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Module « scipy.interpolate »

Fonction make_splrep - module scipy.interpolate

Signature de la fonction make_splrep

def make_splrep(x, y, *, w=None, xb=None, xe=None, k=3, s=0, t=None, nest=None) 

Description

help(scipy.interpolate.make_splrep)

Find the B-spline representation of a 1D function.

Given the set of data points ``(x[i], y[i])``, determine a smooth spline
approximation of degree ``k`` on the interval ``xb <= x <= xe``.

Parameters
----------
x, y : array_like, shape (m,)
    The data points defining a curve ``y = f(x)``.
w : array_like, shape (m,), optional
    Strictly positive 1D array of weights, of the same length as `x` and `y`.
    The weights are used in computing the weighted least-squares spline
    fit. If the errors in the y values have standard-deviation given by the
    vector ``d``, then `w` should be ``1/d``.
    Default is ``np.ones(m)``.
xb, xe : float, optional
    The interval to fit.  If None, these default to ``x[0]`` and ``x[-1]``,
    respectively.
k : int, optional
    The degree of the spline fit. It is recommended to use cubic splines,
    ``k=3``, which is the default. Even values of `k` should be avoided,
    especially with small `s` values.
s : float, optional
    The smoothing condition. The amount of smoothness is determined by
    satisfying the conditions::

        sum((w * (g(x)  - y))**2 ) <= s

    where ``g(x)`` is the smoothed fit to ``(x, y)``. The user can use `s`
    to control the tradeoff between closeness to data and smoothness of fit.
    Larger `s` means more smoothing while smaller values of `s` indicate less
    smoothing.
    Recommended values of `s` depend on the weights, `w`. If the weights
    represent the inverse of the standard deviation of `y`, then a good `s`
    value should be found in the range ``(m-sqrt(2*m), m+sqrt(2*m))`` where
    ``m`` is the number of datapoints in `x`, `y`, and `w`.
    Default is ``s = 0.0``, i.e. interpolation.
t : array_like, optional
    The spline knots. If None (default), the knots will be constructed
    automatically.
    There must be at least ``2*k + 2`` and at most ``m + k + 1`` knots.
nest : int, optional
    The target length of the knot vector. Should be between ``2*(k + 1)``
    (the minimum number of knots for a degree-``k`` spline), and
    ``m + k + 1`` (the number of knots of the interpolating spline).
    The actual number of knots returned by this routine may be slightly
    larger than `nest`.
    Default is None (no limit, add up to ``m + k + 1`` knots).

Returns
-------
spl : a `BSpline` instance
    For `s=0`,  ``spl(x) == y``.
    For non-zero values of `s` the `spl` represents the smoothed approximation
    to `(x, y)`, generally with fewer knots.

See Also
--------
generate_knots : is used under the hood for generating the knots
make_splprep : the analog of this routine for parametric curves
make_interp_spline : construct an interpolating spline (``s = 0``)
make_lsq_spline : construct the least-squares spline given the knot vector
splrep : a FITPACK analog of this routine

References
----------
.. [1] P. Dierckx, "Algorithms for smoothing data with periodic and
    parametric splines, Computer Graphics and Image Processing",
    20 (1982) 171-184.
.. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs on
    Numerical Analysis, Oxford University Press, 1993.

Notes
-----
This routine constructs the smoothing spline function, :math:`g(x)`, to
minimize the sum of jumps, :math:`D_j`, of the ``k``-th derivative at the
internal knots (:math:`x_b < t_i < x_e`), where

.. math::

    D_i = g^{(k)}(t_i + 0) - g^{(k)}(t_i - 0)

Specifically, the routine constructs the spline function :math:`g(x)` which
minimizes

.. math::

        \sum_i | D_i |^2 \to \mathrm{min}

provided that

.. math::

       \sum_{j=1}^m (w_j \times (g(x_j) - y_j))^2 \leqslant s ,

where :math:`s > 0` is the input parameter.

In other words, we balance maximizing the smoothness (measured as the jumps
of the derivative, the first criterion), and the deviation of :math:`g(x_j)`
from the data :math:`y_j` (the second criterion).

Note that the summation in the second criterion is over all data points,
and in the first criterion it is over the internal spline knots (i.e.
those with ``xb < t[i] < xe``). The spline knots are in general a subset
of data, see `generate_knots` for details.

Also note the difference of this routine to `make_lsq_spline`: the latter
routine does not consider smoothness and simply solves a least-squares
problem

.. math::

    \sum w_j \times (g(x_j) - y_j)^2 \to \mathrm{min}

for a spline function :math:`g(x)` with a _fixed_ knot vector ``t``.

.. versionadded:: 1.15.0

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