Edit File: function_base.cpython-37.pyc
B ��Fd�� � + @ s� d dl mZmZmZ d dlZd dlZd dlZd dlZd dlZd dl Z d dl m m Z d dlmZmZmZmZ d dl mZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z( d dl)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5 d dl6m7Z7m8Z8m9Z9m:Z:m;Z;m<Z<m=Z= d dl>m?Z?m@Z@ d dlAmBZB d d lCmDZD d d lEmFZFmGZGmHZHmIZImJZJmKZLmMZN d d l)mOZP d d lQmRZR d dlSmTZT ejUd dk �r�eVZWd dlXZYnd dlYZYddddddddddddd d d d d d!d"d#d$d%d&d'd(d)d*d+d,d-d.d/d0d1d2d3d4d5d6d7d8d9d:g+ZZd�d<d �Z[d=d � Z\d>d� Z]d?d@� Z^dAdB� Z_dCdD� Z`dEdF� ZadGdH� ZbdIdJ� ZcdKdL� Zdedebece`eae^e_dM�Zed�dPd#�Zfd�dQd$�Zgd�dRd"�Zhd�dSd!�ZidTd� Zjd�dUd�Zkd�dWd�ZldXd� Zmd�dZd�Znd�d[d9�ZKd�d\d�Zoe*dYfd]d�Zpd^d� Zqd�d`d�ZreDdadb� �Zsdcd � Ztddd � Zud�dfd�ZvdgZwdh�xew�Zydi�xey�Zzdj�xez�Z{dk�xe{�Z|dldm� Z}dndo� Z~dpdq� Zdrds� Z�dtdu� Z�G dvd � d e��Z�d�dwd'�Z�ddee j�e j�fdxd(�Z�dyd/� Z�dzd.� Z�d{d-� Z�d|d,� Z�d}d~dd�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�g Z�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�d�gZ�d�d�� Z�d�d�� Z�d�d�� Z�d�d2� Z�d�d0� Z�d�d+� Z�d�d)� Z�d�d�� Z�d�d�d*�Z�d�d�dZ�d�d�d�Z�d�d�dƄZ�d�d�d1�Z�d�d3� Z�d�d5� Z�d�d�d6�Z�d�d�d7�Z�d�d�d8�Z�dS )�� )�division�absolute_import�print_functionN)�linspace� atleast_1d� atleast_2d� transpose)�ones�zeros�arange� concatenate�array�asarray� asanyarray�empty� empty_like�ndarray�around�floor�ceil�take�dot�where�intp�integer�isscalar�absolute� AxisError) �pi�multiply�add�arctan2� frompyfunc�cos� less_equal�sqrt�sin�mod�exp�log10)�ravel�nonzero�sort� partition�mean�any�sum)� typecodes�number)�diag� )� deprecate)�_insert� add_docstring�digitize�bincount�normalize_axis_index�interp�interp_complex)�_add_newdoc_ufunc)�long)� basestring� �select� piecewise� trim_zeros�copy�iterable� percentile�diff�gradient�angle�unwrap� sort_complex�disp�flip�rot90�extract�place� vectorize�asarray_chkfinite�average� histogram� histogramddr9 r8 �cov�corrcoef�msort�median�sinc�hamming�hanning�bartlett�blackman�kaiser�trapz�i0� add_newdocr7 �meshgrid�delete�insert�appendr; �add_newdoc_ufunc�r r4 c C sL t |�}t|�dkr td��t| �} |d |d ksNt|d |d �| jkrVtd��|d | jks�|d | j k s�|d | jks�|d | j k r�td�|| j���|d; }|dkr�| dd� S |dkr�tt| |d �|d �S td| j�}||d ||d ||d < ||d < |dk�r4t t| |d �|�S tt | |�|d �S dS ) a� Rotate an array by 90 degrees in the plane specified by axes. Rotation direction is from the first towards the second axis. .. versionadded:: 1.12.0 Parameters ---------- m : array_like Array of two or more dimensions. k : integer Number of times the array is rotated by 90 degrees. axes: (2,) array_like The array is rotated in the plane defined by the axes. Axes must be different. Returns ------- y : ndarray A rotated view of `m`. See Also -------- flip : Reverse the order of elements in an array along the given axis. fliplr : Flip an array horizontally. flipud : Flip an array vertically. Notes ----- rot90(m, k=1, axes=(1,0)) is the reverse of rot90(m, k=1, axes=(0,1)) rot90(m, k=1, axes=(1,0)) is equivalent to rot90(m, k=-1, axes=(0,1)) Examples -------- >>> m = np.array([[1,2],[3,4]], int) >>> m array([[1, 2], [3, 4]]) >>> np.rot90(m) array([[2, 4], [1, 3]]) >>> np.rot90(m, 2) array([[4, 3], [2, 1]]) >>> m = np.arange(8).reshape((2,2,2)) >>> np.rot90(m, 1, (1,2)) array([[[1, 3], [0, 2]], [[5, 7], [4, 6]]]) � zlen(axes) must be 2.r r4 zAxes must be different.z*Axes={} out of range for array of ndim={}.� N) �tuple�len� ValueErrorr r �ndim�formatrM r r )�m�k�axesZ axes_list� rs �J/opt/alt/python37/lib64/python3.7/site-packages/numpy/lib/function_base.pyrN 6 s* 7 * c C sj t | d�st| �} td�g| j }ytddd�||< W n& tk r\ td|| jf ��Y nX | t|� S )a� Reverse the order of elements in an array along the given axis. The shape of the array is preserved, but the elements are reordered. .. versionadded:: 1.12.0 Parameters ---------- m : array_like Input array. axis : integer Axis in array, which entries are reversed. Returns ------- out : array_like A view of `m` with the entries of axis reversed. Since a view is returned, this operation is done in constant time. See Also -------- flipud : Flip an array vertically (axis=0). fliplr : Flip an array horizontally (axis=1). Notes ----- flip(m, 0) is equivalent to flipud(m). flip(m, 1) is equivalent to fliplr(m). flip(m, n) corresponds to ``m[...,::-1,...]`` with ``::-1`` at position n. Examples -------- >>> A = np.arange(8).reshape((2,2,2)) >>> A array([[[0, 1], [2, 3]], [[4, 5], [6, 7]]]) >>> flip(A, 0) array([[[4, 5], [6, 7]], [[0, 1], [2, 3]]]) >>> flip(A, 1) array([[[2, 3], [0, 1]], [[6, 7], [4, 5]]]) >>> A = np.random.randn(3,4,5) >>> np.all(flip(A,2) == A[:,:,::-1,...]) True rn N���z5axis=%i is invalid for the %i-dimensional input array)�hasattrr �slicern � IndexErrorrm rk )rp �axis�indexerrs rs rt rM � s = c C s& y t | � W n tk r dS X dS )aw Check whether or not an object can be iterated over. Parameters ---------- y : object Input object. Returns ------- b : bool Return ``True`` if the object has an iterator method or is a sequence and ``False`` otherwise. Examples -------- >>> np.iterable([1, 2, 3]) True >>> np.iterable(2) False FT)�iter� TypeError)�yrs rs rt rE � s c C s | � � t�| j� S )a~ Square root histogram bin estimator. Bin width is inversely proportional to the data size. Used by many programs for its simplicity. Parameters ---------- x : array_like Input data that is to be histogrammed, trimmed to range. May not be empty. Returns ------- h : An estimate of the optimal bin width for the given data. )�ptp�npr% �size)�xrs rs rt �_hist_bin_sqrt� s r� c C s | � � t�| j�d S )a Sturges histogram bin estimator. A very simplistic estimator based on the assumption of normality of the data. This estimator has poor performance for non-normal data, which becomes especially obvious for large data sets. The estimate depends only on size of the data. Parameters ---------- x : array_like Input data that is to be histogrammed, trimmed to range. May not be empty. Returns ------- h : An estimate of the optimal bin width for the given data. g �?)r~ r �log2r� )r� rs rs rt �_hist_bin_sturges s r� c C s | � � d| jd S )aI Rice histogram bin estimator. Another simple estimator with no normality assumption. It has better performance for large data than Sturges, but tends to overestimate the number of bins. The number of bins is proportional to the cube root of data size (asymptotically optimal). The estimate depends only on size of the data. Parameters ---------- x : array_like Input data that is to be histogrammed, trimmed to range. May not be empty. Returns ------- h : An estimate of the optimal bin width for the given data. g @gUUUUUU�?)r~ r� )r� rs rs rt �_hist_bin_rice s r� c C s" dt jd | j d t �| � S )a� Scott histogram bin estimator. The binwidth is proportional to the standard deviation of the data and inversely proportional to the cube root of data size (asymptotically optimal). Parameters ---------- x : array_like Input data that is to be histogrammed, trimmed to range. May not be empty. Returns ------- h : An estimate of the optimal bin width for the given data. g 8@g �?gUUUUUU�?)r r r� �std)r� rs rs rt �_hist_bin_scott5 s r� c C s� | j dkr�t�d| j d | j d | j d �}t�| �}|dkr�| t�| � }t�|||� t�|d|� t�|�}| �� dt�| j � t�dt� |�| � S dS )a� Doane's histogram bin estimator. Improved version of Sturges' formula which works better for non-normal data. See stats.stackexchange.com/questions/55134/doanes-formula-for-histogram-binning Parameters ---------- x : array_like Input data that is to be histogrammed, trimmed to range. May not be empty. Returns ------- h : An estimate of the optimal bin width for the given data. ri g @g �?r@ g ) r� r r% r� r. � true_divide�powerr~ r� r )r� Zsg1�sigma�tempZg1rs rs rt �_hist_bin_doaneJ s ( r� c C s( t jt �| ddg�� }d| | jd S )aB The Freedman-Diaconis histogram bin estimator. The Freedman-Diaconis rule uses interquartile range (IQR) to estimate binwidth. It is considered a variation of the Scott rule with more robustness as the IQR is less affected by outliers than the standard deviation. However, the IQR depends on fewer points than the standard deviation, so it is less accurate, especially for long tailed distributions. If the IQR is 0, this function returns 1 for the number of bins. Binwidth is inversely proportional to the cube root of data size (asymptotically optimal). Parameters ---------- x : array_like Input data that is to be histogrammed, trimmed to range. May not be empty. Returns ------- h : An estimate of the optimal bin width for the given data. �K � g @gUUUUUUտ)r �subtractrF r� )r� Ziqrrs rs rt � _hist_bin_fdl s r� c C s t t| �t| ��S )a� Histogram bin estimator that uses the minimum width of the Freedman-Diaconis and Sturges estimators. The FD estimator is usually the most robust method, but its width estimate tends to be too large for small `x`. The Sturges estimator is quite good for small (<1000) datasets and is the default in the R language. This method gives good off the shelf behaviour. Parameters ---------- x : array_like Input data that is to be histogrammed, trimmed to range. May not be empty. Returns ------- h : An estimate of the optimal bin width for the given data. See Also -------- _hist_bin_fd, _hist_bin_sturges )�minr� r� )r� rs rs rt �_hist_bin_auto� s r� )�autoZdoane�fdZriceZscottr% Zsturges� Fc C s� t | �} |dk r:t |�}t�|j| jk�r2td��|�� }| �� } |dkrz| jdkr^d\}}q�| �� d | �� d }}ndd� |D �\}}||kr�td��t� t� ||g��s�td ��||kr�|d 8 }|d 7 }t |t ��r||t kr�td �|���|dk �rtd ��| }|dk �r<| |k} | | |kM } tj�| ��s<| | }|jdk�rNd }n.t | |�} | �rxtt�|| | ��}nd }|dk�r�t�tj�} n|j} d} t|��st�|��r�|d k �r�td��|dk �rt�|jtj��st�|jtj��st|||d dd�}t|��s�t� || �} ||| }t|||d dd�}�xjt dt | �| �D �]T}| ||| � }|dk�rxd}n|||| � }||k} | ||kM } tj�| ��s�|| }|dk �r�|| }|� t �}|| }||9 }|� tj�}|||k d 8 < ||| k }|| d 8 < |||d k||d k@ }|| d 7 < | j!dk�r�| j"tj#||j"|d�7 _"| j$tj#||j$|d�7 _$n| tj#|||d�� | �7 } �qRW |}�n^t |�}t�|dd� |d d� k��r�td��t� |j| �} |dk�r\x\t dt | �| �D ]H}t%| ||| � �}| tj&|�'|dd� d�|�'|d d�f 7 } �qW n�t(d| d�}x�t dt | �| �D ]�}| ||| � }|||| � }t�)|�}|| }|| }t�*|g|�+� f�}tj&|�'|dd� d�|�'|d d�f } | || 7 } �qzW t�,| �} |dk �rP|�rFt(t�,|�t �} | | | �-� |fS | |fS n2|�rzt(t�,|�t �} | | | �-� |fS | |fS dS )aQ" Compute the histogram of a set of data. Parameters ---------- a : array_like Input data. The histogram is computed over the flattened array. bins : int or sequence of scalars or str, optional If `bins` is an int, it defines the number of equal-width bins in the given range (10, by default). If `bins` is a sequence, it defines the bin edges, including the rightmost edge, allowing for non-uniform bin widths. .. versionadded:: 1.11.0 If `bins` is a string from the list below, `histogram` will use the method chosen to calculate the optimal bin width and consequently the number of bins (see `Notes` for more detail on the estimators) from the data that falls within the requested range. While the bin width will be optimal for the actual data in the range, the number of bins will be computed to fill the entire range, including the empty portions. For visualisation, using the 'auto' option is suggested. Weighted data is not supported for automated bin size selection. 'auto' Maximum of the 'sturges' and 'fd' estimators. Provides good all around performance. 'fd' (Freedman Diaconis Estimator) Robust (resilient to outliers) estimator that takes into account data variability and data size. 'doane' An improved version of Sturges' estimator that works better with non-normal datasets. 'scott' Less robust estimator that that takes into account data variability and data size. 'rice' Estimator does not take variability into account, only data size. Commonly overestimates number of bins required. 'sturges' R's default method, only accounts for data size. Only optimal for gaussian data and underestimates number of bins for large non-gaussian datasets. 'sqrt' Square root (of data size) estimator, used by Excel and other programs for its speed and simplicity. range : (float, float), optional The lower and upper range of the bins. If not provided, range is simply ``(a.min(), a.max())``. Values outside the range are ignored. The first element of the range must be less than or equal to the second. `range` affects the automatic bin computation as well. While bin width is computed to be optimal based on the actual data within `range`, the bin count will fill the entire range including portions containing no data. normed : bool, optional This keyword is deprecated in NumPy 1.6.0 due to confusing/buggy behavior. It will be removed in NumPy 2.0.0. Use the ``density`` keyword instead. If ``False``, the result will contain the number of samples in each bin. If ``True``, the result is the value of the probability *density* function at the bin, normalized such that the *integral* over the range is 1. Note that this latter behavior is known to be buggy with unequal bin widths; use ``density`` instead. weights : array_like, optional An array of weights, of the same shape as `a`. Each value in `a` only contributes its associated weight towards the bin count (instead of 1). If `density` is True, the weights are normalized, so that the integral of the density over the range remains 1. density : bool, optional If ``False``, the result will contain the number of samples in each bin. If ``True``, the result is the value of the probability *density* function at the bin, normalized such that the *integral* over the range is 1. Note that the sum of the histogram values will not be equal to 1 unless bins of unity width are chosen; it is not a probability *mass* function. Overrides the ``normed`` keyword if given. Returns ------- hist : array The values of the histogram. See `density` and `weights` for a description of the possible semantics. bin_edges : array of dtype float Return the bin edges ``(length(hist)+1)``. See Also -------- histogramdd, bincount, searchsorted, digitize Notes ----- All but the last (righthand-most) bin is half-open. In other words, if `bins` is:: [1, 2, 3, 4] then the first bin is ``[1, 2)`` (including 1, but excluding 2) and the second ``[2, 3)``. The last bin, however, is ``[3, 4]``, which *includes* 4. .. versionadded:: 1.11.0 The methods to estimate the optimal number of bins are well founded in literature, and are inspired by the choices R provides for histogram visualisation. Note that having the number of bins proportional to :math:`n^{1/3}` is asymptotically optimal, which is why it appears in most estimators. These are simply plug-in methods that give good starting points for number of bins. In the equations below, :math:`h` is the binwidth and :math:`n_h` is the number of bins. All estimators that compute bin counts are recast to bin width using the `ptp` of the data. The final bin count is obtained from ``np.round(np.ceil(range / h))`. 