y pair to derive an equation that will represent the phenomenon.

I tried interp1 and lsqcurvefit but they don't give good enough results. Could someone suggest an alternative way to interpolate these data?

Thank you!

[Adam Danz]

"I tried interp1 and lsqcurvefit but they don't give good enough results"

Interpolation won't give you the underlying equation. lsqcurvefit() (or lsqnonlin) works pretty well but you've got to supply it with a function and the results depend heavily on the boundaries and initial starting points specified by the user. So, are you trying to find what type of equation would produce these data (without noise) or are you trying to find the parameters of a known equation that results in a best fit of your data?

If you're trying to do that latter, I'd recommend using lsqnonlin() directly (which is what lsqcurvefit is using) and make sure you're supplying a reasonable function, bounds, and starting points.

[Alberto Basaglia]

Adam,

thank you for your answer. I am actually unsure of which equation or function would better fit the dataset, but I know the bounds and starting points.

If you have any suggestion given the dataset shape, please let me know.

Best

[Adam Danz]

I agree with the answer below. The curve fitting toolbox is a good start.

Answer 1

[Asad Mirza]

Since you do not know what could be the best function to fit your data with try out the Curve Fitting App with cftool. It lets you fit data using a wide range of functions, apply bounds, weights, etc. You even have the option of making your own equation to fit it with.

[Alberto Basaglia]

Asad,

thank you for your answers. I used cftools and I think a Gaussian might fit the dataset accurately enough. Do you know if is it possible, and in case how, to set the max/min values of the function?

Thank you!

[Walter Roberson]

Are you looking to set maximum and minimum values of a parameter to be fitted?

If so then click on Fit Options... which will bring up a window listing coefficient names with a column named Lower and another column named Upper. It turns out that the entries in those two columns are clickable, and you can edit the upper and lower bounds there.

Answer 2

[Walter Roberson]

There are an infinite number of equation that fit your data perfectly in theory , even without allowing for the possibility of noise or of imperfect recording of coefficients .

The number of equation you can derive from the data using no more than 2^48 bytes of memory (limit of x64 architecture in all known implementation ) representing your coefficients as IEEE double precision (64 bits) is a bit more limited but probably still over 2^40

With an infinite number of derived equations, the probability that any one of them is the "correct" equation is 1/infinity which is defined as 0.

Therefore you cannot do anything useful in fitting your data unless you can restrict your equation to being one of a finite number of forms, after which you can do parameter fitting for each of the possible forms and make a decision as to which fits best.