SciPy, 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.


x : array_like, shape (M,)

x-coordinates of the M sample points (x[i], y[i]).

y : array_like, shape (M,) or (M, K)

y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array 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 2e-16 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 y-coordinates of the sample points. For gaussian uncertainties, use 1/sigma (not 1/sigma**2).

cov : bool, optional

Return the estimate and the covariance matrix of the estimate If full is True, then cov is not returned.


p : ndarray, shape (deg + 1,) or (deg + 1, K)

Polynomial coefficients, highest power first. If y was 2-D, the coefficients for k-th data set are in p[:,k].

residuals, rank, singular_values, rcond

Present only if full = True. Residuals of the least-squares fit, the effective rank of the scaled Vandermonde coefficient matrix, its singular values, and the specified value of rcond. For more details, see linalg.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 2-D array, then the covariance matrix for the `k-th data set are in V[:,:,k]



The rank of the coefficient matrix in the least-squares 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

Compute polynomial values.
Computes a least-squares fit.
Computes spline fits.


Any masked values in x is propagated in y, and vice-versa.


[R52]Wikipedia, “Curve fitting”,
[R53]Wikipedia, “Polynomial interpolation”,


>>> 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])

It is convenient to use poly1d objects for dealing with polynomials:

>>> p = np.poly1d(z)
>>> p(0.5)
>>> p(3.5)
>>> p(10)

High-order polynomials may oscillate wildly:

>>> p30 = np.poly1d(np.polyfit(x, y, 30))
/... RankWarning: Polyfit may be poorly conditioned...
>>> p30(4)
>>> p30(5)
>>> p30(4.5)


>>> 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)

(Source code, png, pdf)


Previous topic

Next topic