问题
I am having trouble understanding the error estimates between the variables when performing linear regression. Referring to How to find error on slope and intercept using numpy.polyfit say I am fitting a straight line with numpy.polyfit (code below).
As mentioned in the question in the link, the square root of the diagonals of the covariance matrix are the estimated standard-deviation for each of the fitted coefficients, and so np.sqrt(V[0][0])) is the standard deviation of the slope. My question(s) is(are): how the standard deviation of y should be represented? Should it be the addition of the uncertainties in quadrature, i.e., y +/- np.sqrt(np.sqrt(V[0][0])**2+np.sqrt(V[1][1])**2) ? Or perhaps I could only represent it by the standard deviation of the residuals (which would be np.sqrt(S)/(len(y)-1))? Finally, is it possible to obtain the residuals from the covariance matrix?
PS: thanks for the heads-up on adding an "answer" to a question.
import numpy as np
# Data for testing
x = np.array([0.24580423, 0.59642861, 0.35879163, 0.37891011, 0.02445137,
0.23830957, 0.38793433, 0.68054104, 0.83934083, 0.76073689])
y = np.array([0.61502838, 1.01772738, 1.35351035, 1.32799754, 0.23326104,
0.89275698, 0.689498 , 1.48300835, 2.324673 , 1.52208752])
p, V = np.polyfit(x, y, 1, cov=True)
p2, S, *rest = np.polyfit(x, y, 1, full=True)
来源：https://stackoverflow.com/questions/56644806/error-of-slope-using-numpy-polyfit-and-dependent-variable