Message: 4
Date: Fri, 6 Feb 2009 17:20:27 +0100
From: "Harbinson, Jeremy" <address@hidden>
Subject: Re: fitting functions which contain 'i': more details
To: <address@hidden>
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<address@hidden>
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Hi,
The function I need to fit is one that describes the frequency
dependence of the electrical impedance (Z) of biological tissue in
terms of four parameters:
capacitance,
two limiting resistances, and
alpha, which basically compensates for the fact that biological
materials do not behave as ideal Resistor-Capacitor systems.
It is difficult to clearly write the function here as it has lots of
subscripts etc, but here is the function in Latex:
Z\left(freq\right)=R_{\infty}+\frac{R_{0}-R_{\infty}}{1+i\omega C
\left(R_{0}-R_{\infty}\right)^{\alpha}}
C is capacitance, and R0 and Rinfinity are the limiting resistances.
If C is non-zero the function returns an complex number that
contains the real and imaginary parts of the impedance. If the real
and imaginary parts of the impedance are plotted against each other
(real on x, imaginary on y) the result is a so-called Cole-Cole
plot. I think it is also sometimes called a Nyquist diagram.
The data I collect is the complex impedance (Z) as a function of
frequency (omega). I would like to estimate the parameters R0,
Rinfinity, C and alpha from this data by the fitting the above
function to it. So I need some fitting tool that is happy working
with complex numbers. I have a recollection that this is/was
possible with a function in the basic Matlab package, but I do not
know which.
An alternative approach to fitting the function above is to fit the
Cole-Cole plot (the real and imaginary components plotted against
each other) with a circle, as the locus of the points on a Cole-Cole
plot is a chord or semicircle whose centre is shifted away from the
origin of the graph. Problems with this approach are that the least-
squares fitting routine does not (so far as I know) fit parametric
equations (maybe fsolve does?) and real data often deviates from the
chord/semicircle of the ideal Cole-Cole plot (typical biology - we
are not ideal). These distorted responses are relatively easy to
deal with by tweaking the basic function shown above, but not so
easy to work with via the graphical approach of the Cole-Cole plot.
Hope this clarifies things a bit,
all the best,
Jeremy