1000 lines
38 KiB
Python
1000 lines
38 KiB
Python
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# Copyright Anne M. Archibald 2008
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# Released under the scipy license
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import numpy as np
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from heapq import heappush, heappop
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import scipy.sparse
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__all__ = ['minkowski_distance_p', 'minkowski_distance',
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'distance_matrix',
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'Rectangle', 'KDTree']
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def minkowski_distance_p(x, y, p=2):
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"""
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Compute the pth power of the L**p distance between two arrays.
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For efficiency, this function computes the L**p distance but does
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not extract the pth root. If `p` is 1 or infinity, this is equal to
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the actual L**p distance.
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Parameters
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----------
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x : (M, K) array_like
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Input array.
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y : (N, K) array_like
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Input array.
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p : float, 1 <= p <= infinity
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Which Minkowski p-norm to use.
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Examples
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--------
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>>> from scipy.spatial import minkowski_distance_p
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>>> minkowski_distance_p([[0,0],[0,0]], [[1,1],[0,1]])
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array([2, 1])
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"""
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x = np.asarray(x)
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y = np.asarray(y)
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# Find smallest common datatype with float64 (return type of this function) - addresses #10262.
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# Don't just cast to float64 for complex input case.
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common_datatype = np.promote_types(np.promote_types(x.dtype, y.dtype), 'float64')
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# Make sure x and y are NumPy arrays of correct datatype.
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x = x.astype(common_datatype)
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y = y.astype(common_datatype)
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if p == np.inf:
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return np.amax(np.abs(y-x), axis=-1)
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elif p == 1:
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return np.sum(np.abs(y-x), axis=-1)
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else:
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return np.sum(np.abs(y-x)**p, axis=-1)
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def minkowski_distance(x, y, p=2):
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"""
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Compute the L**p distance between two arrays.
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Parameters
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----------
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x : (M, K) array_like
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Input array.
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y : (N, K) array_like
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Input array.
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p : float, 1 <= p <= infinity
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Which Minkowski p-norm to use.
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Examples
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--------
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>>> from scipy.spatial import minkowski_distance
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>>> minkowski_distance([[0,0],[0,0]], [[1,1],[0,1]])
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array([ 1.41421356, 1. ])
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"""
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x = np.asarray(x)
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y = np.asarray(y)
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if p == np.inf or p == 1:
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return minkowski_distance_p(x, y, p)
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else:
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return minkowski_distance_p(x, y, p)**(1./p)
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class Rectangle(object):
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"""Hyperrectangle class.
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Represents a Cartesian product of intervals.
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"""
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def __init__(self, maxes, mins):
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"""Construct a hyperrectangle."""
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self.maxes = np.maximum(maxes,mins).astype(float)
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self.mins = np.minimum(maxes,mins).astype(float)
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self.m, = self.maxes.shape
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def __repr__(self):
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return "<Rectangle %s>" % list(zip(self.mins, self.maxes))
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def volume(self):
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"""Total volume."""
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return np.prod(self.maxes-self.mins)
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def split(self, d, split):
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"""
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Produce two hyperrectangles by splitting.
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In general, if you need to compute maximum and minimum
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distances to the children, it can be done more efficiently
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by updating the maximum and minimum distances to the parent.
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Parameters
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----------
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d : int
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Axis to split hyperrectangle along.
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split : float
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Position along axis `d` to split at.
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"""
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mid = np.copy(self.maxes)
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mid[d] = split
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less = Rectangle(self.mins, mid)
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mid = np.copy(self.mins)
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mid[d] = split
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greater = Rectangle(mid, self.maxes)
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return less, greater
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def min_distance_point(self, x, p=2.):
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"""
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Return the minimum distance between input and points in the hyperrectangle.
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Parameters
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----------
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x : array_like
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Input.
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p : float, optional
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Input.
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"""
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return minkowski_distance(0, np.maximum(0,np.maximum(self.mins-x,x-self.maxes)),p)
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def max_distance_point(self, x, p=2.):
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"""
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Return the maximum distance between input and points in the hyperrectangle.
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Parameters
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----------
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x : array_like
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Input array.
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p : float, optional
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Input.
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"""
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return minkowski_distance(0, np.maximum(self.maxes-x,x-self.mins),p)
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def min_distance_rectangle(self, other, p=2.):
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"""
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Compute the minimum distance between points in the two hyperrectangles.
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Parameters
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----------
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other : hyperrectangle
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Input.
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p : float
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Input.
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"""
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return minkowski_distance(0, np.maximum(0,np.maximum(self.mins-other.maxes,other.mins-self.maxes)),p)
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def max_distance_rectangle(self, other, p=2.):
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"""
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Compute the maximum distance between points in the two hyperrectangles.
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Parameters
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----------
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other : hyperrectangle
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Input.
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p : float, optional
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Input.
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"""
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return minkowski_distance(0, np.maximum(self.maxes-other.mins,other.maxes-self.mins),p)
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class KDTree(object):
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"""
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kd-tree for quick nearest-neighbor lookup
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This class provides an index into a set of k-D points which
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can be used to rapidly look up the nearest neighbors of any point.
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Parameters
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----------
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data : (N,K) array_like
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The data points to be indexed. This array is not copied, and
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so modifying this data will result in bogus results.
