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.
Parameters: 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.
cov : bool, optional
Return the estimate and the covariance matrix of the estimate If full is True, then cov is not returned.
Returns: p : ndarray, shape (M,) or (M, 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]
Warns: RankWarning
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
- polyval
- Computes polynomial values.
- linalg.lstsq
- Computes a least-squares fit.
- scipy.interpolate.UnivariateSpline
- Computes spline fits.
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 is badly conditioned. This implies that the best fit is not well-defined 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
[R58] Wikipedia, “Curve fitting”, http://en.wikipedia.org/wiki/Curve_fitting [R59] Wikipedia, “Polynomial interpolation”, http://en.wikipedia.org/wiki/Polynomial_interpolation Examples
>>> 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) 0.6143849206349179 >>> p(3.5) -0.34732142857143039 >>> p(10) 22.579365079365115
High-order polynomials may oscillate wildly:
>>> p30 = np.poly1d(np.polyfit(x, y, 30)) /... RankWarning: Polyfit may be poorly conditioned... >>> p30(4) -0.80000000000000204 >>> p30(5) -0.99999999999999445 >>> p30(4.5) -0.10547061179440398
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()
(Source code, png, pdf)