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""" ===================================================== 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. mahalanobis -- the Mahalanobis distance. minkowski -- the Minkowski distance. seuclidean -- the normalized Euclidean distance. sqeuclidean -- the squared Euclidean distance. wminkowski -- (deprecated) alias of `minkowski`.
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. kulsinski -- the Kulsinski 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.
'braycurtis', 'canberra', 'cdist', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice', 'directed_hausdorff', 'euclidean', 'hamming', 'is_valid_dm', 'is_valid_y', 'jaccard', 'kulsinski', 'mahalanobis', 'matching', 'minkowski', 'num_obs_dm', 'num_obs_y', 'pdist', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'squareform', 'wminkowski', 'yule' ]
""" Convert legacy positional arguments to keyword arguments for pdist/cdist. """ if not args: return kwargs
if (callable(metric) and metric not in [ braycurtis, canberra, chebyshev, cityblock, correlation, cosine, dice, euclidean, hamming, jaccard, kulsinski, mahalanobis, matching, minkowski, rogerstanimoto, russellrao, seuclidean, sokalmichener, sokalsneath, sqeuclidean, yule, wminkowski]): raise TypeError('When using a custom metric arguments must be passed' 'as keyword (i.e., ARGNAME=ARGVALUE)')
if func_name == 'pdist': old_arg_names = ['p', 'w', 'V', 'VI'] else: old_arg_names = ['p', 'V', 'VI', 'w']
num_args = len(args) warnings.warn('%d metric parameters have been passed as positional.' 'This will raise an error in a future version.' 'Please pass arguments as keywords(i.e., ARGNAME=ARGVALUE)' % num_args, DeprecationWarning)
if num_args > 4: raise ValueError('Deprecated %s signature accepts only 4' 'positional arguments (%s), %d given.' % (func_name, ', '.join(old_arg_names), num_args))
for old_arg, arg in zip(old_arg_names, args): if old_arg in kwargs: raise TypeError('%s() got multiple values for argument %s' % (func_name, old_arg)) kwargs[old_arg] = arg return kwargs
"""Copy the array if its base points to a parent array.""" if a.base is not None: return a.copy() return a
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)
X2 = X - X.mean(axis=1, keepdims=True) _distance_wrap.pdist_cosine_double_wrap(X2, dm, **kwargs)
return np.ascontiguousarray(X, dtype=out_type)
# Filtering out old default keywords for k in args_blacklist: if k in kwargs: del kwargs[k] warnings.warn('Got unexpected kwarg %s. This will raise an error' ' in a future version.' % k, DeprecationWarning)
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.find_common_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)
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.find_common_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)
if metric_name is not None: # get supported types types = _METRICS[metric_name].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 = _METRICS[metric_name].validator if _validate_kwargs: kwargs = _validate_kwargs(np.vstack([XA, XB]), mA + mB, n, **kwargs) else: typ = None return XA, XB, typ, 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)) CV = np.atleast_2d(np.cov(X.astype(np.double).T)) VI = np.linalg.inv(CV).T.copy() kwargs["VI"] = _convert_to_double(VI) return kwargs
if 'p' not in kwargs: kwargs['p'] = 2. return kwargs
if metric_name is not None: # get supported types types = _METRICS[metric_name].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 = _METRICS[metric_name].validator if _validate_kwargs: kwargs = _validate_kwargs(X, m, n, **kwargs) else: typ = None return X, typ, kwargs
V = kwargs.pop('V', None) if V is None: V = np.var(X.astype(np.double), axis=0, ddof=1) else: V = np.asarray(V, order='c') if V.dtype != np.double: raise TypeError('Variance vector V must contain doubles.') 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
# XXX Is order='c' really necessary? u = np.asarray(u, dtype=dtype, order='c').squeeze() # Ensure values such as u=1 and u=[1] still return 1-D arrays. u = np.atleast_1d(u) if u.ndim > 1: raise ValueError("Input vector should be 1-D.") return u
w = _validate_vector(w, dtype=dtype) if np.any(w < 0): raise ValueError("Input weights should be all non-negative") return w
w = kwargs.pop('w', None) if w is None: raise ValueError('weighted minkowski requires a weight ' 'vector `w` to be given.') kwargs['w'] = _convert_to_double(w) if 'p' not in kwargs: kwargs['p'] = 2. return kwargs
""" Compute the directed Hausdorff distance between two N-D arrays.