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leafsize : int, optional
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The number of points at which the algorithm switches over to
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brute-force. Has to be positive.
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Raises
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------
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RuntimeError
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The maximum recursion limit can be exceeded for large data
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sets. If this happens, either increase the value for the `leafsize`
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parameter or increase the recursion limit by::
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>>> import sys
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>>> sys.setrecursionlimit(10000)
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See Also
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--------
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cKDTree : Implementation of `KDTree` in Cython
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Notes
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-----
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The algorithm used is described in Maneewongvatana and Mount 1999.
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The general idea is that the kd-tree is a binary tree, each of whose
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nodes represents an axis-aligned hyperrectangle. Each node specifies
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an axis and splits the set of points based on whether their coordinate
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along that axis is greater than or less than a particular value.
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During construction, the axis and splitting point are chosen by the
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"sliding midpoint" rule, which ensures that the cells do not all
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become long and thin.
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The tree can be queried for the r closest neighbors of any given point
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(optionally returning only those within some maximum distance of the
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point). It can also be queried, with a substantial gain in efficiency,
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for the r approximate closest neighbors.
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For large dimensions (20 is already large) do not expect this to run
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significantly faster than brute force. High-dimensional nearest-neighbor
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queries are a substantial open problem in computer science.
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The tree also supports all-neighbors queries, both with arrays of points
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and with other kd-trees. These do use a reasonably efficient algorithm,
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but the kd-tree is not necessarily the best data structure for this
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sort of calculation.
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"""
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def __init__(self, data, leafsize=10):
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self.data = np.asarray(data)
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if self.data.dtype.kind == 'c':
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raise TypeError("KDTree does not work with complex data")
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self.n, self.m = np.shape(self.data)
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self.leafsize = int(leafsize)
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if self.leafsize < 1:
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raise ValueError("leafsize must be at least 1")
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self.maxes = np.amax(self.data,axis=0)
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self.mins = np.amin(self.data,axis=0)
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self.tree = self.__build(np.arange(self.n), self.maxes, self.mins)
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class node(object):
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def __lt__(self, other):
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return id(self) < id(other)
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def __gt__(self, other):
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return id(self) > id(other)
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def __le__(self, other):
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return id(self) <= id(other)
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def __ge__(self, other):
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return id(self) >= id(other)
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def __eq__(self, other):
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return id(self) == id(other)
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class leafnode(node):
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def __init__(self, idx):
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self.idx = idx
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self.children = len(idx)
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class innernode(node):
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def __init__(self, split_dim, split, less, greater):
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self.split_dim = split_dim
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self.split = split
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self.less = less
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self.greater = greater
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self.children = less.children+greater.children
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def __build(self, idx, maxes, mins):
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if len(idx) <= self.leafsize:
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return KDTree.leafnode(idx)
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else:
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data = self.data[idx]
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# maxes = np.amax(data,axis=0)
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# mins = np.amin(data,axis=0)
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d = np.argmax(maxes-mins)
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maxval = maxes[d]
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minval = mins[d]
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if maxval == minval:
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# all points are identical; warn user?
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return KDTree.leafnode(idx)
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data = data[:,d]
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# sliding midpoint rule; see Maneewongvatana and Mount 1999
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# for arguments that this is a good idea.
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split = (maxval+minval)/2
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less_idx = np.nonzero(data <= split)[0]
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greater_idx = np.nonzero(data > split)[0]
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if len(less_idx) == 0:
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split = np.amin(data)
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less_idx = np.nonzero(data <= split)[0]
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greater_idx = np.nonzero(data > split)[0]
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if len(greater_idx) == 0:
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split = np.amax(data)
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less_idx = np.nonzero(data < split)[0]
