← Back
Editing: distance.cpython-311.pyc
� d�cf` � �V � d Z g d�ZddlZddlZddlZddlmZmZm Z m Z ddl m Z ddl mZ ddlmZ dd lmZ d d lmZ d d lmZ dd lmZ d dlmZ d� Zd� Zd� Zd� Z dd�Z dd�Z d� Z d� Z d� Z!d� Z"d� Z#d� Z$d� Z%dd �Z&ej' fd �Z(d�d �Z)d�d �Z*dd �Z+dd!�Z,d�d#�Z-dd$�Z.dd%�Z/dd&�Z0 ed'� � dd(�� � Z1dd)�d*�Z2d+� Z3dd,�Z4d-� Z5dd.�Z6dd/�Z7dd0�Z8ddd1d2�d3�Z9dd4�Z:dd5�Z;dd6�Z<dd7�Z=dd8�Z>dd9�Z? e e ej' �:� � Z@ e e eA�:� � ZBee_C ee_D ejE d"�;� � G d<� d=� � � � ZF ejE d"�;� � G d>� d?� � � � ZG ejE d"�;� � G d@� dA� � � � ZH ejE d"�;� � G dB� dC� � � � ZI ejE d"�;� � G dD� dE� � � � ZJ eJdFdFhe7ejK ejL �G� � eJdHdHhe8ejM ejN �G� � eJdIh dJ�e6ejO ejP �G� � eJdKh dL�e4ejQ ejR �G� � eJdMdMdNhe- eFdM� � eHdM� � �G� � eJdOdOdPhe. eFdO� � eHdO� � �G� � eJdQdQhdRge; eFdQ� � eHdQ� � �S� � eJdTh dU�e+ejS ejT �G� � eJdVh dW�dXdRge!e/ eGdVdV� � eIdVdV� � �Y� � eJdZh d[�dXdRge0 eFdZ� � eHdZ� � �S� � eJd\d\d]he9 eFd\� � eHd\� � �G� � eJd^d^hdRge1 eFd^� � eHd^� � �S� � eJd_d_hdRge2 eFd_� � eHd_� � �S� � eJd`h da�e"e5 eFd`� � eHd`� � �b� � eJdch dd�e#e*ejU ejV �b� � eJdedehdRge< eFde� � eHde� � �S� � eJdfdfhdRge= eFdf� � eHdf� � �S� � eJdgh dh�e%e3 eFdg� � eHdg� � �b� � eJdidihdRge> eFdi� � eHdi� � �S� � eJdjdjhdRge? eFdj� � eHdj� � �S� � eJdkh dl�e,ejW ejX �G� � eJdmdmhdRge: eFdm� � eHdm� � �S� � gZYdn� eYD � � ZZ e[do� eYD � � � � Z\ e]eZ�^ � � � � Z_dp� eYD � � Z`d�ddq�dr�Zad�dt�Zbd�dw�Zcd�dx�Zddy� Zedz� Zfd{� Zgd|� Zhd}� Zid�ddq�d~�ZjdS )�a� 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. kulsinski -- the Kulsinski 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� kulsinski� kulczynski1� mahalanobis� minkowski� num_obs_dm� num_obs_y�pdist�rogerstanimoto� russellrao� seuclidean� sokalmichener� sokalsneath� sqeuclidean� squareform�yule� N)�List�Optional�Set�Callable)�partial)�_asarray_validated� )�_distance_wrap)� _hausdorff� )�norm)�rel_entr)�_distance_pybind)� _deprecatedc �< � | j �| � � � S | S )z4Copy the array if its base points to a parent array.)�base�copy)�as �8/usr/lib/python3/dist-packages/scipy/spatial/distance.py�_copy_array_if_base_presentr4 � s � ��v���v�v�x�x�� �H� c � � | | � dd�� � z } ||� dd�� � z }t j | ||fi |�� d S �Nr' T��axis�keepdims)�meanr( �cdist_cosine_double_wrap)�XA�XB�dm�kwargss r3 �_correlation_cdist_wraprA � sY � � �b�g�g�1�t�g�,�,� ,�B� �b�g�g�1�t�g�,�,� ,�B��+�B��B�A�A�&�A�A�A�A�Ar5 c �` � | | � dd�� � z }t j ||fi |�� d S r7 )r; r( �pdist_cosine_double_wrap)�Xr? r@ �X2s r3 �_correlation_pdist_wraprF � s= � � �Q�V�V��T�V� *� *� *�B��+�B��=�=�f�=�=�=�=�=r5 c �. � t j | |�� � S )N��dtype)�np�ascontiguousarray)rD �out_types r3 �_convert_to_typerM � s � � � ��� 2� 2� 2�2r5 c � � | j |j cxk r t k rhn ne|�c| }| }||z � � � }||z � � � }| |z � � � }| |z � � � }n�t j t | j |j � � } | � | � � } |� | � � }d| z }d|z }|� ||z }|| z } ||z � � � }||z � � � }| |z � � � }| |z � � � }||||fS �N� �?�rI �bool�sumrJ � result_type�int�astype) �u�v�w�not_u�not_v�nff�nft�ntf�nttrI s r3 �_nbool_correspond_allr` � sA � ��w�!�'�!�!�!�!�T�!�!�!�!�!�a�i��������u�}�!�!�#�#���q�y�o�o� � ���5�y�o�o� � ���1�u�k�k�m�m�����s�A�G�Q�W�5�5�� �H�H�U�O�O�� �H�H�U�O�O���a����a��� �=���I�E��A��A��u�}�!�!�#�#���q�y�o�o� � ���5�y�o�o� � ���1�u�k�k�m�m�� ��c�3� � r5 c �� � | j |j cxk r t k r:n n7|�5| }| }||z � � � }| |z � � � }n�t j t | j |j � � }| � |� � } |� |� � }d| z }d|z }|� ||z }|| z } ||z � � � }| |z � � � }||fS rO rQ )rW rX rY rZ r[ r] r^ rI s r3 �_nbool_correspond_ft_tfrb � s� � ��w�!�'�!�!�!�!�T�!�!�!�!�!