Least squares fit of Legendre series to data.

Return the coefficients of a Legendre series 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 * L_1(x) + ... + c_n * L_n(x),

where `n` is `deg`.

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 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 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. 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 (M,) or (M, K) Legendre coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. If `deg` is specified as a list, coefficients for terms not included in the fit are set equal to zero in the returned `coef`.

`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`.

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 -------- chebfit, polyfit, lagfit, hermfit, hermefit legval : Evaluates a Legendre series. legvander : Vandermonde matrix of Legendre series. legweight : Legendre weight function (= 1). linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits.

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

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

where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined 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, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. 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.

Fits using Legendre series are usually better conditioned than fits using power series, but much can 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.

References ---------- .. `1`

Wikipedia, 'Curve fitting', https://en.wikipedia.org/wiki/Curve_fitting

Examples --------