Help fitting data to an implicit equation

2021 1 13
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更新时间：2021 1 19
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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

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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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mura0087
mura0087 2021-1-15
编辑：mura0087 2021-1-15
Thank you!
Here is the error function I am using:
function thiserr = err(x,y,t)
b = x(1);
k = x(2);
thiserr = sum( (((1-y).^(1-b)./y) - exp(-k*t)).^2 );
end
And here is how I am calling the function:
fun = @(x)err(x,yvalues,tvalues); %define the function
x0=rand(2,1);%provide a random starting point for your parameters.
options = optimset('MaxFunEvals',1e1000000000000);
bestx=fminsearch(fun,x0);
I set options "MaxFunEvals" to a very high number because I keep getting the following error:
Exiting: Maximum number of function evaluations has been exceeded
- increase MaxFunEvals option.
Current function value: NaN
UPDATED- I believe the I am getting MaxFunEvals because the function does not fit my data. Fminsearch is doing its job just fine, this is a bad model. Thanks for your help
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

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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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