Re: How to fit complex valued data?

• To: mathgroup at smc.vnet.net
• Subject: [mg64131] Re: [mg64096] How to fit complex valued data?
• From: Darren Glosemeyer <darreng at wolfram.com>
• Date: Thu, 2 Feb 2006 00:05:59 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

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

At 04:34 AM 2/1/2006 -0500, James McCambridge wrote:

>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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