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greater_idx = np.nonzero(data >= split)[0]
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if len(less_idx) == 0:
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# _still_ zero? all must have the same value
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if not np.all(data == data[0]):
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raise ValueError("Troublesome data array: %s" % data)
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split = data[0]
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less_idx = np.arange(len(data)-1)
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greater_idx = np.array([len(data)-1])
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lessmaxes = np.copy(maxes)
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lessmaxes[d] = split
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greatermins = np.copy(mins)
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greatermins[d] = split
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return KDTree.innernode(d, split,
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self.__build(idx[less_idx],lessmaxes,mins),
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self.__build(idx[greater_idx],maxes,greatermins))
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def __query(self, x, k=1, eps=0, p=2, distance_upper_bound=np.inf):
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side_distances = np.maximum(0,np.maximum(x-self.maxes,self.mins-x))
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if p != np.inf:
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side_distances **= p
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min_distance = np.sum(side_distances)
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else:
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min_distance = np.amax(side_distances)
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# priority queue for chasing nodes
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# entries are:
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# minimum distance between the cell and the target
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# distances between the nearest side of the cell and the target
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# the head node of the cell
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q = [(min_distance,
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tuple(side_distances),
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self.tree)]
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# priority queue for the nearest neighbors
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# furthest known neighbor first
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# entries are (-distance**p, i)
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neighbors = []
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if eps == 0:
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epsfac = 1
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elif p == np.inf:
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epsfac = 1/(1+eps)
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else:
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epsfac = 1/(1+eps)**p
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if p != np.inf and distance_upper_bound != np.inf:
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distance_upper_bound = distance_upper_bound**p
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while q:
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min_distance, side_distances, node = heappop(q)
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if isinstance(node, KDTree.leafnode):
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# brute-force
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data = self.data[node.idx]
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ds = minkowski_distance_p(data,x[np.newaxis,:],p)
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for i in range(len(ds)):
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if ds[i] < distance_upper_bound:
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if len(neighbors) == k:
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heappop(neighbors)
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heappush(neighbors, (-ds[i], node.idx[i]))
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if len(neighbors) == k:
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distance_upper_bound = -neighbors[0][0]
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else:
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# we don't push cells that are too far onto the queue at all,
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# but since the distance_upper_bound decreases, we might get
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# here even if the cell's too far
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if min_distance > distance_upper_bound*epsfac:
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# since this is the nearest cell, we're done, bail out
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break
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# compute minimum distances to the children and push them on
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if x[node.split_dim] < node.split:
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near, far = node.less, node.greater
|
||
|
|
else:
|
||
|
|
near, far = node.greater, node.less
|
||
|
|
|
||
|
|
# near child is at the same distance as the current node
|
||
|
|
heappush(q,(min_distance, side_distances, near))
|
||
|
|
|
||
|
|
# far child is further by an amount depending only
|
||
|
|
# on the split value
|
||
|
|
sd = list(side_distances)
|
||
|
|
if p == np.inf:
|
||
|
|
min_distance = max(min_distance, abs(node.split-x[node.split_dim]))
|
||
|
|
elif p == 1:
|
||
|
|
sd[node.split_dim] = np.abs(node.split-x[node.split_dim])
|
||
|
|
min_distance = min_distance - side_distances[node.split_dim] + sd[node.split_dim]
|
||
|
|
else:
|
||
|
|
sd[node.split_dim] = np.abs(node.split-x[node.split_dim])**p
|
||
|
|
min_distance = min_distance - side_distances[node.split_dim] + sd[node.split_dim]
|
||
|
|
|
||
|
|
# far child might be too far, if so, don't bother pushing it
|
||
|
|
if min_distance <= distance_upper_bound*epsfac:
|
||
|
|
heappush(q,(min_distance, tuple(sd), far))
|
||
|
|
|
||
|
|
if p == np.inf:
|
||
|
|
return sorted([(-d,i) for (d,i) in neighbors])
|
||
|
|
else:
|
||
|
|
return sorted([((-d)**(1./p),i) for (d,i) in neighbors])
|
||
|
|
|
||
|
|
def query(self, x, k=1, eps=0, p=2, distance_upper_bound=np.inf):
|
||
|
|
"""
|
||
|
|
Query the kd-tree for nearest neighbors
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like, last dimension self.m
|
||
|
|
An array of points to query.
|
||
|
|
k : int, optional
|
||
|
|
The number of nearest neighbors to return.
|
||
|
|
eps : nonnegative float, optional
|
||
|
|
Return approximate nearest neighbors; the kth returned value
|
||
|
|
is guaranteed to be no further than (1+eps) times the
|
||
|
|
distance to the real kth nearest neighbor.
|
||
|
|
p : float, 1<=p<=infinity, optional
|
||
|
|
Which Minkowski p-norm to use.
|
||
|
|
1 is the sum-of-absolute-values "Manhattan" distance
|
||
|
|
2 is the usual Euclidean distance
|
||
|
|
infinity is the maximum-coordinate-difference distance
|
||
|
|
distance_upper_bound : nonnegative float, optional
|
||
|
|
Return only neighbors within this distance. This is used to prune
|
||
|
|
tree searches, so if you are doing a series of nearest-neighbor
|
||
|
|
queries, it may help to supply the distance to the nearest neighbor
|
||
|
|
of the most recent point.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
d : float or array of floats
|
||
|
|
The distances to the nearest neighbors.
|
||
|
|
If x has shape tuple+(self.m,), then d has shape tuple if
|
||
|
|
k is one, or tuple+(k,) if k is larger than one. Missing
|
||
|
|
neighbors (e.g. when k > n or distance_upper_bound is
|
||
|
|
given) are indicated with infinite distances. If k is None,
|
||
|
|
then d is an object array of shape tuple, containing lists
|
||
|
|
of distances. In either case the hits are sorted by distance
|
||
|
|
(nearest first).
