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scipy.interpolate.RectSphereBivariateSpline.__call__

# scipy.interpolate.RectSphereBivariateSpline

class `scipy.interpolate.``RectSphereBivariateSpline`(u, v, r, s=0.0, pole_continuity=False, pole_values=None, pole_exact=False, pole_flat=False)

  Bivariate spline approximation over a rectangular mesh on a sphere.

  Can be used for smoothing data.

  New in version 0.11.0.

  Parameters:

  u : array_like

  > 1-D array of latitude coordinates in strictly ascending order. Coordinates must be given in radians and lie within the interval (0, pi).

  v : array_like

  > 1-D array of longitude coordinates in strictly ascending order. Coordinates must be given in radians, and must lie within (0, 2pi).

  r : array_like

  > 2-D array of data with shape `(u.size, v.size)`.

  s : float, optional

  > Positive smoothing factor defined for estimation condition (`s=0` is for interpolation).

  pole_continuity : bool or (bool, bool), optional

  > Order of continuity at the poles `u=0` (`pole_continuity[0]`) and `u=pi` (`pole_continuity[1]`). The order of continuity at the pole will be 1 or 0 when this is True or False, respectively. Defaults to False.

  pole_values : float or (float, float), optional

  > Data values at the poles `u=0` and `u=pi`. Either the whole parameter or each individual element can be None. Defaults to None.

  pole_exact : bool or (bool, bool), optional

  > Data value exactness at the poles `u=0` and `u=pi`. If True, the value is considered to be the right function value, and it will be fitted exactly. If False, the value will be considered to be a data value just like the other data values. Defaults to False.

  pole_flat : bool or (bool, bool), optional

  > For the poles at `u=0` and `u=pi`, specify whether or not the approximation has vanishing derivatives. Defaults to False.

  See also

  `RectBivariateSpline`
    bivariate spline approximation over a rectangular mesh

  Notes

  Currently, only the smoothing spline approximation (`iopt[0] = 0` and `iopt[0] = 1` in the FITPACK routine) is supported. The exact least-squares spline approximation is not implemented yet.

  When actually performing the interpolation, the requested v values must lie within the same length 2pi interval that the original v values were chosen from.

  For more information, see the FITPACK site about this function.

  Examples

  Suppose we have global data on a coarse grid

      >>> lats = np.linspace(10, 170, 9) * np.pi / 180.
      >>> lons = np.linspace(0, 350, 18) * np.pi / 180.
      >>> data = np.dot(np.atleast_2d(90. - np.linspace(-80., 80., 18)).T,
      ...               np.atleast_2d(180. - np.abs(np.linspace(0., 350., 9)))).T

  We want to interpolate it to a global one-degree grid

      >>> new_lats = np.linspace(1, 180, 180) * np.pi / 180
      >>> new_lons = np.linspace(1, 360, 360) * np.pi / 180
      >>> new_lats, new_lons = np.meshgrid(new_lats, new_lons)

  We need to set up the interpolator object

      >>> from scipy.interpolate import RectSphereBivariateSpline
      >>> lut = RectSphereBivariateSpline(lats, lons, data)

  Finally we interpolate the data. The `RectSphereBivariateSpline` object only takes 1-D arrays as input, therefore we need to do some reshaping.

      >>> data_interp = lut.ev(new_lats.ravel(),
      ...                      new_lons.ravel()).reshape((360, 180)).T

  Looking at the original and the interpolated data, one can see that the interpolant reproduces the original data very well:

      >>> import matplotlib.pyplot as plt
      >>> fig = plt.figure()
      >>> ax1 = fig.add_subplot(211)
      >>> ax1.imshow(data, interpolation='nearest')
      >>> ax2 = fig.add_subplot(212)
      >>> ax2.imshow(data_interp, interpolation='nearest')
      >>> plt.show()

  (Source code)

  Chosing the optimal value of `s` can be a delicate task. Recommended values for `s` depend on the accuracy of the data values. If the user has an idea of the statistical errors on the data, she can also find a proper estimate for `s`. By assuming that, if she specifies the right `s`, the interpolator will use a spline `f(u,v)` which exactly reproduces the function underlying the data, she can evaluate `sum((r(i,j)-s(u(i),v(j)))**2)` to find a good estimate for this `s`. For example, if she knows that the statistical errors on her `r(i,j)`-values are not greater than 0.1, she may expect that a good `s` should have a value not larger than `u.size * v.size * (0.1)**2`.

  If nothing is known about the statistical error in `r(i,j)`, `s` must be determined by trial and error. The best is then to start with a very large value of `s` (to determine the least-squares polynomial and the corresponding upper bound `fp0` for `s`) and then to progressively decrease the value of `s` (say by a factor 10 in the beginning, i.e. `s = fp0 / 10, fp0 / 100, ...` and more carefully as the approximation shows more detail) to obtain closer fits.

  The interpolation results for different values of `s` give some insight into this process:

      >>> fig2 = plt.figure()
      >>> s = [3e9, 2e9, 1e9, 1e8]
      >>> for ii in xrange(len(s)):
      ...     lut = RectSphereBivariateSpline(lats, lons, data, s=s[ii])
      ...     data_interp = lut.ev(new_lats.ravel(),
      ...                          new_lons.ravel()).reshape((360, 180)).T
      ...     ax = fig2.add_subplot(2, 2, ii+1)
      ...     ax.imshow(data_interp, interpolation='nearest')
      ...     ax.set_title("s = %g" % s[ii])
      >>> plt.show()

  Methods

  `__call__`(theta, phi[, dtheta, dphi, grid]) | Evaluate the spline or its derivatives at given positions.
  `ev`(theta, phi[, dtheta, dphi]) | Evaluate the spline at points
  `get_coeffs`() | Return spline coefficients.
  `get_knots`() | Return a tuple (tx,ty) where tx,ty contain knots positions of the spline with respect to x-, y-variable, respectively.
  `get_residual`() | Return weighted sum of squared residuals of the spline