What are the weight values to use in numpy polyfit and what is the error of the fit

Question

I'm trying to do a linear fit to some data in numpy.

Ex (where w is the number of samples I have for that value, i.e. for the point (x=0, y=0) I only have 1 measurement and the value of that measurement is 2.2, but for the point (1,1) I have 2 measurements with a value of 3.5.

x = np.array([0, 1, 2, 3])
y = np.array([2.2, 3.5, 4.6, 5.2])
w = np.array([1, 2, 2, 1])

z = np.polyfit(x, y, 1, w = w)

So, now the question is: is it correct to use w=w in polyfit for these cases or should I use w = sqrt(w) of what should I use?

Also, how can I get the fit error from polyfit?

Solution

If you have normally distributed measurements, then your uncertainty in each value would be proportional to 1/sqrt(n) where n is the number of measurements. You want to weigh your fit by the inverse of your uncertainty, so your second guess is best: w=np.sqrt(n)

To get the covariance on your parameters, also give cov=True.

x = np.array([0, 1, 2, 3])
y = np.array([2.2, 3.5, 4.6, 5.2])
n = np.array([1, 2, 2, 1])

p, c = np.polyfit(x, y, 1, w=np.sqrt(n), cov=True)

The diagonals of your cov matrix are the individual variances on each parameter, and of course the off-diagonals are the covariances. So most likely what you want for "fit error" is the square root of these diagonals:

e = np.sqrt(np.diag(c))

Comments

[jbssm]

Thank you very much. This is what I was looking for.

[askewchan]

Happy to help, @jbssm. By the way, when using np.polyfit,np.polyval, np.poly1d, etc., don't combine them with any of the np.polynomial module functions, as they follow different conventions, specifically the return ordering. Normally it's recommended to use the np.polynomial package exclusively, but for some reason it doesn't provide the covariance