Main Content

This example shows how to perform nonlinear least-squares curve fitting using the Problem-Based Optimization Workflow.

The model equation for this problem is

$$y(t)={A}_{1}\mathrm{exp}({r}_{1}t)+{A}_{2}\mathrm{exp}({r}_{2}t),$$

where $${A}_{1}$$, $${A}_{2}$$, $${r}_{1}$$, and $${r}_{2}$$ are the unknown parameters, $$y$$ is the response, and $$t$$ is time. The problem requires data for times `tdata`

and (noisy) response measurements `ydata`

. The goal is to find the best $$A$$ and $$r$$, meaning those values that minimize

$$\sum _{t\in tdata}{(y(t)-ydata)}^{2}.$$

Typically, you have data for a problem. In this case, generate artificial noisy data for the problem. Use `A = [1,2]`

and `r = [-1,-3]`

as the underlying values, and use 200 random values from `0`

to 3 as the time data. Plot the resulting data points.

rng default % For reproducibility A = [1,2]; r = [-1,-3]; tdata = 3*rand(200,1); tdata = sort(tdata); % Increasing times for easier plotting noisedata = 0.05*randn(size(tdata)); % Artificial noise ydata = A(1)*exp(r(1)*tdata) + A(2)*exp(r(2)*tdata) + noisedata; plot(tdata,ydata,'r*') xlabel 't' ylabel 'Response'

The data are noisy. Therefore, the solution probably will not match the original parameters `A`

and `r`

very well.

To find the best-fitting parameters `A`

and `r`

, first define optimization variables with those names.

A = optimvar('A',2); r = optimvar('r',2);

Create an expression for the objective function, which is the sum of squares to minimize.

fun = A(1)*exp(r(1)*tdata) + A(2)*exp(r(2)*tdata); obj = sum((fun - ydata).^2);

Create an optimization problem with the objective function `obj`

.

`lsqproblem = optimproblem("Objective",obj);`

For the problem-based approach, specify the initial point as a structure, with the variable names as the fields of the structure. Specify the initial `A = [1/2,3/2]`

and the initial `r = [-1/2,-3/2]`

.

x0.A = [1/2,3/2]; x0.r = [-1/2,-3/2];

Review the problem formulation.

show(lsqproblem)

OptimizationProblem : Solve for: A, r minimize : sum(arg6) where: arg5 = extraParams{3}; arg6 = (((A(1) .* exp((r(1) .* extraParams{1}))) + (A(2) .* exp((r(2) .* extraParams{2})))) - arg5).^2; extraParams

Solve the problem.

[sol,fval] = solve(lsqproblem,x0)

Solving problem using lsqnonlin. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. <stopping criteria details>

`sol = `*struct with fields:*
A: [2×1 double]
r: [2×1 double]

fval = 0.4724

Plot the resulting solution and the original data.

figure responsedata = evaluate(fun,sol); plot(tdata,ydata,'r*',tdata,responsedata,'b-') legend('Original Data','Fitted Curve') xlabel 't' ylabel 'Response' title("Fitted Response")

The plot shows that the fitted data matches the original noisy data fairly well.

See how closely the fitted parameters match the original parameters `A = [1,2]`

and `r = [-1,-3]`

.

disp(sol.A)

1.1615 1.8629

disp(sol.r)

-1.0882 -3.2256

The fitted parameters are off by about 15% in `A`

and 8% in `r`

.

`fcn2optimexpr`

If your objective function is not composed of elementary functions, you must convert the function to an optimization expression using `fcn2optimexpr`

. See Convert Nonlinear Function to Optimization Expression. For the present example:

fun = @(A,r) A(1)*exp(r(1)*tdata) + A(2)*exp(r(2)*tdata); response = fcn2optimexpr(fun,A,r); obj = sum((response - ydata).^2);

The remainder of the steps in solving the problem are the same. The only other difference is in the plotting routine, where you call `response`

instead of `fun`

:

responsedata = evaluate(response,sol);

For the list of supported functions, see Supported Operations on Optimization Variables and Expressions.