�a�i��������q�y�o�o� � ���5�y�o�o� � �����s�A�G�Q�W�5�5�� �H�H�U�O�O�� �H�H�U�O�O���a����a��� �=���I�E��A��A��q�y�o�o� � ���5�y�o�o� � �� ��:�r5 c �� � |j }| j |v r ||� | j � � n|d }t | |�� � } t ||�� � }|j } | r | | |f||z |fi |��}| |||fS �Nr �rL ��typesrI �indexrM � validator) r= r> �mA�mB�n� metric_infor@ rg �typ�_validate_kwargss r3 �_validate_cdist_inputrp � s� � � � �E�*,�(�e�*;�*;�%�� � �B�H�%�%� &� &��q��C� �"�s� +� +� +�B� �"�s� +� +� +�B� #�,��� B�!�!�2�r�(�B��G�Q�A�A�&�A�A�� �r�3�� � r5 c �� � |� dd � � }|�|S |j dk s|j d |k r t d|j d � d|� �� � �t |� � |d< |S )NrY r' r z-Weights must have same size as input vector. z vs. )�pop�ndim�shape� ValueError�_validate_weights�rD �mrl r@ rY s r3 �_validate_weight_with_sizery � s� � �� � �3�� � �A��y�� ��v��{�{�a�g�a�j�A�o�o�� 1� �G�A�J�1� 1�-.�1� 1� 2� 2� 2� $�A�&�&�F�3�K� �Mr5 c � � |� dt j |fd�� � � � }|j dk s|j d |k r t d|j d |fz � � �t |� � |d< |S )NrY �doublerH r' r z6Weights must have same size as input vector. %d vs. %d)�getrJ �onesrs rt ru rv rw s r3 �_validate_hamming_kwargsr~ � sy � �� � �3�����H�5�5�5�6�6�A��v��{�{�a�g�a�j�A�o�o��Q�UV�U\�]^�U_�ab�Tc�c�d�d�d�#�A�&�&�F�3�K� �Mr5 c �� � |� dd � � }|��||k rt d|||dz fz � � �t | t � � rt j | � � } t j t j | � t j d�� � j � � � � }t j � |� � j � � � }t |� � |d< |S )N�VIz�The number of observations (%d) is too small; the covariance matrix is singular. For observations with %d dimensions, at least %d observations are required.r' F�r1 )rr ru � isinstance�tuplerJ �vstack� atleast_2d�covrV r{ �T�linalg�invr1 �_convert_to_double)rD rx rl r@ r� �CVs r3 � _validate_mahalanobis_kwargsr� � s� � � ���D�$� � �B� �z� ��6�6� � -� 12�1�a�!�e�}� =� >� >� >� �a�� � � �� �!� � �A� �]�2�6�!�(�(�2�9�5�(�"A�"A�"C�D�D� E� E�� �Y�]�]�2� � � � %� %� '� '��%�b�)�)�F�4�L� �Mr5 c �n � t | ||fi |��}d|vrd|d< n|d dk rt d� � �|S )N�p� @r �p must be greater than 0)ry ru )rD rx rl r@ s r3 �_validate_minkowski_kwargsr� � sT � � '��1�a� :� :�6� :� :�F� �&�����s� � � �#�;�!� � � � 7�8�8� 8� �Mr5 c �� � |j }| j |v r ||� | j � � n|d }t | |�� � } |j }|r || ||fi |��}| ||fS rd rf )rD rx rl rm r@ rg rn ro s r3 �_validate_pdist_inputr� s� � � � �E�)*��E�)9�)9�%�� � �A�G�$�$� %� %�u�Q�x�C���S�)�)�)�A� #�,��� 5�!�!�!�Q��4�4�V�4�4�� �c�6�>�r5 c �� � |� dd � � }|�`t | t � � rt j | � � } t j | � t j d�� � dd�� � }n]t j |d�� � }t |j � � dk rt d � � �|j d |k rt d � � �t |� � |d< |S ) N�VFr� r r' )r9 �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.) rr r� r� rJ r� �varrV r{ �asarray�lenrt ru r� )rD rx rl r@ r� s r3 �_validate_seuclidean_kwargsr� s� � �� � �3�� � �A��y� �a�� � � �� �!� � �A� �F�1�8�8�B�I�E�8�2�2��� C� C� C��� �J�q�� $� $� $�� �q�w�<�<�1� � � � 3� 4� 4� 4� �7�1�:��?�?� � -� .� .� .� %�Q�'�'�F�3�K� �Mr5 c �h � t j | |d�� � } | j dk r| S t d� � �)Nr� �rI r� r' zInput vector should be 1-D.)rJ r� rs ru )rW rI s r3 �_validate_vectorr� ) s7 � � � �1�E��-�-�-�A��v��{�{��� �2� 3� 3�3r5 c �v � t | |�� � } t j | dk � � rt d� � �| S )NrH r z(Input weights should be all non-negative)r� rJ �anyru )rY rI s r3 rv rv 1 s= � ���%�(�(�(�A� �v�a�!�e�}�}� E��C�D�D�D� �Hr5 c � � t j | t j d�� � } t j |t j d�� � }| j d |j d k rt d� � �t j | ||� � }|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. v : (O,N) array_like Input array. seed : int or None 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. 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 >>> 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)rJ r� �float64rt ru r) r )rW rX �seed�results r3 r r 8 s| � �x � �1�B�J�c�2�2�2�A� � �1�B�J�c�2�2�2�A��w�q�z�Q�W�Q�Z� � �� -� .� .� .