Distances between pairs are calculated using a Euclidean metric.
Parameters ---------- u : (M,N) ndarray Input array. v : (O,N) ndarray Input array. seed : int or None Local `np.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`
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.
See Also -------- scipy.spatial.procrustes : Another similarity test for two data sets
Examples -------- Find the directed Hausdorff distance between two 2-D arrays of coordinates:
>>> from scipy.spatial.distance import directed_hausdorff >>> 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') result = _hausdorff.directed_hausdorff(u, v, seed) return result
""" 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 : int The order of the norm of the difference :math:`{||u-v||}_p`. 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 < 1: raise ValueError("p must be at least 1") u_v = u - v if w is not None: w = _validate_weights(w) if p == 1: root_w = w if p == 2: # better precision and speed root_w = np.sqrt(w) else: root_w = np.power(w, 1/p) u_v = root_w * u_v dist = norm(u_v, ord=p) return dist
# `minkowski` gained weights in scipy 1.0. Once we're at say version 1.3, # deprecated `wminkowski`. Not done at once because it would be annoying for # downstream libraries that used `wminkowski` and support multiple scipy # versions. """ Compute the weighted Minkowski distance between two 1-D arrays.
The weighted Minkowski distance between `u` and `v`, defined as
.. math::
\\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 : int The order of the norm of the difference :math:`{||u-v||}_p`. w : (N,) array_like The weight vector.
Returns ------- wminkowski : double The weighted Minkowski distance between vectors `u` and `v`.
Notes ----- `wminkowski` is DEPRECATED. It implements a definition where weights are powered. It is recommended to use the weighted version of `minkowski` instead. This function will be removed in a future version of scipy.
Examples -------- >>> from scipy.spatial import distance >>> distance.wminkowski([1, 0, 0], [0, 1, 0], 1, np.ones(3)) 2.0 >>> distance.wminkowski([1, 0, 0], [0, 1, 0], 2, np.ones(3)) 1.4142135623730951 >>> distance.wminkowski([1, 0, 0], [0, 1, 0], 3, np.ones(3)) 1.2599210498948732 >>> distance.wminkowski([1, 1, 0], [0, 1, 0], 1, np.ones(3)) 1.0 >>> distance.wminkowski([1, 1, 0], [0, 1, 0], 2, np.ones(3)) 1.0 >>> distance.wminkowski([1, 1, 0], [0, 1, 0], 3, np.ones(3)) 1.0
""" w = _validate_weights(w) return minkowski(u, v, p=p, w=w**p)
""" 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)
""" Compute the squared Euclidean distance between two 1-D arrays.
The squared Euclidean distance between `u` and `v` is defined as
.. math::
{||u-v||}_2^2
\\left(\\sum{(w_i |(u_i - v_i)|^2)}\\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 ------- 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)
""" 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
Returns ------- correlation : double The correlation distance between 1-D array `u` and `v`.
""" u = _validate_vector(u) v = _validate_vector(v) if w is not None: w = _validate_weights(w) if centered: umu = np.average(u, weights=w) vmu = np.average(v, weights=w) u = u - umu v = v - vmu uv = np.average(u * v, weights=w) uu = np.average(np.square(u), weights=w) vv = np.average(np.square(v), weights=w) dist = 1.0 - uv / np.sqrt(uu * vv) return dist
""" 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)
""" 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) return np.average(u_ne_v, weights=w)
""" 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`.
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 dist = np.double(unequal_nonzero.sum()) / np.double(nonzero.sum()) return dist
""" Compute the Kulsinski dissimilarity between two boolean 1-D arrays.
The Kulsinski dissimilarity between two boolean 1-D arrays `u` and `v`, is defined as
.. math::
\\frac{c_{TF} + c_{FT} - c_{TT} + n} {c_{FT} + c_{TF} + 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 ------- kulsinski : double The Kulsinski distance between vectors `u` and `v`.
Examples -------- >>> from scipy.spatial import distance >>> distance.kulsinski([1, 0, 0], [0, 1, 0]) 1.0 >>> distance.kulsinski([1, 0, 0], [1, 1, 0]) 0.75 >>> distance.kulsinski([1, 0, 0], [2, 1, 0]) 0.33333333333333331 >>> distance.kulsinski([1, 0, 0], [3, 1, 0]) -0.5
""" u = _validate_vector(u) v = _validate_vector(v) if w is None: n = float(len(u)) else: w = _validate_weights(w) n = w.sum() (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w)
return (ntf + nft - ntt + n) / (ntf + nft + n)
""" Return the standardized Euclidean distance between two 1-D arrays.
