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Diffstat (limited to 'numpy/polynomial/chebyshev.py')
| -rw-r--r-- | numpy/polynomial/chebyshev.py | 39 |
1 files changed, 24 insertions, 15 deletions
diff --git a/numpy/polynomial/chebyshev.py b/numpy/polynomial/chebyshev.py index eb0087395..a81085921 100644 --- a/numpy/polynomial/chebyshev.py +++ b/numpy/polynomial/chebyshev.py @@ -1524,9 +1524,16 @@ def chebfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Chebyshev series to data. - Fit a Chebyshev series ``p(x) = p[0] * T_{0}(x) + ... + p[deg] * - T_{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 * T_1(x) + ... + c_n * T_n(x), + + where `n` is `deg`. Parameters ---------- @@ -1537,7 +1544,7 @@ def chebfit(x, y, deg, rcond=None, full=False, w=None): 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 - Degree of the fitting polynomial + Degree of the fitting series rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The @@ -1552,6 +1559,7 @@ def chebfit(x, y, deg, rcond=None, full=False, w=None): ``(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 @@ -1578,30 +1586,31 @@ def chebfit(x, y, deg, rcond=None, full=False, w=None): See Also -------- + polyfit, legfit, lagfit, hermfit, hermefit chebval : Evaluates a Chebyshev series. chebvander : Vandermonde matrix of Chebyshev series. - polyfit : least squares fit using polynomials. + chebweight : Chebyshev weight function. 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 Chebyshev series - ``T(x)`` that minimizes the squared error + The solution is the coefficients of the Chebyshev series `p` that + minimizes the sum of the weighted squared errors - ``E = \\sum_j |y_j - T(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 |