Least squares polynomial fit.
Fit a polynomial p(x) = p * 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,)
y : array_like, shape (M,) or (M, K)
deg : int
rcond : float, optional
full : bool, optional
p : ndarray, shape (M,) or (M, K)
residuals, rank, singular_values, rcond : present only if full = True
The solution minimizes the squared error
in the equations:
x**n * p[n] + ... + x * p + p = y x**n * p[n] + ... + x * p + p = y ... x[k]**n * p[n] + ... + x[k] * p + p = 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.
|||Wikipedia, “Curve fitting”, http://en.wikipedia.org/wiki/Curve_fitting|
|||Wikipedia, “Polynomial interpolation”, http://en.wikipedia.org/wiki/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) 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
>>> 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) >>> plt.show()