Help fitting data to an implicit equation

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mura0087
mura0087 2021-1-13
评论： mura0087 2021-1-19
Hello:
I need to fit some data to the following implicit equation:
((1-y)^(1-b)/y)=exp(-kt)
t is a vector containing time values and y is a vector containing current values. for each series of data y vs t, I need to determine b and k
b has to be between 0 and 1, and k needs to be greater than 0.
I have both the optimization and the curve fitting toolboxes.
Any suggestions on what tools to use (lsqcurvefit? something else? would be very appreciated)
Thanks!

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Jeff Miller
Jeff Miller 2021-1-14
编辑：Jeff Miller 2021-1-14
I would suggest using fminsearch. The error function to be minimized would be something like:
function thiserr = err(x,y,t)
b = x(1);
k = x(2);
thiserr = sum( (((1-y).^(1-b)./y) - exp(-kt))^2 );
end
You should be able to find examples of how to use fminsearch if you need more detail on how to call it. In your case y and t are "extra parameters". Look here for information on how to handle that.
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Jeff Miller
Jeff Miller 2021-1-16
You are welcome. That function value NaN is a bad sign. It means thiserr is NaN for all values of b and k that fminsearch has checked. You don't have any y=0 values, do you? Dividing by 0 would cause nans for all b and k.
mura0087
mura0087 2021-1-16
No y=0 values. I even scaled the y values so that everything was nonzero and less than 1, and it still did not fit. I think I need a new function!
I did get this fminsearch method to a fit a different function to different data (that I generated, so I knew the function to fit it to) as a sanity check. I had not fit using the optimization toolbox before, but I now see this is a pretty elegant method for fitting, thanks again for suggesting it!

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John D'Errico
John D'Errico 2021-1-16
My thought would be the lazy solution. If your model is:
((1-y)^(1-b)/y)=exp(-kt)
then log the model. That is, we know that
(1-b)*log(1-y) + k*t = log(y)
With one more step, this reduces to
-b*log(1-y) + k*t = log(y) - log(1-y)
You can compute the parameters k and b using a simple linear regression now. Thus, if y and t are column vectors, we have:
bk = [-log(1-y),t] \ (log(y) - log(1-y));
so bk is a vector of length 2, contining the estimates for b and k respectively. If you find that b or k are estimated to be something outside of the valid region, then I would first consider if this is a reasonable model, but then you could just use lsqlin to estimate them, since lsqlin does provide bound constraints.

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