The documentation HERE for numpy.polyfit defines the optional input vector w as the "weights to apply to the y-coordinates". This definition is unclear and does not seem the standard definition for the weights in least-squares fits (see e.g. HERE).
How does one calculate the weights to use in numpy.polyfit, in the common situation where one has the 1σ error (standard deviation) of the input measurements?
The relation between w and the 1σ errors in numpy.polyfit is
w = 1/sigma
which is different from what everybody will expect.
Edit: following my comment on Github, the numpy.polyfit 1.16 documentation now explicitly states "use 1/sigma (not 1/sigma**2)" to avoid people thinking there is a typo in the formula.
In least-squares fitting one generally defines the weights vector in such a way that the fit minimizes the squared error (see e.g. Wikipedia or NIST)
chi2 = np.sum(weights*(p(x) - y)**2)
In the common situation where the 1σ errors (standard deviations) "sigma" are known one has the familiar textbook relation where the weights are the reciprocal of the variance
weights = 1/sigma**2
However the numpy.polyfit documentation defines the weight as "weights to apply to the y-coordinates". This definition is not quite correct. The weights apply to the fit residuals, not only to the y-coordinates.
More importantly, looking at the math in the Numpy (v1.9.1) code it appears that the Numpy code solves the linear problem below in the last-squares sense, where the w vector does indeed multiply the y-coordinate
(vander*w[:, np.newaxis]).dot(x) == y*w
But solving the above array expression in the least-squares sense is equivalent to minimizing the expression below with w inside the parenthesis
chi2 = np.sum((w*(vander.dot(x) - y))**2)
Or, using the notation of the Numpy documentation
chi2 = np.sum((w*(p(x) - y))**2)
in such a way that the relation between w and the 1σ errors is
w = 1/sigma
which is different from what everybody will expect.