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Is there any way to use implicit functions when fitting them to data with NLINFIT? Details: I need to fit a model to some data with NLINFIT. The problem is that the model to fit is an implicit function, like this:

y= x*(1-exp(a*(h*y-T)))

('x' is the independent variable, 'a' and 'h' are the parameters to be calculated and 'T' is a constant). How can I write a function that returns 'y' when it depends on the values of 'y' itself? I saw a (could-be) solution involving FSOLVE, but I don't have the corresponding toolbox.

Just in case, this comes from integrating (for 't' between 0 and T) the ODE:

dN/dt= a*N /(1+ a*h*N)

given that:

y= N(0)-N(T)

Thanks for any suggestions

Tom Lane
on 10 Aug 2012

NLINFIT wants a response vector Y and a function with unknown parameters. You could try supplying the response vector

Y = zeros(size(y))

and the function

(-y) + x*(1-exp(a*(h*y-T)))

and see if that works. I'd expect it to work. You'd pack x and y together as columns in your X matrix, and unpack them to compute the function value.

Star Strider
on 13 Aug 2012

The nlinfit function does not allow parameter constraints. Use lsqcurvefit if you need to do that.

Star Strider
on 10 Aug 2012

When I integrate to get N(t) as:

n = dsolve( diff(N) == a*N / (1 + a*h*N), 'IgnoreAnalyticConstraints', true )

in the Symbolic Math Toolbox, the solution is:

N(t) = [0; lambertw(0, a*h*exp(C3 + a*t))/(a*h)]

and then when I have it create an implicit function for the solution, produces:

Nt = @(C3,a,h,t)[0.0; lambertw(0,a.*h.*exp(C3+a.*t))./(a.*h)];

adding N(0) = N0 as an initial condition:

n = dsolve( diff(N) == a*N / (1 + a*h*N), N(0) == N0, 'IgnoreAnalyticConstraints', true )

and integrating yields:

N(t) = lambertw(0, N0*a*h*exp(a*t)*exp(N0*a*h))/(a*h)

and as an anonymous function:

Nt0 = @(N0,a,h,t) lambertw(0,N0.*a.*h.*exp(a.*t).*exp(N0.*a.*h))./(a.*h);

The lambertw function exists in MATLAB (although I admit I've not heard of it until now).

You need to combine parameters N0, a, and h into one vector for curve-fitting purposes in nlinfit or lsqcurvefit. (I do not have the Curve Fitting Toolbox, and use the Statistics and Optimization Toolboxes functions.)

This at least solves your problem of having y on both sides of the equation you want to fit.

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