EVOLUTION-MANAGER

Edit File: rbf.py

"""rbf - Radial basis functions for interpolation/smoothing scattered Nd data. Written by John Travers <jtravs@gmail.com>, February 2007 Based closely on Matlab code by Alex Chirokov Additional, large, improvements by Robert Hetland Some additional alterations by Travis Oliphant Interpolation with multi-dimensional target domain by Josua Sassen Permission to use, modify, and distribute this software is given under the terms of the SciPy (BSD style) license. See LICENSE.txt that came with this distribution for specifics. NO WARRANTY IS EXPRESSED OR IMPLIED. USE AT YOUR OWN RISK. Copyright (c) 2006-2007, Robert Hetland <hetland@tamu.edu> Copyright (c) 2007, John Travers <jtravs@gmail.com> Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. * Neither the name of Robert Hetland nor the names of any contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. """ from __future__ import division, print_function, absolute_import import sys import numpy as np from scipy import linalg from scipy._lib.six import callable, get_method_function, get_function_code from scipy.special import xlogy from scipy.spatial.distance import cdist, pdist, squareform __all__ = ['Rbf'] class Rbf(object): """ Rbf(*args) A class for radial basis function interpolation of functions from n-dimensional scattered data to an m-dimensional domain. Parameters ---------- *args : arrays x, y, z, ..., d, where x, y, z, ... are the coordinates of the nodes and d is the array of values at the nodes function : str or callable, optional The radial basis function, based on the radius, r, given by the norm (default is Euclidean distance); the default is 'multiquadric':: 'multiquadric': sqrt((r/self.epsilon)**2 + 1) 'inverse': 1.0/sqrt((r/self.epsilon)**2 + 1) 'gaussian': exp(-(r/self.epsilon)**2) 'linear': r 'cubic': r**3 'quintic': r**5 'thin_plate': r**2 * log(r) If callable, then it must take 2 arguments (self, r). The epsilon parameter will be available as self.epsilon. Other keyword arguments passed in will be available as well. epsilon : float, optional Adjustable constant for gaussian or multiquadrics functions - defaults to approximate average distance between nodes (which is a good start). smooth : float, optional Values greater than zero increase the smoothness of the approximation. 0 is for interpolation (default), the function will always go through the nodal points in this case. norm : str, callable, optional A function that returns the 'distance' between two points, with inputs as arrays of positions (x, y, z, ...), and an output as an array of distance. E.g., the default: 'euclidean', such that the result is a matrix of the distances from each point in ``x1`` to each point in ``x2``. For more options, see documentation of `scipy.spatial.distances.cdist`. mode : str, optional Mode of the interpolation, can be '1-D' (default) or 'N-D'. When it is '1-D' the data `d` will be considered as one-dimensional and flattened internally. When it is 'N-D' the data `d` is assumed to be an array of shape (n_samples, m), where m is the dimension of the target domain. Attributes ---------- N : int The number of data points (as determined by the input arrays). di : ndarray The 1-D array of data values at each of the data coordinates `xi`. xi : ndarray The 2-D array of data coordinates. function : str or callable The radial basis function. See description under Parameters. epsilon : float Parameter used by gaussian or multiquadrics functions. See Parameters. smooth : float Smoothing parameter. See description under Parameters. norm : str or callable The distance function. See description under Parameters. mode : str Mode of the interpolation. See description under Parameters. nodes : ndarray A 1-D array of node values for the interpolation. A : internal property, do not use Examples -------- >>> from scipy.interpolate import Rbf >>> x, y, z, d = np.random.rand(4, 50) >>> rbfi = Rbf(x, y, z, d) # radial basis function interpolator instance >>> xi = yi = zi = np.linspace(0, 1, 20) >>> di = rbfi(xi, yi, zi) # interpolated values >>> di.shape (20,) """ # Available radial basis functions that can be selected as strings; # they all start with _h_ (self._init_function relies on that) def _h_multiquadric(self, r): return np.sqrt((1.0/self.epsilon*r)**2 + 1) def _h_inverse_multiquadric(self, r): return 