Using an anonymous function in lsqcurvefit function

Hello,

I am trying to learn how to use an anonymous function in lsqcurvefit. While I was reading documentation on lsqcurvefit I just a question on the format. The below is a simple example.

Here is the observations (input).

xdata = ...
 [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
 [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];

Fit the model using the starting point x0 = [100,-1].

x0 = [100,-1];
x = lsqcurvefit(@(x,xdata) x(1)*exp(x(2)*xdata),x0,xdata,ydata)

My question is why should it be

x = lsqcurvefit(@(x,xdata) x(1)*exp(x(2)*xdata),x0,xdata,ydata)

instead of

x = lsqcurvefit(@(x) x(1)*exp(x(2)*xdata),x0,xdata,ydata)

I can follow the flow. But I don't understand why I should input both x and xdata into the anonymous function as @(x,xdata). This might sound naive but xdata is listed as input after x0 in the lsqcurvefit already, so I thought lsqcurvefit would know what xdata is in the anonymous function. If I run @(x) instead of @(x,data), I got this error message: Too many input arguments. Could someone please explain this?

Thank you!

Answer 1

[Matt J]

I believe the reason is that in curve fitting scenarios you will normally want to do things like this:

fun = @(x,xdata) x(1)*exp(x(2)*xdata);

x_optimal = lsqcurvefit(fun,x0,xdata,ydata)


hold on
 plot(xdata,fun(x_optimal,xdata),'x')
 fplot(@(z)fun(x_optimal,z),xlim);
hold off

In other words, when you are performing the curve fit, you usually want to view the model function as a function of the unknown parameters with xdata held fixed. However, after you have performed the fit and obtained an x_optimal, the opposite is true. You will now want to view the model function as a function of xdata with fixed x=x_optimal, and query the function at various alternative xdata values (for plotting and other purposes). The makers of lsqcurvefit recognized these different use cases and decided that having you work in terms of a two-argument model function would be the user-friendliest way to accommodate them.

[Jason]

Ah. I see.

So you are saying when performing the curve fit, x is unkown while xdata is known, so we are trying to estimate x based upon known xdata. Therefore, if I don't input xdata into 'fun', both x and xdata will remain as unkown to lsqcurvefit after all. And therefore, I need to provide xdata to the anonymous function, so that lsqcurvefit function will recognize the 'fun' with fixed xdata and unkonwn x, right?

But when dealing with a normal function, the situation is the opposite since known parameters are constant usually, and xdata are independent variables that change.

Is my understanding correct?

[Matt J]

So, I should provide xdata to lsqcurvefit, right?

The reason you must provide xdata to lsqcurvefit as the 3rd input argument and the reason fun(x,xdata) must be provided as a two-argument function is purely because this is the input syntax that lsqcurvefit was designed to expect. The expectations of lsqcurvefit, like any function, are laid out in its documentation.The reason the makers of lsqcurvefit designed it that way, moreover, is because they felt that it would be the most convenient to you, as I described above.

If you are asking whether it had to be designed that way, the answer is no. The input syntax is solely connected with what MathWorks programmers thought would be convenient in curve fitting situations. Below is an alternative way to do the same parameter estimation using the 1-argument version of the model function that you posted. Here, we use lsqnonlin, which has different input syntax expectations.

xdata = ...
 [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
 [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];

fun=@(x) x(1)*exp(x(2)*xdata-ydata; %1-argument version

x = lsqnonlin(fun,x0)

[Jason]

So it is about syntax issue. I see. Thank you so much for your thorough explanation!

[Matt J]

You're quite welcome, but please Accept-click the answer if you consider your question resolved.