� � *�1�a�� 6� 6�F� �Mr5 c �j � t | � � } t |� � }|dk rt d� � �| |z }|�gt |� � }|dk r|}nJ|dk rt j |� � }n/|t j k r|dk }nt j |d|z � � }||z }t ||�� � }|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� ru rv rJ �sqrt�inf�powerr+ )rW rX r� rY �u_v�root_w�dists r3 r r � s� � �d ����A�����A��A�v�v��3�4�4�4� �a�%�C��}� �a� � �� ��6�6��F�F� �!�V�V��W�Q�Z�Z�F�F� �"�&�[�[��1�f�F�F��X�a��1��%�%�F��s�l�� ��� � � �D� �Kr5 c �( � t | |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* )r� rY )r �rW rX rY s r3 r r � s � �H �Q��Q�!� $� $� $�$r5 c � � d\ }}t | d� � r$t j | j t j � � s t j }t |d� � r$t j |j t j � � s t j }t | |�� � } t ||�� � }| |z }|}|�t |� � }||z }t j ||� � S )a% 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 )NNrI rH ) �hasattrrJ � issubdtyperI �inexactr� r� rv �dot)rW rX rY �utype�vtyper� �u_v_ws r3 r r s� � �L �L�E�5� �A�w� � � �B�M�!�'�2�:�$F�$F� �� �� �A�w� � � �B�M�!�'�2�:�$F�$F� �� ����%�(�(�(�A���%�(�(�(�A� �a�%�C� �E��}� �a� � ���C��� �6�#�u� � � r5 Tc � � t | � � } t |� � }|�t |� � }|r6t j | |�� � }t j ||�� � }| |z } ||z }t j | |z |�� � }t j t j | � � |�� � }t j t j |� � |�� � }d|t j ||z � � z z } t j | � � S )ar 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`. N��weightsrP )r� rv rJ �average�squarer� �abs) rW rX rY �centered�umu�vmu�uv�uu�vvr� s r3 r r A s� � �B ����A�����A��}� �a� � ��� ��j��A�&�&�&���j��A�&�&�&�� ��G�� ��G�� ��A��E�1� %� %� %�B� ��B�I�a�L�L�!� ,� ,� ,�B� ��B�I�a�L�L�!� ,� ,� ,�B� ��b�g�b�2�g�&�&�&� &�D� �6�$�<�<�r5 c �` � t dt 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 r F)rY r� r� )�max�minr r� s r3 r r s s0 � �V �q�#�k�!�Q�!�e�<�<�<�c�B�B� C� C�Cr5 c �� � t | � � } t |� � }| j |j k rt d� � �| |k }|�t |� � }t j ||�� � 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.Nr� )r� rt ru rv rJ r� )rW rX rY �u_ne_vs r3 r r � sm � �X ����A�����A��w�!�'����A�B�B�B� �!�V�F��}� �a� � �� �:�f�a� (� (� (�(r5 c � � t | � � } t |� � }t j | dk |dk � � }t j | |k |� � }|�t |� � }||z }||z }t j |� � � � � }t j |� � � � � }|dk r||z ndS )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 )r� rJ � bitwise_or� bitwise_andrv r{ rS )rW rX rY �nonzero�unequal_nonzeror2 �bs r3 r r � s� � �x ����A�����A��m�A��F�A��F�+�+�G��n�a�1�f�w�7�7�O��}� �a� � ���g�+���o�-�� � �/�%�%�'�'�(�(�A� � �'�+�+�-�-� � �A��1�f�f�A��E�E�!�#r5 z�Kulsinski has been deprecated from scipy.spatial.distance in SciPy 1.9.0 and it will be removed in SciPy 1.11.0. It is superseded by scipy.spatial.distance.kulczynski1.c � � t | � � } t |� � }|� t t | � � � � }n#t |� � }|� � � }t | ||�� � \ }}}}||z |z |z ||z |z z S )a@ 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`. .. deprecated:: 0.12.0 `kulsinski` has been deprecated from `scipy.spatial.distance` in SciPy 1.9.0 and it will be removed in SciPy 1.11.0. It is superseded by `scipy.spatial.distance.kulczynski1`. 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 N�rY )r� �floatr� rv rS r` )rW rX rY rl r\ r] r^ r_ s r3 r r ! s� � �h ����A�����A��y� �#�a�&�&�M�M��� �a� � �� �E�E�G�G��0��A��;�;�;��S�#�s�C� �#�I��O�a� �C�#�I��M� 2�2r5 r� c � � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}|||z z 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� rv r` )rW rX rY �_r] r^ r_ s r3 r r a s] � �v ����A�����A��}� �a� � ��.�q�!�q�9�9�9��Q��S�#� �#��)� � r5 c �, � t | � � } t |� � }t |t j �� � }|j d | j d k s | j d |j d k rt d� � �t | |d|z �� � S )a] 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 rH r z7V must be a 1-D array of the same dimension as u and v.r' r� )r� rJ r� rt � TypeErrorr )rW rX r� s r3 r r � s� � �@ ����A�����A���"�*�-�-�-�A��w�q�z�Q�W�Q�Z� � �1�7�1�:�����#;�#;�� &� '� '� '� �Q��Q�q�S� !