numpy.polynomial.polynomial.polyfit(x, y, deg, rcond=None, full=False, w=None)
[source]
Least-squares fit of a polynomial to data.
Return the coefficients of a polynomial of degree deg
that is the least squares fit to the data values y
given at points x
. If y
is 1-D the returned coefficients will also be 1-D. If y
is 2-D multiple fits are done, one for each column of y
, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form
where n
is deg
.
Parameters: |
x : array_like, shape ( x-coordinates of the y : array_like, shape ( y-coordinates of the sample points. Several sets of sample points sharing the same x-coordinates can be (independently) fit with one call to deg : int or 1-D array_like Degree(s) of the fitting polynomials. If rcond : float, optional Relative condition number of the fit. Singular values smaller than full : bool, optional Switch determining the nature of the return value. When w : array_like, shape ( Weights. If not None, the contribution of each point New in version 1.5.0. |
---|---|
Returns: |
coef : ndarray, shape ( Polynomial coefficients ordered from low to high. If [residuals, rank, singular_values, rcond] : list These values are only returned if resid – sum of squared residuals of the least squares fit rank – the numerical rank of the scaled Vandermonde matrix sv – singular values of the scaled Vandermonde matrix rcond – value of For more details, see |
Raises: |
RankWarning Raised if the matrix in the least-squares fit is rank deficient. The warning is only raised if >>> import warnings >>> warnings.simplefilter('ignore', RankWarning) |
See also
chebfit
, legfit
, lagfit
, hermfit
, hermefit
polyval
polyvander
linalg.lstsq
scipy.interpolate.UnivariateSpline
The solution is the coefficients of the polynomial p
that minimizes the sum of the weighted squared errors
where the are the weights. This problem is solved by setting up the (typically) over-determined matrix equation:
where V
is the weighted pseudo Vandermonde matrix of x
, c
are the coefficients to be solved for, w
are the weights, and y
are the observed values. This equation is then solved using the singular value decomposition of V
.
If some of the singular values of V
are so small that they are neglected (and full
== False
), a RankWarning
will be raised. This means that the coefficient values may be poorly determined. Fitting to a lower order polynomial will usually get rid of the warning (but may not be what you want, of course; if you have independent reason(s) for choosing the degree which isn’t working, you may have to: a) reconsider those reasons, and/or b) reconsider the quality of your data). The rcond
parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error.
Polynomial fits using double precision tend to “fail” at about (polynomial) degree 20. Fits using Chebyshev or Legendre series are generally better conditioned, but much can still depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate, splines may be a good alternative.
>>> from numpy.polynomial import polynomial as P >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1] >>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise" >>> c, stats = P.polyfit(x,y,3,full=True) >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1 array([ 0.01909725, -1.30598256, -0.00577963, 1.02644286]) >>> stats # note the large SSR, explaining the rather poor results [array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, 0.28853036]), 1.1324274851176597e-014]
Same thing without the added noise
>>> y = x**3 - x >>> c, stats = P.polyfit(x,y,3,full=True) >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1 array([ -1.73362882e-17, -1.00000000e+00, -2.67471909e-16, 1.00000000e+00]) >>> stats # note the minuscule SSR [array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, 0.50443316, 0.28853036]), 1.1324274851176597e-014]
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https://docs.scipy.org/doc/numpy-1.12.0/reference/generated/numpy.polynomial.polynomial.polyfit.html