MATLAB: How to constrain a fitted curve through specific points like the origin in MATLAB

constraincurvefittingfminconMATLABoptimizationOptimization Toolboxpolyfit

I would like to use the POLYFIT function or the Curve Fitting Toolbox to impose linear constraints on fitted curves to force them to pass through specific points like the origin.

Best Answer

Constraining a fitted curve so that it passes through specific points requires the use of a linear constraint. Neither the POLYFIT function nor the Curve Fitting Toolbox allows specifying linear constraints. Performing this operation requires the use of the LSQLIN function in the Optimization Toolbox.

Consider the data created by the following commands:

c = [1 -2 1 -1];  
x = linspace(-2,4);  
y = c(1)*x.^3+c(2)*x.^2+c(3)*x+c(4) + randn(1,100);
plot(x,y,'.b-')

You can view the unconstrained fit to a third-order polynomial (using POLYFIT) via:

hold on
c = polyfit(x,y,3);  
yhat = c(1)*x.^3+c(2)*x.^2+c(3)*x+c(4);
plot(x,yhat,'r','linewidth',2)

However, if you wish to constrain the fit to go through a specific point, for example (x0, y0) where:

x0 = 1;
y0 = 10;

use the LSQLIN function in the Optimization Toolbox to solve the linear least-squares problem with a linear constraint, as in the following example:

x = x(:); %reshape the data into a column vector
y = y(:);
% 'C' is the Vandermonde matrix for 'x'
n = 3; % Degree of polynomial to fit
V(:,n+1) = ones(length(x),1,class(x));
for j = n:-1:1
     V(:,j) = x.*V(:,j+1);
end
C = V;
% 'd' is the vector of target values, 'y'.
d = y;
% There are no inequality constraints in this case, i.e., 
A = [];
b = [];
% We use linear equality constraints to force the curve to hit the required point. In
% this case, 'Aeq' is the Vandermoonde matrix for 'x0'
Aeq = x0.^(n:-1:0);
% and 'beq' is the value the curve should take at that point
beq = y0;
p = lsqlin( C, d, A, b, Aeq, beq )
% We can then use POLYVAL to evaluate the fitted curve
yhat = polyval( p, x );
% Plot original data
plot(x,y,'.b-') 
hold on
% Plot point to go through
plot(x0,y0,'gx','linewidth',4) 
% Plot fitted data
plot(x,yhat,'r','linewidth',2) 
hold off

Related Solutions

MATLAB: Please help!! Need to constrain/force LSQNONLIN to pass through a given point e.g. the origin

Hi Chris

From what I understand, you would like to fit the nonlinear model with your data set, and you would like to know how to impose the nonlinear equality constraint

Fit_simp(x, 0.3, 1) = 0.

In order to achieve data fitting with constraints, I suggest you use the function fmincon (See Documentation) , which allows you to specify nonlinear constraints.

You may consider the following steps:

1. Revise the objective function as follows

function diff = fit_simp(x,X,Y)
A=x(1);
B=x(2);
diff = norm(A*log(X+B) + A*log(1+B) - Y);
end

The new function returns a scalar as the data fitting error. You may save the function as a MATLAB file under the same name as the function. See the documentation for the function norm.

2. Create the nonlinear constraint function

function [c,ceq] = nlcon(x, X, Y)
A=x(1);
B=x(2);
c = [];
ceq = A*log(X+B) + A*log(1+B) - Y;
end

This function has two outputs, c is the output of inequality nonlinear constraints and ceq is the output of the equality constraints. Since we only have equality constraints here, I set c=[]. Again, save this function in a MATLAB file.

3. In the main MATLAB script or in the command window, try the following code (Assuming X and Y are column vectors which contain your data set)

>> X0=[1,0];
>> x=fmincon(@(x) fit_simp(x,X,Y),X0,[],[],[],[],[],[],@(x) nlcon(x,0.3,1));

Then, you should have the expected answer. If you would like to have better data fitting accuracy, you can always specify the options (See Documentation) for fmincon.

For more information regrading nonlinear constraints, you can refer to the link below:

http://www.mathworks.com/help/optim/ug/writing-constraints.html#brhkghv-16

-Yu

MATLAB: How to use lsqlin for linear contraint through points and constrained derivative at those points

A general curve fit, IF the curve is nonlinear, will not always be trivial to enforce a constraint. You would need to use a tool like fmincon.

But if you are asking how to fit a polynomial, as those links talk about, then is there some (ANY?) rational reason why you think you need to use a polynomial for this?

The most common reason is you simply don't know of any other fitting tool. The answer to that is to use my SLM toolbox , which lets you do EXACTLY as you desire, plus, it has many more abilities.

Yes, it is trivial to use lsqlin for this purpose. But a sufficiently high order polynomial to give you any real flexibility will be too high an order to cause problems in the fit. High order polynomials are NEVER a good choice.

Do I really need to show you how to solve this using a polynomial fit? Sigh. It is just a bad idea in general. Use SLM instead. Really. You WILL be happier with the result.

Assume the polynomial has order (highest degree) n, and that at some given point a, f(a)=v, and f'(a)=d.

I'll make up some random data (crap data, but who cares?) This is just an example. I'll do it as a quadratic polynomial.

x = rand(10,1);
y = rand(10,1);
n = 2;
a = 0.5;
v = 0.6;
d = 0.1;
p = n:-1:0;
M = bsxfun(@power,x(:),p);
Aeq = [a.^p;p.*a.^max(0,p-1)];
Beq = [v;d];
coef = lsqlin(M,y(:),[],[],Aeq,Beq);
coef
coef =
     1.0934
   -0.99337
    0.82334

Do ya trust me?

polyval(coef,a)
ans =
        0.6
polyval(polyder(coef),a)
ans =
        0.1

Got lucky I guess. Even so? Use SLM. You WILL be happier with the result.

Best Answer

Related Solutions

MATLAB: Please help!! Need to constrain/force LSQNONLIN to pass through a given point e.g. the origin

MATLAB: How to use lsqlin for linear contraint through points and constrained derivative at those points

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