� !� !�!r5 c � � t | � � } t |� � }t | |z � � }|�t |� � }||z }|� � � 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 )r� r� rv rS )rW rX rY �l1_diffs r3 r r � sW � �J ����A�����A��!�a�%�j�j�G��}� �a� � ���g�+�� �;�;�=�=�r5 c �� � t | � � } t |� � }t j |� � }| |z }t j t j ||� � |� � }t j |� � 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� rJ r� r� r� )rW rX r� �deltarx s r3 r r � sb � �N ����A�����A� � �r� � �B� ��E�E� ��r�v�e�R� � �%�(�(�A� �7�1�:�:�r5 c � � t | � � } t |� � }|�Bt |� � }|dk }|� � � |j k r| | } || }t t | |z � � � � 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� rv rS �sizer� r� )rW rX rY � has_weights r3 r r + sy � �D ����A�����A��}� �a� � ����U� � �>�>� � �a�f� $� $��*� �A��*� �A� �s�1�q�5�z�z�?�?�r5 c �& � t | � � } t |t j �� � }t | |z � � }t | |z � � }|�t |� � }||z }||z }|� � � |� � � z 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 rH )r� rJ r� r� rv rS )rW rX rY r� �l1_sums r3 r r X s� � �J ����A���"�*�-�-�-�A��!�a�%�j�j�G� ��Q��Z�Z�F��}� �a� � ���g�+���V��� �;�;�=�=�6�:�:�<�<� '�'r5 c �| � t | � � } t |t j �� � }|�t |� � }t j d�� � 5 t | |z � � }t | � � }t |� � }|||z z }|�||z }t j |� � }ddd� � n # 1 swxY w 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 rH N�ignore)�invalid)r� rJ r� rv �errstater� �nansum)rW rX rY �abs_uv�abs_u�abs_v�ds r3 r r � s� � �P ����A���"�*�-�-�-�A��}� �a� � �� ��X� &� &� &� � ��Q��U�����A�����A���� �e�e�m� $�� �=��A��A� �I�a�L�L��� � � � � � � � � � ���� � � � � �Hs �AB1�1B5�8B5Fr8 c �� � t j | � � } t j |� � }| t j | |d�� � z } |t j ||d�� � z }| |z dz }t | |� � }t ||� � }t j |||�� � }t j |||�� � } || z } |�| t j |� � z } t j | dz � � 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]) Tr8 r� )rJ r� rS r, �logr� ) r� �qr0 r9 r: rx �left�right�left_sum� right_sum�jss r3 r r � s� � �R � �1� � �A� � �1� � �A� �B�F�1�4�$� /� /� /�/�A� �B�F�1�4�$� /� /� /�/�A� �Q��#� �A� �A�q�>�>�D� �Q��N�N�E��v�d���9�9�9�H���u�4�(�;�;�;�I� �I� �B� �� �b�f�T�l�l��� �7�2��8� � � r5 c �� � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}||z }|dk rdS t d|z ||z |z z � � 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� �r� rv r` r� )rW rX rY r\ r] r^ r_ �half_Rs r3 r r s� � �L ����A�����A��}� �a� � ��0��A��;�;�;��S�#�s�C� �3�Y�F� ��{�{��s��S�6�\�S�3�Y��%7�8�9�9�9r5 c �b � t | � � } t |� � }|�t |� � }| j |j cxk r t k r n n|�| |z � � � }n�t j t | j |j � � }| � |� � } |� |� � }|�| |z � � � }n| |z |z � � � }t | ||�� � \ }}t ||z t j d|z |z |z � � z � � 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� r� ) r� rv rI rR rS rJ rT rU rV rb r� �array)rW rX rY r_ rI r] r^ s r3 r r I s � �^ ����A�����A��}� �a� � ���w�!�'�!�!�!�!�T�!�!�!�!�!�a�i��1�u�k�k�m�m�����s�A�G�Q�W�5�5�� �H�H�U�O�O�� �H�H�U�O�O�� �9��q�5�+�+�-�-�C�C��q�5�1�9�/�/�#�#�C�(��A��3�3�3�J�S�#� �#��)�r�x��c� �C��#�(=� >� >�>� ?� ?�?r5 c �� � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}t d||z z � � t ||z d||z z z � � z 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� r� r� �rW rX rY r\ r] r^ r_ s r3 r r � s� � �T ����A�����A��}� �a� � ��0��A��;�;�;��S�#�s�C� ���c� �"� #� #�e�C�#�I���c� �9J�,K�&L�&L� L�Lr5 c � � t | � � } t |� � }| j |j cxk r t k r9n n6|�4| |z � � � }t t | � � � � }ns|�4| |z � � � }t t | � � � � }n=t |� � }| |z |z � � � }|� � � }t ||z � � |z 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 )r� rI rR rS r� r� rv )rW rX rY r_ rl s r3 r r � s� � �T ����A�����A��w�!�'�!�!�!�!�T�!�!�!�!�!