"""Module for RBF interpolation.""" import warnings from itertools import combinations_with_replacement import numpy as np from numpy.linalg import LinAlgError from scipy.spatial import KDTree from scipy.special import comb from scipy.linalg.lapack import dgesv # type: ignore[attr-defined] from ._rbfinterp_pythran import _build_system, _evaluate, _polynomial_matrix __all__ = ["RBFInterpolator"] # These RBFs are implemented. _AVAILABLE = { "linear", "thin_plate_spline", "cubic", "quintic", "multiquadric", "inverse_multiquadric", "inverse_quadratic", "gaussian" } # The shape parameter does not need to be specified when using these RBFs. _SCALE_INVARIANT = {"linear", "thin_plate_spline", "cubic", "quintic"} # For RBFs that are conditionally positive definite of order m, the interpolant # should include polynomial terms with degree >= m - 1. Define the minimum # degrees here. These values are from Chapter 8 of Fasshauer's "Meshfree # Approximation Methods with MATLAB". The RBFs that are not in this dictionary # are positive definite and do not need polynomial terms. _NAME_TO_MIN_DEGREE = { "multiquadric": 0, "linear": 0, "thin_plate_spline": 1, "cubic": 1, "quintic": 2 } def _monomial_powers(ndim, degree): """Return the powers for each monomial in a polynomial. Parameters ---------- ndim : int Number of variables in the polynomial. degree : int Degree of the polynomial. Returns ------- (nmonos, ndim) int ndarray Array where each row contains the powers for each variable in a monomial. """ nmonos = comb(degree + ndim, ndim, exact=True) out = np.zeros((nmonos, ndim), dtype=int) count = 0 for deg in range(degree + 1): for mono in combinations_with_replacement(range(ndim), deg): # `mono` is a tuple of variables in the current monomial with # multiplicity indicating power (e.g., (0, 1, 1) represents x*y**2) for var in mono: out[count, var] += 1 count += 1 return out def _build_and_solve_system(y, d, smoothing, kernel, epsilon, powers): """Build and solve the RBF interpolation system of equations. Parameters ---------- y : (P, N) float ndarray Data point coordinates. d : (P, S) float ndarray Data values at `y`. smoothing : (P,) float ndarray Smoothing parameter for each data point. kernel : str Name of the RBF. epsilon : float Shape parameter. powers : (R, N) int ndarray The exponents for each monomial in the polynomial. Returns ------- coeffs : (P + R, S) float ndarray Coefficients for each RBF and monomial. shift : (N,) float ndarray Domain shift used to create the polynomial matrix. scale : (N,) float ndarray Domain scaling used to create the polynomial matrix. """ lhs, rhs, shift, scale = _build_system( y, d, smoothing, kernel, epsilon, powers ) _, _, coeffs, info = dgesv(lhs, rhs, overwrite_a=True, overwrite_b=True) if info < 0: raise ValueError(f"The {-info}-th argument had an illegal value.") elif info > 0: msg = "Singular matrix." nmonos = powers.shape[0] if nmonos > 0: pmat = _polynomial_matrix((y - shift)/scale, powers) rank = np.linalg.matrix_rank(pmat) if rank < nmonos: msg = ( "Singular matrix. The matrix of monomials evaluated at " "the data point coordinates does not have full column " f"rank ({rank}/{nmonos})." ) raise LinAlgError(msg) return shift, scale, coeffs class RBFInterpolator: """Radial basis function (RBF) interpolation in N dimensions. Parameters ---------- y : (P, N) array_like Data point coordinates. d : (P, ...) array_like Data values at `y`. 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 (P,) 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 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() """ def __init__(self, y, d, neighbors=None, smoothing=0.0, kernel="thin_plate_spline", epsilon=None, degree=None): y = np.asarray(y, dtype=float, order="C") if y.ndim != 2: raise ValueError("`y` must be a 2-dimensional array.") ny, ndim = y.shape d_dtype = complex if np.iscomplexobj(d) else float d = np.asarray(d, dtype=d_dtype, order="C") if d.shape[0] != ny: raise ValueError( f"Expected the first axis of `d` to have length {ny}." ) d_shape = d.shape[1:] d = d.reshape((ny, -1)) # If `d` is complex, convert it to a float array with twice as many # columns. Otherwise, the