The standardized Euclidean distance between `u` and `v`.
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)
""" 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()
""" 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 : ndarray 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)
""" 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 The weights for each value in `u` and `v`. Default is None, which gives each value a weight of 1.0
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))
""" 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()
""" 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) olderr = np.seterr(invalid='ignore') try: 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) finally: np.seterr(**olderr) return d
""" 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) return float(2.0 * ntf * nft / np.array(ntt * nff + ntf * nft))
"use spatial.distance.hamming instead.") """ Compute the Hamming distance between two boolean 1-D arrays.
This is a deprecated synonym for :func:`hamming`. """ return hamming(u, v, w=w)
""" 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,) ndarray, bool Input 1-D array. v : (N,) ndarray, 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`.
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.find_common_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))
""" 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)))
""" 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
""" 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 u.dtype == v.dtype == bool and w is None: ntt = (u & v).sum() nff = (~u & ~v).sum() elif w is None: ntt = (u * v).sum() nff = ((1.0 - u) * (1.0 - v)).sum() else: w = _validate_weights(w) ntt = (u * v * w).sum() nff = ((1.0 - u) * (1.0 - v) * w).sum() (nft, ntf) = _nbool_correspond_ft_tf(u, v) return float(2.0 * (ntf + nft)) / float(ntt + nff + 2.0 * (ntf + nft))
""" 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
# adding python-only wrappers to _distance_wrap module
# Registry of implemented metrics: # Dictionary with the following structure: # { # metric_name : MetricInfo(aka, types=[double], validator=None) # } # # Where: # `metric_name` must be equal to python metric name # # MetricInfo is a named tuple with fields: # 'aka' : [list of aliases], # # 'validator': f(X, m, n, **kwargs) # function that check kwargs and # # computes default values. # # 'types': [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 #}
'braycurtis': MetricInfo(aka=['braycurtis']), 'canberra': MetricInfo(aka=['canberra']), 'chebyshev': MetricInfo(aka=['chebychev', 'chebyshev', 'cheby', 'cheb', 'ch']), 'cityblock': MetricInfo(aka=['cityblock', 'cblock', 'cb', 'c']), 'correlation': MetricInfo(aka=['correlation', 'co']), 'cosine': MetricInfo(aka=['cosine', 'cos']), 'dice': MetricInfo(aka=['dice'], types=['bool']), 'euclidean': MetricInfo(aka=['euclidean', 'euclid', 'eu', 'e']), 'hamming': MetricInfo(aka=['matching', 'hamming', 'hamm', 'ha', 'h'], types=['double', 'bool']), 'jaccard': MetricInfo(aka=['jaccard', 'jacc', 'ja', 'j'], types=['double', 'bool']), 'kulsinski': MetricInfo(aka=['kulsinski'], types=['bool']), 'mahalanobis': MetricInfo(aka=['mahalanobis', 'mahal', 'mah'], validator=_validate_mahalanobis_kwargs), 'minkowski': MetricInfo(aka=['minkowski', 'mi', 'm', 'pnorm'], validator=_validate_minkowski_kwargs), 'rogerstanimoto': MetricInfo(aka=['rogerstanimoto'], types=['bool']), 'russellrao': MetricInfo(aka=['russellrao'], types=['bool']), 'seuclidean': MetricInfo(aka=['seuclidean', 'se', 's'], validator=_validate_seuclidean_kwargs), 'sokalmichener': MetricInfo(aka=['sokalmichener'], types=['bool']), 'sokalsneath': MetricInfo(aka=['sokalsneath'], types=['bool']), 'sqeuclidean': MetricInfo(aka=['sqeuclidean', 'sqe', 'sqeuclid']), 'wminkowski': MetricInfo(aka=['wminkowski', 'wmi', 'wm', 'wpnorm'], validator=_validate_wminkowski_kwargs), 'yule': MetricInfo(aka=['yule'], types=['bool']), }
for name, info in _METRICS.items() for alias in info.aka)
""" Pairwise distances between observations in n-dimensional space.
