How to force a exponential fit to go through a preestablish point (x,y)?

Hi, I have some data X and Y and I want to fit it to an exponential function but giving it an initial point (Xo,Yo):

f = fit(x,y,'exp1','StartPoint',[0 2])

Is there any way to do it? I think I've tried everything I could.

Answer 1

[Torsten]

x0 = ...;  % x coordinate of preestablished point
y0 = ...;  % y coordinate of preestablished point 
g = @(p,x)y0*exp(-p*(x-x0));
f = fit(x,y,g)
plot(f,x,y)

[Maria Victoria Moragues Gomez]

So simple... Thank you so much!

Answer 2

[John D'Errico]

'StartPoint' refers to the initial parameter guess for the model, since this is an optimization problem. Not a point the curve must pass through.

The exp1 model refers to this model:

fittype('exp1')
ans = 
     General model Exp1:
     ans(a,b,x) = a*exp(b*x)

Now, you want that curve to pass through the point

y0 = a*exp(b*x0)

So you can reduce the model by one parameter, using the above information. That is,

a = y0/exp(b*x0)

Therefore, your model is now:

y = yo/exp(b*x0) * exp(b*x)

We can rewrite it in a slightly less complicated form:

y = yo*exp(b*(x - x0))

See that there is n0 a term out front. The model is just as I wrote it, with only b as an unknown, and x0 and y0 as knowns. So we could use that model, now with x0 and y0 fixed, in custom model. Easy enough to do.

ft = fittype('y=y0*exp(b*(x-x0))','independent','x','dependent','y',...
   'problem',{'x0','y0'},'coefficients','b');

x0 = 0;
y0 = 2;
mdl = fit(x,y,ft,'problem',[x0,y0]);

Best is is you provide some intelligent value for b as a starting guess, thus use 'StartPoint' in fit.

Since you did not provide any data, I'll make some up.

n = 10;
x = rand(n,1);
y = 2*exp(x) + randn(n,1)/100;
plot(x,y,'o')
grid on

<</matlabcentral/answers/uploaded_files/136822/untitled.jpg>>

We know the curve should pass though the point (0,2).

ft = fittype('y0*exp(b*(x-x0))','independent','x',...
   'problem',{'x0','y0'},'coefficients','b');

x0 = 0;
y0 = 2;
mdl = fit(x,y,ft,'problem',{x0,y0},'StartPoint',0.5);

mdl
mdl = 
   General model:
   mdl(x) = y0*exp(b*(x-x0))
   Coefficients (with 95% confidence bounds):
     b =       1.002  (0.9999, 1.005)
   Problem parameters:
     x0 =           0
     y0 =           2

I guessed the initial value for b as 0.5. In general, always provide intelligent starting values for an optimization.

[Shaunak Chatterjee]

What to do for fittype exp2?