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