numpy.polyfit¶

numpy.
polyfit
(x, y, deg, rcond=None, full=False, w=None, cov=False)[source]¶ Least squares polynomial fit.
Fit a polynomial
p(x) = p[0] * x**deg + ... + p[deg]
of degree deg to points (x, y). Returns a vector of coefficients p that minimises the squared error in the order deg, deg1, … 0.The
Polynomial.fit
class method is recommended for new code as it is more stable numerically. See the documentation of the method for more information.Parameters:  x : array_like, shape (M,)
xcoordinates of the M sample points
(x[i], y[i])
. y : array_like, shape (M,) or (M, K)
ycoordinates of the sample points. Several data sets of sample points sharing the same xcoordinates can be fitted at once by passing in a 2Darray that contains one dataset per column.
 deg : int
Degree of the fitting polynomial
 rcond : float, optional
Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e16 in most cases.
 full : bool, optional
Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
 w : array_like, shape (M,), optional
Weights to apply to the ycoordinates of the sample points. For gaussian uncertainties, use 1/sigma (not 1/sigma**2).
 cov : bool or str, optional
If given and not False, return not just the estimate but also its covariance matrix. By default, the covariance are scaled by chi2/sqrt(Ndof), i.e., the weights are presumed to be unreliable except in a relative sense and everything is scaled such that the reduced chi2 is unity. This scaling is omitted if
cov='unscaled'
, as is relevant for the case that the weights are 1/sigma**2, with sigma known to be a reliable estimate of the uncertainty.
Returns:  p : ndarray, shape (deg + 1,) or (deg + 1, K)
Polynomial coefficients, highest power first. If y was 2D, the coefficients for kth data set are in
p[:,k]
. residuals, rank, singular_values, rcond
Present only if
full
= True. Residuals of the leastsquares fit, the effective rank of the scaled Vandermonde coefficient matrix, its singular values, and the specified value of rcond. For more details, seelinalg.lstsq
. V : ndarray, shape (M,M) or (M,M,K)
Present only if
full
= False and cov`=True. The covariance matrix of the polynomial coefficient estimates. The diagonal of this matrix are the variance estimates for each coefficient. If y is a 2D array, then the covariance matrix for the `kth data set are inV[:,:,k]
Warns:  RankWarning
The rank of the coefficient matrix in the leastsquares fit is deficient. The warning is only raised if
full
= False.The warnings can be turned off by
>>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning)
See also
polyval
 Compute polynomial values.
linalg.lstsq
 Computes a leastsquares fit.
scipy.interpolate.UnivariateSpline
 Computes spline fits.
Notes
The solution minimizes the squared error
in the equations:
x[0]**n * p[0] + ... + x[0] * p[n1] + p[n] = y[0] x[1]**n * p[0] + ... + x[1] * p[n1] + p[n] = y[1] ... x[k]**n * p[0] + ... + x[k] * p[n1] + p[n] = y[k]
The coefficient matrix of the coefficients p is a Vandermonde matrix.
polyfit
issues aRankWarning
when the leastsquares fit is badly conditioned. This implies that the best fit is not welldefined due to numerical error. The results may be improved by lowering the polynomial degree or by replacing x by x  x.mean(). The rcond parameter can also be set to a value smaller than its default, but the resulting fit may be spurious: including contributions from the small singular values can add numerical noise to the result.Note that fitting polynomial coefficients is inherently badly conditioned when the degree of the polynomial is large or the interval of sample points is badly centered. The quality of the fit should always be checked in these cases. When polynomial fits are not satisfactory, splines may be a good alternative.
References
[1] Wikipedia, “Curve fitting”, https://en.wikipedia.org/wiki/Curve_fitting [2] Wikipedia, “Polynomial interpolation”, https://en.wikipedia.org/wiki/Polynomial_interpolation Examples
>>> import warnings >>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0]) >>> y = np.array([0.0, 0.8, 0.9, 0.1, 0.8, 1.0]) >>> z = np.polyfit(x, y, 3) >>> z array([ 0.08703704, 0.81349206, 1.69312169, 0.03968254]) # may vary
It is convenient to use
poly1d
objects for dealing with polynomials:>>> p = np.poly1d(z) >>> p(0.5) 0.6143849206349179 # may vary >>> p(3.5) 0.34732142857143039 # may vary >>> p(10) 22.579365079365115 # may vary
Highorder polynomials may oscillate wildly:
>>> with warnings.catch_warnings(): ... warnings.simplefilter('ignore', np.RankWarning) ... p30 = np.poly1d(np.polyfit(x, y, 30)) ... >>> p30(4) 0.80000000000000204 # may vary >>> p30(5) 0.99999999999999445 # may vary >>> p30(4.5) 0.10547061179440398 # may vary
Illustration:
>>> import matplotlib.pyplot as plt >>> xp = np.linspace(2, 6, 100) >>> _ = plt.plot(x, y, '.', xp, p(xp), '', xp, p30(xp), '') >>> plt.ylim(2,2) (2, 2) >>> plt.show()