|
||
|
|
i : integer or array of integers
|
||
|
|
The locations of the neighbors in self.data. i is the same
|
||
|
|
shape as d.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy import spatial
|
||
|
|
>>> x, y = np.mgrid[0:5, 2:8]
|
||
|
|
>>> tree = spatial.KDTree(list(zip(x.ravel(), y.ravel())))
|
||
|
|
>>> tree.data
|
||
|
|
array([[0, 2],
|
||
|
|
[0, 3],
|
||
|
|
[0, 4],
|
||
|
|
[0, 5],
|
||
|
|
[0, 6],
|
||
|
|
[0, 7],
|
||
|
|
[1, 2],
|
||
|
|
[1, 3],
|
||
|
|
[1, 4],
|
||
|
|
[1, 5],
|
||
|
|
[1, 6],
|
||
|
|
[1, 7],
|
||
|
|
[2, 2],
|
||
|
|
[2, 3],
|
||
|
|
[2, 4],
|
||
|
|
[2, 5],
|
||
|
|
[2, 6],
|
||
|
|
[2, 7],
|
||
|
|
[3, 2],
|
||
|
|
[3, 3],
|
||
|
|
[3, 4],
|
||
|
|
[3, 5],
|
||
|
|
[3, 6],
|
||
|
|
[3, 7],
|
||
|
|
[4, 2],
|
||
|
|
[4, 3],
|
||
|
|
[4, 4],
|
||
|
|
[4, 5],
|
||
|
|
[4, 6],
|
||
|
|
[4, 7]])
|
||
|
|
>>> pts = np.array([[0, 0], [2.1, 2.9]])
|
||
|
|
>>> tree.query(pts)
|
||
|
|
(array([ 2. , 0.14142136]), array([ 0, 13]))
|
||
|
|
>>> tree.query(pts[0])
|
||
|
|
(2.0, 0)
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = np.asarray(x)
|
||
|
|
if x.dtype.kind == 'c':
|
||
|
|
raise TypeError("KDTree does not work with complex data")
|
||
|
|
if np.shape(x)[-1] != self.m:
|
||
|
|
raise ValueError("x must consist of vectors of length %d but has shape %s" % (self.m, np.shape(x)))
|
||
|
|
if p < 1:
|
||
|
|
raise ValueError("Only p-norms with 1<=p<=infinity permitted")
|
||
|
|
retshape = np.shape(x)[:-1]
|
||
|
|
if retshape != ():
|
||
|
|
if k is None:
|
||
|
|
dd = np.empty(retshape,dtype=object)
|
||
|
|
ii = np.empty(retshape,dtype=object)
|
||
|
|
elif k > 1:
|
||
|
|
dd = np.empty(retshape+(k,),dtype=float)
|
||
|
|
dd.fill(np.inf)
|
||
|
|
ii = np.empty(retshape+(k,),dtype=int)
|
||
|
|
ii.fill(self.n)
|
||
|
|
elif k == 1:
|
||
|
|
dd = np.empty(retshape,dtype=float)
|
||
|
|
dd.fill(np.inf)
|
||
|
|
ii = np.empty(retshape,dtype=int)
|
||
|
|
ii.fill(self.n)
|
||
|
|
else:
|
||
|
|
raise ValueError("Requested %s nearest neighbors; acceptable numbers are integers greater than or equal to one, or None")
|
||
|
|
for c in np.ndindex(retshape):
|
||
|
|
hits = self.__query(x[c], k=k, eps=eps, p=p, distance_upper_bound=distance_upper_bound)
|
||
|
|
if k is None:
|
||
|
|
dd[c] = [d for (d,i) in hits]
|
||
|
|
ii[c] = [i for (d,i) in hits]
|
||
|
|
elif k > 1:
|
||
|
|
for j in range(len(hits)):
|
||
|
|
dd[c+(j,)], ii[c+(j,)] = hits[j]
|
||
|
|
elif k == 1:
|
||
|
|
if len(hits) > 0:
|
||
|
|
dd[c], ii[c] = hits[0]
|
||
|
|
else:
|
||
|
|
dd[c] = np.inf
|
||
|
|
ii[c] = self.n
|
||
|
|
return dd, ii
|
||
|
|
else:
|
||
|
|
hits = self.__query(x, k=k, eps=eps, p=p, distance_upper_bound=distance_upper_bound)
|
||
|
|
if k is None:
|
||
|
|
return [d for (d,i) in hits], [i for (d,i) in hits]
|
||
|
|
elif k == 1:
|
||
|
|
if len(hits) > 0:
|
||
|
|
return hits[0]
|
||
|
|
else:
|
||
|
|
return np.inf, self.n
|
||
|
|
elif k > 1:
|
||
|
|
dd = np.empty(k,dtype=float)
|
||
|
|
dd.fill(np.inf)
|
||
|
|
ii = np.empty(k,dtype=int)
|
||
|
|
ii.fill(self.n)
|
||
|
|
for j in range(len(hits)):
|
||
|
|
dd[j], ii[j] = hits[j]
|
||
|
|
return dd, ii
|
||
|
|
else:
|
||
|
|
raise ValueError("Requested %s nearest neighbors; acceptable numbers are integers greater than or equal to one, or None")
|
||
|
|
|
||
|
|
def __query_ball_point(self, x, r, p=2., eps=0):
|
||
|
|
R = Rectangle(self.maxes, self.mins)
|
||
|
|
|
||
|
|
def traverse_checking(node, rect):
|
||
|
|
if rect.min_distance_point(x, p) > r / (1. + eps):
|
||
|
|
return []
|
||
|
|
elif rect.max_distance_point(x, p) < r * (1. + eps):
|
||
|
|
return traverse_no_checking(node)
|
||
|
|
elif isinstance(node, KDTree.leafnode):
|
||
|
|
d = self.data[node.idx]
|
||
|
|
return node.idx[minkowski_distance(d, x, p) <= r].tolist()
|
||
|
|
else:
|
||
|
|
less, greater = rect.split(node.split_dim, node.split)
|
||
|
|
return traverse_checking(node.less, less) + \
|
||
|
|
traverse_checking(node.greater, greater)
|
||
|
|
|
||
|
|
def traverse_no_checking(node):
|
||
|
|
if isinstance(node, KDTree.leafnode):
|
||
|
|
return node.idx.tolist()
|
||
|
|
else:
|
||
|
|
return traverse_no_checking(node.less) + \
|
||
|
|
traverse_no_checking(node.greater)
|
||
|
|
|
||
|
|
return traverse_checking(self.tree, R)
|
||
|
|
|
||
|
|
def query_ball_point(self, x, r, p=2., eps=0):
|
||
|
|
"""Find all points within distance r of point(s) x.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like, shape tuple + (self.m,)
|
||
|
|
The point or points to search for neighbors of.
