How do I use MatLab to find a line of best fit with a predetermined y intercept using regression.

I have data on changes in cardiac output with temperature changes from several studies. This data is expressed as CO at Tn / CO at T0, so at baseline temperature the result = 1. I want to find a line of best fit for this data, but the y intercept must be 1 for it to make sense.

How do I use Matlab find a line of best fit with least squares regression for this data, with the y intercept constrained to 1? It is easily done with Microsoft Excel, but I am trying to become more familiar with stats on Matlab.

Answer 1

[Star Strider]

Perhaps —

x = 0:250;
y = randn(size(x));

B0 = x(:) \ (y(:)-1)

B0 = -0.0054

fitline = x(:) * B0 + 1;

figure
plot(x, y, '.')
hold on
plot(x, fitline, '-r')
hold off
grid

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[Chris Joyce]

I'm not sure if that is the solution. The MatLab help for matrix left division (\) states "The symbolic operator \ does not compute least-squares solutions", which is what I am really after.

[John D'Errico]

@Chris Joyce

You misunderstand. Let me show the help for sym/backslash.

help sym/mldivide

 MLDIVIDE Symbolic matrix left division.
    MLDIVIDE(A,B) overloads symbolic A \ B.
    X = A\B solves the symbolic linear equations A*X = B.
    Warning messages are produced if X does not exist or is not unique.
    Rectangular matrices A are allowed, but the equations must be
    consistent; a least squares solution is not computed.
 
    See also SYM/MRDIVIDE, SYM/LDIVIDE, SYM/RDIVIDE, SYM/QUOREM.

And, Yes, it states that the symbolic version of backslash does not produce a least squares solution, but that is only because this is the symbolic solver. The symbolic toolbox does not generate least squares solutions in any case, since they are essentially never an exact solution to a system of equations.

HOWEVER...

What @Star Strider wrote is indeed exactly the solution you need. It solves for the linear least squares solution to the problem

y = 1 + B0*x

where the y-intercept is fixed at 1.

Just because the symbolic solver does not produce that result is not relevant, since this is a problem of numerical linear algebra. And that is what MLDIVIDE does when applied here. So you can happily trust what Star has given you. Of course, you might not trust me either. (We could go down a long way into this rabbit hole.) But honestly, Star was correct. Trust him. Would I buy a used car from him? Well, that might be another question completely. :)

[Star Strider]

The arithmetic operator does compute a least-squares solution. See: mldivide, \ and specifically the third bullet in Description for details.

[Chris Joyce]

Ok. I will accept the opinions of those who know a lot more about this than I do. I will try using this and get back to you if I have difficulty implementing it. Thanks for your help.

[Star Strider]

Our pleasure!

[Chris Joyce]

It works perfectly, giving me exactly the same answers as EXCEL.

Thanks again

[Star Strider]

My pleasure!

If my Answer helped you solve your problem, please Accept it!

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