""" Distance computations (:mod:`scipy.spatial.distance`) ===================================================== .. sectionauthor:: Damian Eads Function reference ------------------ Distance matrix computation from a collection of raw observation vectors stored in a rectangular array. .. autosummary:: :toctree: generated/ pdist -- pairwise distances between observation vectors. cdist -- distances between two collections of observation vectors squareform -- convert distance matrix to a condensed one and vice versa directed_hausdorff -- directed Hausdorff distance between arrays Predicates for checking the validity of distance matrices, both condensed and redundant. Also contained in this module are functions for computing the number of observations in a distance matrix. .. autosummary:: :toctree: generated/ is_valid_dm -- checks for a valid distance matrix is_valid_y -- checks for a valid condensed distance matrix num_obs_dm -- # of observations in a distance matrix num_obs_y -- # of observations in a condensed distance matrix Distance functions between two numeric vectors ``u`` and ``v``. Computing distances over a large collection of vectors is inefficient for these functions. Use ``pdist`` for this purpose. .. autosummary:: :toctree: generated/ braycurtis -- the Bray-Curtis distance. canberra -- the Canberra distance. chebyshev -- the Chebyshev distance. cityblock -- the Manhattan distance. correlation -- the Correlation distance. cosine -- the Cosine distance. euclidean -- the Euclidean distance. jensenshannon -- the Jensen-Shannon distance. mahalanobis -- the Mahalanobis distance. minkowski -- the Minkowski distance. seuclidean -- the normalized Euclidean distance. sqeuclidean -- the squared Euclidean distance. Distance functions between two boolean vectors (representing sets) ``u`` and ``v``. As in the case of numerical vectors, ``pdist`` is more efficient for computing the distances between all pairs. .. autosummary:: :toctree: generated/ dice -- the Dice dissimilarity. hamming -- the Hamming distance. jaccard -- the Jaccard distance. kulczynski1 -- the Kulczynski 1 distance. rogerstanimoto -- the Rogers-Tanimoto dissimilarity. russellrao -- the Russell-Rao dissimilarity. sokalmichener -- the Sokal-Michener dissimilarity. sokalsneath -- the Sokal-Sneath dissimilarity. yule -- the Yule dissimilarity. :func:`hamming` also operates over discrete numerical vectors. """ # Copyright (C) Damian Eads, 2007-2008. New BSD License. __all__ = [ 'braycurtis', 'canberra', 'cdist', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice', 'directed_hausdorff', 'euclidean', 'hamming', 'is_valid_dm', 'is_valid_y', 'jaccard', 'jensenshannon', 'kulczynski1', 'mahalanobis', 'minkowski', 'num_obs_dm', 'num_obs_y', 'pdist', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'squareform', 'yule' ] import math import warnings import numpy as np import dataclasses from typing import Optional, Callable from functools import partial from scipy._lib._util import _asarray_validated from . import _distance_wrap from . import _hausdorff from ..linalg import norm from ..special import rel_entr from . import _distance_pybind def _copy_array_if_base_present(a): """Copy the array if its base points to a parent array.""" if a.base is not None: return a.copy() return a def _correlation_cdist_wrap(XA, XB, dm, **kwargs): XA = XA - XA.mean(axis=1, keepdims=True) XB = XB - XB.mean(axis=1, keepdims=True) _distance_wrap.cdist_cosine_double_wrap(XA, XB, dm, **kwargs) def _correlation_pdist_wrap(X, dm, **kwargs): X2 = X - X.mean(axis=1, keepdims=True) _distance_wrap.pdist_cosine_double_wrap(X2, dm, **kwargs) def _convert_to_type(X, out_type): return np.ascontiguousarray(X, dtype=out_type) def _nbool_correspond_all(u, v, w=None): if u.dtype == v.dtype == bool and w is None: not_u = ~u not_v = ~v nff = (not_u & not_v).sum() nft = (not_u & v).sum() ntf = (u & not_v).sum() ntt = (u & v).sum() else: dtype = np.result_type(int, u.dtype, v.dtype) u = u.astype(dtype) v = v.astype(dtype) not_u = 1.0 - u not_v = 1.0 - v if w is not None: not_u = w * not_u u = w * u nff = (not_u * not_v).sum() nft = (not_u * v).sum() ntf = (u * not_v).sum() ntt = (u * v).sum() return (nff, nft, ntf, ntt) def _nbool_correspond_ft_tf(u, v, w=None): if u.dtype == v.dtype == bool and w is None: not_u = ~u not_v = ~v nft = (not_u & v).sum() ntf = (u & not_v).sum() else: dtype = np.result_type(int, u.dtype, v.dtype) u = u.astype(dtype) v = v.astype(dtype) not_u = 1.0 - u not_v = 1.0 - v if w is not None: not_u = w * not_u u = w * u nft = (not_u * v).sum() ntf = (u * not_v).sum() return (nft, ntf) def _validate_cdist_input(XA, XB, mA, mB, n, metric_info, **kwargs): # get supported types types = metric_info.types # choose best type typ = types[types.index(XA.dtype)] if XA.dtype in types else types[0] # validate data XA = _convert_to_type(XA, out_type=typ) XB = _convert_to_type(XB, out_type=typ) # validate kwargs _validate_kwargs = metric_info.validator if _validate_kwargs: kwargs = _validate_kwargs((XA, XB), mA + mB, n, **kwargs) return XA, XB, typ, kwargs def _validate_weight_with_size(X, m, n, **kwargs): w = kwargs.pop('w', None) if w is None: return kwargs if w.ndim != 1 or w.shape[0] != n: raise ValueError("Weights must have same size as input vector. " f"{w.shape[0]} vs. {n}") kwargs['w'] = _validate_weights(w) return kwargs def _validate_hamming_kwargs(X, m, n, **kwargs): w = kwargs.get('w', np.ones((n,), dtype='double')) if w.ndim != 1 or w.shape[0] != n: raise ValueError( "Weights must have same size as input vector. %d vs. %d" % (w.shape[0], n) ) kwargs['w'] = _validate_weights(w) return kwargs def _validate_mahalanobis_kwargs(X, m, n, **kwargs): VI = kwargs.pop('VI', None) if VI is None: if m <= n: # There are fewer observations than the dimension of # the observations. raise ValueError("The number of observations (%d) is too " "small; the covariance matrix is " "singular. For observations with %d " "dimensions, at least %d observations " "are required." % (m, n, n + 1)) if isinstance(X, tuple): X = np.vstack(X) CV = np.atleast_2d(np.cov(X.astype(np.float64, copy=False).T)) VI = np.linalg.inv(CV).T.copy() kwargs["VI"] = _convert_to_double(VI) return kwargs def _validate_minkowski_kwargs(X, m, n, **kwargs): kwargs = _validate_weight_with_size(X, m, n, **kwargs) if 'p' not in kwargs: kwargs['p'] = 2. else: if kwargs['p'] <= 0: raise ValueError("p must be greater than 0") return kwargs def _validate_pdist_input(X, m, n, metric_info, **kwargs): # get supported types types = metric_info.types # choose best type typ = types[types.index(X.dtype)] if X.dtype in types else types[0] # validate data X = _convert_to_type(X, out_type=typ) # validate kwargs _validate_kwargs = metric_info.validator if _validate_kwargs: kwargs = _validate_kwargs(X, m, n, **kwargs) return X, typ, kwargs def _validate_seuclidean_kwargs(X, m, n, **kwargs): V = kwargs.pop('V', None) if V is None: if isinstance(X, tuple): X = np.vstack(X) V = np.var(X.astype(np.float64, copy=False), axis=0, ddof=1) else: V = np.asarray(V, order='c') if len(V.shape) != 1: raise ValueError('Variance vector V must ' 'be one-dimensional.') if V.shape[0] != n: raise ValueError('Variance vector V must be of the same ' 'dimension as the vectors on which the distances ' 'are computed.') kwargs['V'] = _convert_to_double(V) return kwargs def _validate_vector(u, dtype=None): # XXX Is order='c' really necessary? u = np.asarray(u, dtype=dtype, order='c') if u.ndim == 1: return u raise ValueError("Input vector should be 1-D.") def _validate_weights(w, dtype=np.float64): w = _validate_vector(w, dtype=dtype) if np.any(w < 0): raise ValueError("Input weights should be all non-negative") return w def directed_hausdorff(u, v, seed=0): """ Compute the directed Hausdorff distance between two 2-D arrays. Distances between pairs are calculated using a Euclidean metric. Parameters ---------- u : (M,N) array_like Input array with M points in N dimensions. v : (O,N) array_like Input array with O points in N dimensions. seed : int or None, optional Local `numpy.random.RandomState` seed. Default is 0, a random shuffling of u and v that guarantees reproducibility. Returns ------- d : double The directed Hausdorff distance between arrays `u` and `v`, index_1 : int index of point contributing to Hausdorff pair in `u` index_2 : int index of point contributing to Hausdorff pair in `v` Raises ------ ValueError An exception is thrown if `u` and `v` do not have the same number of columns. See Also -------- scipy.spatial.procrustes : Another similarity test for two data sets Notes ----- Uses the early break technique and the random sampling approach described by [1]_. Although worst-case performance is ``O(m * o)`` (as with the brute force algorithm), this is unlikely in practice