|
||
|
|
r : positive float
|
||
|
|
The radius of points to return.
|
||
|
|
p : float, optional
|
||
|
|
Which Minkowski p-norm to use. Should be in the range [1, inf].
|
||
|
|
eps : nonnegative float, optional
|
||
|
|
Approximate search. Branches of the tree are not explored if their
|
||
|
|
nearest points are further than ``r / (1 + eps)``, and branches are
|
||
|
|
added in bulk if their furthest points are nearer than
|
||
|
|
``r * (1 + eps)``.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
results : list or array of lists
|
||
|
|
If `x` is a single point, returns a list of the indices of the
|
||
|
|
neighbors of `x`. If `x` is an array of points, returns an object
|
||
|
|
array of shape tuple containing lists of neighbors.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
If you have many points whose neighbors you want to find, you may save
|
||
|
|
substantial amounts of time by putting them in a KDTree and using
|
||
|
|
query_ball_tree.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy import spatial
|
||
|
|
>>> x, y = np.mgrid[0:5, 0:5]
|
||
|
|
>>> points = np.c_[x.ravel(), y.ravel()]
|
||
|
|
>>> tree = spatial.KDTree(points)
|
||
|
|
>>> tree.query_ball_point([2, 0], 1)
|
||
|
|
[5, 10, 11, 15]
|
||
|
|
|
||
|
|
Query multiple points and plot the results:
|
||
|
|
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> points = np.asarray(points)
|
||
|
|
>>> plt.plot(points[:,0], points[:,1], '.')
|
||
|
|
>>> for results in tree.query_ball_point(([2, 0], [3, 3]), 1):
|
||
|
|
... nearby_points = points[results]
|
||
|
|
... plt.plot(nearby_points[:,0], nearby_points[:,1], 'o')
|
||
|
|
>>> plt.margins(0.1, 0.1)
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = np.asarray(x)
|
||
|
|
if x.dtype.kind == 'c':
|
||
|
|
raise TypeError("KDTree does not work with complex data")
|
||
|
|
if x.shape[-1] != self.m:
|
||
|
|
raise ValueError("Searching for a %d-dimensional point in a "
|
||
|
|
"%d-dimensional KDTree" % (x.shape[-1], self.m))
|
||
|
|
if len(x.shape) == 1:
|
||
|
|
return self.__query_ball_point(x, r, p, eps)
|
||
|
|
else:
|
||
|
|
retshape = x.shape[:-1]
|
||
|
|
result = np.empty(retshape, dtype=object)
|
||
|
|
for c in np.ndindex(retshape):
|
||
|
|
result[c] = self.__query_ball_point(x[c], r, p=p, eps=eps)
|
||
|
|
return result
|
||
|
|
|
||
|
|
def query_ball_tree(self, other, r, p=2., eps=0):
|
||
|
|
"""Find all pairs of points whose distance is at most r
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
other : KDTree instance
|
||
|
|
The tree containing points to search against.
|
||
|
|
r : float
|
||
|
|
The maximum distance, has to be positive.
|
||
|
|
p : float, optional
|
||
|
|
Which Minkowski norm to use. `p` has to meet the condition
|
||
|
|
``1 <= p <= infinity``.
|
||
|
|
eps : float, optional
|
||
|
|
Approximate search. Branches of the tree are not explored
|
||
|
|
if their nearest points are further than ``r/(1+eps)``, and
|
||
|
|
branches are added in bulk if their furthest points are nearer
|
||
|
|
than ``r * (1+eps)``. `eps` has to be non-negative.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
results : list of lists
|
||
|
|
For each element ``self.data[i]`` of this tree, ``results[i]`` is a
|
||
|
|
list of the indices of its neighbors in ``other.data``.
