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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.
   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.
)
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    N)OptionalCallable)partial)_asarray_validated   )_distance_wrap)
_hausdorff   )norm)rel_entr)_distance_pybindc                 C   s   | j dur|  S | S )z4Copy the array if its base points to a parent array.N)basecopy)a r-   R/var/www/html/django/DPS/env/lib/python3.9/site-packages/scipy/spatial/distance.py_copy_array_if_base_present~   s    
r/   c                 K   s>   | | j ddd } ||j ddd }tj| ||fi | d S Nr#   Taxiskeepdims)meanr$   Zcdist_cosine_double_wrap)XAXBdmkwargsr-   r-   r.   _correlation_cdist_wrap   s    r9   c                 K   s*   | | j ddd }tj||fi | d S r0   )r4   r$   Zpdist_cosine_double_wrap)Xr7   r8   ZX2r-   r-   r.   _correlation_pdist_wrap   s    r;   c                 C   s   t j| |dS )Ndtype)npascontiguousarray)r:   out_typer-   r-   r.   _convert_to_type   s    rA   c           
      C   s   | j |j   krtkrbn nF|d u rb|  }| }||@  }||@  }| |@  }| |@  }n~tt| j |j }	| |	} ||	}d|  }d| }|d ur|| }||  } ||  }||  }| |  }| |  }||||fS N      ?r=   boolsumr>   result_typeintastype)
uvwnot_unot_vnffnftntfnttr=   r-   r-   r.   _nbool_correspond_all   s(    $

rS   c                 C   s   | j |j   krtkrJn n.|d u rJ|  }| }||@  }| |@  }nftt| j |j }| |} ||}d|  }d| }|d ur|| }||  } ||  }| |  }||fS rB   rD   )rJ   rK   rL   rM   rN   rP   rQ   r=   r-   r-   r.   _nbool_correspond_ft_tf   s     $

rT   c           
      K   sr   |j }| j|v r ||| j n|d }t| |d} t||d}|j}	|	rf|	| |f|| |fi |}| |||fS Nr   r@   typesr=   indexrA   	validator)
r5   r6   mAmBnmetric_infor8   rX   typ_validate_kwargsr-   r-   r.   _validate_cdist_input   s    "ra   c                 K   sZ   | dd }|d u r|S |jdks0|jd |krJtd|jd  d| t||d< |S )NrL   r#   r   z-Weights must have same size as input vector. z vs. )popndimshape
ValueError_validate_weightsr:   mr]   r8   rL   r-   r-   r.   _validate_weight_with_size   s    ri   c                 K   sV   | dtj|fdd}|jdks0|jd |krFtd|jd |f t||d< |S )NrL   doubler<   r#   r   z6Weights must have same size as input vector. %d vs. %d)getr>   Zonesrc   rd   re   rf   rg   r-   r-   r.   _validate_hamming_kwargs   s    rl   c                 K   s   | dd }|d u rv||kr2td|||d f t| trFt| } tt| jtj	ddj
}tj|j
 }t||d< |S )NVIzThe number of observations (%d) is too small; the covariance matrix is singular. For observations with %d dimensions, at least %d observations are required.r#   Fr+   )rb   re   
isinstancetupler>   vstack
atleast_2dZcovrI   float64Tlinalginvr+   _convert_to_double)r:   rh   r]   r8   rm   ZCVr-   r-   r.   _validate_mahalanobis_kwargs   s    

rx   c                 K   s>   t | ||fi |}d|vr&d|d< n|d dkr:td|S )Np       @r   p must be greater than 0)ri   re   )r:   rh   r]   r8   r-   r-   r.   _validate_minkowski_kwargs   s    
r|   c                 K   s\   |j }| j|v r ||| j n|d }t| |d} |j}|rR|| ||fi |}| ||fS rU   rW   )r:   rh   r]   r^   r8   rX   r_   r`   r-   r-   r.   _validate_pdist_input  s    "r}   c                 K   s   | dd }|d u rFt| tr(t| } tj| jtjddddd}n:tj|dd}t	|j
dkrjtd	|j
d |krtd
t||d< |S )NVFrn   r   r#   )r2   Zddofcorderz*Variance vector V must be one-dimensional.zcVariance vector V must be of the same dimension as the vectors on which the distances are computed.)rb   ro   rp   r>   rq   varrI   rs   asarraylenrd   re   rw   )r:   rh   r]   r8   r~   r-   r-   r.   _validate_seuclidean_kwargs  s    

