package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Compute distance between each pair of the two collections of inputs.

See Notes for common calling conventions.

Parameters ---------- XA : ndarray An :math:`m_A` by :math:`n` array of :math:`m_A` original observations in an :math:`n`-dimensional space. Inputs are converted to float type. XB : ndarray An :math:`m_B` by :math:`n` array of :math:`m_B` original observations in an :math:`n`-dimensional space. Inputs are converted to float type. metric : str or callable, optional The distance metric to use. If a string, the distance function can be 'braycurtis', 'canberra', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice', 'euclidean', 'hamming', 'jaccard', 'jensenshannon', 'kulsinski', 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'wminkowski', 'yule'. *args : tuple. Deprecated. Additional arguments should be passed as keyword arguments **kwargs : dict, optional Extra arguments to `metric`: refer to each metric documentation for a list of all possible arguments.

Some possible arguments:

p : scalar The p-norm to apply for Minkowski, weighted and unweighted. Default: 2.

w : ndarray The weight vector for metrics that support weights (e.g., Minkowski).

V : ndarray The variance vector for standardized Euclidean. Default: var(vstack(XA, XB), axis=0, ddof=1)

VI : ndarray The inverse of the covariance matrix for Mahalanobis. Default: inv(cov(vstack(XA, XB.T))).T

out : ndarray The output array If not None, the distance matrix Y is stored in this array. Note: metric independent, it will become a regular keyword arg in a future scipy version

Returns ------- Y : ndarray A :math:`m_A` by :math:`m_B` distance matrix is returned. For each :math:`i` and :math:`j`, the metric ``dist(u=XAi, v=XBj)`` is computed and stored in the :math:`ij` th entry.

Raises ------ ValueError An exception is thrown if `XA` and `XB` do not have the same number of columns.

Notes ----- The following are common calling conventions:

1. ``Y = cdist(XA, XB, 'euclidean')``

Computes the distance between :math:`m` points using Euclidean distance (2-norm) as the distance metric between the points. The points are arranged as :math:`m` :math:`n`-dimensional row vectors in the matrix X.

2. ``Y = cdist(XA, XB, 'minkowski', p=2.)``

Computes the distances using the Minkowski distance :math:`||u-v||_p` (:math:`p`-norm) where :math:`p \geq 1`.

3. ``Y = cdist(XA, XB, 'cityblock')``

Computes the city block or Manhattan distance between the points.

4. ``Y = cdist(XA, XB, 'seuclidean', V=None)``

Computes the standardized Euclidean distance. The standardized Euclidean distance between two n-vectors ``u`` and ``v`` is

.. math::

\sqrt\sum {(u_i-v_i)^2 / V[x_i]

}

.

V is the variance vector; Vi 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 - \fracu \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

})}
               {{ ||(u - \bar{u})|| }_2 { ||(v - \bar{v})|| }_2}

   where :math:`\bar{v}` is the mean of the elements of vector v,
   and :math:`x \cdot y` is the dot product of :math:`x` and :math:`y`.


8. ``Y = cdist(XA, XB, 'hamming')``

   Computes the normalized Hamming distance, or the proportion of
   those vector elements between two n-vectors ``u`` and ``v``
   which disagree. To save memory, the matrix ``X`` can be of type
   boolean.

9. ``Y = cdist(XA, XB, 'jaccard')``

   Computes the Jaccard distance between the points. Given two
   vectors, ``u`` and ``v``, the Jaccard distance is the
   proportion of those elements ``u[i]`` and ``v[i]`` that
   disagree where at least one of them is non-zero.

10. ``Y = cdist(XA, XB, 'chebyshev')``

   Computes the Chebyshev distance between the points. The
   Chebyshev distance between two n-vectors ``u`` and ``v`` is the
   maximum norm-1 distance between their respective elements. More
   precisely, the distance is given by

   .. math::

      d(u,v) = \max_i { |u_i-v_i| }.

11. ``Y = cdist(XA, XB, 'canberra')``

   Computes the Canberra distance between the points. The
   Canberra distance between two points ``u`` and ``v`` is

   .. math::

     d(u,v) = \sum_i \frac{ |u_i-v_i| }
                          { |u_i|+|v_i| }.

12. ``Y = cdist(XA, XB, 'braycurtis')``

   Computes the Bray-Curtis distance between the points. The
   Bray-Curtis distance between two points ``u`` and ``v`` is


   .. math::

        d(u,v) = \frac{\sum_i (|u_i-v_i|)}
                      {\sum_i (|u_i+v_i|)}

13. ``Y = cdist(XA, XB, 'mahalanobis', VI=None)``

   Computes the Mahalanobis distance between the points. The
   Mahalanobis distance between two points ``u`` and ``v`` is
   :math:`\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI``
   variable) is the inverse covariance. If ``VI`` is not None,
   ``VI`` will be used as the inverse covariance matrix.

