问题
I use simple polyfit of order 2 to fit a line in sample data:
np.polyfit(x, y, 2)
which returns the coefficients.
Now I want to find uncertainty of the fitted line, and tried to use cov argument, which returns 3x3 covariance matrix:
np.polyfit(x, y, 2, cov=True)
But I'm not sure how to calculate the uncertainty, which according my Google search should be calculated by squaring the diagonal of covariance matrix.
回答1:
This problem is addressed by "Estimating Errors in Least-Squares Fitting" by P.H. Richter, 1995, TDA Progress Report 42-122.
From the report, this paragraph may already be sufficient to you
The first instance considered above, namely, determining the error of one or more fitting parameters, has a straightforward answer given in terms of the diagonal elements of the covariance matrix of the fit, and is well known.
The diagonal elements you are interested in are for example:
x = linspace(0,1,1000)
# comment and uncomment the last term to see how the fit appears in the figure,
# and how the covariances of the single polynomial coefficients vary in turn.
y = cos(x)*x**2+x+sin(x-1.) #+(x*1.3)**6
p,cov = polyfit(x,y,2,cov=True)
plot(x,y,'b')
plot(x,polyval(p,x),'r')
print sqrt(diag(cov))
More in general, the reference addresses how this error in the polynomial coefficients is also an error of the dependent variable y as a function of the independent variable x. From the report:
It is the purpose of this article to discuss the above errors and, in particular, to present results that will permit one to determine the standard error of the fit as a function of the independent variable, as well as to establish confidence limits for these errors.
回答2:
For your convenience I made a fully working example for Python 3 based on gg349's answer.
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0,1,1000)
# comment and uncomment the last term to see how the fit appears in the figure,
# and how the covariances of the single polynomial coefficients vary in turn.
y = np.cos(x) * x**2 + x + np.sin(x - 1.) \
# + (x * 1.3)**6
p, cov = np.polyfit(x, y, 2, cov=True)
plt.plot(x, y)
plt.plot(x, np.polyval(p,x))
plt.show()
print(np.sqrt(np.diag(cov)))
来源：https://stackoverflow.com/questions/27757732/find-uncertainty-from-polyfit