|
||
|
|
|
||
|
|
"""
|
||
|
|
results = [[] for i in range(self.n)]
|
||
|
|
|
||
|
|
def traverse_checking(node1, rect1, node2, rect2):
|
||
|
|
if rect1.min_distance_rectangle(rect2, p) > r/(1.+eps):
|
||
|
|
return
|
||
|
|
elif rect1.max_distance_rectangle(rect2, p) < r*(1.+eps):
|
||
|
|
traverse_no_checking(node1, node2)
|
||
|
|
elif isinstance(node1, KDTree.leafnode):
|
||
|
|
if isinstance(node2, KDTree.leafnode):
|
||
|
|
d = other.data[node2.idx]
|
||
|
|
for i in node1.idx:
|
||
|
|
results[i] += node2.idx[minkowski_distance(d,self.data[i],p) <= r].tolist()
|
||
|
|
else:
|
||
|
|
less, greater = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse_checking(node1,rect1,node2.less,less)
|
||
|
|
traverse_checking(node1,rect1,node2.greater,greater)
|
||
|
|
elif isinstance(node2, KDTree.leafnode):
|
||
|
|
less, greater = rect1.split(node1.split_dim, node1.split)
|
||
|
|
traverse_checking(node1.less,less,node2,rect2)
|
||
|
|
traverse_checking(node1.greater,greater,node2,rect2)
|
||
|
|
else:
|
||
|
|
less1, greater1 = rect1.split(node1.split_dim, node1.split)
|
||
|
|
less2, greater2 = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse_checking(node1.less,less1,node2.less,less2)
|
||
|
|
traverse_checking(node1.less,less1,node2.greater,greater2)
|
||
|
|
traverse_checking(node1.greater,greater1,node2.less,less2)
|
||
|
|
traverse_checking(node1.greater,greater1,node2.greater,greater2)
|
||
|
|
|
||
|
|
def traverse_no_checking(node1, node2):
|
||
|
|
if isinstance(node1, KDTree.leafnode):
|
||
|
|
if isinstance(node2, KDTree.leafnode):
|
||
|
|
for i in node1.idx:
|
||
|
|
results[i] += node2.idx.tolist()
|
||
|
|
else:
|
||
|
|
traverse_no_checking(node1, node2.less)
|
||
|
|
traverse_no_checking(node1, node2.greater)
|
||
|
|
else:
|
||
|
|
traverse_no_checking(node1.less, node2)
|
||
|
|
traverse_no_checking(node1.greater, node2)
|
||
|
|
|
||
|
|
traverse_checking(self.tree, Rectangle(self.maxes, self.mins),
|
||
|
|
other.tree, Rectangle(other.maxes, other.mins))
|
||
|
|
return results
|
||
|
|
|
||
|
|
def query_pairs(self, r, p=2., eps=0):
|
||
|
|
"""
|
||
|
|
Find all pairs of points within a distance.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
r : positive float
|
||
|
|
The maximum distance.
|
||
|
|
p : float, optional
|
||
|
|
Which Minkowski norm to use. `p` has to meet the condition
|
||
|
|
``1 <= p <= infinity``.
|
||
|
|
eps : float, optional
|
||
|
|
Approximate search. Branches of the tree are not explored
|
||
|
|
if their nearest points are further than ``r/(1+eps)``, and
|
||
|
|
branches are added in bulk if their furthest points are nearer
|
||
|
|
than ``r * (1+eps)``. `eps` has to be non-negative.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
results : set
|
||
|
|
Set of pairs ``(i,j)``, with ``i < j``, for which the corresponding
|
||
|
|
positions are close.
|
||
|
|
|
||
|
|
"""
|
||
|
|
results = set()
|
||
|
|
|
||
|
|
def traverse_checking(node1, rect1, node2, rect2):
|
||
|
|
if rect1.min_distance_rectangle(rect2, p) > r/(1.+eps):
|
||
|
|
return
|
||
|
|
elif rect1.max_distance_rectangle(rect2, p) < r*(1.+eps):
|
||
|
|
traverse_no_checking(node1, node2)
|
||
|
|
elif isinstance(node1, KDTree.leafnode):
|
||
|
|
if isinstance(node2, KDTree.leafnode):
|
||
|
|
# Special care to avoid duplicate pairs
|
||
|
|
if id(node1) == id(node2):
|
||
|
|
d = self.data[node2.idx]
|
||
|
|
for i in node1.idx:
|
||
|
|
for j in node2.idx[minkowski_distance(d,self.data[i],p) <= r]:
|
||
|
|
if i < j:
|
||
|
|
results.add((i,j))
|
||
|
|
else:
|
||
|
|
d = self.data[node2.idx]
|
||
|
|
for i in node1.idx:
|
||
|
|
for j in node2.idx[minkowski_distance(d,self.data[i],p) <= r]:
|
||
|
|
if i < j:
|
||
|
|
results.add((i,j))
|
||
|
|
elif j < i:
|
||
|
|
results.add((j,i))
|
||
|
|
else:
|
||
|
|
less, greater = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse_checking(node1,rect1,node2.less,less)
|
||
|
|
traverse_checking(node1,rect1,node2.greater,greater)
|
||
|
|
elif isinstance(node2, KDTree.leafnode):
|
||
|
|
less, greater = rect1.split(node1.split_dim, node1.split)
|
||
|
|
traverse_checking(node1.less,less,node2,rect2)
|
||
|
|
traverse_checking(node1.greater,greater,node2,rect2)
|
||
|
|
else:
|
||
|
|
less1, greater1 = rect1.split(node1.split_dim, node1.split)
|
||
|
|
less2, greater2 = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse_checking(node1.less,less1,node2.less,less2)
|
||
|
|
traverse_checking(node1.less,less1,node2.greater,greater2)
|
||
|
|
|
||
|
|
# Avoid traversing (node1.less, node2.greater) and
|
||
|
|
# (node1.greater, node2.less) (it's the same node pair twice
|
||
|
|
# over, which is the source of the complication in the
|
||
|
|
# original KDTree.query_pairs)
|
||
|
|
if id(node1) != id(node2):
|
||
|
|
traverse_checking(node1.greater,greater1,node2.less,less2)
|
||
|
|
|
||
|
|
traverse_checking(node1.greater,greater1,node2.greater,greater2)
|
||
|
|
|
||
|
|
def traverse_no_checking(node1, node2):
|
||
|
|
if isinstance(node1, KDTree.leafnode):
|
||
|
|
