This example shows how to perform nonlinear
fitting of complex-valued data. While most Optimization Toolbox™ solvers
and algorithms operate only on real-valued data, the
works on both real-valued and complex-valued data.
Do not set the
FunValCheck option to
using complex data. The solver errors.
The data model is a simple exponential:
The x is input data, y is the response, and v is a complex-valued vector of coefficients. The goal is to estimate v from x and noisy observations y.
Artificial Data with Noise
Generate artificial data for the model. Take the complex coefficient
vector v as
Take the observations x as exponentially distributed.
Add complex-valued noise to the responses y.
rng default % for reproducibility N = 100; % number of observations v0 = [2;3+4i;-.5+.4i]; % coefficient vector xdata = -log(rand(N,1)); % exponentially distributed noisedata = randn(N,1).*exp((1i*randn(N,1))); % complex noise cplxydata = v0(1) + v0(2).*exp(v0(3)*xdata) + noisedata;
Fit the Model to Recover the Coefficient Vector
The difference between the response predicted by the data model
and an observation (
xdata for x and
cplxydata for y) is:
objfcn = @(v)v(1)+v(2)*exp(v(3)*xdata) - cplxydata;
fit the model to the data. This example first uses
Because the data is complex, set the
opts = optimoptions(@lsqnonlin,... 'Algorithm','levenberg-marquardt','Display','off'); x0 = (1+1i)*[1;1;1]; % arbitrary initial guess [vestimated,resnorm,residuals,exitflag,output] = lsqnonlin(objfcn,x0,,,opts); vestimated,resnorm,exitflag,output.firstorderopt
vestimated = 2.1581 + 0.1351i 2.7399 + 3.8012i -0.5338 + 0.4660i resnorm = 100.9933 exitflag = 3 ans = 0.0013
lsqnonlin recovers the complex coefficient
vector to about one significant digit. The norm of the residual is
sizable, indicating that the noise keeps the model from fitting all
the observations. The exit flag is
3, not the preferable
because the first-order optimality measure is about
Alternative: Use lsqcurvefit
To fit using
lsqcurvefit, write the model
to give just the responses, not the responses minus the response data.
objfcn = @(v,xdata)v(1)+v(2)*exp(v(3)*xdata);
lsqcurvefit options and syntax.
opts = optimoptions(@lsqcurvefit,opts); % reuse the options [vestimated,resnorm] = lsqcurvefit(objfcn,x0,xdata,cplxydata,,,opts)
vestimated = 2.1581 + 0.1351i 2.7399 + 3.8012i -0.5338 + 0.4660i resnorm = 100.9933
The results match those from
because the underlying algorithms are identical. Use whichever solver
you find more convenient.
Alternative: Split Real and Imaginary Parts
To use the
such as when you want to include bounds, you must split the real and
complex parts of the coefficients into separate variables. For this
problem, split the coefficients as follows:
Write the response function for
function yout = cplxreal(v,xdata) yout = zeros(length(xdata),2); % allocate yout expcoef = exp(v(5)*xdata(:)); % magnitude coscoef = cos(v(6)*xdata(:)); % real cosine term sincoef = sin(v(6)*xdata(:)); % imaginary sin term yout(:,1) = v(1) + expcoef.*(v(3)*coscoef - v(4)*sincoef); yout(:,2) = v(2) + expcoef.*(v(4)*coscoef + v(3)*sincoef);
Save this code as the file
your MATLAB® path.
Split the response data into its real and imaginary parts.
ydata2 = [real(cplxydata),imag(cplxydata)];
The coefficient vector
v now has six
dimensions. Initialize it as all ones, and solve the problem using
x0 = ones(6,1); [vestimated,resnorm,residuals,exitflag,output] = ... lsqcurvefit(@cplxreal,x0,xdata,ydata2); vestimated,resnorm,exitflag,output.firstorderopt
vestimated = 2.1582 0.1351 2.7399 3.8012 -0.5338 0.4660 resnorm = 100.9933 exitflag = 3 ans = 0.0018
Interpret the six-element vector
a three-element complex vector, and you see that the solution is virtually
the same as the previous solutions.