Re: forcing fitted curve through a point

[Rafael Laboissiere]

* M Devoto <address@hidden> [2007-10-25 18:41]:

> is there a way to force a fitted curve through a given point? I'm using 
> polyfit to find the two coefficients of a curve. I want to force it through 
> (1,1). Any ideas?

Bad news and good news.  The bad news, first: I am afraid you will not be
able to do it directly with polyfit.  The good news: polyfit is just a
wrapper around the the left matrix division operator ("\") , so it is easy
to do it yourself.  Here is the cookbook:

Let us say that you want to fit y against x using the following polynomial
model (2nd order for the sake of simplicity, but the technique can be
generalized to any order):

    y = a * x * x + b * x + c

Note that you have three parameters to estimate (a, b, and c).  Now, if you
force the curve to pass through (1, 1), then we are effectively introducing
a constraint that will reduce the number of parameters by one. Indeed, this
impose:

    c = 1 - a  - b

which yields:

    y = a * (x * x - 1) + b * (x - 1) + 1

This is easily solved in Octave.  Let us say that x and y are column
vectors.  Then, you should just do:

    coeff = [(x .*  x - 1), (x - 1)] \ (y - 1)
    a = coeff (1)
    b = coeff (2)
    c = 1 - a - b    

and it is done.

-- 
Rafael