if isinstance(node2, KDTree.leafnode):
|
||
|
|
# Special care to avoid duplicate pairs
|
||
|
|
if id(node1) == id(node2):
|
||
|
|
for i in node1.idx:
|
||
|
|
for j in node2.idx:
|
||
|
|
if i < j:
|
||
|
|
results.add((i,j))
|
||
|
|
else:
|
||
|
|
for i in node1.idx:
|
||
|
|
for j in node2.idx:
|
||
|
|
if i < j:
|
||
|
|
results.add((i,j))
|
||
|
|
elif j < i:
|
||
|
|
results.add((j,i))
|
||
|
|
else:
|
||
|
|
traverse_no_checking(node1, node2.less)
|
||
|
|
traverse_no_checking(node1, node2.greater)
|
||
|
|
else:
|
||
|
|
# Avoid traversing (node1.less, node2.greater) and
|
||
|
|
# (node1.greater, node2.less) (it's the same node pair twice
|
||
|
|
# over, which is the source of the complication in the
|
||
|
|
# original KDTree.query_pairs)
|
||
|
|
if id(node1) == id(node2):
|
||
|
|
traverse_no_checking(node1.less, node2.less)
|
||
|
|
traverse_no_checking(node1.less, node2.greater)
|
||
|
|
traverse_no_checking(node1.greater, node2.greater)
|
||
|
|
else:
|
||
|
|
traverse_no_checking(node1.less, node2)
|
||
|
|
traverse_no_checking(node1.greater, node2)
|
||
|
|
|
||
|
|
traverse_checking(self.tree, Rectangle(self.maxes, self.mins),
|
||
|
|
self.tree, Rectangle(self.maxes, self.mins))
|
||
|
|
return results
|
||
|
|
|
||
|
|
def count_neighbors(self, other, r, p=2.):
|
||
|
|
"""
|
||
|
|
Count how many nearby pairs can be formed.
|
||
|
|
|
||
|
|
Count the number of pairs (x1,x2) can be formed, with x1 drawn
|
||
|
|
from self and x2 drawn from ``other``, and where
|
||
|
|
``distance(x1, x2, p) <= r``.
|
||
|
|
This is the "two-point correlation" described in Gray and Moore 2000,
|
||
|
|
"N-body problems in statistical learning", and the code here is based
|
||
|
|
on their algorithm.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
other : KDTree instance
|
||
|
|
The other tree to draw points from.
|
||
|
|
r : float or one-dimensional array of floats
|
||
|
|
The radius to produce a count for. Multiple radii are searched with
|
||
|
|
a single tree traversal.
|
||
|
|
p : float, 1<=p<=infinity, optional
|
||
|
|
Which Minkowski p-norm to use
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
result : int or 1-D array of ints
|
||
|
|
The number of pairs. Note that this is internally stored in a numpy
|
||
|
|
int, and so may overflow if very large (2e9).
|
||
|
|
|
||
|
|
"""
|
||
|
|
def traverse(node1, rect1, node2, rect2, idx):
|
||
|
|
min_r = rect1.min_distance_rectangle(rect2,p)
|
||
|
|
max_r = rect1.max_distance_rectangle(rect2,p)
|
||
|
|
c_greater = r[idx] > max_r
|
||
|
|
result[idx[c_greater]] += node1.children*node2.children
|
||
|
|
idx = idx[(min_r <= r[idx]) & (r[idx] <= max_r)]
|
||
|
|
if len(idx) == 0:
|
||
|
|
return
|
||
|
|
|
||
|
|
if isinstance(node1,KDTree.leafnode):
|
||
|
|
if isinstance(node2,KDTree.leafnode):
|
||
|
|
ds = minkowski_distance(self.data[node1.idx][:,np.newaxis,:],
|
||
|
|
other.data[node2.idx][np.newaxis,:,:],
|
||
|
|
p).ravel()
|
||
|
|
ds.sort()
|
||
|
|
result[idx] += np.searchsorted(ds,r[idx],side='right')
|
||
|
|
else:
|
||
|
|
less, greater = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse(node1, rect1, node2.less, less, idx)
|
||
|
|
traverse(node1, rect1, node2.greater, greater, idx)
|
||
|
|
else:
|
||
|
|
if isinstance(node2,KDTree.leafnode):
|
||
|
|
less, greater = rect1.split(node1.split_dim, node1.split)
|
||
|
|
traverse(node1.less, less, node2, rect2, idx)
|
||
|
|
traverse(node1.greater, greater, node2, rect2, idx)
|
||
|
|
else:
|
||
|
|
less1, greater1 = rect1.split(node1.split_dim, node1.split)
|
||
|
|
less2, greater2 = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse(node1.less,less1,node2.less,less2,idx)
|
||
|
|
traverse(node1.less,less1,node2.greater,greater2,idx)
|
||
|
|
traverse(node1.greater,greater1,node2.less,less2,idx)
|
||
|
|
traverse(node1.greater,greater1,node2.greater,greater2,idx)
|
||
|
|
|
||
|
|
R1 = Rectangle(self.maxes, self.mins)
|
||
|
|
R2 = Rectangle(other.maxes, other.mins)
|
||
|
|
if np.shape(r) == ():
|
||
|
|
r = np.array([r])
|
||
|
|
result = np.zeros(1,dtype=int)
|
||
|
|
traverse(self.tree, R1, other.tree, R2, np.arange(1))
|
||
|
|
return result[0]
|
||
|
|
elif len(np.shape(r)) == 1:
|
||
|
|
r = np.asarray(r)
|
||
|
|
n, = r.shape
|
||
|
|
result = np.zeros(n,dtype=int)
|
||
|
|
traverse(self.tree, R1, other.tree, R2, np.arange(n))
|
||
|
|
return result
|
||
|
|
else:
|
||
|
|
raise ValueError("r must be either a single value or a one-dimensional array of values")
|
||
|
|
|
||
|
|
def sparse_distance_matrix(self, other, max_distance, p=2.):
|
||
|
|
"""
|
||
|
|
Compute a sparse distance matrix
|
||
|
|
|
||
|
|
Computes a distance matrix between two KDTrees, leaving as zero
|
||
|
|
any distance greater than max_distance.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
other : KDTree
|
||
|
|
|
||
|
|
max_distance : positive float
|
||
|
|
|
||
|
|
p : float, optional
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
result : dok_matrix
|
||
|
|
Sparse matrix representing the results in "dictionary of keys" format.