r   c                 C   s*   t j| |dd} | jdkr| S tdd S )Nr   r=   r   r#   zInput vector should be 1-D.)r>   r   rc   re   )rJ   r=   r-   r-   r.   _validate_vector)  s    
r   c                 C   s&   t | |d} t| dk r"td| S )Nr<   r   z(Input weights should be all non-negative)r   r>   anyre   )rL   r=   r-   r-   r.   rf   1  s    rf   c                 C   sR   t j| t jdd} t j|t jdd}| jd |jd kr@tdt| ||}|S )a  
    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)

    r   r   r#   z/u and v need to have the same number of columns)r>   r   rs   rd   re   r%   r	   )rJ   rK   seedresultr-   r-   r.   r	   8  s    \r	   c                 C   s   t | } t |}|dkr td| | }|durt|}|dkrF|}n8|dkrZt|}n$|tjkrn|dk}nt|d| }|| }t||d}|S )aA  
    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

    r   r{   Nr#   r&   )ord)r   re   rf   r>   sqrtinfpowerr'   )rJ   rK   ry   rL   u_vZroot_wdistr-   r-   r.   r     s"    2

r   c                 C   s   t | |d|dS )a'  
    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

    r&   )ry   rL   )r   rJ   rK   rL   r-   r-   r.   r
     s    $r
   c                 C   s   d\}}t | dr"t| jtjs(tj}t |drBt|jtjsHtj}t| |d} t||d}| | }|}|durt|}|| }t||S )a   
    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

    )NNr=   r<   N)	hasattrr>   Z
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    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
    NrC           rz   )	r   rf   rF   r>   r   r4   mathr   Zclip)rJ   rK   rL   centeredZumuZvmuZvwZuwZuvuuvvr   r-   r-   r.   r   ?  s,    2
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r   c                 C   s   t | ||ddS )a  
    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

    F)rL   r   )r   r   r-   r-   r.   r     s    *r   c                 C   sr   t | } t |}| j|jkr$td| |k}|durht|}|j| jkrPtd||  }t||S t|S )a  
    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

    z&The 1d arrays must have equal lengths.Nz/'w' should have the same length as 'u' and 'v'.)r   rd   re   rf   rF   r>   r   r4   )rJ   rK   rL   Zu_ne_vr-   r-   r.   r     s    ,r   c                 C   s   t | } t |}t| dk|dk}t| |k|}|durTt|}|| }|| }t| }t| }|dkr|| S dS )a  
    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

    r   N)r   r>   Z
bitwise_orZbitwise_andrf   rs   rF   )rJ   rK   rL   ZnonzeroZunequal_nonzeror,   br-   r-   r.   r     s    <r   rL   c                C   sB   t | } t |}|dur t|}t| ||d\}}}}|||  S )aJ  
    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

    Nr   )r   rf   rS   )rJ   rK   rL   _rP   rQ   rR   r-   r-   r.   r   =  s    ;r   c                 C   s`   t | } t |}t |tjd}|jd | jd ksF| jd |jd krNtdt| |d| dS )aa  
    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

    r<   r   z7V must be a 1-D array of the same dimension as u and v.r#   r   )r   r>   rs   rd   	TypeErrorr
   )rJ   rK   r~   r-   r-   r.   r     s    '(r   c                 C   s<   t | } t |}t| | }|dur4t|}|| }| S )aD  
    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

    N)r   absrf   rF   )rJ   rK   rL   l1_diffr-   r-   r.   r     s    %r   c                 C   s@   t | } t |}t|}| | }tt|||}t|S )a  
    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

    )r   r>   rr   r   r   )rJ   rK   rm   deltarh   r-   r-   r.   r     s    '
r   c                 C   sV   t | } t |}|durFt|}|dk}| |jk rF| | } || }tt| | S )a  
    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

    Nr   )r   rf   rF   sizemaxr   )rJ   rK   rL   Z
has_weightr-   r-   r.   r     s    "r   c                 C   s^   t | } t |tjd}t| | }t| | }|durNt|}|| }|| }| |  S )a{  
    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

    r<   N)r   r>   rs   r   rf   rF   )rJ   rK   rL   r   Zl1_sumr-   r-   r.   r   ;  s    %r   c                 C   s   t | } t |tjd}|dur&t|}tjddR t| | }t| }t|}|||  }|durl|| }t|}W d   n1 s0    Y  |S )a{  
    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