'Auto' (maximum of the 'Sturges' and 'FD' estimators) A compromise to get a good value. For small datasets the Sturges value will usually be chosen, while larger datasets will usually default to FD. Avoids the overly conservative behaviour of FD and Sturges for small and large datasets respectively. Switchover point is usually :math:`a.size \approx 1000`. 'FD' (Freedman Diaconis Estimator) .. math:: h = 2 \frac{IQR}{n^{1/3}} The binwidth is proportional to the interquartile range (IQR) and inversely proportional to cube root of a.size. Can be too conservative for small datasets, but is quite good for large datasets. The IQR is very robust to outliers. 'Scott' .. math:: h = \sigma \sqrt[3]{\frac{24 * \sqrt{\pi}}{n}} The binwidth is proportional to the standard deviation of the data and inversely proportional to cube root of ``x.size``. Can be too conservative for small datasets, but is quite good for large datasets. The standard deviation is not very robust to outliers. Values are very similar to the Freedman-Diaconis estimator in the absence of outliers. 'Rice' .. math:: n_h = 2n^{1/3} The number of bins is only proportional to cube root of ``a.size``. It tends to overestimate the number of bins and it does not take into account data variability. 'Sturges' .. math:: n_h = \log _{2}n+1 The number of bins is the base 2 log of ``a.size``. This estimator assumes normality of data and is too conservative for larger, non-normal datasets. This is the default method in R's ``hist`` method. 'Doane' .. math:: n_h = 1 + \log_{2}(n) + \log_{2}(1 + \frac{|g_1|}{\sigma_{g_1}}) g_1 = mean[(\frac{x - \mu}{\sigma})^3] \sigma_{g_1} = \sqrt{\frac{6(n - 2)}{(n + 1)(n + 3)}} An improved version of Sturges' formula that produces better estimates for non-normal datasets. This estimator attempts to account for the skew of the data. 'Sqrt' .. math:: n_h = \sqrt n The simplest and fastest estimator. Only takes into account the data size. Examples -------- >>> np.histogram([1, 2, 1], bins=[0, 1, 2, 3]) (array([0, 2, 1]), array([0, 1, 2, 3])) >>> np.histogram(np.arange(4), bins=np.arange(5), density=True) (array([ 0.25, 0.25, 0.25, 0.25]), array([0, 1, 2, 3, 4])) >>> np.histogram([[1, 2, 1], [1, 0, 1]], bins=[0,1,2,3]) (array([1, 4, 1]), array([0, 1, 2, 3])) >>> a = np.arange(5) >>> hist, bin_edges = np.histogram(a, density=True) >>> hist array([ 0.5, 0. , 0.5, 0. , 0. , 0.5, 0. , 0.5, 0. , 0.5]) >>> hist.sum() 2.4999999999999996 >>> np.sum(hist * np.diff(bin_edges)) 1.0 .. versionadded:: 1.11.0 Automated Bin Selection Methods example, using 2 peak random data with 2000 points: >>> import matplotlib.pyplot as plt >>> rng = np.random.RandomState(10) # deterministic random data >>> a = np.hstack((rng.normal(size=1000), ... rng.normal(loc=5, scale=2, size=1000))) >>> plt.hist(a, bins='auto') # arguments are passed to np.histogram >>> plt.title("Histogram with 'auto' bins") >>> plt.show() Nz(weights should have the same shape as a.r )g g �?g c S s g | ] }|d �qS )g rs )�.0�mirs rs rt � <listcomp>� s z histogram.<locals>.<listcomp>z/max must be larger than min in range parameter.z range parameter must be finite.g �?z"{0} not a valid estimator for binszMAutomated estimation of the number of bins is not supported for weighted datar4 i z$`bins` should be a positive integer.T)Zendpoint�c)�weightsZ minlengthru z!bins must increase monotonically.�left�right)�dtype).r r r/ �shaperm r* r� r� �max�all�isfinite� isinstancer? �_hist_bin_selectorsro r| � logical_and�reduce�intr r� r rE r �can_cast�double�complexr r r rl �astype�float�kind�realr9 �imagr, Zr_Z searchsortedr �argsortr �cumsumrG r0 ) �a�bins�range�normedr� ZdensityZmn�mx�b�keep�widthZntypeZBLOCK�nZnormZ bin_edges�iZtmp_aZtmp_wZ tmp_a_data�indicesZ decrementZ increment�saZzeroZ sorting_index�swZcwZ bin_indexZdbrs rs rt rT � s� X c C s y| j \}}W n, ttfk r: t| �j} | j \}}Y nX t|t�}|dg }|dg } |dk rjt|�}y t|�} | |kr�td��W n t k r� ||g }Y nX |dkr�|dkr�t |�} t |�} n(t t | �d�t��} t t | �d�t��} nRt�t�|���s td��t |�} t |�} x&t|�D ]} || \| | < | | < �q&W xHtt| ��D ]8} | | | | k�rR| | d | | < | | d | | < �qRW t�| jtj��r�| j}nt}x�t|�D ]�} t|| ��r|| dk �r�td| ��|| d || < t| | | | || d |d �|| < n&t|| |�|| < t|| �d || < t|| �| | < t�t�| | �dk��r�td ���q�W t|�}|dk�r�t� |d �|fS i }x0t|�D ]$} t | dd�| f || �|| < �q�W x�t|�D ]�} | | �� }t� |��s�tt |� �d }| dd�| f || d k}t | dd�| f |�t || d |�k}|| t ||@ �d d8 < �q�W t |t��!d �}|�"� }t |t�}x@td|d �D ].} |||| ||| d d� �#� 7 }�q�W |||d 7 }t|�dk�rt |d t�|fS t$||�}tt|��}|||< |�!t%|��}xFt|j&�D ]8} |�"� | }|�'| |�}|| || || < ||< �q>W |t(dd �g }|| }|�r�|�)� }x>t|�D ]2} t |t�} || d | | < || | �!| � }�q�W || }|j |d k�� �rt*d ��||fS )aQ Compute the multidimensional histogram of some data. Parameters ---------- sample : array_like The data to be histogrammed. It must be an (N,D) array or data that can be converted to such. The rows of the resulting array are the coordinates of points in a D dimensional polytope. bins : sequence or int, optional The bin specification: * A sequence of arrays describing the bin edges along each dimension. * The number of bins for each dimension (nx, ny, ... =bins) * The number of bins for all dimensions (nx=ny=...=bins). range : sequence, optional A sequence of lower and upper bin edges to be used if the edges are not given explicitly in `bins`. Defaults to the minimum and maximum values along each dimension. normed : bool, optional If False, returns the number of samples in each bin. If True, returns the bin density ``bin_count / sample_count / bin_volume``. weights : (N,) array_like, optional An array of values `w_i` weighing each sample `(x_i, y_i, z_i, ...)`. Weights are normalized to 1 if normed is True. If normed is False, the values of the returned histogram are equal to the sum of the weights belonging to the samples falling into each bin. Returns ------- H : ndarray The multidimensional histogram of sample x. See normed and weights for the different possible semantics. edges : list A list of D arrays describing the bin edges for each dimension. See Also -------- histogram: 1-D histogram histogram2d: 2-D histogram Examples -------- >>> r = np.random.randn(100,3) >>> H, edges = np.histogramdd(r, bins = (5, 8, 4)) >>> H.shape, edges[0].size, edges[1].size, edges[2].size ((5, 8, 4), 6, 9, 5) NzFThe dimension of bins must be equal to the dimension of the sample x.r z range parameter must be finite.g �?r4 z;Element at index %s in `bins` should be a positive integer.ri )r� zOFound bin edge of size <= 0. Did you specify `bins` withnon-monotonic sequence?