How do I constrain a fitted curve through specific points like the origin in MATLAB?
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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);
You can view the unconstrained fit to a third-order polynomial (using POLYFIT) via:
c = polyfit(x,y,3);
yhat = c(1)*x.^3+c(2)*x.^2+c(3)*x+c(4);
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);
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 point to go through
% Plot fitted data
More Answers (2)
John D'Errico on 27 Mar 2015
Well, actually, this problem can be solve quite trivially in a way that does not require lsqlin, or anything fancy. There really is no requirement for a linear constraint, since the constraint will be that the constant parameter in the polynomial model is just zero. So use backslash to estimate the model coefficients, and just leave out the constant term. Note that this works ONLY to force a point through the origin, since the point (x,y) = (0,0) in a polynomial model implies that the constant term MUST be identically zero.
So, given a model of the form
y = a1*x^3 + a*x^2 + a3*x
we can fit it simply as
a123 = [x.^3, x.^2, x]\y;
where x and y are column vectors of data.
You can even use polyval to evaluate the polynomial, simply by appending a zero to the end of that vector f coefficients, thus
P = [a123;0];
If you wanted to force the mode to pass through the point (x0,y0), where these values are not the origin, then we need to work slightly more, but not a lot. While you can do so with lsqlin, if you don't have that tool (or my own LSE, as found on the file exchange) just do this simple transformation:
xtr = x - x0;
a123 = [xtr.^3, xtr.^2, xtr]\(y - y0);
Essentially, we have now fit the model
y - y0 = a1*(x - x0)^3 + a*(x - x0)^2 + a3*(x - x0)
You can see that when x == x0, y must be predicted as y0.
Evaluation of this model is just as easy. You can still use polyval, just remember to subtract off x0 from x first.
P = [a123;y0];
ypred = polyval(P,xpred - x0);
What happens when we have a more complicated problem, where maybe you need to force TWO points on the curve? Well, then, you can use either lsqlin or my own LSE. (Note that LSE runs in R2015a with no warning messages.)
Or, if you have a trigonometric model? If the model has nonlinear parameters in it, then you will need to use a nonlinear optimization. And depending on the model, to force it through a given point, that may require a nonlinear tool that can handle an equality constraint, so possibly fmincon. But sometimes, again, depending on the model, there may be simpler solutions. It all depends on the model.