�a�i��1�u�k�k�m�m�� �#�a�&�&�M�M��� ���1�u�k�k�m�m�� �#�a�&�&�M�M��� �a� � ���1�u�q�y�o�o� � �� �E�E�G�G�� ��S��>�>�A� � r5 c �� � t | � � } t |� � }|�t |� � }t | ||�� � \ }}}}t d||z z � � t ||z d||z z z � � z 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� r� r� r� s r3 r r � s� � �V ����A�����A��}� �a� � ��.�q�!�q�9�9�9��C��c�3� ���c� �"� #� #�e�C�#�I��s�S�y�8I�,I�&J�&J� J�Jr5 c � � t | � � } t |� � }| j |j cxk r t k r n n|�| |z � � � }nC|�| |z � � � }n)t |� � }| |z |z � � � }t | ||�� � \ }}t j |d||z z z � � }|� � � st d� � �t d||z z � � |z 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� r� zNSokal-Sneath dissimilarity is not defined for vectors that are entirely false.) r� rI rR rS rv rb rJ r� r� ru r� )rW rX rY r_ r] r^ �denoms r3 r r ( s � �V ����A�����A��w�!�'�!�!�!�!�T�!�!�!�!�!�a�i��1�u�k�k�m�m��� ���1�u�k�k�m�m��� �a� � ���1�u�q�y�o�o� � ��(��A��3�3�3�J�S�#� �H�S�3�#��)�,�,� -� -�E� �9�9�;�;� =�� <� =� =� =� ���c� �"� #� #�e� +�+r5 re )�frozenc �& � e Zd ZU eed< dd�d�ZdS )�CDistMetricWrapper� metric_nameN��outc � � t j |� � }t j |� � }|j \ }}|j \ }}| j } t | } t |||||| fi |��\ }}} }|� dd � � } | �| j } t ||f| || d�|��S t |t j ||f� � }t t d| � d| � d�� � } ||||fi |�� |S )NrY ��metricr rY �cdist_r� �_wrap) rJ rK rt r �_METRICSrp rr � dist_func�_cdist_callable�_prepare_out_argumentr{ �getattrr( )�selfr= r> r r@ rj rl rk r� r rm rn rY r r? �cdist_fns r3 �__call__zCDistMetricWrapper.__call__p s+ � � � !�"� %� %�� � !�"� %� %������A�����A��&� � �{�+� � 3� ��B��A�{� 6� 6�.4� 6� 6���B��V� �J�J�s�D� !� !�� �=� �*�F�"��B�?� %�3�!�?� ?�7=�?� ?� ?� #�3�� �B��8� <� <���>�+L�K�+L�+L�#�+L�+L�+L�M�M�����R��&�&�v�&�&�&�� r5 ��__name__� __module__� __qualname__�str�__annotations__r � r5 r3 r� r� l s> � � � � � � �����&*� � � � � � � r5 r� c �0 � e Zd ZU eed< eed< dd�d�ZdS )�CDistWeightedMetricWrapperr �weighted_metricNr c � � t j |� � }t j |� � }|j \ }}|j \ }}| j } t |||||t | fi |��\ }}} }t |t j ||f� � } |� dd � � } | � | j } | |d< t t d| � d| � d�� � } | ||| fi |�� | S )NrY r r� r ) rJ rK rt r rp r r r{ rr r r r( )r r= r> r r@ rj rl rk r� r rn r? rY r s r3 r z#CDistWeightedMetricWrapper.__call__� s � � � !�"� %� %�� � !�"� %� %������A�����A��&� � 3� ��B��A�x� � 4� @� @�8>� @� @���B��V� "�3�� �B��8� <� <�� �J�J�s�D� !� !�� �=� �.�K��F�3�K� �>�+L�K�+L�+L�#�+L�+L�+L�M�M�����R��&�&�v�&�&�&�� r5 r r r5 r3 r r � sJ � � � � � � ���������&*� � � � � � � r5 r c �& � e Zd ZU eed< dd�d�ZdS )�PDistMetricWrapperr Nr c � � t j |� � }|j \ }}| j }t | }t ||||fi |��\ }}}||dz z dz } |� dd � � } | �|j } t |f| || d�|��S t |t j | f� � } t t d|� d|� d�� � } | || fi |�� | S )Nr' r* rY r �pdist_r� r ) rJ rK rt r r r� rr r �_pdist_callabler r{ r r( )r rD r r@ rx rl r rm rn �out_sizerY r r? �pdist_fns r3 r zPDistMetricWrapper.__call__� s � � � �� #� #���w���1��&� � �{�+� �.� �q�!�[�,� ,�$*�,� ,���3����Q��K�A�%�� �J�J�s�D� !� !�� �=� �*�F�"��:� �c�Q�:� :�28�:� :� :� #�3�� �H�;� ?� ?���>�+L�K�+L�+L�#�+L�+L�+L�M�M�����B�!�!�&�!�!�!�� r5 r r r5 r3 r r � s> � � � � � � �����!%� � � � � � � r5 r c �0 � e Zd ZU eed< eed< dd�d�ZdS )�PDistWeightedMetricWrapperr r Nr c �p � t j |� � }|j \ }}| j }t |||t | fi |��\ }}}||dz z dz }t |t j |f� � } |� dd � � } | � | j }| |d< t t d|� d|� d�� � } | || fi |�� | S )Nr' r* rY r r� r ) rJ rK rt r r� r r r{ rr r r r( ) r rD r r@ rx rl r rn r r? rY r! s r3 r z#PDistWeightedMetricWrapper.__call__� s� � � � �� #� #���w���1��&� �.� �q�!�X�k�*�6� 6�.4�6� 6���3����Q��K�A�%�� "�3�� �H�;� ?� ?�� �J�J�s�D� !� !�� �=� �.�K��F�3�K� �>�+L�K�+L�+L�#�+L�+L�+L�M�M�����B�!�!�&�!�!�!�� r5 r r r5 r3 r# r# � sJ � � � � � � ���������!%� � � � � � � r5 r# c � � e Zd ZU eed< ee ed< eed< eed< eed< dZee ed< e j d� � � � Z e e ed < d Z eed < dS ) � MetricInfo�canonical_name�akar � cdist_func� pdist_funcNri c � � dgS )Nr{ r r r5 r3 �<lambda>zMetricInfo.