numpy.polyfit — NumPy v2.0 Manual (2024)

Least squares polynomial fit.

Fit a polynomial p(x) = p[0] * x**deg + ... + p[deg] of degree degto points (x, y). Returns a vector of coefficients p that minimisesthe squared error in the order deg, deg-1, … 0.

The Polynomial.fit classmethod is recommended for new code as it is more stable numerically. Seethe documentation of the method for more information.

Parameters:

xarray_like, shape (M,)

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

yarray_like, shape (M,) or (M, K)

y-coordinates of the sample points. Several data sets of samplepoints sharing the same x-coordinates can be fitted at once bypassing in a 2D-array that contains one dataset per column.

degint

Degree of the fitting polynomial

rcondfloat, optional

Relative condition number of the fit. Singular values smaller thanthis relative to the largest singular value will be ignored. Thedefault value is len(x)*eps, where eps is the relative precision ofthe float type, about 2e-16 in most cases.

fullbool, optional

Switch determining nature of return value. When it is False (thedefault) just the coefficients are returned, when True diagnosticinformation from the singular value decomposition is also returned.

warray_like, shape (M,), optional

Weights. If not None, the weight w[i] applies to the unsquaredresidual y[i] - y_hat[i] at x[i]. Ideally the weights arechosen so that the errors of the products w[i]*y[i] all have thesame variance. When using inverse-variance weighting, usew[i] = 1/sigma(y[i]). The default value is None.

covbool or str, optional

If given and not False, return not just the estimate but also itscovariance matrix. By default, the covariance are scaled bychi2/dof, where dof = M - (deg + 1), i.e., the weights are presumedto be unreliable except in a relative sense and everything is scaledsuch that the reduced chi2 is unity. This scaling is omitted ifcov='unscaled', as is relevant for the case that the weights arew = 1/sigma, with sigma known to be a reliable estimate of theuncertainty.

Returns:

pndarray, shape (deg + 1,) or (deg + 1, K)

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

residuals, rank, singular_values, rcond

These values are only returned if full == True

See Also

Least squares fitting with Numpy and Scipyleast_squares — SciPy v1.14.0 Manual

• residuals – sum of squared residuals of the least squares fit

• rank – the effective rank of the scaled Vandermonde

  coefficient matrix

• singular_values – singular values of the scaled Vandermonde

  coefficient matrix

• rcond – value of rcond.

For more details, see numpy.linalg.lstsq.

Vndarray, shape (deg + 1, deg + 1) or (deg + 1, deg + 1, K)

Present only if full == False and cov == True. The covariancematrix of the polynomial coefficient estimates. The diagonal ofthis matrix are the variance estimates for each coefficient. If yis a 2-D array, then the covariance matrix for the k-th data setare in V[:,:,k]

Warns:

RankWarning

The rank of the coefficient matrix in the least-squares fit isdeficient. The warning is only raised if full == False.

The warnings can be turned off by

>>> import warnings>>> warnings.simplefilter('ignore', np.exceptions.RankWarning)

Notes

The solution minimizes the squared error

\[E = \sum_{j=0}^k |p(x_j) - y_j|^2\]

in the equations:

x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0]x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1]...x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k]

The coefficient matrix of the coefficients p is a Vandermonde matrix.

polyfit issues a RankWarning when the least-squares fit isbadly conditioned. This implies that the best fit is not well-defined dueto numerical error. The results may be improved by lowering the polynomialdegree or by replacing x by x - x.mean(). The rcond parametercan also be set to a value smaller than its default, but the resultingfit may be spurious: including contributions from the small singularvalues can add numerical noise to the result.

Note that fitting polynomial coefficients is inherently badly conditionedwhen the degree of the polynomial is large or the interval of sample pointsis badly centered. The quality of the fit should always be checked in thesecases. When polynomial fits are not satisfactory, splines may be a goodalternative.

References

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)>>> zarray([ 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

High-order polynomials may oscillate wildly:

>>> with warnings.catch_warnings():...  warnings.simplefilter('ignore', np.exceptions.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()