Classe « RBFInterpolator »

Informations générales

Héritage

builtins.object
    RBFInterpolator

Définition

class RBFInterpolator(builtins.object):

help(RBFInterpolator)

Radial basis function (RBF) interpolation in N dimensions.

Parameters
----------
y : (npoints, ndims) array_like
    2-D array of data point coordinates.
d : (npoints, ...) array_like
    N-D array of data values at `y`. The length of `d` along the first
    axis must be equal to the length of `y`. Unlike some interpolators, the
    interpolation axis cannot be changed.
neighbors : int, optional
    If specified, the value of the interpolant at each evaluation point
    will be computed using only this many nearest data points. All the data
    points are used by default.
smoothing : float or (npoints, ) array_like, optional
    Smoothing parameter. The interpolant perfectly fits the data when this
    is set to 0. For large values, the interpolant approaches a least
    squares fit of a polynomial with the specified degree. Default is 0.
kernel : str, optional
    Type of RBF. This should be one of

        - 'linear'               : ``-r``
        - 'thin_plate_spline'    : ``r**2 * log(r)``
        - 'cubic'                : ``r**3``
        - 'quintic'              : ``-r**5``
        - 'multiquadric'         : ``-sqrt(1 + r**2)``
        - 'inverse_multiquadric' : ``1/sqrt(1 + r**2)``
        - 'inverse_quadratic'    : ``1/(1 + r**2)``
        - 'gaussian'             : ``exp(-r**2)``

    Default is 'thin_plate_spline'.
epsilon : float, optional
    Shape parameter that scales the input to the RBF. If `kernel` is
    'linear', 'thin_plate_spline', 'cubic', or 'quintic', this defaults to
    1 and can be ignored because it has the same effect as scaling the
    smoothing parameter. Otherwise, this must be specified.
degree : int, optional
    Degree of the added polynomial. For some RBFs the interpolant may not
    be well-posed if the polynomial degree is too small. Those RBFs and
    their corresponding minimum degrees are

        - 'multiquadric'      : 0
        - 'linear'            : 0
        - 'thin_plate_spline' : 1
        - 'cubic'             : 1
        - 'quintic'           : 2

    The default value is the minimum degree for `kernel` or 0 if there is
    no minimum degree. Set this to -1 for no added polynomial.

Notes
-----
An RBF is a scalar valued function in N-dimensional space whose value at
:math:`x` can be expressed in terms of :math:`r=||x - c||`, where :math:`c`
is the center of the RBF.

An RBF interpolant for the vector of data values :math:`d`, which are from
locations :math:`y`, is a linear combination of RBFs centered at :math:`y`
plus a polynomial with a specified degree. The RBF interpolant is written
as

.. math::
    f(x) = K(x, y) a + P(x) b,

where :math:`K(x, y)` is a matrix of RBFs with centers at :math:`y`
evaluated at the points :math:`x`, and :math:`P(x)` is a matrix of
monomials, which span polynomials with the specified degree, evaluated at
:math:`x`. The coefficients :math:`a` and :math:`b` are the solution to the
linear equations

.. math::
    (K(y, y) + \lambda I) a + P(y) b = d

and

.. math::
    P(y)^T a = 0,

where :math:`\lambda` is a non-negative smoothing parameter that controls
how well we want to fit the data. The data are fit exactly when the
smoothing parameter is 0.

The above system is uniquely solvable if the following requirements are
met:

    - :math:`P(y)` must have full column rank. :math:`P(y)` always has full
      column rank when `degree` is -1 or 0. When `degree` is 1,
      :math:`P(y)` has full column rank if the data point locations are not
      all collinear (N=2), coplanar (N=3), etc.
    - If `kernel` is 'multiquadric', 'linear', 'thin_plate_spline',
      'cubic', or 'quintic', then `degree` must not be lower than the
      minimum value listed above.
    - If `smoothing` is 0, then each data point location must be distinct.

When using an RBF that is not scale invariant ('multiquadric',
'inverse_multiquadric', 'inverse_quadratic', or 'gaussian'), an appropriate
shape parameter must be chosen (e.g., through cross validation). Smaller
values for the shape parameter correspond to wider RBFs. The problem can
become ill-conditioned or singular when the shape parameter is too small.

The memory required to solve for the RBF interpolation coefficients
increases quadratically with the number of data points, which can become
impractical when interpolating more than about a thousand data points.
To overcome memory limitations for large interpolation problems, the
`neighbors` argument can be specified to compute an RBF interpolant for
each evaluation point using only the nearest data points.

.. versionadded:: 1.7.0

See Also
--------
NearestNDInterpolator
LinearNDInterpolator
CloughTocher2DInterpolator

References
----------
.. [1] Fasshauer, G., 2007. Meshfree Approximation Methods with Matlab.
    World Scientific Publishing Co.

.. [2] http://amadeus.math.iit.edu/~fass/603_ch3.pdf

.. [3] Wahba, G., 1990. Spline Models for Observational Data. SIAM.

.. [4] http://pages.stat.wisc.edu/~wahba/stat860public/lect/lect8/lect8.pdf

Examples
--------
Demonstrate interpolating scattered data to a grid in 2-D.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import RBFInterpolator
>>> from scipy.stats.qmc import Halton

>>> rng = np.random.default_rng()
>>> xobs = 2*Halton(2, seed=rng).random(100) - 1
>>> yobs = np.sum(xobs, axis=1)*np.exp(-6*np.sum(xobs**2, axis=1))

>>> xgrid = np.mgrid[-1:1:50j, -1:1:50j]
>>> xflat = xgrid.reshape(2, -1).T
>>> yflat = RBFInterpolator(xobs, yobs)(xflat)
>>> ygrid = yflat.reshape(50, 50)

>>> fig, ax = plt.subplots()
>>> ax.pcolormesh(*xgrid, ygrid, vmin=-0.25, vmax=0.25, shading='gouraud')
>>> p = ax.scatter(*xobs.T, c=yobs, s=50, ec='k', vmin=-0.25, vmax=0.25)
>>> fig.colorbar(p)
>>> plt.show()