<lambda>� s � �(�� r5 )�default_factoryrg T�requires_contiguous_out)r r r r r r# r$ ri r" � dataclasses�fieldrg r! r. rR r r5 r3 r& r& � s� � � � � � � � ���� �S��M�M�M������������� %)�I�x��!�(�(�(� )�{�(�9K�9K�L�L�L�E�4��9�L�L�L�$(��T�(�(�(�(�(r5 r& r )r' r( r r) r* r r > �ch�cheb�cheby� chebychevr r > r� �cb�cblockr r �cor �cosr rR )r' r( rg r r) r* r > �e�eu�euclidr r > �h�ha�hammr �matchingr{ )r' r( rg ri r r) r* r > �j�ja�jaccr r r� r r r > �mah�mahalr )r' r( ri r r) r* r > rx �mi�pnormr r r r > �s�ser r r r > �sqe�sqeuclidr r c � � i | ] }|j |�� S r �r' ��.0�infos r3 � <dictcomp>rP � s � � @� @� @�$�D� �� @� @� @r5 c # �2 K � | ]}|j D ]}||fV � � �d S �N)r( )rN rO �aliass r3 � <genexpr>rT � sQ � � � � ,� ,� �"&�(�,� ,� � �T�]� ,� ,� ,� ,� ,� ,� ,r5 c �$ � i | ] }d |j z |��S )�test_rL rM s r3 rP rP � s" � �O�O�O���4�.�.��O�O�Or5 r c �$ � t | dddd�� � } | j }t |� � dk rt d� � �|\ }}t |� � rRt |dd� � }t � |d� � }|�t | |||fi |��\ } } }t | f||d �|��S t |t � � r�|� � � }t � |d� � }|�|j } | | fd |i|��S |� d � � r`t � |d� � }|�t d |d d� � �� � �t | |||fi |��\ } } }t | f|j |d �|��S t d|z � � �t# d� � �)aj! 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'. **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. 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') FT)� sparse_ok� objects_ok�mask_ok� check_finiter* z%A 2-dimensional array must be passed.r �UnknownCustomMetricN�r r r rV � Unknown "Test" Distance Metric: � �Unknown Distance Metric: %s�>2nd argument metric must be a string identifier or a function.)r& rt r� ru �callabler � _METRIC_ALIASr| r� r r� r �lowerr* � startswith� _TEST_METRICSr r� ) rD r r r@ rG rx rl �mstrrm rn r! s r3 r r � s � �T �1��$��(-� /� /� /�A� ��A� �1�v�v��{�{��@�A�A�A� �D�A�q����� *��v�z�+@�A�A��#�'�'��d�3�3� � � "� 2��1�a�� 0� 0�(.� 0� 0�N�A�s�F� �q�C��S�C�C�F�C�C�C� �F�C� � � *��|�|�~�~��#�'�'��d�3�3� � � "�"�-�H��8�A�1�1�3�1�&�1�1� 1� �_�_�W� %� %� C�'�+�+�D�$�7�7�K��"� �!N�D����H�!N�!N�O�O�O� 2��1�a�� 0� 0�(.� 0� 0�N�A�s�F�"��D�%�/�S�D� D�<B�D� D� D� � :�T� A�B�B� B�� )� *� *� *r5 �noc � � t j | � � } | j }|� � � dk r#t |� � dk rt d� � �n:|� � � dk r"t |� � dk rt d� � �t |� � dk r�|d dk rt j d| j � � � S t t j t j |d dz � � � � � � }||dz z |d dz k rt d � � �t j ||f| j � � � }t | � � } t j || � � |S t |� � dk r�|d |d k rt d � � �|rt | d d �� � |d }|dk rt j g | j � � � S t j ||dz z dz | j � � � }t | � � } t j | |� � |S t dt |� � z � � �)ap 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. �tomatrixr' z8Forcing 'tomatrix' but input X is not a distance vector.�tovectorr* z8Forcing 'tovector' but input X is not a distance matrix.r )r' r' rH 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.TrD ��throw�namez`The first argument must be one or two dimensional array. A %d-dimensional array is not permitted)rJ rK rt rd r� ru �zerosrI rU �ceilr� r4 r( � to_squareform_from_vector_wrapr r� � to_vector_from_squareform_wrap)rD �force�checksrG r� �MrX s r3 r r � s^ � �h � �Q� � �A� ��A� �{�{�}�}� �"�"� �q�6�6�Q�;�;� � 0� 1� 1� 1� � ��� � �*� $� $� �q�6�6�Q�;�;� � 0� 1� 1� 1� �1�v�v��{�{� �Q�4�1�9�9��8�F�!�'�2�2�2� 2� ������!��q��)�)�*�*� +� +�� ��A��;�!�A�$��(� "� "� � O� P� P� P� �H�a��V�1�7� +� +� +�� (�� *� *�� �5�a��;�;�;� �� �Q���1��� �Q�4�1�Q�4�<�<� � B�C�C� C� � 1� ���C� 0� 0� 0� 0� �a�D�� ��6�6��8�B�a�g�.�.�.� .