� ru zInternal Shape Error)+r� �AttributeErrorrm r �Tr r� r rl r| r r r r r� r� r� r r� r� r � issubdtyper� �inexactr r rG r/ r8 �isinfr) r r �reshaper� �prodr9 r, r� �swapaxesrw r0 � RuntimeError) �sampler� r� r� r� �N�DZnbinZedgesZdedges�MZsminZsmaxr� Zedge_dtZNcountZmindiff�decimalZnot_smaller_than_edgeZon_edgeZhist�ni�xyZ flatcountr� �j�core�sr� rs rs rt rU = s� 4 & $ & . c C s^ t �| �} |dkr2| �|�}|j�| j|j �}n�t �|�}t| jjt jt jf�rft � | j|jd�}nt � | j|j�}| j |j kr�|dkr�t d��|j dkr�t d��|j d | j | kr�t d��t �|| j d d |j �}|�d |�}|j||d �}t �|d k��rtd ��t j| ||d ��|�| }|�rV|j |j k�rNt �||j ��� }||fS |S dS )a Compute the weighted average along the specified axis. Parameters ---------- a : array_like Array containing data to be averaged. If `a` is not an array, a conversion is attempted. axis : None or int or tuple of ints, optional Axis or axes along which to average `a`. The default, axis=None, will average over all of the elements of the input array. If axis is negative it counts from the last to the first axis. .. versionadded:: 1.7.0 If axis is a tuple of ints, averaging is performed on all of the axes specified in the tuple instead of a single axis or all the axes as before. weights : array_like, optional An array of weights associated with the values in `a`. Each value in `a` contributes to the average according to its associated weight. The weights array can either be 1-D (in which case its length must be the size of `a` along the given axis) or of the same shape as `a`. If `weights=None`, then all data in `a` are assumed to have a weight equal to one. returned : bool, optional Default is `False`. If `True`, the tuple (`average`, `sum_of_weights`) is returned, otherwise only the average is returned. If `weights=None`, `sum_of_weights` is equivalent to the number of elements over which the average is taken. Returns ------- average, [sum_of_weights] : array_type or double Return the average along the specified axis. When returned is `True`, return a tuple with the average as the first element and the sum of the weights as the second element. The return type is `Float` if `a` is of integer type, otherwise it is of the same type as `a`. `sum_of_weights` is of the same type as `average`. Raises ------ ZeroDivisionError When all weights along axis are zero. See `numpy.ma.average` for a version robust to this type of error. TypeError When the length of 1D `weights` is not the same as the shape of `a` along axis. See Also -------- mean ma.average : average for masked arrays -- useful if your data contains "missing" values Examples -------- >>> data = range(1,5) >>> data [1, 2, 3, 4] >>> np.average(data) 2.5 >>> np.average(range(1,11), weights=range(10,0,-1)) 4.0 >>> data = np.arange(6).reshape((3,2)) >>> data array([[0, 1], [2, 3], [4, 5]]) >>> np.average(data, axis=1, weights=[1./4, 3./4]) array([ 0.75, 2.75, 4.75]) >>> np.average(data, weights=[1./4, 3./4]) Traceback (most recent call last): ... TypeError: Axis must be specified when shapes of a and weights differ. NZf8z;Axis must be specified when shapes of a and weights differ.r4 z81D weights expected when shapes of a and weights differ.r z5Length of weights not compatible with specified axis.)r4 ru )ry r� g z(Weights sum to zero, can't be normalized)r� )r r r. r� �typer� � issubclassr �bool_� result_typer� r| rn rm � broadcast_tor� r0 r/ �ZeroDivisionErrorr rD )r� ry r� �returned�avgZsclZwgtZ result_dtypers rs rt rS s< Q c C s8 t | ||d�} | jjtd kr4t�| ��� s4td��| S )aT Convert the input to an array, checking for NaNs or Infs. Parameters ---------- a : array_like Input data, in any form that can be converted to an array. This includes lists, lists of tuples, tuples, tuples of tuples, tuples of lists and ndarrays. Success requires no NaNs or Infs. dtype : data-type, optional By default, the data-type is inferred from the input data. order : {'C', 'F'}, optional Whether to use row-major (C-style) or column-major (Fortran-style) memory representation. Defaults to 'C'. Returns ------- out : ndarray Array interpretation of `a`. No copy is performed if the input is already an ndarray. If `a` is a subclass of ndarray, a base class ndarray is returned. Raises ------ ValueError Raises ValueError if `a` contains NaN (Not a Number) or Inf (Infinity). See Also -------- asarray : Create and array. asanyarray : Similar function which passes through subclasses. ascontiguousarray : Convert input to a contiguous array. asfarray : Convert input to a floating point ndarray. asfortranarray : Convert input to an ndarray with column-major memory order. fromiter : Create an array from an iterator. fromfunction : Construct an array by executing a function on grid positions. Examples -------- Convert a list into an array. If all elements are finite ``asarray_chkfinite`` is identical to ``asarray``. >>> a = [1, 2] >>> np.asarray_chkfinite(a, dtype=float) array([1., 2.]) Raises ValueError if array_like contains Nans or Infs. >>> a = [1, 2, np.inf] >>> try: ... np.asarray_chkfinite(a) ... except ValueError: ... print('ValueError') ... ValueError )r� �orderZAllFloatz#array must not contain infs or NaNs)r r� �charr1 r r� r� rm )r� r� r� rs rs rt rR � s < c O s� t | �} t|�}t|�s4t|d t�sft|d t�sft|�s`| jdkr`| jdkr`dd� |D �}n|g}t|t d�}t|�}d}| jdkr�| d } d}||d k�rt j j |dd �}| jdkr�t � || g�}nBd d� |D �}|d }x td|�D ]} ||| O }q�W |�| � |d7 }t| j| j�} xft|�D ]Z} || } t| tj��sX| | || < n0| || } | jdk�r0| | f|�|�| || < �q0W |�r�| �� } | S ) a� Evaluate a piecewise-defined function. Given a set of conditions and corresponding functions, evaluate each function on the input data wherever its condition is true. Parameters ---------- x : ndarray or scalar The input domain. condlist : list of bool arrays or bool scalars Each boolean array corresponds to a function in `funclist`. Wherever `condlist[i]` is True, `funclist[i](x)` is used as the output value. Each boolean array in `condlist` selects a piece of `x`, and should therefore be of the same shape as `x`. The length of `condlist` must correspond to that of `funclist`. If one extra function is given, i.e. if ``len(funclist) - len(condlist) == 1``, then that extra function is the default value, used wherever all conditions are false. funclist : list of callables, f(x,*args,**kw), or scalars Each function is evaluated over `x` wherever its corresponding condition is True. It should take an array as input and give an array or a scalar value as output. If, instead of a callable, a scalar is provided then a constant function (``lambda x: scalar``) is assumed. args : tuple, optional Any further arguments given to `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., 1, 'a')``, then each function is called as ``f(x, 1, 'a')``. kw : dict, optional Keyword arguments used in calling `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., alpha=1)``, then each function is called as ``f(x, alpha=1)``. Returns ------- out : ndarray The output is the same shape and type as x and is found by calling the functions in `funclist` on the appropriate portions of `x`, as defined by the boolean arrays in `condlist`. Portions not covered by any condition have a default value of 0. See Also -------- choose, select, where Notes ----- This is similar to choose or select, except that functions are evaluated on elements of `x` that satisfy the corresponding condition from `condlist`. The result is:: |-- |funclist[0](x[condlist[0]]) out = |funclist[1](x[condlist[1]]) |... |funclist[n2](x[condlist[n2]]) |-- Examples -------- Define the sigma function, which is -1 for ``x < 0`` and +1 for ``x >= 0``. >>> x = np.linspace(-2.5, 2.5, 6) >>> np.piecewise(x, [x < 0, x >= 0], [-1, 1]) array([-1., -1., -1., 1., 1., 1.]) Define the absolute value, which is ``-x`` for ``x <0`` and ``x`` for ``x >= 0``. >>> np.piecewise(x, [x < 0, x >= 0], [lambda x: -x, lambda x: x]) array([ 2.5, 1.5, 0.5, 0.5, 1.5, 2.5]) Apply the same function to a scalar value. >>> y = -2 >>> np.piecewise(y, [y < 0, y >= 0], [lambda x: -x, lambda x: x]) array(2) r r4 c S s g | ] }|g�qS rs rs )r� r� rs rs rt r� s z piecewise.