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

.. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n,

where `n` is `deg`.

Parameters ---------- x : array_like, shape (`M`,) x-coordinates of the `M` sample (data) points ``(x`i`

, y`i`

)``. y : array_like, shape (`M`,) or (`M`, `K`) y-coordinates of the sample points. Several sets of sample points sharing the same x-coordinates can be (independently) fit with one call to `polyfit` by passing in for `y` a 2-D array that contains one data set per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than `rcond`, relative to the largest singular value, will be ignored. The default value is ``len(x)*eps``, where `eps` is the relative precision of the platform's float type, about 2e-16 in most cases. full : bool, optional Switch determining the nature of the return value. When ``False`` (the default) just the coefficients are returned; when ``True``, diagnostic information from the singular value decomposition (used to solve the fit's matrix equation) is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the contribution of each point ``(x`i`

,y`i`

)`` to the fit is weighted by `w`i`

`. Ideally the weights are chosen so that the errors of the products ``w`i`

*y`i`

`` all have the same variance. The default value is None.

.. versionadded:: 1.5.0

Returns ------- coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`) Polynomial coefficients ordered from low to high. If `y` was 2-D, the coefficients in column `k` of `coef` represent the polynomial fit to the data in `y`'s `k`-th column.

`residuals, rank, singular_values, rcond`

: list These values are only returned if `full` = True

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 `rcond`.

For more details, see `linalg.lstsq`.

Raises ------ RankWarning Raised if the matrix in the least-squares fit is rank 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 -------- chebfit, legfit, lagfit, hermfit, hermefit polyval : Evaluates a polynomial. polyvander : Vandermonde matrix for powers. linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits.

Notes ----- The solution is the coefficients of the polynomial `p` that minimizes the sum of the weighted squared errors

.. math :: E = \sum_j w_j^2 * |y_j - p(x_j)|^2,

where the :math:`w_j` are the weights. This problem is solved by setting up the (typically) over-determined matrix equation:

.. math :: V(x) * c = w * y,

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.

Examples -------- >>> np.random.seed(123) >>> 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) >>> np.random.seed(123) >>> 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`

) # may vary >>> stats # note the large SSR, explaining the rather poor results ```
array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, # may vary
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(`-6.36925336e-18, -1.00000000e+00, -4.08053781e-16, 1.00000000e+00`

) >>> stats # note the minuscule SSR ```
array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, # may vary
0.50443316, 0.28853036]), 1.1324274851176597e-014
```