� �H�a�1�q�5�k�a�'�q�w� 7� 7� 7�� (�� *� *�� �5�a��;�;�;���� &�),�Q���0� 1� 1� 1r5 r� �Dc � � t j | d�� � } d} | j }t | j � � dk r#|rt d|z � � �t d� � �|dk r�| | j k � � � s#|rt d|z � � �t d � � �| t d |d � � t d |d � � f d k � � � s#|rt d |z � � �t d � � �n�| | j z |k � � � s(|rt d ||fz � � �t d|z � � �| t d |d � � t d |d � � f |k � � � s(|rt d||fz � � �t d|z � � �n?# t $ r2}|r� |r!t j t |� � � � d}Y d}~nd}~ww xY w|S )a� 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`. 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 '%s' must be symmetric within tolerance %5.5f.z9Distance matrix must be symmetric within tolerance %5.5f.zKDistance matrix '%s' diagonal must be close to zero within tolerance %5.5f.FN) rJ r� rt r� ru r� �all�range� Exception�warnings�warnr )rv �tolrm rn �warning�validrG r9 s r3 r r J s� � �J � �1�C� � � �A� �E�-� �G�� �q�w�<�<�1� � �� 9� � #?�BF�"G� H� H� H� !� "8� 9� 9� 9� �#�:�:����H�>�>�#�#� K�� K�$� '2�59�&:� ;� ;� ;� %�%I�J�J�J��e�A�q��t�n�n�e�A�q��t�n�n�4�5��:�?�?�A�A� O�� O�$� '1�48�&9� :� :� :� %�%M�N�N�N� O� ���G�s�N�'�'�)�)� @�� @�$� 'J�(,�c�{�&3� 4� 4� 4� %� &9�;>�&?� @� @� @��e�A�q��t�n�n�e�A�q��t�n�n�4�5��<�A�A�C�C� ,�� ,�$� 'O�(,�c�{�&3� 4� 4� 4� %� 'O�'*�&+� ,� ,� ,��� � � � � � � � � "� �M�#�a�&�&� !� !� !������������ ���� �Ls �F,G � H�(G>�>Hc � � t j | d�� � } d} t | j � � dk r#|rt d|z � � �t d� � �| j d }t t j t j |dz � � � � � � }||dz z dz |k r#|rt d |z � � �t d � � �n?# t $ r2}|r� |r!t j t |� � � � d }Y d }~nd }~ww xY w|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. 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)!FN) rJ r� r� rt ru rU rp r� rz r{ r| r )�yr~ rm rn r rl r� r9 s r3 r r � sj � �2 � �1�C� � � �A� �E�� �q�w�<�<�1� � �� ?� � #L�#'�"(� )� )� )� !� ">� ?� ?� ?� �G�A�J�� ������A����'�'� (� (�� ��Q��K�!�O�� !� !�� H� � #7� ;?�"?� @� @� @� !� "G� H� H� H� "�� � � � � � � � � "� �M�#�a�&�&� !� !� !������������ ���� �Ls �B0C � D�(D�Dc � � t j | d�� � } t | t j dd�� � | j d S )a: 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. r� r� Tr� )r} rm rn r )rJ r� r r� rt )r� s r3 r r � s= � � � �1�C� � � �A���r�v�T��4�4�4�4� �7�1�:�r5 c �B � t j | d�� � } t | dd�� � | j d }|dk rt d� � �t t j t j |dz � � � � � � }||d z z dz |k rt 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`. r� r� T�Yrl 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.)rJ r� r rt ru rU rp r� )r� �kr� s r3 r r � s� � � � �1�C� � � �A��q��3�'�'�'�'� ��� �A��A�v�v�� 5� 6� 6� 6� �B�G�B�G�A��E�N�N� #� #�$�$�A� �Q��U� �a��A� � �� H� I� I� I� �Hr5 c �� � | �t j ||�� � S | j |k rt d� � �| j j st d� � �| j t j k rt d� � �| S )NrH z!Output array has incorrect shape.z"Output array must be C-contiguous.z!Output array must be double type.)rJ �emptyrt ru �flags� c_contiguousrI r{ )r rI �expected_shapes r3 r r sw � � �{��x��e�4�4�4�4� �y�N�"�"��<�=�=�=� �9� !� ?��=�>�>�>� �y�B�I� � ��<�=�=�=� �Jr5 c �0 � | j d }||dz z dz }t |t j |f� � }d}t | j d dz � � D ]A}t |dz | j d � � D ] } || | | | fi |��||< |dz }�!�B|S )Nr r' r* �rt r rJ r{ ry ) rD r r r@ rl r r? r� �ir@ s r3 r r s� � � ��� �A��Q��U� ��!�H� �s�B�I��{� ;� ;�B� �A� �1�7�1�:��>� "� "� � ���q�1�u�a�g�a�j�)�)� � �A��F�1�Q�4��1��0�0��0�0�B�q�E� ��F�A�A� � �Ir5 c �� � | j d }|j d }t |t j ||f� � }t |� � D ]/}t |� � D ] } || | || fi |��||| f< � �0|S )Nr r� ) r= r> r r r@ rj rk r? r� r@ s r3 r r & s� � � ��!��B� ��!��B� �s�B�I��B�x� 8� 8�B� �2�Y�Y� 6� 6���r��� 6� 6�A� �v�b��e�R��U�5�5�f�5�5�B�q�!�t�H�H� 6� �Ir5 c � � t j | � � } t j |� � }| j }|j }t |� � dk rt d� � �t |� � dk rt d� � �|d |d k rt d� � �|d }|d }|d } t |� � rVt |dd� � } t � | d � � } | �t | |||| | fi |��\ } }} }t | |f||d �|��S t |t � � r�|� � � } t � | d � � } | �| j } | | |fd |i|��S | � d � � rdt � | d � � } | �t d | dd � � �� � �t | |||| | fi |��\ } }} }t | |f| j |d �|��S t d| z � � �t% d� � �)a!