when running lsqcurvefit to fit the complex function: F = a1*exp(-1i*x*a2) + a3*exp(-1i*x*a4), i get a2 and a4 as complex estimates, however these values should be real. how to solve this?

when runing lsqcurvefit to fit the complex function: F = a1*exp(-1i*x*a2) + a3*exp(-1i*x*a4), i get a2 and a4 as complex estimates, however these values should be real. how to solve this?

Answer 1

[Walter Roberson]

lsqcurvefit defines that the xdata and ydata must be real valued. In your model

   F = a1*exp(-1i*x*a2) + a3*exp(-1i*x*a4)

with x and a2 and a4 real-valued, -1i*x*a2 and -1i*x*a4 are guaranteed to be complex valued (or 0), and exp() of a complex value is complex.

If a1 and a2 are real-valued then the two summands would be complex, and it is quite unlikely that their imaginary components will exactly balance out to give a real value -- but ydata must be real.

If a1 and a2 are complex valued then although it is possible for the a1*exp() terms and a3*exp() terms to happen to be real-valued, it is not likely.

If you are counting on complex components happening to cancel out or happening to multiply to give real values, you should probably be using a different formulation of your model.

[joseph chahin]

thanks. Any idea how to refomulate such model?

[Walter Roberson]

Please describe more clearly what you want to do.

Are you trying to find real-valued a1, a2, a3, a4, with real-valued x and y, such that

 a1*exp(-1i*x*a2) + a3*exp(-1i*x*a4) - y

is real-valued ? So, in other words,

imag(a1*exp(-1i*x*a2)) == -imag(a3*exp(-1i*x*a4))

so that the imaginary part exactly cancels to 0?

[joseph chahin]

I have a complex data y which should exhibit:

 y = a1*exp(-1i*xdata*a2) + a3*exp(-1i*xdata*a4)

where a2 and a4 should be real positive values (a1 and a3 can be complex). Now my function is defined as:

F = myfun(a,xdata) 
F = a1*exp(-1i*xdata*a2) + a3*exp(-1i*xdata*a4)

when performing:

[p,resnorm] = lsqcurvefit(@myfun,a0,xdata,(ydata), [],[],opts);

i am getting a2 and a4 as complex values whereas these should be real and positive. please let me know if is not clear? thx.

Answer 2

[Matt J]

Convert F to an equivalent real-valued function (but with higher dimensional output) as follows,

F_modified=@(a,xdata) [real(F(a,xdata(:)))  ; imag(F(a,xdata(:)))];

Similarly, convert your ydata to a real equivalent,

ydata_modified = [real(ydata(:)); imag(ydata(:))];

Pass the modified inputs to lsqcurvefit in place of the original.

[joseph chahin]

do you mean to evaluate the fitting for real and imaginary parts separatly? this means that i can implement two functions F_real and F_imag and combine the results (F=F_real+i*F_imag). However in your case you define only one function. correct?

[Matt J]

do you mean to evaluate the fitting for real and imaginary parts separatly?

I mean that the model function F(a,xdata) that you will pass to lsqcurvefit will return the concatenation of the real and the imaginary part [F_real;F_imag]. Similarly, your ydata will be [y_real;y_imag].

[joseph chahin]

not succeed to implement what you mean. could you please provide steps to follow. Thx in advance. This is what i have till now:

    %%%mfile for model 

    function F  = myfunLSQ(p, xdata)

    F           = p(1)*exp( -1i*xdata*(p(2))) + p(3)*exp(-1i*xdata*p(4)) ;

    end

    %%%main program 

    opts        = optimset('TolFun', 1e-12,'TolX', 1e-12,'MaxFunEvals',400 );

    [p,resnorm] = lsqcurvefit(@myfunLSQ,p0,xdata,ydata, [],[],opts)

where should F_modified be defined? is it a new mfile, or added in the main program?

[Matt J]

where should F_modified be defined? is it a new mfile, or added in the main program?

Either one is possible, but below I define it in the main program.

F=@myfunLSQ;
F_modified=@(a,xdata) [real(F(a,xdata(:)))  ; imag(F(a,xdata(:)))];
ydata_modified = [real(ydata(:)); imag(ydata(:))];

    [p,resnorm] = lsqcurvefit(F_modified,p0,xdata,ydata_modified)

[Torsten]

And start with complex-valued p(1) and p(3) and real-valued p(2) and p(4) as initial guess for the parameters.