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|dur|
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d S )a  
    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])

    Tr1   rz   N)r>   r   rF   r(   logr   )ry   qr*   r2   r3   rh   leftrightZleft_sumZ	right_sumjsr-   r-   r.   r     s    I



r   c                 C   sf   t | } t |}|dur t|}t| ||d\}}}}|| }|dkrJdS td| || |  S dS )a  
    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

    Nr   r   r   rz   r   rf   rS   float)rJ   rK   rL   rO   rP   rQ   rR   Zhalf_Rr-   r-   r.   r     s    &r   c                 C   s   t | } t |}|dur t|}| j|j  kr8tkrRn n|du rR| |@  }nLtt| j|j}| |} ||}|du r| |  }n| | |  }t	| ||d\}}t
|| td| | |  S )a  
    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

    Nr   rz   )r   rf   r=   rE   rF   r>   rG   rH   rI   rT   r   array)rJ   rK   rL   rR   r=   rP   rQ   r-   r-   r.   r   ,  s    /$

r   c                 C   s^   t | } t |}|dur t|}t| ||d\}}}}td||  t|| d||    S )aI  
    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

    Nr   rz   r   rJ   rK   rL   rO   rP   rQ   rR   r-   r-   r.   r   m  s    *r   c                 C   s   t | } t |}| j|j  kr(tkrNn n"|du rN| |@  }tt| }nB|du rp| |  }tt| }n t|}| | |  }| }t|| | S )a  
    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

    N)r   r=   rE   rF   r   r   rf   )rJ   rK   rL   rR   r]   r-   r-   r.   r     s    *$r   c                 C   s^   t | } t |}|dur t|}t| ||d\}}}}td||  t|| d||    S )aM  
    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

    Nr   rz   r   r   r-   r-   r.   r     s    +r   c                 C   s   t | } t |}| j|j  kr(tkrBn n|du rB| |@  }n.|du rX| |  }nt|}| | |  }t| ||d\}}t|d||   }| st	dt
d||  | S )aF  
    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

    Nr   rz   zNSokal-Sneath dissimilarity is not defined for vectors that are entirely false.)r   r=   rE   rF   rf   rT   r>   r   r   re   r   )rJ   rK   rL   rR   rP   rQ   denomr-   r-   r.   r     s    +$r   rV   )frozenc                   @   s$   e Zd ZU eed< ddddZdS )CDistMetricWrappermetric_nameNoutc                K   s   t |}t |}|j\}}|j\}}| j}	t|	 }
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zCDistMetricWrapper.__call____name__
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r   c                   @   s$   e Zd ZU eed< ddddZdS )PDistMetricWrapperr   Nr   c                K   s   t |}|j\}}| j}t| }t||||fi |\}}}||d  d }	|dd }
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r   c                   @   sr   e Zd ZU eed< ee ed< eed< eed< eed< dZee ed< e	j
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r   )r   r   r   r   r   >   Zchebr   Z	chebychevchZcheby>   Zcblockr   cbr   cocosrE   )r   r   rX   r   r   r   >   Zeuclidr
   eue>   r   ZhahZmatchingZhammrj   )r   r   rX   rZ   r   r   r   >   jjaZjaccr   r   >   Zmahr   Zmahal)r   r   rZ   r   r   r   >   rh   Zpnormr   mi>   r   ses>   Zsqeuclidr   Zsqec                 C   s   i | ]}|j |qS r-   r   .0infor-   r-   r.   
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|trJ| }t|d}|dur|j}
|
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|i|S |dr<t|d}|du rtd|dd  t| |||fi |\} }	}t	| f|j|d	|S td| ntddS )a%  
    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<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 ``m
        * i + j - ((i + 2) * (i + 1)) // 2``.

    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` (:math:`p`-norm) where :math:`p > 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.])