& 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]]) r* z!XA must be a 2-dimensional array.z!XB must be a 2-dimensional array.r' zHXA and XB must have the same number of columns (i.e. feature dimension.)r r �UnknownNr] r rV r^ r_ r` ra )rJ r� rt r� ru rb r rc r| rp r r� r rd r) re rf r r� )r= r> r r r@ rG �sBrj rk rl rg rm rn r s r3 r r 0 s� � �Z ��B� � �B� ��B� � �B� ��A� ��B� �1�v�v��{�{��<�=�=�=� �2�w�w�!�|�|��<�=�=�=���t�r�!�u�}�}�� 5� 6� 6� 6� �1��B� �A��B� �!��A����� *��v�z�9�5�5��#�'�'��d�3�3� � � "�"7��B��B��;�#:� #:�28�#:� #:� �B��C�� �r�2�H�f�#�H�H��H�H�H� �F�C� � � *��|�|�~�~��#�'�'��d�3�3� � � "�"�-�H��8�B��6�6��6�v�6�6� 6� �_�_�W� %� %� C�'�+�+�D�$�7�7�K��"� �!N�D����H�!N�!N�O�O�O�"7��B��B��;�#:� #:�28�#:� #:� �B��C��"��B�I� *� 4�#�I� I�AG�I� I� I� � :�T� A�B�B� B�� )� *� *� *r5 rR )r )r* N)NT)r )rh T)r� Frv F)FFN)k�__doc__�__all__r{ �numpyrJ r/ �typingr! r" r# r$ � functoolsr% �scipy._lib._utilr& � r( r) r� r+ �specialr, r- �_lib.deprecationr. r4 rA rF rM r` rb rp ry r~ r� r� r� r� r� r{ rv r r r r r r r r r r r r r r r r r r r r r r r r� rR �_convert_to_bool� pdist_correlation_double_wrap� cdist_correlation_double_wrap� dataclassr� r r r# r& �cdist_braycurtis�pdist_braycurtis�cdist_canberra�pdist_canberra�cdist_chebyshev�pdist_chebyshev�cdist_cityblock�pdist_cityblock�cdist_euclidean�pdist_euclidean�cdist_minkowski�pdist_minkowski�cdist_sqeuclidean�pdist_sqeuclidean� _METRIC_INFOSr �dictrc �list�keys�_METRICS_NAMESrf r r r r r r r r r r r r5 r3 �<module>r� s~ ��G� G�V � � ��D ���� � � � � � � � � 0� 0� 0� 0� 0� 0� 0� 0� 0� 0� 0� 0� � � � � � � /� /� /� /� /� /� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � *� *� *� *� *� *� � � �B� B� B� >� >� >� 3� 3� 3� � � � �0� � � �( � � � � � �� � �� � �&� � � � � � � � �&4� 4� 4� 4� "�y� � � � �b� b� b� b�JD� D� D� D�N$%� $%� $%� $%�N3 � 3 � 3 � 3 �l/� /� /� /�d+D� +D� +D� +D�\3)� 3)� 3)� 3)�lG$� G$� G$� G$�T �� H� I� I�:3� :3� :3�I� I�:3�z � A � A � A � A � A �H&"� &"� &"�R+� +� +� +�\,� ,� ,�^*� *� *� *�Z-(� -(� -(� -(�`4 � 4 � 4 � 4 �nU �1�u� U � U � U � U � U �p/:� /:� /:� /:�d>@� >@� >@� >@�B/M� /M� /M� /M�d6 � 6 � 6 � 6 �r0K� 0K� 0K� 0K�f9,� 9,� 9,� 9,�x �W� -�� �B�B�B� ��7�+�d�;�;�;� � 0G�� ,�/F�� ,� ���d�#�#�#�� � � � � � � $�#��4 ���d�#�#�#�� � � � � � � $�#��2 ���d�#�#�#�� � � � � � � $�#��0 ���d�#�#�#�� � � � � � � $�#��0 ���d�#�#�#�)� )� )� )� )� )� )� $�#�)�0 �J�#� �N� �#�4�#�4� � � � �J�!� �L��#�2�#�2� � � � �J�"� =� =� =��#�3�#�3� � � � �J�"� .� .� .��#�3�#�3� � � � �J�$� �D� !� �%�%�m�4�4�%�%�m�4�4� � � � �J� � �u� ��%�%�h�/�/�%�%�h�/�/� � � � �J� � �H��h��%�%�f�-�-�%�%�f�-�-� � � � �J�"� .� .� .��#�3�#�3� � � � �J� � 6� 6� 6��� �*��-�-�i��C�C�-�-�i��C�C�� � � �J� � *� *� *��� ��%�%�i�0�0�%�%�i�0�0� � � � �J�&� �d� #� �%�%�o�6�6�%�%�o�6�6� � � � �J�"� �M��h��%�%�k�2�2�%�%�k�2�2� � � � �J�$� �O��h� �%�%�m�4�4�%�%�m�4�4� � � � �J�$� +� +� +�.� �%�%�m�4�4�%�%�m�4�4� � � � �J�"� -� -� -�,��#�3�#�3� � � � �J�'� � ��h� �%�%�&6�7�7�%�%�&6�7�7� � � � �J�#� �N��h� �%�%�l�3�3�%�%�l�3�3� � � � �J�#� %� %� %�-� �%�%�l�3�3�%�%�l�3�3� � � � �J�&� � ��h� �%�%�o�6�6�%�%�o�6�6� � � � �J�$� �O��h� �%�%�m�4�4�%�%�m�4�4� � � � �J�$� .� .� .� �#�5�#�5� � � � �J� � �H��h��%�%�f�-�-�%�%�f�-�-� � � �Ci� �V A� @�-� @� @� @���� ,� ,�!.�,� ,� ,� ,� ,� � ��h�m�m�o�o�&�&��O�O� �O�O�O� �o*�� o*� o*� o*� o*� o*�d u1� u1� u1� u1�pU� U� U� U�p6� 6� 6� 6�r� � �* � � �: � � � � � �� � �X*�T� X*� X*� X*� X*� X*� X*� X*r5
Save File
Cancel