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Diffstat (limited to 'numpy/polynomial/polynomial.py')
| -rw-r--r-- | numpy/polynomial/polynomial.py | 48 |
1 files changed, 28 insertions, 20 deletions
diff --git a/numpy/polynomial/polynomial.py b/numpy/polynomial/polynomial.py index 73090244c..99a555e71 100644 --- a/numpy/polynomial/polynomial.py +++ b/numpy/polynomial/polynomial.py @@ -1122,10 +1122,16 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None): """ Least-squares fit of a polynomial to data. - Fit a polynomial ``c0 + c1*x + c2*x**2 + ... + c[deg]*x**deg`` to - points (`x`, `y`). Returns a 1-d (if `y` is 1-d) or 2-d (if `y` is 2-d) - array of coefficients representing, from lowest order term to highest, - the polynomial(s) which minimize the total square error. + 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 ---------- @@ -1134,7 +1140,7 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None): 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 + call to `polyfit` by passing in for `y` a 2-D array that contains one data set per column. deg : int Degree of the polynomial(s) to be fit. @@ -1154,12 +1160,13 @@ def polyfit(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 ------- coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`) - Polynomial coefficients ordered from low to high. If `y` was 2-d, + 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. @@ -1181,27 +1188,27 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None): See Also -------- + chebfit, legfit, lagfit, hermfit, hermefit polyval : Evaluates a polynomial. polyvander : Vandermonde matrix for powers. - chebfit : least squares fit using Chebyshev series. linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- - The solutions are the coefficients ``c[i]`` of the polynomial ``P(x)`` - that minimizes the total squared error: + The solution is the coefficients of the polynomial `p` that minimizes + the sum of the weighted squared errors - .. math :: 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 the (typically) over-determined - matrix equation: + 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 = 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 observed - values. This equation is then solved using the singular value + 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 @@ -1216,10 +1223,11 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None): contributions from roundoff error. Polynomial fits using double precision tend to "fail" at about - (polynomial) degree 20. Fits using Chebyshev 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. + (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 -------- |