    FT)Z	sparse_okZ
objects_okZmask_okZcheck_finiter&   z%A 2-dimensional array must be passed.r   ZUnknownCustomMetricNr   r   r   r    Unknown "Test" Distance Metric:    Unknown Distance Metric: %s>2nd argument metric must be a string identifier or a function.)r"   rd   r   re   callabler   _METRIC_ALIASrk   r}   r   ro   r   lowerr   
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
r   noc                 C   s  t | } | j}| dkr2t|dkrRtdn | dkrRt|dkrRtdt|dkr|d dkrzt jd| jd	S tt 	t 
|d d }||d  |d d krtd
t j||f| jd	}t| } t||  |S t|dkrt|d |d kr
td|rt| ddd |d }|dkr@t jg | jd	S t j||d  d | jd	}t| } t| | |S tdt| dS )a)  
    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])
    Ztomatrixr#   z8Forcing 'tomatrix' but input X is not a distance vector.Ztovectorr&   z8Forcing 'tovector' but input X is not a distance matrix.r   )r#   r#   r<   z_Incompatible vector size. It must be a binomial coefficient n choose 2 for some integer n >= 2.z#The matrix argument must be square.Tr:   thrownamez`The first argument must be one or two dimensional array. A %d-dimensional array is not permittedN)r>   r?   rd   r   r   re   Zzerosr=   rH   ceilr   r/   r$   Zto_squareform_from_vector_wrapr   r   Zto_vector_from_squareform_wrap)r:   forceZchecksr   r   MrK   r-   r-   r.   r     sB    V


r   r   Dc              
   C   s  t j| dd} d}z:| j}t| jdkrD|r<td| ntd|dkr| | jk st|rltd| ntd	| td
|d
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 f |k sL|r>td| d|ddntdj| W nJ t	y } z0|rh |rt
jt|dd d}W Y d}~n
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    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

    r   r   Tr&   zADistance matrix '%s' must have shape=2 (i.e. be two-dimensional).z<Distance matrix must have shape=2 (i.e. be two-dimensional).r   z'Distance matrix '%s' must be symmetric.z"Distance matrix must be symmetric.r   z+Distance matrix '%s' diagonal must be zero.z&Distance matrix diagonal must be zero.zDistance matrix 'z%' must be symmetric within tolerance z5.5f.z9Distance matrix must be symmetric within tolerance %5.5f.z2' diagonal must be close to zero within tolerance zMDistance matrix '{}' diagonal must be close to zero within tolerance {:5.5f}.
stacklevelFN)r>   r   rd   r   re   rt   allrangeformat	Exceptionwarningswarnr   )r   tolr   r   warningvalidr   r   r-   r-   r.   r   .	  s\    D(

*

r   c              
   C   s   t j| dd} d}z|t| jdkr<|r4td| ntd| jd }tt t |d }||d  d |kr|rtd	| ntd
W nD ty } z,|r |rt	j
t|dd d}W Y d}~n
d}~0 0 |S )a  
    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

    r   r   Tr#   zKCondensed distance matrix '%s' must have shape=1 (i.e. be one-dimensional).zFCondensed distance matrix must have shape=1 (i.e. be one-dimensional).r   r&   z|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)!zxLength n of condensed distance matrix must be a binomial coefficient, i.e. there must be a k such that (k \choose 2)=n)!r   FN)r>   r   r   rd   re   rH   r   r   r  r  r  r   )yr  r   r   r  r]   r   r   r-   r-   r.   r   	  s0    0
r   c                 C   s*   t j| dd} t| t jddd | jd S )aM  
    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
    r   r   Tr   )r  r   r   r   )r>   r   r   r   rd   )r   r-   r-   r.   r   	  s    r   c                 C   sn   t j| dd} t| ddd | jd }|dkr6tdtt t |d }||d	  d |krjtd
|S )a  
    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
    r   r   TYr   r   zLThe number of observations cannot be determined on an empty distance matrix.r&   r#   z^Invalid condensed distance matrix passed. Must be some k where k=(n choose 2) for some n >= 2.)r>   r   r   rd   re   rH   r   r   )r	  kr   r-   r-   r.   r   
  s    
r   c                 C   sP   | d u rt j||dS | j|kr(td| jjs8td| jt jkrLtd| S )Nr<   z!Output array has incorrect shape.z"Output array must be C-contiguous.z!Output array must be double type.)r>   emptyrd   re   flagsc_contiguousr=   rs   )r   r=   Zexpected_shaper-   r-   r.   r   6
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    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]])

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