See Notes for common calling conventions.
Parameters ---------- X : ndarray 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', 'kulsinski', 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'. *args : tuple. Deprecated. Additional arguments should be passed as keyword arguments **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
out : ndarray. The output array If not None, condensed distance matrix Y is stored in this array. Note: metric independent, it will become a regular keyword arg in a future scipy version
Returns ------- Y : ndarray Returns a condensed distance matrix Y. For each :math:`i` and :math:`j` (where :math:`i<j<m`),where m is the number of original observations. The metric ``dist(u=X[i], v=X[j])`` is computed and stored in entry ``ij``.
See Also -------- squareform : converts between condensed distance matrices and square distance matrices.
Notes ----- See ``squareform`` for information on how to calculate the index of this entry or to convert the condensed distance matrix to a redundant square matrix.
The following are common calling conventions.
1. ``Y = pdist(X, 'euclidean')``
Computes the distance between m points using Euclidean distance (2-norm) as the distance metric between the points. The points are arranged as m n-dimensional row vectors in the matrix X.
2. ``Y = pdist(X, 'minkowski', p=2.)``
Computes the distances using the Minkowski distance :math:`||u-v||_p` (p-norm) where :math:`p \\geq 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, '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|}
11. ``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|}
12. ``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|}}
13. ``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.
14. ``Y = pdist(X, 'yule')``
Computes the Yule distance between each pair of boolean vectors. (see yule function documentation)
15. ``Y = pdist(X, 'matching')``
Synonym for 'hamming'.
16. ``Y = pdist(X, 'dice')``
Computes the Dice distance between each pair of boolean vectors. (see dice function documentation)
17. ``Y = pdist(X, 'kulsinski')``
Computes the Kulsinski distance between each pair of boolean vectors. (see kulsinski function documentation)
18. ``Y = pdist(X, 'rogerstanimoto')``
Computes the Rogers-Tanimoto distance between each pair of boolean vectors. (see rogerstanimoto function documentation)
19. ``Y = pdist(X, 'russellrao')``
Computes the Russell-Rao distance between each pair of boolean vectors. (see russellrao function documentation)
20. ``Y = pdist(X, 'sokalmichener')``
Computes the Sokal-Michener distance between each pair of boolean vectors. (see sokalmichener function documentation)
21. ``Y = pdist(X, 'sokalsneath')``
Computes the Sokal-Sneath distance between each pair of boolean vectors. (see sokalsneath function documentation)
22. ``Y = pdist(X, 'wminkowski', p=2, w=w)``
Computes the weighted Minkowski distance between each pair of vectors. (see wminkowski function documentation)
23. ``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')
""" # 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) kwargs = _args_to_kwargs_xdist(args, kwargs, metric, "pdist")
X = np.asarray(X, order='c')
s = X.shape if len(s) != 2: raise ValueError('A 2-dimensional array must be passed.')
m, n = s out = kwargs.pop("out", None) if out is None: dm = np.empty((m * (m - 1)) // 2, dtype=np.double) else: if out.shape != (m * (m - 1) // 2,): 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.double: raise ValueError("Output array must be double type.") dm = out
# compute blacklist for deprecated kwargs if(metric in _METRICS['minkowski'].aka or metric in _METRICS['wminkowski'].aka or metric in ['test_minkowski', 'test_wminkowski'] or metric in [minkowski, wminkowski]): kwargs_blacklist = ["V", "VI"] elif(metric in _METRICS['seuclidean'].aka or metric == 'test_seuclidean' or metric == seuclidean): kwargs_blacklist = ["p", "w", "VI"] elif(metric in _METRICS['mahalanobis'].aka or metric == 'test_mahalanobis' or metric == mahalanobis): kwargs_blacklist = ["p", "w", "V"] else: kwargs_blacklist = ["p", "V", "VI"]
_filter_deprecated_kwargs(kwargs, kwargs_blacklist)
if callable(metric): mstr = getattr(metric, '__name__', 'UnknownCustomMetric') metric_name = _METRIC_ALIAS.get(mstr, None)
if metric_name is not None: X, typ, kwargs = _validate_pdist_input(X, m, n, metric_name, **kwargs)
k = 0 for i in xrange(0, m - 1): for j in xrange(i + 1, m): dm[k] = metric(X[i], X[j], **kwargs) k = k + 1
elif isinstance(metric, string_types): mstr = metric.lower()
