MATLAB: How to fit a complex valued function to the data

curvedatafitttingimaginaryOptimization Toolbox

I am trying to find parameters which would fit a complex valued function to my data, which is real. The function that I want to use is:

 y = c*x*1i + x^2;

Best Answer

The Curve Fitting Toolbox does not accept complex valued functions. To carry out the fit, use the LSQNONLIN function or the FMINSEARCH function. The objective function passed to LSQNONLIN or FMINSEARCH should take the real and complex parts of the parameters separately and return the real and complex parts of the residuals separately.

For example, create the function 'myfun.m'.

function F = myfun(c,xdata,ydata)
% c is a two-element vector whose components are the real
% and imaginary parts of the parameter respectively
errors = (c(1) + c(2)*1i)*xdata*1i + xdata.^2 - ydata;
F = [real(errors(:)); imag(errors(:))];

Note that here, xdata and ydata may be complex, but the input and output are real.

To use this with LSQNONLIN:

[x,resnorm] = lsqnonlin(@(c)myfun(c,xdata,ydata),[1  3])

To use this with FMINSEARCH, add

F = sum(F.^2);

to the bottom of 'myfunc.m' and replace 'lsqnonlin' with 'fminsearch' in the above syntax.

Related Solutions

MATLAB: Lsqcurvefit

Since the distance between two complex numbers a+i*b and c+i*d is (a-c)^2 + (b-d)^2, you could treat each complex number as a pair of independent real numbers. Wrap fun in a function rfun that returns the real and complex parts of the output. For example, assuming rfun returns a column vector,

function y = rfun(x)
  y = rfun(x);
  y = [real(y); imag(y)];
end

Similarly, you could replace ydata by

rdata = [real(ydata); imag(ydata)].

Then you have the first two inputs for lsqcurvefit.

MATLAB: How to use Curve Fitting Toolbox for fitting complex data points

Here is an example showing how to approximate the function:

f(x, y, z, w) = (x + iy) + (w + iz)

To accomplish this, we can use the "lsqcurvefit" function from the Optimization Toolbox. The reason being Curve Fitting Toolbox does not handle complex data as mentioned here:

https://www.mathworks.com/help/curvefit/fit.html#bto2vuv-1-x

Also, not all the functions in the Optimization Toolbox handle complex data as well as discussed here:

https://www.mathworks.com/help/optim/ug/complex-numbers-in-optimization-toolbox-solvers.html

The following documentation link describes some approaches for handling complex data using the functions in Optimization Toolbox:

https://www.mathworks.com/help/optim/ug/fit-model-to-complex-data.html

The example shown here is based on splitting real and imaginary part of the complex data as discussed in the above link under the Section

"Alternative: Split Real and Imaginary Parts"

%% Create target data
tmpData = 100*rand(10, 4);
targetData = tmpData(:, 1) + 3i*tmpData(:, 2) + 0.5*tmpData(:, 3) - 2*tmpData(:, 4);
% goal is to approximate x + yi + w + zi

%% Split target data into real and imaginary parts
uData = [real(targetData) imag(targetData)];

%% Input data
inputData = ones(10, 4);
% these are all ones because the function to approximate can be written as:
% x * 1 + y * 1 * 1i + z * 1 + w * 1 * 1i
% we want to find x, y, z, w
% so the input data are all ones

%% Starting assumption for x, y, z, w values
coeff0 = zeros(size(inputData));

%% Create options for the function call
opts = optimoptions(@lsqcurvefit, 'Algorithm', 'levenberg-marquardt');

%% Call the optimization function
coeffVals = lsqcurvefit(@curveFunc, coeff0, inputData, uData, [], [], opts);

%% Get the predicted data based on output of the optimization routine
predictedData = curveFunc(coeffVals, inputData);

%% Plot the original data and predicted data
figure; hold on;
plot(uData(:, 1), uData(:, 2), 'g*');
plot(predictedData(:, 1), predictedData(:, 2), 'bo');

function uout = curveFunc(v, xdata)
    % this part is broken down to show how the "curveFunc" relates to the function that we are trying to approximate
    tmpUout = v(:, 1) .* xdata(:, 1); % a values
    tmpUout = tmpUout + 1i * v(:, 2) .* xdata(:, 2); % b values
    tmpUout = tmpUout + v(:, 3) .* xdata(:, 3); % c values
    tmpUout = tmpUout + 1i * v(:, 4) .* xdata(:, 4); % d values
    uout = [real(tmpUout) imag(tmpUout)];
end

Best Answer

Related Solutions

MATLAB: Lsqcurvefit

MATLAB: How to use Curve Fitting Toolbox for fitting complex data points

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