Linear regression with more than two unknowns

Assuming that I have 10 values of X(measured):Xm and 10 values of Y(measured): Ym. I know the model of Y=f(X) has the form: Y=(a+b)*X+c. Can I determine a, b and c using the cftool or can I only determine a+b and c ?

Answer 1

[Star Strider]

The objective function you supply to any curve-fitting function determines the parameters that will be estimated, so in your example, a, b, and c will all be uniquely estimated. Be sure to get the 95% confidence intervals so you know if they are significant. (I suspect one will not be.) Combining parameters such as you want is generally not recommended, unless a and b (in this example) are used elsewhere in the function either independently or as a factor or argument as (a-b) or some such.

[Image Analyst]

How can he get a and b each uniquely? As far as I know he can only get the sum a+b, and there are lots of a and b that will sum to whatever coefficient it comes up with.

coeffs = polyfit(Xm, Ym, 1);
a_plus_b = coeffs(1); % Inifinite # of a & b that will sum to coeffs(1).
c = coeffs(2);

[Star Strider]

The regression function will estimate all the parameters requested of it, but it estimates the individual elements of summed parameters like (a+b) here as best it can. (I used nlinfit here because I’m more familiar with it than the linear solvers.)

f = @(b,x) (b(1) + b(2)).*x + b(3);
b = [3  5  7];
x = 1:10;
y = f(b,x)+rand(1,10)-.5;
[b,R,J,CovB,MSE,ErrorModelInfo] = nlinfit(x, y, f, zeros(3,1));
b
ci = nlparci(b,R,'covar', CovB)

produces:

b =

       4.0149e+000
       4.0149e+000
       6.9965e+000

ci =

       3.9709e+000     4.0588e+000
       3.9709e+000     4.0588e+000
       6.4511e+000     7.5419e+000

Changing the objective function to:

f2 = @(b,x) (b(1) + b(2)).*x + b(3).*b(2).^b(1);

will provide unique solutions:

b =

       5.3045e+000
       2.5513e+000
       6.0922e+000

ci =

       5.0843e+000     5.5246e+000
       2.4551e+000     2.6474e+000
       6.0469e+000     6.1375e+000