# NOTE: C-version still does not support weights if "w" in kwargs and not mstr.startswith("test_"): if(mstr in _METRICS['seuclidean'].aka or mstr in _METRICS['mahalanobis'].aka): raise ValueError("metric %s incompatible with weights" % mstr) # need to use python version for weighting kwargs['out'] = out mstr = "test_%s" % mstr
metric_name = _METRIC_ALIAS.get(mstr, None)
if metric_name is not None: X, typ, kwargs = _validate_pdist_input(X, m, n, metric_name, **kwargs)
# get pdist wrapper pdist_fn = getattr(_distance_wrap, "pdist_%s_%s_wrap" % (metric_name, typ)) pdist_fn(X, dm, **kwargs) return dm
elif mstr in ['old_cosine', 'old_cos']: warnings.warn('"old_cosine" is deprecated and will be removed in ' 'a future version. Use "cosine" instead.', DeprecationWarning) X = _convert_to_double(X) norms = np.einsum('ij,ij->i', X, X, dtype=np.double) np.sqrt(norms, out=norms) nV = norms.reshape(m, 1) # The numerator u * v nm = np.dot(X, X.T) # The denom. ||u||*||v|| de = np.dot(nV, nV.T) dm = 1.0 - (nm / de) dm[xrange(0, m), xrange(0, m)] = 0.0 dm = squareform(dm) elif mstr.startswith("test_"): if mstr in _TEST_METRICS: dm = pdist(X, _TEST_METRICS[mstr], **kwargs) else: raise ValueError('Unknown "Test" Distance Metric: %s' % mstr[5:]) else: raise ValueError('Unknown Distance Metric: %s' % mstr) else: raise TypeError('2nd argument metric must be a string identifier ' 'or a function.') return dm
""" Convert a vector-form distance vector to a square-form distance matrix, and vice-versa.
Parameters ---------- X : ndarray 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 d-by-d symmetric distance matrix X, ``v = squareform(X)`` returns a ``d * (d-1) / 2`` (or :math:`{n \\choose 2}`) sized vector v.
:math:`v[{n \\choose 2}-{n-i \\choose 2} + (j-i-1)]` is the distance between points i and j. If X is non-square or asymmetric, an error is returned.
2. X = squareform(v)
Given a ``d*(d-1)/2`` sized v for some integer ``d >= 2`` encoding distances as described, ``X = squareform(v)`` returns a d by d 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.
"""
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))
""" 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 : ndarray 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`.
""" 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[xrange(0, s[0]), xrange(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(('Distance matrix \'%s\' must be ' 'symmetric within tolerance %5.5f.') % (name, tol)) else: raise ValueError('Distance matrix must be symmetric within' ' tolerance %5.5f.' % tol) if not (D[xrange(0, s[0]), xrange(0, s[0])] <= tol).all(): if name: raise ValueError(('Distance matrix \'%s\' diagonal must be' ' close to zero within tolerance %5.5f.') % (name, tol)) else: raise ValueError(('Distance matrix \'%s\' diagonal must be' ' close to zero within tolerance %5.5f.') % tol) except Exception as e: if throw: raise if warning: warnings.warn(str(e)) valid = False return valid
""" 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 : ndarray 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.
""" 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)) valid = False return valid
""" Return the number of original observations that correspond to a square, redundant distance matrix.
Parameters ---------- d : ndarray The target distance matrix.
Returns ------- num_obs_dm : int The number of observations in the redundant distance matrix.
""" d = np.asarray(d, order='c') is_valid_dm(d, tol=np.inf, throw=True, name='d') return d.shape[0]
""" Return the number of original observations that correspond to a condensed distance matrix.
Parameters ---------- Y : ndarray Condensed distance matrix.
Returns ------- n : int The number of observations in the condensed distance matrix `Y`.
""" 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
""" Compute distance between each pair of the two collections of inputs.
See Notes for common calling conventions.
Parameters ---------- XA : ndarray 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 : ndarray 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', 'kulsinski', 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'wminkowski', 'yule'. *args : tuple. Deprecated. Additional arguments should be passed as keyword arguments **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(vstack([XA, XB]), axis=0, ddof=1)
VI : ndarray 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. Note: metric independent, it will become a regular keyword arg in a future scipy version
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 \\geq 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, '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|}.