as the input data would have to require the algorithm to explore every single point interaction, and after the algorithm shuffles the input points at that. The best case performance is O(m), which is satisfied by selecting an inner loop distance that is less than cmax and leads to an early break as often as possible. The authors have formally shown that the average runtime is closer to O(m). .. versionadded:: 0.19.0 References ---------- .. [1] A. A. Taha and A. Hanbury, "An efficient algorithm for calculating the exact Hausdorff distance." IEEE Transactions On Pattern Analysis And Machine Intelligence, vol. 37 pp. 2153-63, 2015. Examples -------- Find the directed Hausdorff distance between two 2-D arrays of coordinates: >>> from scipy.spatial.distance import directed_hausdorff >>> import numpy as np >>> u = np.array([(1.0, 0.0), ... (0.0, 1.0), ... (-1.0, 0.0), ... (0.0, -1.0)]) >>> v = np.array([(2.0, 0.0), ... (0.0, 2.0), ... (-2.0, 0.0), ... (0.0, -4.0)]) >>> directed_hausdorff(u, v)[0] 2.23606797749979 >>> directed_hausdorff(v, u)[0] 3.0 Find the general (symmetric) Hausdorff distance between two 2-D arrays of coordinates: >>> max(directed_hausdorff(u, v)[0], directed_hausdorff(v, u)[0]) 3.0 Find the indices of the points that generate the Hausdorff distance (the Hausdorff pair): >>> directed_hausdorff(v, u)[1:] (3, 3) """ u = np.asarray(u, dtype=np.float64, order='c') v = np.asarray(v, dtype=np.float64, order='c') if u.shape[1] != v.shape[1]: raise ValueError('u and v need to have the same ' 'number of columns') result = _hausdorff.directed_hausdorff(u, v, seed) return result def minkowski(u, v, p=2, w=None): """ Compute the Minkowski distance between two 1-D arrays. The Minkowski distance between 1-D arrays `u` and `v`, is defined as .. math:: {\\|u-v\\|}_p = (\\sum{|u_i - v_i|^p})^{1/p}. \\left(\\sum{w_i(|(u_i - v_i)|^p)}\\right)^{1/p}. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. p : scalar The order of the norm of the difference :math:`{\\|u-v\\|}_p`. Note that for :math:`0 < p < 1`, the triangle inequality only holds with an additional multiplicative factor, i.e. it is only a quasi-metric. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- minkowski : double The Minkowski distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.minkowski([1, 0, 0], [0, 1, 0], 1) 2.0 >>> distance.minkowski([1, 0, 0], [0, 1, 0], 2) 1.4142135623730951 >>> distance.minkowski([1, 0, 0], [0, 1, 0], 3) 1.2599210498948732 >>> distance.minkowski([1, 1, 0], [0, 1, 0], 1) 1.0 >>> distance.minkowski([1, 1, 0], [0, 1, 0], 2) 1.0 >>> distance.minkowski([1, 1, 0], [0, 1, 0], 3) 1.0 """ u = _validate_vector(u) v = _validate_vector(v) if p <= 0: raise ValueError("p must be greater than 0") u_v = u - v if w is not None: w = _validate_weights(w) if p == 1: root_w = w elif p == 2: # better precision and speed root_w = np.sqrt(w) elif p == np.inf: root_w = (w != 0) else: root_w = np.power(w, 1/p) u_v = root_w * u_v dist = norm(u_v, ord=p) return dist def euclidean(u, v, w=None): """ Computes the Euclidean distance between two 1-D arrays. The Euclidean distance between 1-D arrays `u` and `v`, is defined as .. math:: {\\|u-v\\|}_2 \\left(\\sum{(w_i |(u_i - v_i)|^2)}\\right)^{1/2} Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- euclidean : double The Euclidean distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.euclidean([1, 0, 0], [0, 1, 0]) 1.4142135623730951 >>> distance.euclidean([1, 1, 0], [0, 1, 0]) 1.0 """ return minkowski(u, v, p=2, w=w) def sqeuclidean(u, v, w=None): """ Compute the squared Euclidean distance between two 1-D arrays. The squared Euclidean distance between `u` and `v` is defined as .. math:: \\sum_i{w_i |u_i - v_i|^2} Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- sqeuclidean : double The squared Euclidean distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.sqeuclidean([1, 0, 0], [0, 1, 0]) 2.0 >>> distance.sqeuclidean([1, 1, 0], [0, 1, 0]) 1.0 """ # Preserve float dtypes, but convert everything else to np.float64 # for stability. utype, vtype = None, None if not (hasattr(u, "dtype") and np.issubdtype(u.dtype, np.inexact)): utype = np.float64 if not (hasattr(v, "dtype") and np.issubdtype(v.dtype, np.inexact)): vtype = np.float64 u = _validate_vector(u, dtype=utype) v = _validate_vector(v, dtype=vtype) u_v = u - v u_v_w = u_v # only want weights applied once if w is not None: w = _validate_weights(w) u_v_w = w * u_v return np.dot(u_v, u_v_w) def correlation(u, v, w=None, centered=True): """ Compute the correlation distance between two 1-D arrays. The correlation distance between `u` and `v`, is defined as .. math:: 1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})} {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2} where :math:`\\bar{u}` is the mean of the elements of `u` and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 centered : bool, optional If True, `u` and `v` will be centered. Default is True. Returns ------- correlation : double The correlation distance between 1-D array `u` and `v`. Examples -------- Find the correlation between two arrays. >>> from scipy.spatial.distance import correlation >>> correlation([1, 0, 1], [1, 1, 0]) 1.5 Using a weighting array, the correlation can be calculated as: >>> correlation([1, 0, 1], [1, 1, 0], w=[0.9, 0.1, 0.1]) 1.1 If centering is not needed, the correlation can be calculated as: >>> correlation([1, 0, 1], [1, 1, 0], centered=False) 0.5 """ u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) w = w / w.sum() if centered: if w is not None: umu = np.dot(u, w) vmu = np.dot(v, w) else: umu = np.mean(u) vmu = np.mean(v) u = u - umu v = v - vmu if w is not None: vw = v * w uw = u * w else: vw, uw = v, u uv = np.dot(u, vw) uu = np.dot(u, uw) vv = np.dot(v, vw) dist = 1.0 - uv / math.sqrt(uu * vv) # Clip the result to avoid rounding error return np.clip(dist, 0.0, 2.0) def cosine(u, v, w=None): """ Compute the Cosine distance between 1-D arrays. The Cosine distance between `u` and `v`, is defined as .. math:: 1 - \\frac{u \\cdot v} {\\|u\\|_2 \\|v\\|_2}. where :math:`u \\cdot v` is the dot product of :math:`u` and :math:`v`. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- cosine : double The Cosine distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.cosine([1, 0, 0], [0, 1, 0]) 1.0 >>> distance.cosine([100, 0, 0], [0, 1, 0]) 1.0 >>> distance.cosine([1, 1, 0], [0, 1, 0]) 0.29289321881345254 """ # cosine distance is also referred to as 'uncentered correlation', # or 'reflective correlation' return correlation(u, v, w=w, centered=False) def hamming(u, v, w=None): """ Compute the Hamming distance between two 1-D arrays. The Hamming distance between 1-D arrays `u` and `v`, is simply the proportion of disagreeing components in `u` and `v`. If `u` and `v` are boolean vectors, the Hamming distance is .. math:: \\frac{c_{01} + c_{10}}{n} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- hamming : double The Hamming distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.hamming([1, 0, 0], [0, 1, 0]) 0.66666666666666663 >>> distance.hamming([1, 0, 0], [1, 1, 0]) 0.33333333333333331 >>> distance.hamming([1, 0, 0], [2, 0, 0]) 0.33333333333333331 >>> distance.hamming([1, 0, 0], [3, 0, 0]) 0.33333333333333331 """ u = _validate_vector(u) v = _validate_vector(v) if u.shape != v.shape: raise ValueError('The 1d arrays must have equal lengths.') u_ne_v = u != v if w is not None: w = _validate_weights(w) if w.shape != u.shape: raise ValueError("'w' should have the same length as 'u' and 'v'.") w = w / w.sum() return np.dot(u_ne_v, w) return np.mean(u_ne_v) def jaccard(u, v, w=None): """ Compute the Jaccard-Needham dissimilarity between two boolean 1-D arrays. The Jaccard-Needham dissimilarity between 1-D boolean arrays `u` and `v`, is defined as .. math:: \\frac{c_{TF} + c_{FT}} {c_{TT} + c_{FT} + c_{TF}} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : (N,) array_like, bool Input array. v : (N,) array_like, bool Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- jaccard : double The Jaccard distance between vectors `u` and `v`. Notes ----- When both `u` and `v` lead to a `0/0` division i.e. there is no overlap between the items in the vectors the returned distance is 0. See the Wikipedia page on the Jaccard index [1]_, and this paper [2]_. .. versionchanged:: 1.2.0 Previously, when `u` and `v` lead to a `0/0` division, the function would return NaN. This was changed to return 0 instead. References ---------- .. [1] https://en.wikipedia.org/wiki/Jaccard_index .. [2] S. Kosub, "A note on the triangle inequality for the Jaccard distance", 2016, :arxiv:`1612.02696` Examples -------- >>> from scipy.spatial import distance >>> distance.jaccard([1, 0, 0], [0, 1, 0]) 1.0 >>> distance.jaccard([1, 0, 0], [1, 1, 0]) 0.5 >>> distance.jaccard([1, 0, 0], [1, 2, 0]) 0.5 >>> distance.jaccard([1, 0, 0], [1, 1, 1]) 0.66666666666666663 """ u = _validate_vector(u) v = _validate_vector(v) nonzero = np.bitwise_or(u != 0, v != 0) unequal_nonzero = np.bitwise_and((u != v), nonzero) if w is not None: w = _validate_weights(w) nonzero = w * nonzero unequal_nonzero = w * unequal_nonzero a = np.float64(unequal_nonzero.sum()) b = np.float64(nonzero.sum()) return (a / b) if b != 0 else 0 def kulczynski1(u, v, *, w=None): """ Compute the Kulczynski 1 dissimilarity between two boolean 1-D arrays. The Kulczynski 1 dissimilarity between two boolean 1-D arrays `u` and `v` of length ``n``, is defined as .. math:: \\frac{c_{11}} {c_{01} + c_{10}} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k \\in {0, 1, ..., n-1}`. Parameters ---------- u : (N,) array_like, bool Input array. v : (N,) array_like, bool Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- kulczynski1 : float The Kulczynski 1 distance between vectors `u` and `v`. Notes ----- This measure has a minimum value of 0 and no upper limit. It is un-defined when there are no non-matches. .. versionadded:: 1.8.0 References ---------- .. [1] Kulczynski S. et al. Bulletin International de l'Academie Polonaise des Sciences et des Lettres, Classe des Sciences Mathematiques et Naturelles, Serie B (Sciences Naturelles). 1927; Supplement II: 57-203. Examples -------- >>> from scipy.spatial import distance >>> distance.kulczynski1([1, 0, 0], [0, 1, 0]) 0.0 >>> distance.kulczynski1([True, False, False], [True, True, False]) 1.0 >>> distance.kulczynski1([True, False, False], [True]) 0.5 >>> distance.kulczynski1([1, 0, 0], [3, 1, 0]) -3.0 """ u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) (_, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w) return ntt / (ntf + nft) def seuclidean(u, v, V): """ Return the standardized Euclidean distance between two 1-D arrays. The standardized Euclidean distance between two n-vectors `u` and `v` is .. math:: \\sqrt{\\sum\\limits_i \\frac{1}{V_i} \\left(u_i-v_i \\right)^2} ``V`` is the variance vector; ``V[I]`` is the variance computed over all the i-th components of the points. If not passed, it is automatically computed. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. V : (N,) array_like `V` is an 1-D array of component variances. It is usually computed among a larger collection vectors. Returns ------- seuclidean : double The standardized Euclidean distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [0.1, 0.1, 0.1]) 4.4721359549995796 >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [1, 0.1, 0.1]) 3.3166247903553998 >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [10, 0.1, 0.1]) 3.1780497164141406 """ u = _validate_vector(u) v = _validate_vector(v) V = _validate_vector(V, dtype=np.float64) if V.shape[0] != u.shape[0] or u.shape[0] != v.shape[0]: raise TypeError('V must be a 1-D array of the same dimension ' 'as u and v.') return euclidean(u, v, w=1/V) def cityblock(u, v, w=None): """ Compute the City Block (Manhattan) distance. Computes the Manhattan distance between two 1-D arrays `u` and `v`, which is defined as .. math:: \\sum_i {\\left| u_i - v_i \\right|}. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- cityblock : double The City Block (Manhattan) distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.cityblock([1, 0, 0], [0, 1, 0]) 2 >>> distance.cityblock([1, 0, 0], [0, 2, 0]) 3 >>> distance.cityblock([1, 0, 0], [1, 1, 0]) 1 """ u = _validate_vector(u) v = _validate_vector(v) l1_diff = abs(u - v) if w is not None: w = _validate_weights(w) l1_diff = w * l1_diff return l1_diff.sum() def mahalanobis(u, v, VI): """ Compute the Mahalanobis distance between two 1-D arrays. The Mahalanobis distance between 1-D arrays `u` and `v`, is defined as .. math:: \\sqrt{ (u-v) V^{-1} (u-v)^T } where ``V`` is the covariance matrix. Note that the argument `VI` is the inverse of ``V``. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. VI : array_like The inverse of the covariance matrix. Returns ------- mahalanobis : double The Mahalanobis distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> iv = [[1, 0.5, 0.5], [0.5, 1, 0.5], [0.5, 0.5, 1]] >>> distance.mahalanobis([1, 0, 0], [0, 1, 0], iv) 1.0 >>> distance.mahalanobis([0, 2, 0], [0, 1, 0], iv) 1.0 >>> distance.mahalanobis([2, 0, 0], [0, 1, 0], iv) 1.7320508075688772 """ u = _validate_vector(u) v = _validate_vector(v) VI = np.atleast_2d(VI) delta = u - v m = np.dot(np.dot(delta, VI), delta) return np.sqrt(m) def chebyshev(u, v, w=None): """ Compute the Chebyshev distance. Computes the Chebyshev distance between two 1-D arrays `u` and `v`, which is defined as .. math:: \\max_i {|u_i-v_i|}. Parameters ---------- u : (N,) array_like Input vector. v : (N,) array_like Input vector. w : (N,) array_like, optional Unused, as 'max' is a weightless operation. Here for API consistency. Returns ------- chebyshev : double The Chebyshev distance between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.chebyshev([1, 0, 0], [0, 1, 0]) 1 >>> distance.chebyshev([1, 1, 0], [0, 1, 0]) 1 """ u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) has_weight = w > 0 if has_weight.sum() < w.size: u = u[has_weight] v = v[has_weight] return max(abs(u - v)) def braycurtis(u, v, w=None): """ Compute the Bray-Curtis distance between two 1-D arrays. Bray-Curtis distance is defined as .. math:: \\sum{|u_i-v_i|} / \\sum{|u_i+v_i|} The Bray-Curtis distance is in the range [0, 1] if all coordinates are positive, and is undefined if the inputs are of length zero. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- braycurtis : double The Bray-Curtis distance between 1-D arrays `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.braycurtis([1, 0, 0], [0, 1, 0]) 1.0 >>> distance.braycurtis([1, 1, 0], [0, 1, 0]) 0.33333333333333331 """ u = _validate_vector(u) v = _validate_vector(v, dtype=np.float64) l1_diff = abs(u - v) l1_sum = abs(u + v) if w is not None: w = _validate_weights(w) l1_diff = w * l1_diff l1_sum = w * l1_sum return l1_diff.sum() / l1_sum.sum() def canberra(u, v, w=None): """ Compute the Canberra distance between two 1-D arrays. The Canberra distance is defined as .. math:: d(u,v) = \\sum_i \\frac{|u_i-v_i|} {|u_i|+|v_i|}. Parameters ---------- u : (N,) array_like Input array. v : (N,) array_like Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- canberra : double The Canberra distance between vectors `u` and `v`. Notes ----- When `u[i]` and `v[i]` are 0 for given i, then the fraction 0/0 = 0 is used in the calculation. Examples -------- >>> from scipy.spatial import distance >>> distance.canberra([1, 0, 0], [0, 1, 0]) 2.0 >>> distance.canberra([1, 1, 0], [0, 1, 0]) 1.0 """ u = _validate_vector(u) v = _validate_vector(v, dtype=np.float64) if w is not None: w = _validate_weights(w) with np.errstate(invalid='ignore'): abs_uv = abs(u - v) abs_u = abs(u) abs_v = abs(v) d = abs_uv / (abs_u + abs_v) if w is not None: d = w * d d = np.nansum(d) return d def jensenshannon(p, q, base=None, *, axis=0, keepdims=False): """ Compute the Jensen-Shannon distance (metric) between two probability arrays. This is the square root of the Jensen-Shannon divergence. The Jensen-Shannon distance between two probability vectors `p` and `q` is defined as, .. math:: \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}} where :math:`m` is the pointwise mean of :math:`p` and :math:`q` and :math:`D` is the Kullback-Leibler divergence. This routine will normalize `p` and `q` if they don't sum to 1.0. Parameters ---------- p : (N,) array_like left probability vector q : (N,) array_like right probability vector base : double, optional the base of the logarithm used to compute the output if not given, then the routine uses the default base of scipy.stats.entropy. axis : int, optional Axis along which the Jensen-Shannon distances are computed. The default is 0. .. versionadded:: 1.7.0 keepdims : bool, optional If this is set to `True`, the reduced axes are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array. Default is False. .. versionadded:: 1.7.0 Returns ------- js : double or ndarray The Jensen-Shannon distances between `p` and `q` along the `axis`. Notes ----- .. versionadded:: 1.2.0 Examples -------- >>> from scipy.spatial import distance >>> import numpy as np >>> distance.jensenshannon([1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 2.0) 1.0 >>> distance.jensenshannon([1.0, 0.0], [0.5, 0.5]) 0.46450140402245893 >>> distance.jensenshannon([1.0, 0.0, 0.0], [1.0, 0.0, 0.0]) 0.0 >>> a = np.array([[1, 2, 3, 4], ... [5, 6, 7, 8], ... [9, 10, 11, 12]]) >>> b = np.array([[13, 14, 15, 16], ... [17, 18, 19, 20], ... [21, 22, 23, 24]]) >>> distance.jensenshannon(a, b, axis=0) array([0.1954288, 0.1447697, 0.1138377, 