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
where n is deg.
Since numpy version 1.7.0, legfit also supports NA. If any of the elements of x, y, or w are NA, then the corresponding rows of the linear least squares problem (see Notes) are set to 0. If y is 2-D, then an NA in any row of y invalidates that whole row.
x : array_like, shape (M,)
y : array_like, shape (M,) or (M, K)
deg : int
rcond : float, optional
full : bool, optional
w : array_like, shape (M,), optional
coef : ndarray, shape (M,) or (M, K)
[residuals, rank, singular_values, rcond] : present when full = True
chebfit, polyfit, lagfit, hermfit, hermefit
The solution is the coefficients of the Legendre series p that minimizes the sum of the weighted squared errors
where are the weights. This problem is solved by setting up as the (typically) overdetermined 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, 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.
|[R62]||Wikipedia, “Curve fitting”, http://en.wikipedia.org/wiki/Curve_fitting|