11. ``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|}.
12. ``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|)}
13. ``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.
14. ``Y = cdist(XA, XB, 'yule')``
Computes the Yule distance between the boolean vectors. (see `yule` function documentation)
15. ``Y = cdist(XA, XB, 'matching')``
Synonym for 'hamming'.
16. ``Y = cdist(XA, XB, 'dice')``
Computes the Dice distance between the boolean vectors. (see `dice` function documentation)
17. ``Y = cdist(XA, XB, 'kulsinski')``
Computes the Kulsinski distance between the boolean vectors. (see `kulsinski` function documentation)
18. ``Y = cdist(XA, XB, 'rogerstanimoto')``
Computes the Rogers-Tanimoto distance between the boolean vectors. (see `rogerstanimoto` function documentation)
19. ``Y = cdist(XA, XB, 'russellrao')``
Computes the Russell-Rao distance between the boolean vectors. (see `russellrao` function documentation)
20. ``Y = cdist(XA, XB, 'sokalmichener')``
Computes the Sokal-Michener distance between the boolean vectors. (see `sokalmichener` function documentation)
21. ``Y = cdist(XA, XB, 'sokalsneath')``
Computes the Sokal-Sneath distance between the vectors. (see `sokalsneath` function documentation)
22. ``Y = cdist(XA, XB, 'wminkowski', p=2., w=w)``
Computes the weighted Minkowski distance between the vectors. (see `wminkowski` 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 >>> 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.
kwargs = _args_to_kwargs_xdist(args, kwargs, metric, "cdist")
XA = np.asarray(XA, order='c') XB = np.asarray(XB, order='c')
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] out = kwargs.pop("out", None) if out is None: dm = np.empty((mA, mB), dtype=np.double) else: if out.shape != (mA, mB): 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.double: raise ValueError("Output array must be double type.") dm = out
# compute blacklist for deprecated kwargs if(metric in _METRICS['minkowski'].aka or metric in _METRICS['wminkowski'].aka or metric in ['test_minkowski', 'test_wminkowski'] or metric in [minkowski, wminkowski]): kwargs_blacklist = ["V", "VI"] elif(metric in _METRICS['seuclidean'].aka or metric == 'test_seuclidean' or metric == seuclidean): kwargs_blacklist = ["p", "w", "VI"] elif(metric in _METRICS['mahalanobis'].aka or metric == 'test_mahalanobis' or metric == mahalanobis): kwargs_blacklist = ["p", "w", "V"] else: kwargs_blacklist = ["p", "V", "VI"]
_filter_deprecated_kwargs(kwargs, kwargs_blacklist)
if callable(metric):
mstr = getattr(metric, '__name__', 'Unknown') metric_name = _METRIC_ALIAS.get(mstr, None)
XA, XB, typ, kwargs = _validate_cdist_input(XA, XB, mA, mB, n, metric_name, **kwargs)
for i in xrange(0, mA): for j in xrange(0, mB): dm[i, j] = metric(XA[i], XB[j], **kwargs)
elif isinstance(metric, string_types): mstr = metric.lower()
# NOTE: C-version still does not support weights if "w" in kwargs and not mstr.startswith("test_"): if(mstr in _METRICS['seuclidean'].aka or mstr in _METRICS['mahalanobis'].aka): raise ValueError("metric %s incompatible with weights" % mstr) # need to use python version for weighting kwargs['out'] = out mstr = "test_%s" % mstr
metric_name = _METRIC_ALIAS.get(mstr, None) if metric_name is not None: XA, XB, typ, kwargs = _validate_cdist_input(XA, XB, mA, mB, n, metric_name, **kwargs) # get cdist wrapper cdist_fn = getattr(_distance_wrap, "cdist_%s_%s_wrap" % (metric_name, typ)) cdist_fn(XA, XB, dm, **kwargs) return dm
elif mstr.startswith("test_"): if mstr in _TEST_METRICS: dm = cdist(XA, XB, _TEST_METRICS[mstr], **kwargs) else: raise ValueError('Unknown "Test" Distance Metric: %s' % mstr[5:]) else: raise ValueError('Unknown Distance Metric: %s' % mstr) else: raise TypeError('2nd argument metric must be a string identifier ' 'or a function.') return dm |