From c462637f9b398600d25ca449aef8586d8d9d6210 Mon Sep 17 00:00:00 2001 From: Charles Harris Date: Tue, 27 Dec 2011 17:54:09 -0700 Subject: DOC: Document xxxfit functions in the polynomial package modules. --- numpy/polynomial/legendre.py | 39 ++++++++++++++++++++++++--------------- 1 file changed, 24 insertions(+), 15 deletions(-) (limited to 'numpy/polynomial/legendre.py') diff --git a/numpy/polynomial/legendre.py b/numpy/polynomial/legendre.py index 319fb505b..da2c2d846 100644 --- a/numpy/polynomial/legendre.py +++ b/numpy/polynomial/legendre.py @@ -1326,9 +1326,16 @@ def legfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Legendre series to data. - Fit a Legendre series ``p(x) = p[0] * P_{0}(x) + ... + p[deg] * - P_{deg}(x)`` of degree `deg` to points `(x, y)`. Returns a vector of - coefficients `p` that minimises the squared error. + 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 ---------- @@ -1355,6 +1362,8 @@ def legfit(x, y, deg, rcond=None, full=False, w=None): 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) @@ -1379,31 +1388,31 @@ def legfit(x, y, deg, rcond=None, full=False, w=None): See Also -------- + chebfit, polyfit, lagfit, hermfit, hermefit legval : Evaluates a Legendre series. legvander : Vandermonde matrix of Legendre series. - polyfit : least squares fit using polynomials. - chebfit : least squares fit using Chebyshev 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 are the coefficients ``c[i]`` of the Legendre series - ``P(x)`` that minimizes the squared error + The solution is the coefficients of the Legendre series `p` that + minimizes the sum of the weighted squared errors - ``E = \\sum_j |y_j - P(x_j)|^2``. + .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, - This problem is solved by setting up as the overdetermined matrix - equation + where :math:`w_j` are the weights. This problem is solved by setting up + as the (typically) overdetermined matrix equation - ``V(x)*c = y``, + .. math:: V(x) * c = w * y, - where ``V`` is the Vandermonde matrix of `x`, the elements of ``c`` are - the coefficients to be solved for, and the elements of `y` are the + 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``. + decomposition of `V`. - If some of the singular values of ``V`` are so small that they are + 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 coeficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be -- cgit v1.2.1