How do I fit two equations that explain two parts of a curve?

Hi,my equations are: y(t)={(a/b)*(1-exp(-b(x-x_start)),for x_start <= x < x_end %equ1 and {c*exp(-b(x-xend)),for x > x_end %equ2

Here I do not know a,b and c. The idea is that since both equations have b in common, I would like to know how fitting these equations I can get one single value for b. I already fitted these equations individually, but I obtained two values of b, which are different from each fit.

[John D'Errico]

I predict that your next question will be why is my curvefit not continuous at the break point? I.e., at x_end, the pair of functions will not be continuous.

Answer 1

[Sean de Wolski]

Break the data into two pieces and do the curve fit

xlow = x(x<x_end);
ylow = y*x<x_end);
xhigh = x(x>x_end); % Note you have nothing for x==x_end
yhigh = y(x>x_end);

Now do the curve fit - either with the Curve Fitting App, lsqcurvefit or fitnlm.

[mdjupin]

Thanks Sean. I want to make sure that the common variable (here 'b') in both equations is fitted to the same value. Does your suggestion do this?

[Sean de Wolski]

No, you would have to solve for one and use that value of b for both with mine (i.e. not solve for it in the second equation). In order to find b that works for both equations, I don't know how to do that as there is a discontinuity in the fitting function. A global optim solver would probably be best doing both curve fits with it at once..

[mdjupin]

Hi Sean, could you tell me more about global optimum solver. Thank you.

[Torsten]

Use lsqnonlin and define the f_i as

f_i = y_i -(p(1)/p(2))*(1-exp(-p(2)(x_i - x_start)) for x_start <= x_i < x_end

and

f_i = y_i -{p(3)*exp(-p(2)(x_i-xend)) for x_i > x_end

(x_i,y_i) are your measurement data.

Best wishes

Torsten.

[mdjupin]

@ Torsten thank you. But this does not work out, due to lsqnonlin fit only the first equation in myfuntion.m. I also tried: myfuntion.m

funtion err = myfuntion(p) err = sum(y_i -(p(1)/p(2))*(1-exp(-p(2)(x_i - x_start)))-sum(y_i -{p(3)*exp(-p(2)(x_i-xend)))

But this also failed. Any other option?

[Torsten]

You programmed it as

function call

xdata=[...];
ydata=[...];
xstart=...;
xend=...;

p0=[1 1 1];
p = lsqnonlin(@(p)myfun(xdata,ydata,xstart,xend,p),p0);

function F = myfun(xdata,ydata,xstart,xend,p)

for i=1:length(xdata)

if (xdata(i) >= xstart) && (xdata(i) < xend)
F(i) = ydata(i)-(p(1)/p(2))*(1-exp(-p(2)(xdata(i)-xstart)); 
else
F(i)=ydata(i)-(p(3)*exp(-p(2)(xdata(i)-xend));
end

end

?

Best wishes

Torsten.

[mdjupin]

Thanks Torsten, this does work. I would advise not to use more than two equations, because the fitting get harder.

Cheers.

mdjupin

Answer 2

[Joep]

This is very simple solution which works in some cases one thing you need to noticed is the 1-based indexing of matlab.

a=10; b=20; c=-1; d=-3;

for x=1:a
 t(x)=x;
 y(x)=c*x;
end
for x=a:b
 t(x)=t;
 y(x)=d*x.^2;
end
figure
plot(t,y)

[mdjupin]

Thanks Joep. Here I do not know a,b and c. The idea is that since both equations have b in common, I would like to know how fitting these equations I can get one single value for b. I already fitted these equations individually, but I obtained two values of b, which are different from each fit.