0.0927636]) >>> distance.jensenshannon(a, b, axis=1) array([0.1402339, 0.0399106, 0.0201815]) """ p = np.asarray(p) q = np.asarray(q) p = p / np.sum(p, axis=axis, keepdims=True) q = q / np.sum(q, axis=axis, keepdims=True) m = (p + q) / 2.0 left = rel_entr(p, m) right = rel_entr(q, m) left_sum = np.sum(left, axis=axis, keepdims=keepdims) right_sum = np.sum(right, axis=axis, keepdims=keepdims) js = left_sum + right_sum if base is not None: js /= np.log(base) return np.sqrt(js / 2.0) def yule(u, v, w=None): """ Compute the Yule dissimilarity between two boolean 1-D arrays. The Yule dissimilarity is defined as .. math:: \\frac{R}{c_{TT} * c_{FF} + \\frac{R}{2}} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n` and :math:`R = 2.0 * c_{TF} * c_{FT}`. Parameters ---------- u : (N,) array_like, bool Input array. v : (N,) array_like, bool Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- yule : double The Yule dissimilarity between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.yule([1, 0, 0], [0, 1, 0]) 2.0 >>> distance.yule([1, 1, 0], [0, 1, 0]) 0.0 """ u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w) half_R = ntf * nft if half_R == 0: return 0.0 else: return float(2.0 * half_R / (ntt * nff + half_R)) def dice(u, v, w=None): """ Compute the Dice dissimilarity between two boolean 1-D arrays. The Dice dissimilarity between `u` and `v`, is .. math:: \\frac{c_{TF} + c_{FT}} {2c_{TT} + c_{FT} + c_{TF}} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : (N,) array_like, bool Input 1-D array. v : (N,) array_like, bool Input 1-D array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- dice : double The Dice dissimilarity between 1-D arrays `u` and `v`. Notes ----- This function computes the Dice dissimilarity index. To compute the Dice similarity index, convert one to the other with similarity = 1 - dissimilarity. Examples -------- >>> from scipy.spatial import distance >>> distance.dice([1, 0, 0], [0, 1, 0]) 1.0 >>> distance.dice([1, 0, 0], [1, 1, 0]) 0.3333333333333333 >>> distance.dice([1, 0, 0], [2, 0, 0]) -0.3333333333333333 """ u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) if u.dtype == v.dtype == bool and w is None: ntt = (u & v).sum() else: dtype = np.result_type(int, u.dtype, v.dtype) u = u.astype(dtype) v = v.astype(dtype) if w is None: ntt = (u * v).sum() else: ntt = (u * v * w).sum() (nft, ntf) = _nbool_correspond_ft_tf(u, v, w=w) return float((ntf + nft) / np.array(2.0 * ntt + ntf + nft)) def rogerstanimoto(u, v, w=None): """ Compute the Rogers-Tanimoto dissimilarity between two boolean 1-D arrays. The Rogers-Tanimoto dissimilarity between two boolean 1-D arrays `u` and `v`, is defined as .. math:: \\frac{R} {c_{TT} + c_{FF} + R} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`. Parameters ---------- u : (N,) array_like, bool Input array. v : (N,) array_like, bool Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- rogerstanimoto : double The Rogers-Tanimoto dissimilarity between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.rogerstanimoto([1, 0, 0], [0, 1, 0]) 0.8 >>> distance.rogerstanimoto([1, 0, 0], [1, 1, 0]) 0.5 >>> distance.rogerstanimoto([1, 0, 0], [2, 0, 0]) -1.0 """ u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w) return float(2.0 * (ntf + nft)) / float(ntt + nff + (2.0 * (ntf + nft))) def russellrao(u, v, w=None): """ Compute the Russell-Rao dissimilarity between two boolean 1-D arrays. The Russell-Rao dissimilarity between two boolean 1-D arrays, `u` and `v`, is defined as .. math:: \\frac{n - c_{TT}} {n} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : (N,) array_like, bool Input array. v : (N,) array_like, bool Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- russellrao : double The Russell-Rao dissimilarity between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.russellrao([1, 0, 0], [0, 1, 0]) 1.0 >>> distance.russellrao([1, 0, 0], [1, 1, 0]) 0.6666666666666666 >>> distance.russellrao([1, 0, 0], [2, 0, 0]) 0.3333333333333333 """ u = _validate_vector(u) v = _validate_vector(v) if u.dtype == v.dtype == bool and w is None: ntt = (u & v).sum() n = float(len(u)) elif w is None: ntt = (u * v).sum() n = float(len(u)) else: w = _validate_weights(w) ntt = (u * v * w).sum() n = w.sum() return float(n - ntt) / n def sokalmichener(u, v, w=None): """ Compute the Sokal-Michener dissimilarity between two boolean 1-D arrays. The Sokal-Michener dissimilarity between boolean 1-D arrays `u` and `v`, is defined as .. math:: \\frac{R} {S + R} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n`, :math:`R = 2 * (c_{TF} + c_{FT})` and :math:`S = c_{FF} + c_{TT}`. Parameters ---------- u : (N,) array_like, bool Input array. v : (N,) array_like, bool Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- sokalmichener : double The Sokal-Michener dissimilarity between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.sokalmichener([1, 0, 0], [0, 1, 0]) 0.8 >>> distance.sokalmichener([1, 0, 0], [1, 1, 0]) 0.5 >>> distance.sokalmichener([1, 0, 0], [2, 0, 0]) -1.0 """ u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) nff, nft, ntf, ntt = _nbool_correspond_all(u, v, w=w) return float(2.0 * (ntf + nft)) / float(ntt + nff + 2.0 * (ntf + nft)) def sokalsneath(u, v, w=None): """ Compute the Sokal-Sneath dissimilarity between two boolean 1-D arrays. The Sokal-Sneath dissimilarity between `u` and `v`, .. math:: \\frac{R} {c_{TT} + R} where :math:`c_{ij}` is the number of occurrences of :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`. Parameters ---------- u : (N,) array_like, bool Input array. v : (N,) array_like, bool Input array. w : (N,) array_like, optional The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0 Returns ------- sokalsneath : double The Sokal-Sneath dissimilarity between vectors `u` and `v`. Examples -------- >>> from scipy.spatial import distance >>> distance.sokalsneath([1, 0, 0], [0, 1, 0]) 1.0 >>> distance.sokalsneath([1, 0, 0], [1, 1, 0]) 0.66666666666666663 >>> distance.sokalsneath([1, 0, 0], [2, 1, 0]) 0.0 >>> distance.sokalsneath([1, 0, 0], [3, 1, 0]) -2.0 """ u = _validate_vector(u) v = _validate_vector(v) if u.dtype == v.dtype == bool and w is None: ntt = (u & v).sum() elif w is None: ntt = (u * v).sum() else: w = _validate_weights(w) ntt = (u * v * w).sum() (nft, ntf) = _nbool_correspond_ft_tf(u, v, w=w) denom = np.array(ntt + 2.0 * (ntf + nft)) if not denom.any(): raise ValueError('Sokal-Sneath dissimilarity is not defined for ' 'vectors that are entirely false.') return float(2.0 * (ntf + nft)) / denom _convert_to_double = partial(_convert_to_type, out_type=np.float64) _convert_to_bool = partial(_convert_to_type, out_type=bool) # adding python-only wrappers to _distance_wrap module _distance_wrap.pdist_correlation_double_wrap = _correlation_pdist_wrap _distance_wrap.cdist_correlation_double_wrap = _correlation_cdist_wrap @dataclasses.dataclass(frozen=True) class CDistMetricWrapper: metric_name: str def __call__(self, XA, XB, *, out=None, **kwargs): XA = np.ascontiguousarray(XA) XB = np.ascontiguousarray(XB) mA, n = XA.shape mB, _ = XB.shape metric_name = self.metric_name metric_info = _METRICS[metric_name] XA, XB, typ, kwargs = _validate_cdist_input( XA, XB, mA, mB, n, metric_info, **kwargs) w = kwargs.pop('w', None) if w is not None: metric = metric_info.dist_func return _cdist_callable( XA, XB, metric=metric, out=out, w=w, **kwargs) dm = _prepare_out_argument(out, np.float64, (mA, mB)) # get cdist wrapper cdist_fn = getattr(_distance_wrap, f'cdist_{metric_name}_{typ}_wrap') cdist_fn(XA, XB, dm, **kwargs) return dm @dataclasses.dataclass(frozen=True) class PDistMetricWrapper: metric_name: str def __call__(self, X, *, out=None, **kwargs): X = np.ascontiguousarray(X) m, n = X.shape metric_name = self.metric_name metric_info = _METRICS[metric_name] X, typ, kwargs = _validate_pdist_input( X, m, n, metric_info, **kwargs) out_size = (m * (m - 1)) // 2 w = kwargs.pop('w', None) if w is not None: metric = metric_info.dist_func return _pdist_callable( X, metric=metric, out=out, w=w, **kwargs) dm = _prepare_out_argument(out, np.float64, (out_size,)) # get pdist wrapper pdist_fn = getattr(_distance_wrap, f'pdist_{metric_name}_{typ}_wrap') pdist_fn(X, dm, **kwargs) return dm @dataclasses.dataclass(frozen=True) class MetricInfo: # Name of python distance function canonical_name: str # All aliases, including canonical_name aka: set[str] # unvectorized distance function dist_func: Callable # Optimized cdist function cdist_func: Callable # Optimized pdist function pdist_func: Callable # function that checks kwargs and computes default values: # f(X, m, n, **kwargs) validator: Optional[Callable] = None # list of supported types: # X (pdist) and XA (cdist) are used to choose the type. if there is no # match the first type is used. Default double