14. ``Y = cdist(XA, XB, 'yule')``

   Computes the Yule distance between the boolean
   vectors. (see `yule` function documentation)

15. ``Y = cdist(XA, XB, 'matching')``

   Synonym for 'hamming'.

16. ``Y = cdist(XA, XB, 'dice')``

   Computes the Dice distance between the boolean vectors. (see
   `dice` function documentation)

17. ``Y = cdist(XA, XB, 'kulsinski')``

   Computes the Kulsinski distance between the boolean
   vectors. (see `kulsinski` function documentation)

18. ``Y = cdist(XA, XB, 'rogerstanimoto')``

   Computes the Rogers-Tanimoto distance between the boolean
   vectors. (see `rogerstanimoto` function documentation)

19. ``Y = cdist(XA, XB, 'russellrao')``

   Computes the Russell-Rao distance between the boolean
   vectors. (see `russellrao` function documentation)

20. ``Y = cdist(XA, XB, 'sokalmichener')``

   Computes the Sokal-Michener distance between the boolean
   vectors. (see `sokalmichener` function documentation)

21. ``Y = cdist(XA, XB, 'sokalsneath')``

   Computes the Sokal-Sneath distance between the vectors. (see
   `sokalsneath` function documentation)


22. ``Y = cdist(XA, XB, 'wminkowski', p=2., w=w)``

   Computes the weighted Minkowski distance between the
   vectors. (see `wminkowski` function documentation)

23. ``Y = cdist(XA, XB, f)``

   Computes the distance between all pairs of vectors in X
   using the user supplied 2-arity function f. For example,
   Euclidean distance between the vectors could be computed
   as follows::

     dm = cdist(XA, XB, lambda u, v: np.sqrt(((u-v)**2).sum()))

   Note that you should avoid passing a reference to one of
   the distance functions defined in this library. For example,::

     dm = cdist(XA, XB, sokalsneath)

   would calculate the pair-wise distances between the vectors in
   X using the Python function `sokalsneath`. This would result in
   sokalsneath being called :math:`{n \choose 2}` times, which
   is inefficient. Instead, the optimized C version is more
   efficient, and we call it using the following syntax::

     dm = cdist(XA, XB, 'sokalsneath')

Examples
--------
Find the Euclidean distances between four 2-D coordinates:

>>> from scipy.spatial import distance
>>> coords = [(35.0456, -85.2672),
...           (35.1174, -89.9711),
...           (35.9728, -83.9422),
...           (36.1667, -86.7833)]
>>> distance.cdist(coords, coords, 'euclidean')
array([[ 0.    ,  4.7044,  1.6172,  1.8856],
       [ 4.7044,  0.    ,  6.0893,  3.3561],
       [ 1.6172,  6.0893,  0.    ,  2.8477],
       [ 1.8856,  3.3561,  2.8477,  0.    ]])


Find the Manhattan distance from a 3-D point to the corners of the unit
cube:

>>> a = np.array([[0, 0, 0],
...               [0, 0, 1],
...               [0, 1, 0],
...               [0, 1, 1],
...               [1, 0, 0],
...               [1, 0, 1],
...               [1, 1, 0],
...               [1, 1, 1]])
>>> b = np.array([[ 0.1,  0.2,  0.4]])
>>> distance.cdist(a, b, 'cityblock')
array([[ 0.7],
       [ 0.9],
       [ 1.3],
       [ 1.5],
       [ 1.5],
       [ 1.7],
       [ 2.1],
       [ 2.3]])

Performs k-means on a set of observation vectors forming k clusters.

The k-means algorithm adjusts the classification of the observations into clusters and updates the cluster centroids until the position of the centroids is stable over successive iterations. In this implementation of the algorithm, the stability of the centroids is determined by comparing the absolute value of the change in the average Euclidean distance between the observations and their corresponding centroids against a threshold. This yields a code book mapping centroids to codes and vice versa.

Parameters ---------- obs : ndarray Each row of the M by N array is an observation vector. The columns are the features seen during each observation. The features must be whitened first with the `whiten` function.

k_or_guess : int or ndarray The number of centroids to generate. A code is assigned to each centroid, which is also the row index of the centroid in the code_book matrix generated.