<locals>.<listcomp>)r� FNT)ry c S s g | ]}t |td ��qS ))r� )r �bool)r� r� rs rs rt r� 1 s )r rl r r� �listr r� rn r r� r � logical_orr� �vstackr� rf r r� r� � collections�Callable�squeeze) r� �condlist�funclist�args�kw�n2r� �zerodZtotlistrq r} �item�valsrs rs rt rB � sD W c C s� t | �t |�krtd��t | �dkrBtjdtdd� t�|�d S dd� |D �}|�t�|�� tj|� }tj | � } tj |� }d }xZt t | ��D ]J}| | }|j j tj k r�t�|j tj�r�| | �t�| |< d }q�td ��q�W |r�d }tj|tdd� |d jdk�r| d j}nt� | d |d �d j}t�||d |�} |ddd � }| ddd � } x(t|| �D ]\} }tj| | |d� �qfW | S )a� Return an array drawn from elements in choicelist, depending on conditions. Parameters ---------- condlist : list of bool ndarrays The list of conditions which determine from which array in `choicelist` the output elements are taken. When multiple conditions are satisfied, the first one encountered in `condlist` is used. choicelist : list of ndarrays The list of arrays from which the output elements are taken. It has to be of the same length as `condlist`. default : scalar, optional The element inserted in `output` when all conditions evaluate to False. Returns ------- output : ndarray The output at position m is the m-th element of the array in `choicelist` where the m-th element of the corresponding array in `condlist` is True. See Also -------- where : Return elements from one of two arrays depending on condition. take, choose, compress, diag, diagonal Examples -------- >>> x = np.arange(10) >>> condlist = [x<3, x>5] >>> choicelist = [x, x**2] >>> np.select(condlist, choicelist) array([ 0, 1, 2, 0, 0, 0, 36, 49, 64, 81]) z7list of cases must be same length as list of conditionsr zIselect with an empty condition list is not possibleand will be deprecatedri )� stacklevelrs c S s g | ]}t �|��qS rs )r r )r� �choicers rs rt r� x s zselect.<locals>.<listcomp>FTz6invalid entry in choicelist: should be boolean ndarrayz�select condlists containing integer ndarrays is deprecated and will be removed in the future. Use `.astype(bool)` to convert to bools.ru ���N)r )rl rm �warnings�warn�DeprecationWarningr r rf r� �broadcast_arraysr� r� r� r� r� r r� r� rn r� �full�zip�copyto) r� Z choicelist�defaultr� Zdeprecated_intsr� �cond�msgZ result_shape�resultr� rs rs rt rA F sB & �Kc C s t | |dd�S )a� Return an array copy of the given object. Parameters ---------- a : array_like Input data. order : {'C', 'F', 'A', 'K'}, optional Controls the memory layout of the copy. 'C' means C-order, 'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous, 'C' otherwise. 'K' means match the layout of `a` as closely as possible. (Note that this function and :meth:`ndarray.copy` are very similar, but have different default values for their order= arguments.) Returns ------- arr : ndarray Array interpretation of `a`. Notes ----- This is equivalent to: >>> np.array(a, copy=True) #doctest: +SKIP Examples -------- Create an array x, with a reference y and a copy z: >>> x = np.array([1, 2, 3]) >>> y = x >>> z = np.copy(x) Note that, when we modify x, y changes, but not z: >>> x[0] = 10 >>> x[0] == y[0] True >>> x[0] == z[0] False T)r� rD )r )r� r� rs rs rt rD � s ,c O s� t �| �} | j}|�dd�}|dkr2tt|��}n t�||�}t|�}t|�}|dkrbdg| }n�|dkr�t �|d �dkr�|| }n�||k�r"t |�}x�t |�D ]z\}} t �| �dkr�q�nt �| �dkr�t d��t| �| j || kr�t d��t � | �} | | d k�� �r| d } | ||< q�W ntd��|�d d�} |�rTtd �d �|�� ����| d k�rft d ��g } td�g| } td�g| }td�g| }td�g| }| jj}|dk�r�d}|dk�r�| jj�dd�}n|dk�r�| j}| jjdk�r�| �d�}n| }�x�t |�D �]�\}}|j | | d k �r4t d��t j||d�}t �|| �dk}tdd�| |< tdd�||< tdd�||< td d�||< |�r�| | | | d|| || < n�|| dd� }|| dd� }| ||| }|| || }|||| }t j|td�}d||< | |_ |_ |_ || | || | || | || < | dk�r�d| |< d||< d||< |�r�|| n || d } || || | || < d| |< d||< d||< |�r�|| n || d } || || | || < �n�d| |< d||< d||< d ||< |�rDd|| }d|| }d || }nT|| d }|| d }d| | ||| }|| || }| ||| }||| ||| ||| || < d| |< d ||< d||< d||< |�r d || }d || }d || }nR|| d }|| d }|||| }|| || }d| | ||| }||| ||| ||| || < | � |� td�| |< td�||< td�||< td�||< �qW |dk�r�| d S | S dS )!a Return the gradient of an N-dimensional array. The gradient is computed using second order accurate central differences in the interior points and either first or second order accurate one-sides (forward or backwards) differences at the boundaries. The returned gradient hence has the same shape as the input array. Parameters ---------- f : array_like An N-dimensional array containing samples of a scalar function. varargs : list of scalar or array, optional Spacing between f values. Default unitary spacing for all dimensions. Spacing can be specified using: 1. single scalar to specify a sample distance for all dimensions. 2. N scalars to specify a constant sample distance for each dimension. i.e. `dx`, `dy`, `dz`, ... 3. N arrays to specify the coordinates of the values along each dimension of F. The length of the array must match the size of the corresponding dimension 4. Any combination of N scalars/arrays with the meaning of 2. and 3. If `axis` is given, the number of varargs must equal the number of axes. Default: 1. edge_order : {1, 2}, optional Gradient is calculated using N-th order accurate differences at the boundaries. Default: 1. .. versionadded:: 1.9.1 axis : None or int or tuple of ints, optional Gradient is calculated only along the given axis or axes The default (axis = None) is to calculate the gradient for all the axes of the input array. axis may be negative, in which case it counts from the last to the first axis. .. versionadded:: 1.11.0 Returns ------- gradient : ndarray or list of ndarray A set of ndarrays (or a single ndarray if there is only one dimension) corresponding to the derivatives of f with respect to each dimension. Each derivative has the same shape as f. Examples -------- >>> f = np.array([1, 2, 4, 7, 11, 16], dtype=np.float) >>> np.gradient(f) array([ 1. , 1.5, 2.5, 3.5, 4.5, 5. ]) >>> np.gradient(f, 2) array([ 0.5 , 0.75, 1.25, 1.75, 2.25, 2.5 ]) Spacing can be also specified with an array that represents the coordinates of the values F along the dimensions. For instance a uniform spacing: >>> x = np.arange(f.size) >>> np.gradient(f, x) array([ 1. , 1.5, 2.5, 3.5, 4.5, 5. ]) Or a non uniform one: >>> x = np.array([0., 1., 1.5, 3.5, 4., 6.], dtype=np.float) >>> np.gradient(f, x) array([ 1. , 3. , 3.5, 6.7, 6.9, 2.5]) For two dimensional arrays, the return will be two arrays ordered by axis. In this example the first array stands for the gradient in rows and the second one in columns direction: >>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=np.float)) [array([[ 2., 2., -1.], [ 2., 2., -1.]]), array([[ 1. , 2.5, 4. ], [ 1. , 1. , 1. ]])] In this example the spacing is also specified: uniform for axis=0 and non uniform for axis=1 >>> dx = 2. >>> y = [1., 1.5, 3.5] >>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=np.float), dx, y) [array([[ 1. , 1. , -0.5], [ 1. , 1. , -0.5]]), array([[ 2. , 2. , 2. ], [ 2. , 1.7, 0.5]])] It is possible to specify how boundaries are treated using `edge_order` >>> x = np.array([0, 1, 2, 3, 4]) >>> f = x**2 >>> np.gradient(f, edge_order=1) array([ 1., 2., 4., 6., 7.]) >>> np.gradient(f, edge_order=2) array([-0., 2., 4., 6., 8.]) The `axis` keyword can be used to specify a subset of axes of which the gradient is calculated >>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=np.float), axis=0) array([[ 2., 2., -1.], [ 2., 2., -1.]]) Notes ----- Assuming that :math:`f\in C^{3}` (i.e., :math:`f` has at least 3 continuous derivatives) and let be :math:`h_{*}` a non homogeneous stepsize, the spacing the finite difference coefficients are computed by minimising the consistency error :math:`\eta_{i}`: .. math:: \eta_{i} = f_{i}^{\left(1\right)} - \left[ \alpha f\left(x_{i}\right) + \beta f\left(x_{i} + h_{d}\right) + \gamma f\left(x_{i}-h_{s}\right) \right] By substituting :math:`f(x_{i} + h_{d})` and :math:`f(x_{i} - h_{s})` with their Taylor series expansion, this translates into solving the following the linear system: .. math:: \left\{ \begin{array}{r} \alpha+\beta+\gamma=0 \\ -\beta h_{d}+\gamma h_{s}=1 \\ \beta h_{d}^{2}+\gamma h_{s}^{2}=0 \end{array} \right. The resulting approximation of :math:`f_{i}^{(1)}` is the following: .. math:: \hat f_{i}^{(1)} = \frac{ h_{s}^{2}f\left(x_{i} + h_{d}\right) + \left(h_{d}^{2} - h_{s}^{2}\right)f\left(x_{i}\right) - h_{d}^{2}f\left(x_{i}-h_{s}\right)} { h_{s}h_{d}\left(h_{d} + h_{s}\right)} + \mathcal{O}\left(\frac{h_{d}h_{s}^{2} + h_{s}h_{d}^{2}}{h_{d} + h_{s}}\right) It is worth noting that if :math:`h_{s}=h_{d}` (i.e., data are evenly spaced) we find the standard second order approximation: .. math:: \hat f_{i}^{(1)}= \frac{f\left(x_{i+1}\right) - f\left(x_{i-1}\right)}{2h} + \mathcal{O}\left(h^{2}\right) With a similar procedure the forward/backward approximations used for boundaries can be derived. References ---------- .. [1] Quarteroni A., Sacco R., Saleri F. (2007) Numerical Mathematics (Texts in Applied Mathematics). New York: Springer. .. [2] Durran D. R. (1999) Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. New York: Springer. .. [3] Fornberg B. (1988) Generation of Finite Difference Formulas on Arbitrarily Spaced Grids, Mathematics of Computation 51, no. 184 : 699-706. `PDF <http://www.ams.org/journals/mcom/1988-51-184/ S0025-5718-1988-0935077-0/S0025-5718-1988-0935077-0.pdf>`_. ry Nr g �?r4 z&distances must be either scalars or 1dzGwhen 1d, distances must match the length of the corresponding dimensionzinvalid number of arguments� edge_orderz%"{}" are not valid keyword arguments.z", "ri z)'edge_order' greater than 2 not supported)�f�d�Fr� rp r� r� r� �datetime� timedeltarp )r� rp �int64zlShape of array too small to calculate a numerical gradient, at least (edge_order + 1) elements are required.)r� ru r� g @g ��g ����g �?g �g �?) r r rn �poprk r� �_nx�normalize_axis_tuplerl r� � enumeraterm r� rG r� r| ro �join�keysrw r� r� �name�replace�viewr r r� rf ) r� �varargs�kwargsr� rr Zlen_axesr� �dxr� Z distancesZdiffxr� Zoutvals�slice1�slice2Zslice3Zslice4Zotyper} ry �outZuniform_spacingZdx1Zdx2r� r� r� r� Zdx_0Zdx_nrs rs rt rH � s� / "( ( ( ru c C s� |dkr | S |dk r$t dt|� ��t| �} | j}td�g| }td�g| }tdd�||< tdd�||< t|�}t|�}|dkr�t| | | | |d |d�S | | | | S dS )a6 Calculate the n-th discrete difference along given axis. The first difference is given by ``out[n] = a[n+1] - a[n]`` along the given axis, higher differences are calculated by using `diff` recursively. Parameters ---------- a : array_like Input array n : int, optional The number of times values are differenced. axis : int, optional The axis along which the difference is taken, default is the last axis. Returns ------- diff : ndarray The n-th differences. The shape of the output is the same as `a` except along `axis` where the dimension is smaller by `n`. The type of the output is the same as that of the input. See Also -------- gradient, ediff1d, cumsum Notes ----- For boolean arrays, the preservation of type means that the result will contain `False` when consecutive elements are the same and `True` when they differ. For unsigned integer arrays, the results will also be unsigned. This should not be surprising, as the result is consistent with calculating the difference directly: >>> u8_arr = np.array([1, 0], dtype=np.uint8) >>> np.diff(u8_arr) array([255], dtype=uint8) >>> u8_arr[1,...] - u8_arr[0,...] array(255, np.uint8) If this is not desirable, then the array should be cast to a larger integer type first: >>> i16_arr = u8_arr.astype(np.int16) >>> np.diff(i16_arr) array([-1], dtype=int16) Examples -------- >>> x = np.array([1, 2, 4, 7, 0]) >>> np.diff(x) array([ 1, 2, 3, -7]) >>> np.diff(x, n=2) array([ 1, 1, -10]) >>> x = np.array([[1, 3, 6, 10], [0, 5, 6, 8]]) >>> np.diff(x) array([[2, 3, 4], [5, 1, 2]]) >>> np.diff(x, axis=0) array([[-1, 2, 0, -2]]) r z#order must be non-negative but got Nr4 ru )ry )rm �reprr rn rw rk rG )r� r� ry �ndr r rs rs rt rG 8 s C c C s� t �|�}t �|�r t}t j}n t}t j}|dkr�t| tt t f�rX|| g||||�� � S t| t j �r�| j dkr�|| g||||�� � S || ||||�S �nF|dkr�td��t|�}d}d}d}t| tt t f�r�d}| g} t j| t jd�} t j|t jd�}t j||d�}|j dk�s |j dk�r$td��|jd |jd k�rBtd ��| | } || }t �|�} || }|| }t �|d d� | ||dd� | f�}t �|d d� ||dd� f�}|�r�|| ||||�S || ||||�� � S dS ) a� One-dimensional linear interpolation. Returns the one-dimensional piecewise linear interpolant to a function with given values at discrete data-points. Parameters ---------- x : array_like The x-coordinates of the interpolated values. xp : 1-D sequence of floats The x-coordinates of the data points, must be increasing if argument `period` is not specified. Otherwise, `xp` is internally sorted after normalizing the periodic boundaries with ``xp = xp % period``. fp : 1-D sequence of float or complex The y-coordinates of the data points, same length as `xp`. left : optional float or complex corresponding to fp Value to return for `x < xp[0]`, default is `fp[0]`. right : optional float or complex corresponding to fp Value to return for `x > xp[-1]`, default is `fp[-1]`. period : None or float, optional A period for the x-coordinates. This parameter allows the proper interpolation of angular x-coordinates. Parameters `left` and `right` are ignored if `period` is specified. .. versionadded:: 1.10.0 Returns ------- y : float or complex (corresponding to fp) or ndarray The interpolated values, same shape as `x`. Raises ------ ValueError If `xp` and `fp` have different length If `xp` or `fp` are not 1-D sequences If `period == 0` Notes ----- Does not check that the x-coordinate sequence `xp` is increasing. If `xp` is not increasing, the results are nonsense. A simple check for increasing is:: np.all(np.diff(xp) > 0) Examples -------- >>> xp = [1, 2, 3] >>> fp = [3, 2, 0] >>> np.interp(2.5, xp, fp) 1.0 >>> np.interp([0, 1, 1.5, 2.72, 3.14], xp, fp) array([ 3. , 3. , 2.5 , 0.56, 0. ]) >>> UNDEF = -99.0 >>> np.interp(3.14, xp, fp, right=UNDEF) -99.0 Plot an interpolant to the sine function: >>> x = np.linspace(0, 2*np.pi, 10) >>> y = np.sin(x) >>> xvals = np.linspace(0, 2*np.pi, 50) >>> yinterp = np.interp(xvals, x, y) >>> import matplotlib.pyplot as plt >>> plt.plot(x, y, 'o') [<matplotlib.lines.Line2D object at 0x...