types: list[str] = dataclasses.field(default_factory=lambda: ['double']) # true if out array must be C-contiguous requires_contiguous_out: bool = True # Registry of implemented metrics: _METRIC_INFOS = [ MetricInfo( canonical_name='braycurtis', aka={'braycurtis'}, dist_func=braycurtis, cdist_func=_distance_pybind.cdist_braycurtis, pdist_func=_distance_pybind.pdist_braycurtis, ), MetricInfo( canonical_name='canberra', aka={'canberra'}, dist_func=canberra, cdist_func=_distance_pybind.cdist_canberra, pdist_func=_distance_pybind.pdist_canberra, ), MetricInfo( canonical_name='chebyshev', aka={'chebychev', 'chebyshev', 'cheby', 'cheb', 'ch'}, dist_func=chebyshev, cdist_func=_distance_pybind.cdist_chebyshev, pdist_func=_distance_pybind.pdist_chebyshev, ), MetricInfo( canonical_name='cityblock', aka={'cityblock', 'cblock', 'cb', 'c'}, dist_func=cityblock, cdist_func=_distance_pybind.cdist_cityblock, pdist_func=_distance_pybind.pdist_cityblock, ), MetricInfo( canonical_name='correlation', aka={'correlation', 'co'}, dist_func=correlation, cdist_func=CDistMetricWrapper('correlation'), pdist_func=PDistMetricWrapper('correlation'), ), MetricInfo( canonical_name='cosine', aka={'cosine', 'cos'}, dist_func=cosine, cdist_func=CDistMetricWrapper('cosine'), pdist_func=PDistMetricWrapper('cosine'), ), MetricInfo( canonical_name='dice', aka={'dice'}, types=['bool'], dist_func=dice, cdist_func=_distance_pybind.cdist_dice, pdist_func=_distance_pybind.pdist_dice, ), MetricInfo( canonical_name='euclidean', aka={'euclidean', 'euclid', 'eu', 'e'}, dist_func=euclidean, cdist_func=_distance_pybind.cdist_euclidean, pdist_func=_distance_pybind.pdist_euclidean, ), MetricInfo( canonical_name='hamming', aka={'matching', 'hamming', 'hamm', 'ha', 'h'}, types=['double', 'bool'], validator=_validate_hamming_kwargs, dist_func=hamming, cdist_func=_distance_pybind.cdist_hamming, pdist_func=_distance_pybind.pdist_hamming, ), MetricInfo( canonical_name='jaccard', aka={'jaccard', 'jacc', 'ja', 'j'}, types=['double', 'bool'], dist_func=jaccard, cdist_func=_distance_pybind.cdist_jaccard, pdist_func=_distance_pybind.pdist_jaccard, ), MetricInfo( canonical_name='jensenshannon', aka={'jensenshannon', 'js'}, dist_func=jensenshannon, cdist_func=CDistMetricWrapper('jensenshannon'), pdist_func=PDistMetricWrapper('jensenshannon'), ), MetricInfo( canonical_name='kulczynski1', aka={'kulczynski1'}, types=['bool'], dist_func=kulczynski1, cdist_func=_distance_pybind.cdist_kulczynski1, pdist_func=_distance_pybind.pdist_kulczynski1, ), MetricInfo( canonical_name='mahalanobis', aka={'mahalanobis', 'mahal', 'mah'}, validator=_validate_mahalanobis_kwargs, dist_func=mahalanobis, cdist_func=CDistMetricWrapper('mahalanobis'), pdist_func=PDistMetricWrapper('mahalanobis'), ), MetricInfo( canonical_name='minkowski', aka={'minkowski', 'mi', 'm', 'pnorm'}, validator=_validate_minkowski_kwargs, dist_func=minkowski, cdist_func=_distance_pybind.cdist_minkowski, pdist_func=_distance_pybind.pdist_minkowski, ), MetricInfo( canonical_name='rogerstanimoto', aka={'rogerstanimoto'}, types=['bool'], dist_func=rogerstanimoto, cdist_func=_distance_pybind.cdist_rogerstanimoto, pdist_func=_distance_pybind.pdist_rogerstanimoto, ), MetricInfo( canonical_name='russellrao', aka={'russellrao'}, types=['bool'], dist_func=russellrao, cdist_func=_distance_pybind.cdist_russellrao, pdist_func=_distance_pybind.pdist_russellrao, ), MetricInfo( canonical_name='seuclidean', aka={'seuclidean', 'se', 's'}, validator=_validate_seuclidean_kwargs, dist_func=seuclidean, cdist_func=CDistMetricWrapper('seuclidean'), pdist_func=PDistMetricWrapper('seuclidean'), ), MetricInfo( canonical_name='sokalmichener', aka={'sokalmichener'}, types=['bool'], dist_func=sokalmichener, cdist_func=_distance_pybind.cdist_sokalmichener, pdist_func=_distance_pybind.pdist_sokalmichener, ), MetricInfo( canonical_name='sokalsneath', aka={'sokalsneath'}, types=['bool'], dist_func=sokalsneath, cdist_func=_distance_pybind.cdist_sokalsneath, pdist_func=_distance_pybind.pdist_sokalsneath, ), MetricInfo( canonical_name='sqeuclidean', aka={'sqeuclidean', 'sqe', 'sqeuclid'}, dist_func=sqeuclidean, cdist_func=_distance_pybind.cdist_sqeuclidean, pdist_func=_distance_pybind.pdist_sqeuclidean, ), MetricInfo( canonical_name='yule', aka={'yule'}, types=['bool'], dist_func=yule, cdist_func=_distance_pybind.cdist_yule, pdist_func=_distance_pybind.pdist_yule, ), ] _METRICS = {info.canonical_name: info for info in _METRIC_INFOS} _METRIC_ALIAS = {alias: info for info in _METRIC_INFOS for alias in info.aka} _METRICS_NAMES = list(_METRICS.keys()) _TEST_METRICS = {'test_' + info.canonical_name: info for info in _METRIC_INFOS} def pdist(X, metric='euclidean', *, out=None, **kwargs): """ Pairwise distances between observations in n-dimensional space. See Notes for common calling conventions. Parameters ---------- X : array_like An m by n array of m original observations in an n-dimensional space. metric : str or function, optional The distance metric to use. The distance function can be 'braycurtis', 'canberra', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice', 'euclidean', 'hamming', 'jaccard', 'jensenshannon', 'kulczynski1', 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'. out : ndarray, optional The output array. If not None, condensed distance matrix Y is stored in this array. **kwargs : dict, optional Extra arguments to `metric`: refer to each metric documentation for a list of all possible arguments. Some possible arguments: p : scalar The p-norm to apply for Minkowski, weighted and unweighted. Default: 2. w : ndarray The weight vector for metrics that support weights (e.g., Minkowski). V : ndarray The variance vector for standardized Euclidean. Default: var(X, axis=0, ddof=1) VI : ndarray The inverse of the covariance matrix for Mahalanobis. Default: inv(cov(X.T)).T Returns ------- Y : ndarray Returns a condensed distance matrix Y. For each :math:`i` and :math:`j` (where :math:`i 0` (note that this is only a quasi-metric if :math:`0 < p < 1`). 3. ``Y = pdist(X, 'cityblock')`` Computes the city block or Manhattan distance between the points. 4. ``Y = pdist(X, 'seuclidean', V=None)`` Computes the standardized Euclidean distance. The standardized Euclidean distance between two n-vectors ``u`` and ``v`` is .. math:: \\sqrt{\\sum {(u_i-v_i)^2 / V[x_i]}} V is the variance vector; V[i] is the variance computed over all the i'th components of the points. If not passed, it is automatically computed. 5. ``Y = pdist(X, 'sqeuclidean')`` Computes the squared Euclidean distance :math:`\\|u-v\\|_2^2` between the vectors. 6. ``Y = pdist(X, 'cosine')`` Computes the cosine distance between vectors u and v, .. math:: 1 - \\frac{u \\cdot v} {{\\|u\\|}_2 {\\|v\\|}_2} where :math:`\\|*\\|_2` is the 2-norm of its argument ``*``, and :math:`u \\cdot v` is the dot product of ``u`` and ``v``. 7. ``Y = pdist(X, 'correlation')`` Computes the correlation distance between vectors u and v. This is .. math:: 1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})} {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2} where :math:`\\bar{v}` is the mean of the elements of vector v, and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`. 8. ``Y = pdist(X, 'hamming')`` Computes the normalized Hamming distance, or the proportion of those vector elements between two n-vectors ``u`` and ``v`` which disagree. To save memory, the matrix ``X`` can be of type boolean. 9. ``Y = pdist(X, 'jaccard')`` Computes the Jaccard distance between the points. Given two vectors, ``u`` and ``v``, the Jaccard distance is the proportion of those elements ``u[i]`` and ``v[i]`` that disagree. 10. ``Y = pdist(X, 'jensenshannon')`` Computes the Jensen-Shannon distance between two probability arrays. Given two probability vectors, :math:`p` and :math:`q`, the Jensen-Shannon distance is .. math:: \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}} where :math:`m` is the pointwise mean of :math:`p` and :math:`q` and :math:`D` is the Kullback-Leibler divergence. 11. ``Y = pdist(X, 'chebyshev')`` Computes the Chebyshev distance between the points. The Chebyshev distance between two n-vectors ``u`` and ``v`` is the maximum norm-1 distance between their respective elements. More precisely, the distance is given by .. math:: d(u,v) = \\max_i {|u_i-v_i|} 12. ``Y = pdist(X, 'canberra')`` Computes the Canberra distance between the points. The Canberra distance between two points ``u`` and ``v`` is .. math:: d(u,v) = \\sum_i \\frac{|u_i-v_i|} {|u_i|+|v_i|} 13. ``Y = pdist(X, 'braycurtis')`` Computes the Bray-Curtis distance between the points. The Bray-Curtis distance between two points ``u`` and ``v`` is .. math:: d(u,v) = \\frac{\\sum_i {|u_i-v_i|}} {\\sum_i {|u_i+v_i|}} 14. ``Y = pdist(X, 'mahalanobis', VI=None)`` Computes the Mahalanobis distance between the points. The Mahalanobis distance between two points ``u`` and ``v`` is :math:`\\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI`` variable) is the inverse covariance. If ``VI`` is not None, ``VI`` will be used as the inverse covariance matrix. 