1.0/np.sqrt((1.0/self.epsilon*r)**2 + 1) def _h_gaussian(self, r): return np.exp(-(1.0/self.epsilon*r)**2) def _h_linear(self, r): return r def _h_cubic(self, r): return r**3 def _h_quintic(self, r): return r**5 def _h_thin_plate(self, r): return xlogy(r**2, r) # Setup self._function and do smoke test on initial r def _init_function(self, r): if isinstance(self.function, str): self.function = self.function.lower() _mapped = {'inverse': 'inverse_multiquadric', 'inverse multiquadric': 'inverse_multiquadric', 'thin-plate': 'thin_plate'} if self.function in _mapped: self.function = _mapped[self.function] func_name = "_h_" + self.function if hasattr(self, func_name): self._function = getattr(self, func_name) else: functionlist = [x[3:] for x in dir(self) if x.startswith('_h_')] raise ValueError("function must be a callable or one of " + ", ".join(functionlist)) self._function = getattr(self, "_h_"+self.function) elif callable(self.function): allow_one = False if hasattr(self.function, 'func_code') or \ hasattr(self.function, '__code__'): val = self.function allow_one = True elif hasattr(self.function, "im_func"): val = get_method_function(self.function) elif hasattr(self.function, "__call__"): val = get_method_function(self.function.__call__) else: raise ValueError("Cannot determine number of arguments to " "function") argcount = get_function_code(val).co_argcount if allow_one and argcount == 1: self._function = self.function elif argcount == 2: if sys.version_info[0] >= 3: self._function = self.function.__get__(self, Rbf) else: import new self._function = new.instancemethod(self.function, self, Rbf) else: raise ValueError("Function argument must take 1 or 2 " "arguments.") a0 = self._function(r) if a0.shape != r.shape: raise ValueError("Callable must take array and return array of " "the same shape") return a0 def __init__(self, *args, **kwargs): # `args` can be a variable number of arrays; we flatten them and store # them as a single 2-D array `xi` of shape (n_args-1, array_size), # plus a 1-D array `di` for the values. # All arrays must have the same number of elements self.xi = np.asarray([np.asarray(a, dtype=np.float_).flatten() for a in args[:-1]]) self.N = self.xi.shape[-1] self.mode = kwargs.pop('mode', '1-D') if self.mode == '1-D': self.di = np.asarray(args[-1]).flatten() self._target_dim = 1 elif self.mode == 'N-D': self.di = np.asarray(args[-1]) self._target_dim = self.di.shape[-1] else: raise ValueError("Mode has to be 1-D or N-D.") if not all([x.size == self.di.shape[0] for x in self.xi]): raise ValueError("All arrays must be equal length.") self.norm = kwargs.pop('norm', 'euclidean') self.epsilon = kwargs.pop('epsilon', None) if self.epsilon is None: # default epsilon is the "the average distance between nodes" based # on a bounding hypercube ximax = np.amax(self.xi, axis=1) ximin = np.amin(self.xi, axis=1) edges = ximax - ximin edges = edges[np.nonzero(edges)] self.epsilon = np.power(np.prod(edges)/self.N, 1.0/edges.size) self.smooth = kwargs.pop('smooth', 0.0) self.function = kwargs.pop('function', 'multiquadric') # attach anything left in kwargs to self for use by any user-callable # function or to save on the object returned. for item, value in kwargs.items(): setattr(self, item, value) # Compute weights if self._target_dim > 1: # If we have more than one target dimension, # we first factorize the matrix self.nodes = np.zeros((self.N, self._target_dim), dtype=self.di.dtype) lu, piv = linalg.lu_factor(self.A) for i in range(self._target_dim): self.nodes[:, i] = linalg.lu_solve((lu, piv), self.di[:, i]) else: self.nodes = linalg.solve(self.A, self.di) @property def A(self): # this only exists for backwards compatibility: self.A was available # and, at least technically, public. r = squareform(pdist(self.xi.T, self.norm)) # Pairwise norm return self._init_function(r) - np.eye(self.N)*self.smooth def _call_norm(self, x1, x2): return cdist(x1.T, x2.T, self.norm) def __call__(self, *args): args = [np.asarray(x) for x in args] if not all([x.shape == y.shape for x in args for y in args]): raise ValueError("Array lengths must be equal") if self._target_dim > 1: shp = args[0].shape + (self._target_dim,) else: shp = args[0].shape xa = np.asarray([a.flatten() for a in args], dtype=np.float_) r = self._call_norm(xa, self.xi) return np.dot(self._function(r), self.nodes).reshape(shp)