>] >>> plt.plot(xvals, yinterp, '-x') [<matplotlib.lines.Line2D object at 0x...>] >>> plt.show() Interpolation with periodic x-coordinates: >>> x = [-180, -170, -185, 185, -10, -5, 0, 365] >>> xp = [190, -190, 350, -350] >>> fp = [5, 10, 3, 4] >>> np.interp(x, xp, fp, period=360) array([7.5, 5., 8.75, 6.25, 3., 3.25, 3.5, 3.75]) Complex interpolation >>> x = [1.5, 4.0] >>> xp = [2,3,5] >>> fp = [1.0j, 0, 2+3j] >>> np.interp(x, xp, fp) array([ 0.+1.j , 1.+1.5j]) Nr z period must be a non-zero valueTF)r� r4 z!Data points must be 1-D sequencesz$fp and xp are not of the same lengthru )r r � iscomplexobj�compiled_interp_complexZ complex128�compiled_interp�float64r� r� r� r2 r� r rn rm �absr� r� r ) r� Zxp�fpr� r� ZperiodZ interp_funcZ input_dtypeZ return_arrayZasort_xprs rs rt r; � sL _ ( c C sN |rdt }nd}t| �} t| jjtj�r8| j}| j}nd}| }t ||�| S )a� Return the angle of the complex argument. Parameters ---------- z : array_like A complex number or sequence of complex numbers. deg : bool, optional Return angle in degrees if True, radians if False (default). Returns ------- angle : ndarray or scalar The counterclockwise angle from the positive real axis on the complex plane, with dtype as numpy.float64. See Also -------- arctan2 absolute Examples -------- >>> np.angle([1.0, 1.0j, 1+1j]) # in radians array([ 0. , 1.57079633, 0.78539816]) >>> np.angle(1+1j, deg=True) # in degrees 45.0 � g �?r ) r r r� r� r� r �complexfloatingr� r� r! )�z�deg�factZzimagZzrealrs rs rt rI s c C s� t | �} | j}t| |d�}tdd�g| }tdd�||< t|t dt �t }tj|t|t k|dk@ d� || }tj|dt|�|k d� t | ddd �}| | |� |� ||< |S ) a8 Unwrap by changing deltas between values to 2*pi complement. Unwrap radian phase `p` by changing absolute jumps greater than `discont` to their 2*pi complement along the given axis. Parameters ---------- p : array_like Input array. discont : float, optional Maximum discontinuity between values, default is ``pi``. axis : int, optional Axis along which unwrap will operate, default is the last axis. Returns ------- out : ndarray Output array. See Also -------- rad2deg, deg2rad Notes ----- If the discontinuity in `p` is smaller than ``pi``, but larger than `discont`, no unwrapping is done because taking the 2*pi complement would only make the discontinuity larger. Examples -------- >>> phase = np.linspace(0, np.pi, num=5) >>> phase[3:] += np.pi >>> phase array([ 0. , 0.78539816, 1.57079633, 5.49778714, 6.28318531]) >>> np.unwrap(phase) array([ 0. , 0.78539816, 1.57079633, -0.78539816, 0. ]) )ry Nr4 ri r )r Tr� )rD r� ) r rn rG rw r' r r r� r r r� ) �pZdiscontry r �ddr ZddmodZ ph_correctZuprs rs rt rJ K s ) c C sd t | dd�}|�� t|jjtj�s\|jjdkr:|�d�S |jjdkrP|�d�S |�d�S n|S dS ) a� Sort a complex array using the real part first, then the imaginary part. Parameters ---------- a : array_like Input array Returns ------- out : complex ndarray Always returns a sorted complex array. Examples -------- >>> np.sort_complex([5, 3, 6, 2, 1]) array([ 1.+0.j, 2.+0.j, 3.+0.j, 5.+0.j, 6.+0.j]) >>> np.sort_complex([1 + 2j, 2 - 1j, 3 - 2j, 3 - 3j, 3 + 5j]) array([ 1.+2.j, 2.-1.j, 3.-3.j, 3.-2.j, 3.+5.j]) T)rD ZbhBHr� �g�Gr� N) r r, r� r� r� r r r� r� )r� r� rs rs rt rK � s �fbc C s~ d}|� � }d|kr6x | D ]}|dkr*P q|d }qW t| �}d|krrx*| ddd� D ]}|dkrfP qV|d }qVW | ||� S )a3 Trim the leading and/or trailing zeros from a 1-D array or sequence. Parameters ---------- filt : 1-D array or sequence Input array. trim : str, optional A string with 'f' representing trim from front and 'b' to trim from back. Default is 'fb', trim zeros from both front and back of the array. Returns ------- trimmed : 1-D array or sequence The result of trimming the input. The input data type is preserved. Examples -------- >>> a = np.array((0, 0, 0, 1, 2, 3, 0, 2, 1, 0)) >>> np.trim_zeros(a) array([1, 2, 3, 0, 2, 1]) >>> np.trim_zeros(a, 'b') array([0, 0, 0, 1, 2, 3, 0, 2, 1]) The input data type is preserved, list/tuple in means list/tuple out. >>> np.trim_zeros([0, 1, 2, 0]) [1, 2] r r� g r4 �BNru )�upperrl )ZfiltZtrim�firstr� �lastrs rs rt rC � s ! c C sr yH| � � }|jdkr|S |�� tdg|dd� |dd� kf�}|| S tk rl tt| ��}t|�S X dS )zW This function is deprecated. Use numpy.lib.arraysetops.unique() instead. r Tr4 Nru )�flattenr� r, r r� �sorted�setr )r� �tmp�idx�itemsrs rs rt �unique� s " r- c C s t �t|�tt| ��d �S )a� Return the elements of an array that satisfy some condition. This is equivalent to ``np.compress(ravel(condition), ravel(arr))``. If `condition` is boolean ``np.extract`` is equivalent to ``arr[condition]``. Note that `place` does the exact opposite of `extract`. Parameters ---------- condition : array_like An array whose nonzero or True entries indicate the elements of `arr` to extract. arr : array_like Input array of the same size as `condition`. Returns ------- extract : ndarray Rank 1 array of values from `arr` where `condition` is True. See Also -------- take, put, copyto, compress, place Examples -------- >>> arr = np.arange(12).reshape((3, 4)) >>> arr array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> condition = np.mod(arr, 3)==0 >>> condition array([[ True, False, False, True], [False, False, True, False], [False, True, False, False]], dtype=bool) >>> np.extract(condition, arr) array([0, 3, 6, 9]) If `condition` is boolean: >>> arr[condition] array([0, 3, 6, 9]) r )r r r* r+ )� condition�arrrs rs rt rO � s 0c C s. t | tj�s"tdjt| �jd���t| ||�S )a Change elements of an array based on conditional and input values. Similar to ``np.copyto(arr, vals, where=mask)``, the difference is that `place` uses the first N elements of `vals`, where N is the number of True values in `mask`, while `copyto` uses the elements where `mask` is True. Note that `extract` does the exact opposite of `place`. Parameters ---------- arr : ndarray Array to put data into. mask : array_like Boolean mask array. Must have the same size as `a`. vals : 1-D sequence Values to put into `a`. Only the first N elements are used, where N is the number of True values in `mask`. If `vals` is smaller than N, it will be repeated, and if elements of `a` are to be masked, this sequence must be non-empty. See Also -------- copyto, put, take, extract Examples -------- >>> arr = np.arange(6).reshape(2, 3) >>> np.place(arr, arr>2, [44, 55]) >>> arr array([[ 0, 1, 2], [44, 55, 44]]) z,argument 1 must be numpy.ndarray, not {name})r )r� r r r| ro r� �__name__r6 )r/ �maskr� rs rs rt rP s $ Tc C s< |dkrt j}|r"|�d| � n|�d| � |�� dS )a7 Display a message on a device. Parameters ---------- mesg : str Message to display. device : object Device to write message. If None, defaults to ``sys.stdout`` which is very similar to ``print``. `device` needs to have ``write()`` and ``flush()`` methods. linefeed : bool, optional Option whether to print a line feed or not. Defaults to True. Raises ------ AttributeError If `device` does not have a ``write()`` or ``flush()`` method. Examples -------- Besides ``sys.stdout``, a file-like object can also be used as it has both required methods: >>> from StringIO import StringIO >>> buf = StringIO() >>> np.disp('"Display" in a file', device=buf) >>> buf.getvalue() '"Display" in a file\n' Nz%s z%s)�sys�stdout�write�flush)ZmesgZdeviceZlinefeedrs rs rt rL I s z\w+z(?:{0:}(?:,{0:})*)?z\({}\)z{0:}(?:,{0:})*z ^{0:}->{0:}$c C s2 t �t| �std�| ���tdd� | �d�D ��S )as Parse string signatures for a generalized universal function. Arguments --------- signature : string Generalized universal function signature, e.g., ``(m,n),(n,p)->(m,p)`` for ``np.matmul``. Returns ------- Tuple of input and output core dimensions parsed from the signature, each of the form List[Tuple[str, ...]]