15. ``Y = pdist(X, 'yule')`` Computes the Yule distance between each pair of boolean vectors. (see yule function documentation) 16. ``Y = pdist(X, 'matching')`` Synonym for 'hamming'. 17. ``Y = pdist(X, 'dice')`` Computes the Dice distance between each pair of boolean vectors. (see dice function documentation) 18. ``Y = pdist(X, 'kulczynski1')`` Computes the kulczynski1 distance between each pair of boolean vectors. (see kulczynski1 function documentation) 19. ``Y = pdist(X, 'rogerstanimoto')`` Computes the Rogers-Tanimoto distance between each pair of boolean vectors. (see rogerstanimoto function documentation) 20. ``Y = pdist(X, 'russellrao')`` Computes the Russell-Rao distance between each pair of boolean vectors. (see russellrao function documentation) 21. ``Y = pdist(X, 'sokalmichener')`` Computes the Sokal-Michener distance between each pair of boolean vectors. (see sokalmichener function documentation) 22. ``Y = pdist(X, 'sokalsneath')`` Computes the Sokal-Sneath distance between each pair of boolean vectors. (see sokalsneath function documentation) 23. ``Y = pdist(X, 'kulczynski1')`` Computes the Kulczynski 1 distance between each pair of boolean vectors. (see kulczynski1 function documentation) 24. ``Y = pdist(X, f)`` Computes the distance between all pairs of vectors in X using the user supplied 2-arity function f. For example, Euclidean distance between the vectors could be computed as follows:: dm = pdist(X, lambda u, v: np.sqrt(((u-v)**2).sum())) Note that you should avoid passing a reference to one of the distance functions defined in this library. For example,:: dm = pdist(X, sokalsneath) would calculate the pair-wise distances between the vectors in X using the Python function sokalsneath. This would result in sokalsneath being called :math:`{n \\choose 2}` times, which is inefficient. Instead, the optimized C version is more efficient, and we call it using the following syntax.:: dm = pdist(X, 'sokalsneath') Examples -------- >>> import numpy as np >>> from scipy.spatial.distance import pdist ``x`` is an array of five points in three-dimensional space. >>> x = np.array([[2, 0, 2], [2, 2, 3], [-2, 4, 5], [0, 1, 9], [2, 2, 4]]) ``pdist(x)`` with no additional arguments computes the 10 pairwise Euclidean distances: >>> pdist(x) array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949, 6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558]) The following computes the pairwise Minkowski distances with ``p = 3.5``: >>> pdist(x, metric='minkowski', p=3.5) array([2.04898923, 5.1154929 , 7.02700737, 2.43802731, 4.19042714, 6.03956994, 1. , 4.45128103, 4.10636143, 5.0619695 ]) The pairwise city block or Manhattan distances: >>> pdist(x, metric='cityblock') array([ 3., 11., 10., 4., 8., 9., 1., 9., 7., 8.]) """ # You can also call this as: # Y = pdist(X, 'test_abc') # where 'abc' is the metric being tested. This computes the distance # between all pairs of vectors in X using the distance metric 'abc' but # with a more succinct, verifiable, but less efficient implementation. X = _asarray_validated(X, sparse_ok=False, objects_ok=True, mask_ok=True, check_finite=False) s = X.shape if len(s) != 2: raise ValueError('A 2-dimensional array must be passed.') m, n = s if callable(metric): mstr = getattr(metric, '__name__', 'UnknownCustomMetric') metric_info = _METRIC_ALIAS.get(mstr, None) if metric_info is not None: X, typ, kwargs = _validate_pdist_input( X, m, n, metric_info, **kwargs) return _pdist_callable(X, metric=metric, out=out, **kwargs) elif isinstance(metric, str): mstr = metric.lower() metric_info = _METRIC_ALIAS.get(mstr, None) if metric_info is not None: pdist_fn = metric_info.pdist_func return pdist_fn(X, out=out, **kwargs) elif mstr.startswith("test_"): metric_info = _TEST_METRICS.get(mstr, None) if metric_info is None: raise ValueError(f'Unknown "Test" Distance Metric: {mstr[5:]}') X, typ, kwargs = _validate_pdist_input( X, m, n, metric_info, **kwargs) return _pdist_callable( X, metric=metric_info.dist_func, out=out, **kwargs) else: raise ValueError('Unknown Distance Metric: %s' % mstr) else: raise TypeError('2nd argument metric must be a string identifier ' 'or a function.') def squareform(X, force="no", checks=True): """ Convert a vector-form distance vector to a square-form distance matrix, and vice-versa. Parameters ---------- X : array_like Either a condensed or redundant distance matrix. force : str, optional As with MATLAB(TM), if force is equal to ``'tovector'`` or ``'tomatrix'``, the input will be treated as a distance matrix or distance vector respectively. checks : bool, optional If set to False, no checks will be made for matrix symmetry nor zero diagonals. This is useful if it is known that ``X - X.T1`` is small and ``diag(X)`` is close to zero. These values are ignored any way so they do not disrupt the squareform transformation. Returns ------- Y : ndarray If a condensed distance matrix is passed, a redundant one is returned, or if a redundant one is passed, a condensed distance matrix is returned. Notes ----- 1. ``v = squareform(X)`` Given a square n-by-n symmetric distance matrix ``X``, ``v = squareform(X)`` returns a ``n * (n-1) / 2`` (i.e. binomial coefficient n choose 2) sized vector `v` where :math:`v[{n \\choose 2} - {n-i \\choose 2} + (j-i-1)]` is the distance between distinct points ``i`` and ``j``. If ``X`` is non-square or asymmetric, an error is raised. 2. ``X = squareform(v)`` Given a ``n * (n-1) / 2`` sized vector ``v`` for some integer ``n >= 1`` encoding distances as described, ``X = squareform(v)`` returns a n-by-n distance matrix ``X``. The ``X[i, j]`` and ``X[j, i]`` values are set to :math:`v[{n \\choose 2} - {n-i \\choose 2} + (j-i-1)]` and all diagonal elements are zero. In SciPy 0.19.0, ``squareform`` stopped casting all input types to float64, and started returning arrays of the same dtype as the input. Examples -------- >>> import numpy as np >>> from scipy.spatial.distance import pdist, squareform ``x`` is an array of five points in three-dimensional space. >>> x = np.array([[2, 0, 2], [2, 2, 3], [-2, 4, 5], [0, 1, 9], [2, 2, 4]]) ``pdist(x)`` computes the Euclidean distances between each pair of points in ``x``. The distances are returned in a one-dimensional array with length ``5*(5 - 1)/2 = 10``. >>> distvec = pdist(x) >>> distvec array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949, 6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558]) ``squareform(distvec)`` returns the 5x5 distance matrix. >>> m = squareform(distvec) >>> m array([[0. , 2.23606798, 6.40312424, 7.34846923, 2.82842712], [2.23606798, 0. , 4.89897949, 6.40312424, 1. ], [6.40312424, 4.89897949, 0. , 5.38516481, 4.58257569], [7.34846923, 6.40312424, 5.38516481, 0. , 5.47722558], [2.82842712, 1. , 4.58257569, 5.47722558, 0. ]]) When given a square distance matrix ``m``, ``squareform(m)`` returns the one-dimensional condensed distance vector associated with the matrix. In this case, we recover ``distvec``. >>> squareform(m) array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949, 6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558]) """ X = np.ascontiguousarray(X) s = X.shape if force.lower() == 'tomatrix': if len(s) != 1: raise ValueError("Forcing 'tomatrix' but input X is not a " "distance vector.") elif force.lower() == 'tovector': if len(s) != 2: raise ValueError("Forcing 'tovector' but input X is not a " "distance matrix.") # X = squareform(v) if len(s) == 1: if s[0] == 0: return np.zeros((1, 1), dtype=X.dtype) # Grab the closest value to the square root of the number # of elements times 2 to see if the number of elements # is indeed a binomial coefficient. d = int(np.ceil(np.sqrt(s[0] * 2))) # Check that v is of valid dimensions. if d * (d - 1) != s[0] * 2: raise ValueError('Incompatible vector size. It must be a binomial ' 'coefficient n choose 2 for some integer n >= 2.') # Allocate memory for the distance matrix. M = np.zeros((d, d), dtype=X.dtype) # Since the C code does not support striding using strides. # The dimensions are used instead. X = _copy_array_if_base_present(X) # Fill in the values of the distance matrix. _distance_wrap.to_squareform_from_vector_wrap(M, X) # Return the distance matrix. return M elif len(s) == 2: if s[0] != s[1]: raise ValueError('The matrix argument must be square.') if checks: is_valid_dm(X, throw=True, name='X') # One-side of the dimensions is set here. d = s[0] if d <= 1: return np.array([], dtype=X.dtype) # Create a vector. v = np.zeros((d * (d - 1)) // 2, dtype=X.dtype) # Since the C code does not support striding using strides. # The dimensions are used instead. X = _copy_array_if_base_present(X) # Convert the vector to squareform. _distance_wrap.to_vector_from_squareform_wrap(X, v) return v else: raise ValueError(('The first argument must be one or two dimensional ' 'array. A %d-dimensional array is not ' 'permitted') % len(s)) def is_valid_dm(D, tol=0.0, throw=False, name="D", warning=False): """ Return True if input array is a valid distance matrix. Distance matrices must be 2-dimensional numpy arrays. They must have a zero-diagonal, and they must be symmetric. Parameters ---------- D : array_like The candidate object to test for validity. tol : float, optional The distance matrix should be symmetric. `tol` is the maximum difference between entries ``ij`` and ``ji`` for the distance metric to be considered symmetric. throw : bool, optional An exception is thrown if the distance matrix passed is not valid. name : str, optional The name of the variable to checked. This is useful if throw is set to True so the offending variable can be identified in the exception message when an exception is thrown. warning : bool, optional Instead of throwing an exception, a warning message is raised. Returns ------- valid : bool True if the variable `D` passed is a valid distance matrix. Notes ----- Small numerical differences in `D` and `D.T` and non-zeroness of the diagonal are ignored if they are within the tolerance specified by `tol`. Examples -------- >>> import numpy as np >>> from scipy.spatial.distance import is_valid_dm This matrix is a valid distance matrix. >>> d = np.array([[0.0, 1.1, 1.2, 1.3], ... [1.1, 0.0, 1.0, 1.4], ... [1.2, 1.0, 0.0, 1.5], ... [1.3, 1.4, 1.5, 0.0]]) >>> is_valid_dm(d) True In the following examples, the input is not a valid distance matrix. Not square: >>> is_valid_dm([[0, 2, 2], [2, 0, 2]]) False Nonzero diagonal element: >>> is_valid_dm([[0, 1, 1], [1, 2, 3], [1, 3, 0]]) False Not symmetric: >>> is_valid_dm([[0, 1, 3], [2, 0, 1], [3, 1, 0]]) False """ D = np.asarray(D, order='c') valid = True try: s = D.shape if len(D.shape) != 2: if name: raise ValueError(('Distance matrix \'%s\' must have shape=2 ' '(i.e. be two-dimensional).') % name) else: raise ValueError('Distance matrix must have shape=2 (i.e. ' 'be two-dimensional).') if tol == 0.0: if not (D == D.T).all(): if name: raise ValueError(('Distance matrix \'%s\' must be ' 'symmetric.') % name) else: raise ValueError('Distance matrix must be symmetric.') if not (D[range(0, s[0]), range(0, s[0])] == 0).all(): if name: raise ValueError(('Distance matrix \'%s\' diagonal must ' 'be zero.') % name) else: raise ValueError('Distance matrix diagonal must be zero.') else: if not (D - D.T <= tol).all(): if name: raise ValueError(f'Distance matrix \'{name}\' must be ' f'symmetric within tolerance {tol:5.5f}.') else: raise ValueError('Distance matrix must be symmetric within ' 'tolerance %5.5f.' % tol) if not (D[range(0, s[0]), range(0, s[0])] <= tol).all(): if name: raise ValueError(f'Distance matrix \'{name}\' diagonal must be ' f'close to zero within tolerance {tol:5.5f}.') else: raise ValueError(('Distance matrix \'{}\' diagonal must be close ' 'to zero within tolerance {:5.5f}.').format(*tol)) except Exception as e: if throw: raise if warning: warnings.warn(str(e), stacklevel=2) valid = False return valid def is_valid_y(y, warning=False, throw=False, name=None): """ Return True if the input array is a valid condensed distance matrix. Condensed distance matrices must be 1-dimensional numpy arrays. Their length must be a binomial coefficient :math:`{n \\choose 2}` for some positive integer n. Parameters ---------- y : array_like The condensed distance matrix. warning : bool, optional Invokes a warning if the variable passed is not a valid condensed distance matrix. The warning message explains why the distance matrix is not valid. `name` is used when referencing the offending variable. throw : bool, optional Throws an exception if the variable passed is not a valid condensed distance matrix. name : bool, optional Used when referencing the offending variable in the warning or exception message. Returns ------- bool True if the input array is a valid condensed distance matrix, False otherwise. Examples -------- >>> from scipy.spatial.distance import is_valid_y This vector is a valid condensed distance matrix. The length is 6, which corresponds to ``n = 4``, since ``4*(4 - 1)/2`` is 6. >>> v = [1.0, 1.2, 1.0, 0.5, 1.3, 0.9] >>> is_valid_y(v) True An input vector with length, say, 7, is not a valid condensed distance matrix. >>> is_valid_y([1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7]) False """ y = np.asarray(y, order='c') valid = True try: if len(y.shape) != 1: if name: raise ValueError(('Condensed distance matrix \'%s\' must ' 'have shape=1 (i.e. be one-dimensional).') % name) else: raise ValueError('Condensed distance matrix must have shape=1 ' '(i.e. be one-dimensional).') n = y.shape[0] d = int(np.ceil(np.sqrt(n * 2))) if (d * (d - 1) / 2) != n: if name: raise ValueError(('Length n of condensed distance matrix ' '\'%s\' must be a binomial coefficient, i.e.' 'there must be a k such that ' '(k \\choose 2)=n)!') % name) else: raise ValueError('Length n of condensed distance matrix must ' 'be a binomial coefficient, i.e. there must ' 'be a k such that (k \\choose 2)=n)!') except Exception as e: if throw: raise if warning: warnings.warn(str(e), stacklevel=2) valid = False return valid def num_obs_dm(d): """ Return the number of original observations that correspond to a square, redundant distance matrix. Parameters ---------- d : array_like The target distance matrix. Returns ------- num_obs_dm : int The number of observations in the redundant distance matrix. Examples -------- Find the number of original observations corresponding to a square redundant distance matrix d. >>> from scipy.spatial.distance import num_obs_dm >>> d = [[0, 100, 200], [100, 0, 150], [200, 150, 0]] >>> num_obs_dm(d) 3 """ d = np.asarray(d, order='c') is_valid_dm(d, tol=np.inf, throw=True, name='d') return d.shape[0] def num_obs_y(Y): """ Return the number of original observations that correspond to a condensed distance matrix. Parameters ---------- Y : array_like Condensed distance matrix. Returns ------- n : int The number of observations in the condensed distance matrix `Y`. Examples -------- Find the number of original observations corresponding to a condensed distance matrix Y. >>> from scipy.spatial.distance import num_obs_y >>> Y = [1, 2, 3.5, 7, 10, 4] >>> num_obs_y(Y) 4 """ Y = np.asarray(Y, order='c') is_valid_y(Y, throw=True, name='Y') k = Y.shape[0] if k == 0: raise ValueError("The number of observations cannot be determined on " "an empty distance matrix.") d = int(np.ceil(np.sqrt(k * 2))) if (d * (d - 1) / 2) != k: raise ValueError("Invalid condensed distance matrix passed. Must be " "some k where k=(n choose 2) for some n >= 2.") return d def _prepare_out_argument(out, dtype, expected_shape): if out is None: return np.empty(expected_shape, dtype=dtype) if out.shape != expected_shape: raise ValueError("Output array has incorrect shape.") if not out.flags.c_contiguous: raise ValueError("Output array must be C-contiguous.") if out.dtype != np.float64: raise ValueError("Output array must be double type.") return out def _pdist_callable(X, *, out, metric, **kwargs): n = X.shape[0] out_size = (n * (n - 1)) // 2 dm = _prepare_out_argument(out, np.float64, (out_size,)) k = 0 for i in range(X.shape[0] - 1): for j in range(i + 1, X.shape[0]): dm[k] = metric(X[i], X[j], **kwargs) k += 1 return dm def _cdist_callable(XA, XB, *, out, metric, **kwargs): mA = XA.shape[0] mB = XB.shape[0] dm = _prepare_out_argument(out, np.float64, (mA, mB)) for i in range(mA): for j in range(mB): dm[i, j] = metric(XA[i], XB[j], **kwargs) return dm def cdist(XA, XB, metric='euclidean', *, out=None, **kwargs): """ Compute distance between each pair of the two collections of inputs. See Notes for common calling conventions. Parameters ---------- XA : array_like An :math:`m_A` by :math:`n` array of :math:`m_A` original observations in an :math:`n`-dimensional space. Inputs are converted to float type. XB : array_like An :math:`m_B` by :math:`n` array of :math:`m_B` original observations in an :math:`n`-dimensional space. Inputs are converted to float type. metric : str or callable, optional The distance metric to use. If a string, the distance function can be 'braycurtis', 'canberra', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice', 'euclidean', 'hamming', 'jaccard', 'jensenshannon', 'kulczynski1', 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'. **kwargs : dict, optional Extra arguments to `metric`: refer to each metric documentation for a list of all possible arguments. Some possible arguments: p : scalar The p-norm to apply for Minkowski, weighted and unweighted. Default: 2. w : array_like The weight vector for metrics that support weights (e.g., Minkowski). V : array_like The variance vector for standardized Euclidean. Default: var(vstack([XA, XB]), axis=0, ddof=1) VI : array_like The inverse of the covariance matrix for Mahalanobis. Default: inv(cov(vstack([XA, XB].T))).T out : ndarray The output array If not None, the distance matrix Y is stored in this array. Returns ------- Y : ndarray A :math:`m_A` by :math:`m_B` distance matrix is returned. For each :math:`i` and :math:`j`, the metric ``dist(u=XA[i], v=XB[j])`` is computed and stored in the :math:`ij` th entry. Raises ------ ValueError An exception is thrown if `XA` and `XB` do not have the same number of columns. Notes ----- The following are common calling conventions: 1. ``Y = cdist(XA, XB, 'euclidean')`` Computes the distance between :math:`m` points using Euclidean distance (2-norm) as the distance metric between the points. The points are arranged as :math:`m` :math:`n`-dimensional row vectors in the matrix X. 2. ``Y = cdist(XA, XB, 'minkowski', p=2.)