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Module « scipy.interpolate »

Fonction splrep - module scipy.interpolate

Signature de la fonction splrep

def splrep(x, y, w=None, xb=None, xe=None, k=3, task=0, s=None, t=None, full_output=0, per=0, quiet=1) 

Description

help(scipy.interpolate.splrep)

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

.. legacy:: function

    Specifically, we recommend using `make_splrep` in new code.


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
    The data points defining a curve ``y = f(x)``.
w : array_like, optional
    Strictly positive rank-1 array of weights 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 ``ones(len(x))``.
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.
    Even values of `k` should be avoided especially with small `s` values.
    ``1 <= k <= 5``.
task : {1, 0, -1}, optional
    If ``task==0``, find ``t`` and ``c`` for a given smoothing factor, `s`.

    If ``task==1`` find ``t`` and ``c`` for another value of the smoothing factor,
    `s`. There must have been a previous call with ``task=0`` or ``task=1`` for
    the same set of data (``t`` will be stored an used internally)

    If ``task=-1`` find the weighted least square spline for a given set of
    knots, ``t``. These should be interior knots as knots on the ends will be
    added automatically.
s : float, optional
    A smoothing condition. The amount of smoothness is determined by
    satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s`` where ``g(x)``
    is the smoothed interpolation of ``(x,y)``. The user can use `s` to control
    the tradeoff between closeness 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 : ``s=m-sqrt(2*m)`` if
    weights are supplied. ``s = 0.0`` (interpolating) if no weights are
    supplied.
t : array_like, optional
    The knots needed for ``task=-1``. If given then task is automatically set
    to ``-1``.
full_output : bool, optional
    If non-zero, then return optional outputs.
per : bool, optional
    If non-zero, data points are considered periodic with period ``x[m-1]`` -
    ``x[0]`` and a smooth periodic spline approximation is returned. Values of
    ``y[m-1]`` and ``w[m-1]`` are not used.
    The default is zero, corresponding to boundary condition 'not-a-knot'.
quiet : bool, optional
    Non-zero to suppress messages.

Returns
-------
tck : tuple
    A tuple ``(t,c,k)`` containing the vector of knots, the B-spline
    coefficients, and the degree of the spline.
fp : array, optional
    The weighted sum of squared residuals of the spline approximation.
ier : int, optional
    An integer flag about splrep success. Success is indicated if ``ier<=0``.
    If ``ier in [1,2,3]``, an error occurred but was not raised. Otherwise an
    error is raised.
msg : str, optional
    A message corresponding to the integer flag, `ier`.

See Also
--------
UnivariateSpline, BivariateSpline
splprep, splev, sproot, spalde, splint
bisplrep, bisplev
BSpline
make_interp_spline

Notes
-----
See `splev` for evaluation of the spline and its derivatives. Uses the
FORTRAN routine ``curfit`` from FITPACK.

The user is responsible for assuring that the values of `x` are unique.
Otherwise, `splrep` will not return sensible results.

If provided, knots `t` must satisfy the Schoenberg-Whitney conditions,
i.e., there must be a subset of data points ``x[j]`` such that
``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``.

This routine zero-pads the coefficients array ``c`` to have the same length
as the array of knots ``t`` (the trailing ``k + 1`` coefficients are ignored
by the evaluation routines, `splev` and `BSpline`.) This is in contrast with
`splprep`, which does not zero-pad the coefficients.

The default boundary condition is 'not-a-knot', i.e. the first and second
segment at a curve end are the same polynomial. More boundary conditions are
available in `CubicSpline`.

References
----------
Based on algorithms described in [1]_, [2]_, [3]_, and [4]_:

.. [1] P. Dierckx, "An algorithm for smoothing, differentiation and
   integration of experimental data using spline functions",
   J.Comp.Appl.Maths 1 (1975) 165-184.
.. [2] P. Dierckx, "A fast algorithm for smoothing data on a rectangular
   grid while using spline functions", SIAM J.Numer.Anal. 19 (1982)
   1286-1304.
.. [3] P. Dierckx, "An improved algorithm for curve fitting with spline
   functions", report tw54, Dept. Computer Science,K.U. Leuven, 1981.
.. [4] P. Dierckx, "Curve and surface fitting with splines", Monographs on
   Numerical Analysis, Oxford University Press, 1993.

Examples
--------
You can interpolate 1-D points with a B-spline curve.
Further examples are given in
:ref:`in the tutorial <tutorial-interpolate_splXXX>`.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import splev, splrep
>>> x = np.linspace(0, 10, 10)
>>> y = np.sin(x)
>>> spl = splrep(x, y)
>>> x2 = np.linspace(0, 10, 200)
>>> y2 = splev(x2, spl)
>>> plt.plot(x, y, 'o', x2, y2)
>>> plt.show()

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