Constrained Polynomial Regression

Hey all,

I'm sure someone will have a quick answer for my probably dumb question. Performing polynomial least squares regression of a set of [x,y] data. Using polyfit, polyval, and corrcoef, I solved for the standard regression.

My problem is when I need to solve for a constrained polynomial such as: f(x)=a3*x.^3+a1*x+a0

So what am I missing that leaves me with no understanding how to solve this problem.

[Michael Nemesure]

One way to achieve a polynomial fit with some coefficients constrained is to use the psedo-inverse pinv on an appropriately modified Vandermonde matrix. An example: for a 4th order fit to n data points (x,y) with linear and quadratic coefficents fixed at p1 and p2, compute

pinv([ones(n,1) zeros(n,1) zeros(n,1) X.^3 X.^4])* (Y - p1*X - p2*X.^2)

The resulting column vector will have 0's in the constrained locations to be replaced by p1 and p2.

Answer 1

[Geoff]

So you mean that you want to constrain a cubic polynomial regression such that the x^2 term is zero?

For this I would use fsolve if you have the optimization toolbox.

If you don't, there is always the Nelder-Mead algorithm implemented as the function fminsearch.

You'd go something like this:

constraint = [1,0,1,1];  % Ignore possible x^2 term
polyFunc = @(p) polyval(p.*constraint,x);
objectiveFunc = @(p) (y - polyFunc(p)).^2;
p0 = [1, 0, 2, 3];  % It pays to have a realistic initial guess
p = fminsearch( objectiveFunc, p0 );

Instead of using polyval you could always just write the polynomial formula verbatim. However, this way you get a result that is consistent with other polynomial functions.

Warning: Nelder-Mead can be a hairy beast. Read the documentation, play around, and find ways to tame it.

[Geoff]

Sorry, my mistake!

The objective function should be the SUM of squares.

objectiveFunc = @(p) sum((y - polyFunc(p)).^2);

Also, just because you constrained x^2 term to zero and supplied zero starting value for it, doesn't mean the algorithm won't output a non-zero value. So you might want to multiply the result by the constraint again:

p = p .* constraint;

[Jared]

This way seemed to work like a dream and it uses methods that I can somewhat understand.

Answer 2

[Richard Willey]

Curve Fitting Toolbox allows you to specify constraints for individual regression coefficients.

With this said and done, could you please elaborate why you need to constrain a regression coefficient to be precisely equal to zero rather than simply leaving that term out of your model?

Answer 3

[Mark Selby]

The simple first principles way is as follows...

x=(1:10)'; % invent some data for prediction
y = x.^3 + 2; % and generate some "experimental" data

% create a matrix that contains all you predictors
% note just include the terms you want
vm=[x.^3 ones(size(x))]; % the vandermode matrix 

% then solve the least squares problem.
beta = vm\y %this is essentially what polyfit does.

Alternatively download "polyfitn" from the FEX.... it does what it says on the tin AND your problem with a nice interface and some help.

[Jared]

what do you mean by matrix containing all my predictors?

Lets say this is the start of my code

x=[-5:0.1:5]'; % prediction data

y=(2+0.5*randn(length(x),1)).*x.^3+30*randn(length(x),1); %experimental data

and I want to fit it to a curve of the form

y= a3*x.^3 + a1*x - 200

what do you mean by creating a matrix of my predictors?

[Sean de Wolski]

http://en.wikipedia.org/wiki/Vandermonde_matrix

You create a vandermonde matrix for your above expression (y = a3.*x^3+a1*x-200) such that v, contains the xpressions to solve for) and v*x=y;

V = [x(:).^3, x(:), ones(size(x(:)))]

[Jared]

Nevermind. I'm dumb and apparently knew what you meant. I just called it a Z matrix all the time instead of vandermode.

[Sean de Wolski]

Totally okay. And this question has yielded lots of good edumacational information, so +1.

Answer 4

[Jared]

I am an engineering student learning numerical methods and matlab for the first time. I understand how to do standard polynomial regression however I do not know how to just leave the term out of the model and still solve for the coefficients.

The long way seems to be set the derivatives of the sum of the squares of the residuals to zero with respect to each of the unknown coefficients and I would end up with a column of zeros in my coefficient matrix but I figured there had to be an easier way.

Answer 5

[Richard Willey]

The 12a release of Statistics Toolbox has some very nice new features for regression analysis. There is a new function named LinearModel for linear regression. The function implements a scripting syntax called "Wilkinson's Notation" which is designed for implementing custom linear models. (Wilkinson's Notation" is pretty standard amongst statisticians)

% Create a data set
X = linspace(1, 10, 100);
X = X';
Y = 3*X.^3 + 5*X + 7;

% Fit a linear regression to the model 
myFit = LinearModel.fit(X,Y, 'y~x1:x1:x1 + x1')
plot(myFit)
methods(myFit)

The text string 'y~x1:x1:x1 + x1' translates as y is modeled as function of X^3, X, and a constant. The model excludes the term X^2 which is equivalent to setting the coefficient equal to zero.

Answer 6

[Roland]

For others looking into polynomial regression with constraints:

A general framework for constraint polynomial approximation (value constraints, derivative constraints, coefficient constraints) can be found in a paper of me and my colleagues:

• O’Leary, P., Ritt, R. and Harker, M. (2019) ‘Constrained Polynomial Approximation for Inverse Problems in Engineering’, in Wahab, M. A. (ed.) Proceedings of the 1st International Conference on Numerical Modelling in Engineering. Springer Singapore, pp. 225–244. doi: 10.1007/978-981-13-2273-0_19.

Answer 7

[Kin Sung Chan]

One way is to use the fittype and fix the lower and upper bound. (x,y) are some vectors that you want to fit.

The good thing about this method is you can fit with whatever equation you like.

s = fitoptions('Method','NonlinearLeastSquares',...

'Lower',[10 -10 ],...

'Upper',[10 10 ],...

'MaxIter',10000,...

'TolX',1E-8,...

'Startpoint',[10,0]);

f = fittype('a1*x+b','options',s);

ffit = fit(x, y, f)