`` Computes the distances using the Minkowski distance :math:`\\|u-v\\|_p` (:math:`p`-norm) where :math:`p > 0` (note that this is only a quasi-metric if :math:`0 < p < 1`). 3. ``Y = cdist(XA, XB, 'cityblock')`` Computes the city block or Manhattan distance between the points. 4. ``Y = cdist(XA, XB, 'seuclidean', V=None)`` Computes the standardized Euclidean distance. The standardized Euclidean distance between two n-vectors ``u`` and ``v`` is .. math:: \\sqrt{\\sum {(u_i-v_i)^2 / V[x_i]}}. V is the variance vector; V[i] is the variance computed over all the i'th components of the points. If not passed, it is automatically computed. 5. ``Y = cdist(XA, XB, 'sqeuclidean')`` Computes the squared Euclidean distance :math:`\\|u-v\\|_2^2` between the vectors. 6. ``Y = cdist(XA, XB, 'cosine')`` Computes the cosine distance between vectors u and v, .. math:: 1 - \\frac{u \\cdot v} {{\\|u\\|}_2 {\\|v\\|}_2} where :math:`\\|*\\|_2` is the 2-norm of its argument ``*``, and :math:`u \\cdot v` is the dot product of :math:`u` and :math:`v`. 7. ``Y = cdist(XA, XB, 'correlation')`` Computes the correlation distance between vectors u and v. This is .. math:: 1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})} {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2} where :math:`\\bar{v}` is the mean of the elements of vector v, and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`. 8. ``Y = cdist(XA, XB, 'hamming')`` Computes the normalized Hamming distance, or the proportion of those vector elements between two n-vectors ``u`` and ``v`` which disagree. To save memory, the matrix ``X`` can be of type boolean. 9. ``Y = cdist(XA, XB, 'jaccard')`` Computes the Jaccard distance between the points. Given two vectors, ``u`` and ``v``, the Jaccard distance is the proportion of those elements ``u[i]`` and ``v[i]`` that disagree where at least one of them is non-zero. 10. ``Y = cdist(XA, XB, 'jensenshannon')`` Computes the Jensen-Shannon distance between two probability arrays. Given two probability vectors, :math:`p` and :math:`q`, the Jensen-Shannon distance is .. math:: \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}} where :math:`m` is the pointwise mean of :math:`p` and :math:`q` and :math:`D` is the Kullback-Leibler divergence. 11. ``Y = cdist(XA, XB, 'chebyshev')`` Computes the Chebyshev distance between the points. The Chebyshev distance between two n-vectors ``u`` and ``v`` is the maximum norm-1 distance between their respective elements. More precisely, the distance is given by .. math:: d(u,v) = \\max_i {|u_i-v_i|}. 12. ``Y = cdist(XA, XB, 'canberra')`` Computes the Canberra distance between the points. The Canberra distance between two points ``u`` and ``v`` is .. math:: d(u,v) = \\sum_i \\frac{|u_i-v_i|} {|u_i|+|v_i|}. 13. ``Y = cdist(XA, XB, 'braycurtis')`` Computes the Bray-Curtis distance between the points. The Bray-Curtis distance between two points ``u`` and ``v`` is .. math:: d(u,v) = \\frac{\\sum_i (|u_i-v_i|)} {\\sum_i (|u_i+v_i|)} 14. ``Y = cdist(XA, XB, 'mahalanobis', VI=None)`` Computes the Mahalanobis distance between the points. The Mahalanobis distance between two points ``u`` and ``v`` is :math:`\\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI`` variable) is the inverse covariance. If ``VI`` is not None, ``VI`` will be used as the inverse covariance matrix. 15. ``Y = cdist(XA, XB, 'yule')`` Computes the Yule distance between the boolean vectors. (see `yule` function documentation) 16. ``Y = cdist(XA, XB, 'matching')`` Synonym for 'hamming'. 17. ``Y = cdist(XA, XB, 'dice')`` Computes the Dice distance between the boolean vectors. (see `dice` function documentation) 18. ``Y = cdist(XA, XB, 'kulczynski1')`` Computes the kulczynski distance between the boolean vectors. (see `kulczynski1` function documentation) 19. ``Y = cdist(XA, XB, 'rogerstanimoto')`` Computes the Rogers-Tanimoto distance between the boolean vectors. (see `rogerstanimoto` function documentation) 20. ``Y = cdist(XA, XB, 'russellrao')`` Computes the Russell-Rao distance between the boolean vectors. (see `russellrao` function documentation) 21. ``Y = cdist(XA, XB, 'sokalmichener')`` Computes the Sokal-Michener distance between the boolean vectors. (see `sokalmichener` function documentation) 22. ``Y = cdist(XA, XB, 'sokalsneath')`` Computes the Sokal-Sneath distance between the vectors. (see `sokalsneath` function documentation) 23. ``Y = cdist(XA, XB, f)`` Computes the distance between all pairs of vectors in X using the user supplied 2-arity function f. For example, Euclidean distance between the vectors could be computed as follows:: dm = cdist(XA, XB, lambda u, v: np.sqrt(((u-v)**2).sum())) Note that you should avoid passing a reference to one of the distance functions defined in this library. For example,:: dm = cdist(XA, XB, sokalsneath) would calculate the pair-wise distances between the vectors in X using the Python function `sokalsneath`. This would result in sokalsneath being called :math:`{n \\choose 2}` times, which is inefficient. Instead, the optimized C version is more efficient, and we call it using the following syntax:: dm = cdist(XA, XB, 'sokalsneath') Examples -------- Find the Euclidean distances between four 2-D coordinates: >>> from scipy.spatial import distance >>> import numpy as np >>> coords = [(35.0456, -85.2672), ... (35.1174, -89.9711), ... (35.9728, -83.9422), ... (36.1667, -86.7833)] >>> distance.cdist(coords, coords, 'euclidean') array([[ 0. , 4.7044, 1.6172, 1.8856], [ 4.7044, 0. , 6.0893, 3.3561], [ 1.6172, 6.0893, 0. , 2.8477], [ 1.8856, 3.3561, 2.8477, 0. ]]) Find the Manhattan distance from a 3-D point to the corners of the unit cube: >>> a = np.array([[0, 0, 0], ... [0, 0, 1], ... [0, 1, 0], ... [0, 1, 1], ... [1, 0, 0], ... [1, 0, 1], ... [1, 1, 0], ... [1, 1, 1]]) >>> b = np.array([[ 0.1, 0.2, 0.4]]) >>> distance.cdist(a, b, 'cityblock') array([[ 0.7], [ 0.9], [ 1.3], [ 1.5], [ 1.5], [ 1.7], [ 2.1], [ 2.3]]) """ # You can also call this as: # Y = cdist(XA, XB, 'test_abc') # where 'abc' is the metric being tested. This computes the distance # between all pairs of vectors in XA and XB using the distance metric 'abc' # but with a more succinct, verifiable, but less efficient implementation. XA = np.asarray(XA) XB = np.asarray(XB) s = XA.shape sB = XB.shape if len(s) != 2: raise ValueError('XA must be a 2-dimensional array.') if len(sB) != 2: raise ValueError('XB must be a 2-dimensional array.') if s[1] != sB[1]: raise ValueError('XA and XB must have the same number of columns ' '(i.e. feature dimension.)') mA = s[0] mB = sB[0] n = s[1] if callable(metric): mstr = getattr(metric, '__name__', 'Unknown') metric_info = _METRIC_ALIAS.get(mstr, None) if metric_info is not None: XA, XB, typ, kwargs = _validate_cdist_input( XA, XB, mA, mB, n, metric_info, **kwargs) return _cdist_callable(XA, XB, metric=metric, out=out, **kwargs) elif isinstance(metric, str): mstr = metric.lower() metric_info = _METRIC_ALIAS.get(mstr, None) if metric_info is not None: cdist_fn = metric_info.cdist_func return cdist_fn(XA, XB, out=out, **kwargs) elif mstr.startswith("test_"): metric_info = _TEST_METRICS.get(mstr, None) if metric_info is None: raise ValueError(f'Unknown "Test" Distance Metric: {mstr[5:]}') XA, XB, typ, kwargs = _validate_cdist_input( XA, XB, mA, mB, n, metric_info, **kwargs) return _cdist_callable( XA, XB, metric=metric_info.dist_func, out=out, **kwargs) else: raise ValueError('Unknown Distance Metric: %s' % mstr) else: raise TypeError('2nd argument metric must be a string identifier ' 'or a function.')