The initial k centroids are chosen by randomly selecting observations from the observation matrix. Alternatively, passing a k by N array specifies the initial k centroids.

iter : int, optional The number of times to run k-means, returning the codebook with the lowest distortion. This argument is ignored if initial centroids are specified with an array for the ``k_or_guess`` parameter. This parameter does not represent the number of iterations of the k-means algorithm.

thresh : float, optional Terminates the k-means algorithm if the change in distortion since the last k-means iteration is less than or equal to thresh.

check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default: True

Returns ------- codebook : ndarray A k by N array of k centroids. The i'th centroid codebooki is represented with the code i. The centroids and codes generated represent the lowest distortion seen, not necessarily the globally minimal distortion.

distortion : float The mean (non-squared) Euclidean distance between the observations passed and the centroids generated. Note the difference to the standard definition of distortion in the context of the K-means algorithm, which is the sum of the squared distances.

See Also -------- kmeans2 : a different implementation of k-means clustering with more methods for generating initial centroids but without using a distortion change threshold as a stopping criterion.

whiten : must be called prior to passing an observation matrix to kmeans.

Examples -------- >>> from numpy import array >>> from scipy.cluster.vq import vq, kmeans, whiten >>> import matplotlib.pyplot as plt >>> features = array([ 1.9,2.3], ... [ 1.5,2.5], ... [ 0.8,0.6], ... [ 0.4,1.8], ... [ 0.1,0.1], ... [ 0.2,1.8], ... [ 2.0,0.5], ... [ 0.3,1.5], ... [ 1.0,1.0]) >>> whitened = whiten(features) >>> book = np.array((whitened0,whitened2)) >>> kmeans(whitened,book) (array([ 2.3110306 , 2.86287398], # random [ 0.93218041, 1.24398691]), 0.85684700941625547)

>>> from numpy import random >>> random.seed((1000,2000)) >>> codes = 3 >>> kmeans(whitened,codes) (array([ 2.3110306 , 2.86287398], # random [ 1.32544402, 0.65607529], [ 0.40782893, 2.02786907]), 0.5196582527686241)

>>> # Create 50 datapoints in two clusters a and b >>> pts = 50 >>> a = np.random.multivariate_normal(0, 0, [4, 1], [1, 4], size=pts) >>> b = np.random.multivariate_normal(30, 10, ... [10, 2], [2, 1], ... size=pts) >>> features = np.concatenate((a, b)) >>> # Whiten data >>> whitened = whiten(features) >>> # Find 2 clusters in the data >>> codebook, distortion = kmeans(whitened, 2) >>> # Plot whitened data and cluster centers in red >>> plt.scatter(whitened:, 0, whitened:, 1) >>> plt.scatter(codebook:, 0, codebook:, 1, c='r') >>> plt.show()

Classify a set of observations into k clusters using the k-means algorithm.

The algorithm attempts to minimize the Euclidian distance between observations and centroids. Several initialization methods are included.

Parameters ---------- data : ndarray A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length 'M' array of 'M' one-dimensional observations. k : int or ndarray The number of clusters to form as well as the number of centroids to generate. If `minit` initialization string is 'matrix', or if a ndarray is given instead, it is interpreted as initial cluster to use instead. iter : int, optional Number of iterations of the k-means algorithm to run. Note that this differs in meaning from the iters parameter to the kmeans function. thresh : float, optional (not used yet) minit : str, optional Method for initialization. Available methods are 'random', 'points', '++' and 'matrix':

'random': generate k centroids from a Gaussian with mean and variance estimated from the data.

'points': choose k observations (rows) at random from data for the initial centroids.

'++': choose k observations accordingly to the kmeans++ method (careful seeding)

'matrix': interpret the k parameter as a k by M (or length k array for one-dimensional data) array of initial centroids. missing : str, optional Method to deal with empty clusters. Available methods are 'warn' and 'raise':

'warn': give a warning and continue.

'raise': raise an ClusterError and terminate the algorithm. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default: True

Returns ------- centroid : ndarray A 'k' by 'N' array of centroids found at the last iteration of k-means. label : ndarray labeli is the code or index of the centroid the i'th observation is closest to.

See Also -------- kmeans

References ---------- .. 1 D. Arthur and S. Vassilvitskii, 'k-means++: the advantages of careful seeding', Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2007.

Examples -------- >>> from scipy.cluster.vq import kmeans2 >>> import matplotlib.pyplot as plt

Create z, an array with shape (100, 2) containing a mixture of samples from three multivariate normal distributions.

>>> np.random.seed(12345678) >>> a = np.random.multivariate_normal(0, 6, [2, 1], [1, 1.5], size=45) >>> b = np.random.multivariate_normal(2, 0, [1, -1], [-1, 3], size=30) >>> c = np.random.multivariate_normal(6, 4, [5, 0], [0, 1.2], size=25) >>> z = np.concatenate((a, b, c)) >>> np.random.shuffle(z)

Compute three clusters.

