How to fit complex valued data?

[James McCambridge]

Esteemed Colleagues,

I would like to fit a complex valued data set with FindFit (I'm willing to
try other methods too, but I thought I'd start out with the basics). Is
this possible?

For example, I am interested in fitting the complex permittivity of liquid
polymers vs frequency, which has a functional form:

In[1]:= expr[b_,x_]:=(1+I b x)^-1;

Using this form to generate data, with b = 1.0, I get:

In[2]:= data ={{1,0.5 -0.5 I},{2.,0.2 -0.4 I},{3.,0.1 -0.3 I
},{4.,0.0588235 -0.235294 I},{5.,0.0384615 -0.192308 I},{6.,0.027027
-0.162162 I},{7.,0.02 -0.14 I},{8.,0.0153846 -0.123077 I},{9.,0.0121951
-0.109756 I},{10.,0.00990099 -0.0990099 I}};

Using FindFit, I come up against an error message.

In[3]:= FindFit[data,expr[b,y],{{b,1.}}, y]

has the output

Out[3]:= FindFit::nrlnum: The function value {0.+0. I, 0.+0. I, -1.38778
10^-17+0. I, <<4>>, 0.+0. I, 0.+0. I, 0.+0. I} is not a list of real
numbers with dimensions {10} at {b} = {0.}

The form expr[0.,x] doesn't look poorly behaved, so what gives?

I could separately fit the real and imaginary components, but this often
gives two somewhat different sets of parameters; I would like to obtain
the parameters which optimize the fit to BOTH the real and imaginary
parts.

Your comments and suggestions are greatly appreciated!

Jim McCambridge

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[Darren Glosemeyer]

One possibility is to use FindFit with a non-default NormFunction that
returns a real value even when residuals have imaginary parts. For instance,

FindFit[data, expr[b, y], {{b, 1.}}, y, NormFunction -> (Abs[Norm[#]] &)]

which uses the absolute value of the sum of squared errors rather than the
default sum of squared errors, works fine in this example.

Darren Glosemeyer
Wolfram Research

[Ray Koopman]

NMinimize[Norm[1/(1 + I b*data[[All,1]]) - data[[All,2]]]^2, {b}]

{1.5401072015490602*^-13, {b -> 1.0000000069007826}}

[Bob Hanlon]

You could try fitting the absolute value since it depends on both the real and
imaginary parts. You must test the result since you might get the sign of b
wrong.

expr[b_,x_]:=(1+I b x)^-1;

b=Random[]

0.9831217471019718

data={#,expr[b,#]}&/@Range[10]//N;

Clear[b];

Off[FindFit::fmgz];

FindFit[{#[[1]],Abs[#[[2]]]}&/@data,
ComplexExpand[Abs[expr[b,x]]],b,x]

{b -> 0.9831217471019714}

Bob Hanlon