LHS matrix would need to be converted to # complex and take up 2x more memory than necessary. d = d.view(float) if np.isscalar(smoothing): smoothing = np.full(ny, smoothing, dtype=float) else: smoothing = np.asarray(smoothing, dtype=float, order="C") if smoothing.shape != (ny,): raise ValueError( "Expected `smoothing` to be a scalar or have shape " f"({ny},)." ) kernel = kernel.lower() if kernel not in _AVAILABLE: raise ValueError(f"`kernel` must be one of {_AVAILABLE}.") if epsilon is None: if kernel in _SCALE_INVARIANT: epsilon = 1.0 else: raise ValueError( "`epsilon` must be specified if `kernel` is not one of " f"{_SCALE_INVARIANT}." ) else: epsilon = float(epsilon) min_degree = _NAME_TO_MIN_DEGREE.get(kernel, -1) if degree is None: degree = max(min_degree, 0) else: degree = int(degree) if degree < -1: raise ValueError("`degree` must be at least -1.") elif degree < min_degree: warnings.warn( f"`degree` should not be below {min_degree} when `kernel` " f"is '{kernel}'. The interpolant may not be uniquely " "solvable, and the smoothing parameter may have an " "unintuitive effect.", UserWarning ) if neighbors is None: nobs = ny else: # Make sure the number of nearest neighbors used for interpolation # does not exceed the number of observations. neighbors = int(min(neighbors, ny)) nobs = neighbors powers = _monomial_powers(ndim, degree) # The polynomial matrix must have full column rank in order for the # interpolant to be well-posed, which is not possible if there are # fewer observations than monomials. if powers.shape[0] > nobs: raise ValueError( f"At least {powers.shape[0]} data points are required when " f"`degree` is {degree} and the number of dimensions is {ndim}." ) if neighbors is None: shift, scale, coeffs = _build_and_solve_system( y, d, smoothing, kernel, epsilon, powers ) # Make these attributes private since they do not always exist. self._shift = shift self._scale = scale self._coeffs = coeffs else: self._tree = KDTree(y) self.y = y self.d = d self.d_shape = d_shape self.d_dtype = d_dtype self.neighbors = neighbors self.smoothing = smoothing self.kernel = kernel self.epsilon = epsilon self.powers = powers def __call__(self, x): """Evaluate the interpolant at `x`. Parameters ---------- x : (Q, N) array_like Evaluation point coordinates. Returns ------- (Q, ...) ndarray Values of the interpolant at `x`. """ x = np.asarray(x, dtype=float, order="C") if x.ndim != 2: raise ValueError("`x` must be a 2-dimensional array.") nx, ndim = x.shape if ndim != self.y.shape[1]: raise ValueError( "Expected the second axis of `x` to have length " f"{self.y.shape[1]}." ) if self.neighbors is None: out = _evaluate( x, self.y, self.kernel, self.epsilon, self.powers, self._shift, self._scale, self._coeffs ) else: # Get the indices of the k nearest observation points to each # evaluation point. _, yindices = self._tree.query(x, self.neighbors) if self.neighbors == 1: # `KDTree` squeezes the output when neighbors=1. yindices = yindices[:, None] # Multiple evaluation points may have the same neighborhood of # observation points. Make the neighborhoods unique so that we only # compute the interpolation coefficients once for each # neighborhood. yindices = np.sort(yindices, axis=1) yindices, inv = np.unique(yindices, return_inverse=True, axis=0) # `inv` tells us which neighborhood will be used by each evaluation # point. Now we find which evaluation points will be using each # neighborhood. xindices = [[] for _ in range(len(yindices))] for i, j in enumerate(inv): xindices[j].append(i) out = np.empty((nx, self.d.shape[1]), dtype=float) for xidx, yidx in zip(xindices, yindices): # `yidx` are the indices of the observations in this # neighborhood. `xidx` are the indices of the evaluation points # that are using this neighborhood. xnbr = x[xidx] ynbr = self.y[yidx] dnbr = self.d[yidx] snbr = self.smoothing[yidx] shift, scale, coeffs = _build_and_solve_system( ynbr, dnbr, snbr, self.kernel, self.epsilon, self.powers, ) out[xidx] = _evaluate( xnbr, ynbr, self.kernel, self.epsilon, self.powers, shift, scale, coeffs ) out = out.view(self.d_dtype) out = out.reshape((nx,) + self.d_shape) return out