>>> centroid, label = kmeans2(z, 3, minit='points') >>> centroid array([-0.35770296, 5.31342524], [ 2.32210289, -0.50551972], [ 6.17653859, 4.16719247])

How many points are in each cluster?

>>> counts = np.bincount(label) >>> counts array(52, 27, 21)

Plot the clusters.

>>> w0 = zlabel == 0 >>> w1 = zlabel == 1 >>> w2 = zlabel == 2 >>> plt.plot(w0:, 0, w0:, 1, 'o', alpha=0.5, label='cluster 0') >>> plt.plot(w1:, 0, w1:, 1, 'd', alpha=0.5, label='cluster 1') >>> plt.plot(w2:, 0, w2:, 1, 's', alpha=0.5, label='cluster 2') >>> plt.plot(centroid:, 0, centroid:, 1, 'k*', label='centroids') >>> plt.axis('equal') >>> plt.legend(shadow=True) >>> plt.show()

Python version of vq algorithm.

The algorithm computes the euclidian distance between each observation and every frame in the code_book.

Parameters ---------- obs : ndarray Expects a rank 2 array. Each row is one observation. code_book : ndarray Code book to use. Same format than obs. Should have same number of features (eg columns) than obs. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default: True

Returns ------- code : ndarray codei gives the label of the ith obversation, that its code is code_bookcode[i]. mind_dist : ndarray min_disti gives the distance between the ith observation and its corresponding code.

Notes ----- This function is slower than the C version but works for all input types. If the inputs have the wrong types for the C versions of the function, this one is called as a last resort.

It is about 20 times slower than the C version.

`py_vq2` is deprecated, use `py_vq` instead!

Python version of vq algorithm.

The algorithm computes the euclidian distance between each observation and every frame in the code_book.

Parameters ---------- obs : ndarray Expects a rank 2 array. Each row is one observation. code_book : ndarray Code book to use. Same format than obs. Should have same number of features (eg columns) than obs. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default: True

Returns ------- code : ndarray codei gives the label of the ith obversation, that its code is code_bookcode[i]. mind_dist : ndarray min_disti gives the distance between the ith observation and its corresponding code.

Notes ----- This function is slower than the C version but works for all input types. If the inputs have the wrong types for the C versions of the function, this one is called as a last resort.

It is about 20 times slower than the C version.

Assign codes from a code book to observations.

Assigns a code from a code book to each observation. Each observation vector in the 'M' by 'N' `obs` array is compared with the centroids in the code book and assigned the code of the closest centroid.

The features in `obs` should have unit variance, which can be achieved by passing them through the whiten function. The code book can be created with the k-means algorithm or a different encoding algorithm.

Parameters ---------- obs : ndarray Each row of the 'M' x 'N' array is an observation. The columns are the 'features' seen during each observation. The features must be whitened first using the whiten function or something equivalent. code_book : ndarray The code book is usually generated using the k-means algorithm. Each row of the array holds a different code, and the columns are the features of the code.

>>> # f0 f1 f2 f3 >>> code_book = ... [ 1., 2., 3., 4.], #c0 ... [ 1., 2., 3., 4.], #c1 ... [ 1., 2., 3., 4.] #c2

check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default: True

Returns ------- code : ndarray A length M array holding the code book index for each observation. dist : ndarray The distortion (distance) between the observation and its nearest code.

Examples -------- >>> from numpy import array >>> from scipy.cluster.vq import vq >>> code_book = array([1.,1.,1.], ... [2.,2.,2.]) >>> features = array([ 1.9,2.3,1.7], ... [ 1.5,2.5,2.2], ... [ 0.8,0.6,1.7]) >>> vq(features,code_book) (array(1, 1, 0,'i'), array( 0.43588989, 0.73484692, 0.83066239))

Normalize a group of observations on a per feature basis.

Before running k-means, it is beneficial to rescale each feature dimension of the observation set with whitening. Each feature is divided by its standard deviation across all observations to give it unit variance.

Parameters ---------- obs : ndarray Each row of the array is an observation. The columns are the features seen during each observation.

>>> # f0 f1 f2 >>> obs = [ 1., 1., 1.], #o0 ... [ 2., 2., 2.], #o1 ... [ 3., 3., 3.], #o2 ... [ 4., 4., 4.] #o3

check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default: True

Returns ------- result : ndarray Contains the values in `obs` scaled by the standard deviation of each column.

Examples -------- >>> from scipy.cluster.vq import whiten >>> features = np.array([1.9, 2.3, 1.7], ... [1.5, 2.5, 2.2], ... [0.8, 0.6, 1.7,]) >>> whiten(features) array([ 4.17944278, 2.69811351, 7.21248917], [ 3.29956009, 2.93273208, 9.33380951], [ 1.75976538, 0.7038557 , 7.21248917])