|
||
|
|
|
||
|
|
"""
|
||
|
|
result = scipy.sparse.dok_matrix((self.n,other.n))
|
||
|
|
|
||
|
|
def traverse(node1, rect1, node2, rect2):
|
||
|
|
if rect1.min_distance_rectangle(rect2, p) > max_distance:
|
||
|
|
return
|
||
|
|
elif isinstance(node1, KDTree.leafnode):
|
||
|
|
if isinstance(node2, KDTree.leafnode):
|
||
|
|
for i in node1.idx:
|
||
|
|
for j in node2.idx:
|
||
|
|
d = minkowski_distance(self.data[i],other.data[j],p)
|
||
|
|
if d <= max_distance:
|
||
|
|
result[i,j] = d
|
||
|
|
else:
|
||
|
|
less, greater = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse(node1,rect1,node2.less,less)
|
||
|
|
traverse(node1,rect1,node2.greater,greater)
|
||
|
|
elif isinstance(node2, KDTree.leafnode):
|
||
|
|
less, greater = rect1.split(node1.split_dim, node1.split)
|
||
|
|
traverse(node1.less,less,node2,rect2)
|
||
|
|
traverse(node1.greater,greater,node2,rect2)
|
||
|
|
else:
|
||
|
|
less1, greater1 = rect1.split(node1.split_dim, node1.split)
|
||
|
|
less2, greater2 = rect2.split(node2.split_dim, node2.split)
|
||
|
|
traverse(node1.less,less1,node2.less,less2)
|
||
|
|
traverse(node1.less,less1,node2.greater,greater2)
|
||
|
|
traverse(node1.greater,greater1,node2.less,less2)
|
||
|
|
traverse(node1.greater,greater1,node2.greater,greater2)
|
||
|
|
traverse(self.tree, Rectangle(self.maxes, self.mins),
|
||
|
|
other.tree, Rectangle(other.maxes, other.mins))
|
||
|
|
|
||
|
|
return result
|
||
|
|
|
||
|
|
|
||
|
|
def distance_matrix(x, y, p=2, threshold=1000000):
|
||
|
|
"""
|
||
|
|
Compute the distance matrix.
|
||
|
|
|
||
|
|
Returns the matrix of all pair-wise distances.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : (M, K) array_like
|
||
|
|
Matrix of M vectors in K dimensions.
|
||
|
|
y : (N, K) array_like
|
||
|
|
Matrix of N vectors in K dimensions.
|
||
|
|
p : float, 1 <= p <= infinity
|
||
|
|
Which Minkowski p-norm to use.
|
||
|
|
threshold : positive int
|
||
|
|
If ``M * N * K`` > `threshold`, algorithm uses a Python loop instead
|
||
|
|
of large temporary arrays.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
result : (M, N) ndarray
|
||
|
|
Matrix containing the distance from every vector in `x` to every vector
|
||
|
|
in `y`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy.spatial import distance_matrix
|
||
|
|
>>> distance_matrix([[0,0],[0,1]], [[1,0],[1,1]])
|
||
|
|
array([[ 1. , 1.41421356],
|
||
|
|
[ 1.41421356, 1. ]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
x = np.asarray(x)
|
||
|
|
m, k = x.shape
|
||
|
|
y = np.asarray(y)
|
||
|
|
n, kk = y.shape
|
||
|
|
|
||
|
|
if k != kk:
|
||
|
|
raise ValueError("x contains %d-dimensional vectors but y contains %d-dimensional vectors" % (k, kk))
|
||
|
|
|
||
|
|
if m*n*k <= threshold:
|
||
|
|
return minkowski_distance(x[:,np.newaxis,:],y[np.newaxis,:,:],p)
|
||
|
|
else:
|
||
|
|
result = np.empty((m,n),dtype=float) # FIXME: figure out the best dtype
|
||
|
|
if m < n:
|
||
|
|
for i in range(m):
|
||
|
|
result[i,:] = minkowski_distance(x[i],y,p)
|
||
|
|
else:
|
||
|
|
for j in range(n):
|
||
|
|
result[:,j] = minkowski_distance(x,y[j],p)
|
||
|
|
return result
|