Best wishes

Torsten.

[Matt J]

And start with complex-valued p(1) and p(3) and real-valued p(2) and p(4) as initial guess for the parameters.

No! Do not do that!

[Torsten]

a1 and a3 are supposed to be complex-valued.

If you start with real-valued a1 and a3, they will remain real-valued during the fitting process.

Best wishes

Torsten.

[Matt J]

I missed the part about a1 and a3 being complex-valued. In that case, the whole thing should be modified to a 6 real parameter fit as follows,

      function F  = myfunLSQ(p, xdata)

           Fc = complex(p(1),p(5)) *exp( -1i*xdata(:)*(p(2))) + ...
                complex(p(3),p(6)) *exp(-1i*xdata(:)*p(4)) ;

           F=[real(Fc);imag(Fc)];

      end

   p0=[...some six-dimensional initial vector ...]; %REAL-VALUED

   [p,resnorm] = lsqcurvefit(@myfunLSQ,p0,xdata,...
                            [real(ydata(:));imag(ydata(:))] );

    a1=complex(p(1),p(5));
    a2=p(2);
    a3=complex(p(3),p(6));
    a4=p(4);

[joseph chahin]

Dear Matt, Thank you for your proposal. It is now much clear. I will try it. Br.

[joseph chahin]

Dear Matt, the real value should always be possitive. shall I set this in LB,UB parameters or is there any option to include this condition in the function F. Thx. Br.

[Matt J]

Use LB to enforce the positivity.

Answer 3

[YT]

I haven't used lsqcurvefit myself, but if you want the real value of a complex number, you can use

a2 = real(a2);
a4 = real(a4);

[joseph chahin]

what I mean is that the estimate has to be given as real and not complex. this has to be taken into accout in the estimation process.

Answer 4

[Torsten]

Write your function F as

y = (a11+1i*a12)*(cos(a2*xdata)-1i*sin(a2*xdata))+(a31+1i*a32)*(cos(a4*xdata)-1i*sin(a4*xdata)) =
(a11*cos(a2*xdata)+a12*sin(a2*xdata)+a31*cos(a4*xdata)+a32*sin(a4*xdata))+1i*(-a11*sin(a2*xdata)+a12*cos(a2*xdata)-a31*sin(a4*xdata)+a32*cos(a4*xdata))

and fit simultaneously real(ydata) against

a11*cos(a2*xdata)+a12*sin(a2*xdata)+a31*cos(a4*xdata)+a32*sin(a4*xdata)

and imag(ydata) against

-a11*sin(a2*xdata)+a12*cos(a2*xdata)-a31*sin(a4*xdata)+a32*cos(a4*xdata)

Best wishes

Torsten.

[joseph chahin]

Hi Torsten, thanks a lot for your response. how can i fit it simultaneously real and imag? please see my code above in the comment of Matt. thanks again.

[Torsten]

x0 = [1 1 1 1 1 1];
x = lsqnonlin(@(x)fun(x,xdata,ydata),x0)

function res = fun(x,xdata,ydata)
a11 = x(1);
a12 = x(2);
a2 = x(3);
a31 = x(4);
a32 = x(5);
a4 = x(6);
rydata = real(ydata);
iydata = imag(ydata);
n = numel(xdata);
res(1:n) = a11*cos(a2*xdata(1:n))+a12*sin(a2*xdata(1:n))+a31*cos(a4*xdata(1:n))+a32*sin(a4*xdata(1:n))-rydata(1:n);
res(n+1:2*n) = -a11*sin(a2*xdata(1:n))+a12*cos(a2*xdata(1:n))-a31*sin(a4*xdata(1:n))+a32*cos(a4*xdata(1:n))-iydata(1:n);

[joseph chahin]

Thanks. I will give it a try. I see that you suggested lsqnonlin, however i am using lsqcurvefit. could you please tell why? does it matter?

[Torsten]

I used lsqnonlin because the handling of xdata and ydata is easier.

To use "lsqcurvefit", you will have to double "xdata" as

Xdata = [xdata,xdata];

,supply the corresponding ydata as

Ydata = [real(ydata),imag(ydata)];

and call "lsqcurvefit" as

x = lsqcurvefit(@fun,x0,Xdata,Ydata);

Further, you will have to change res because the subtraction of rydata and iydata is no longer necessary.

From the numerical point of view, both solvers are equivalent ; only the Interface is a bit different.

Best wishes

Torsten.