. z not a valid gufunc signature: {}c s s$ | ] }d d� t �t|�D �V qdS )c S s g | ]}t t�t|���qS rs )rk �re�findall�_DIMENSION_NAME)r� �argrs rs rt r� � s z5_parse_gufunc_signature.<locals>.<genexpr>.<listcomp>N)r6 r7 � _ARGUMENT)r� �arg_listrs rs rt � <genexpr>� s z*_parse_gufunc_signature.<locals>.<genexpr>z->)r6 �match� _SIGNATURErm ro rk �split)� signaturers rs rt �_parse_gufunc_signature{ s rA c C s� |sdS t |�}|j|k r,td|j|f ��|j| d� }xJt||�D ]<\}}|| kr||| | kr�td||| | f ��qH|| |< qHW dS )aO Incrementally check and update core dimension sizes for a single argument. Arguments --------- dim_sizes : Dict[str, int] Sizes of existing core dimensions. Will be updated in-place. arg : ndarray Argument to examine. core_dims : Tuple[str, ...] Core dimensions for this argument. NzR%d-dimensional argument does not have enough dimensions for all core dimensions %rz1inconsistent size for core dimension %r: %r vs %r)rl rn rm r� r� )� dim_sizesr9 � core_dimsZ num_core_dimsZ core_shape�dimr� rs rs rt �_update_dim_sizes� s rE c C st g }i }xTt | |�D ]F\}}t|||� |jt|� }tjj�d|jd|� �}|� |� qW tjjj |� }||fS )a� Parse broadcast and core dimensions for vectorize with a signature. Arguments --------- args : Tuple[ndarray, ...] Tuple of input arguments to examine. input_core_dims : List[Tuple[str, ...]] List of core dimensions corresponding to each input. Returns ------- broadcast_shape : Tuple[int, ...] Common shape to broadcast all non-core dimensions to. dim_sizes : Dict[str, int] Common sizes for named core dimensions. r N) r� rE rn rl r �libZ stride_tricksZ as_stridedr� rf Z_broadcast_shape) r� �input_core_dimsZbroadcast_argsrB r9 rC rn Z dummy_array�broadcast_shapers rs rt �_parse_input_dimensions� s rI c s � �fdd�|D �S )z=Helper for calculating broadcast shapes with core dimensions.c s& g | ] }� t �fd d�|D �� �qS )c 3 s | ]}� | V qd S )Nrs )r� rD )rB rs rt r< � s z/_calculate_shapes.<locals>.<listcomp>.<genexpr>)rk )r� rC )rH rB rs rt r� � s z%_calculate_shapes.<locals>.<listcomp>rs )rH rB �list_of_core_dimsrs )rH rB rt �_calculate_shapes� s rK c C s( t | ||�}tdd� t||�D ��}|S )z/Helper for creating output arrays in vectorize.c s s | ]\}}t j||d �V qdS ))r� N)r r )r� r� r� rs rs rt r< � s z!_create_arrays.<locals>.<genexpr>)rK rk r� )rH rB rJ ZdtypesZshapesZarraysrs rs rt �_create_arrays� s rL c @ s: e Zd ZdZddd�Zdd� Zdd � Zd d � Zd d � ZdS )rQ aC vectorize(pyfunc, otypes=None, doc=None, excluded=None, cache=False, signature=None) Generalized function class. Define a vectorized function which takes a nested sequence of objects or numpy arrays as inputs and returns an single or tuple of numpy array as output. The vectorized function evaluates `pyfunc` over successive tuples of the input arrays like the python map function, except it uses the broadcasting rules of numpy. The data type of the output of `vectorized` is determined by calling the function with the first element of the input. This can be avoided by specifying the `otypes` argument. Parameters ---------- pyfunc : callable A python function or method. otypes : str or list of dtypes, optional The output data type. It must be specified as either a string of typecode characters or a list of data type specifiers. There should be one data type specifier for each output. doc : str, optional The docstring for the function. If `None`, the docstring will be the ``pyfunc.__doc__``. excluded : set, optional Set of strings or integers representing the positional or keyword arguments for which the function will not be vectorized. These will be passed directly to `pyfunc` unmodified. .. versionadded:: 1.7.0 cache : bool, optional If `True`, then cache the first function call that determines the number of outputs if `otypes` is not provided. .. versionadded:: 1.7.0 signature : string, optional Generalized universal function signature, e.g., ``(m,n),(n)->(m)`` for vectorized matrix-vector multiplication. If provided, ``pyfunc`` will be called with (and expected to return) arrays with shapes given by the size of corresponding core dimensions. By default, ``pyfunc`` is assumed to take scalars as input and output. .. versionadded:: 1.12.0 Returns ------- vectorized : callable Vectorized function. Examples -------- >>> def myfunc(a, b): ... "Return a-b if a>b, otherwise return a+b" ... if a > b: ... return a - b ... else: ... return a + b >>> vfunc = np.vectorize(myfunc) >>> vfunc([1, 2, 3, 4], 2) array([3, 4, 1, 2]) The docstring is taken from the input function to `vectorize` unless it is specified: >>> vfunc.__doc__ 'Return a-b if a>b, otherwise return a+b' >>> vfunc = np.vectorize(myfunc, doc='Vectorized `myfunc`') >>> vfunc.__doc__ 'Vectorized `myfunc`' The output type is determined by evaluating the first element of the input, unless it is specified: >>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) <type 'numpy.int32'> >>> vfunc = np.vectorize(myfunc, otypes=[np.float]) >>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) <type 'numpy.float64'> The `excluded` argument can be used to prevent vectorizing over certain arguments. This can be useful for array-like arguments of a fixed length such as the coefficients for a polynomial as in `polyval`: >>> def mypolyval(p, x): ... _p = list(p) ... res = _p.pop(0) ... while _p: ... res = res*x + _p.pop(0) ... return res >>> vpolyval = np.vectorize(mypolyval, excluded=['p']) >>> vpolyval(p=[1, 2, 3], x=[0, 1]) array([3, 6]) Positional arguments may also be excluded by specifying their position: >>> vpolyval.excluded.add(0) >>> vpolyval([1, 2, 3], x=[0, 1]) array([3, 6]) The `signature` argument allows for vectorizing functions that act on non-scalar arrays of fixed length. For example, you can use it for a vectorized calculation of Pearson correlation coefficient and its p-value: >>> import scipy.stats >>> pearsonr = np.vectorize(scipy.stats.pearsonr, ... signature='(n),(n)->(),()') >>> pearsonr([[0, 1, 2, 3]], [[1, 2, 3, 4], [4, 3, 2, 1]]) (array([ 1., -1.]), array([ 0., 0.])) Or for a vectorized convolution: >>> convolve = np.vectorize(np.convolve, signature='(n),(m)->(k)') >>> convolve(np.eye(4), [1, 2, 1]) array([[ 1., 2., 1., 0., 0., 0.], [ 0., 1., 2., 1., 0., 0.], [ 0., 0., 1., 2., 1., 0.], [ 0., 0., 0., 1., 2., 1.]]) See Also -------- frompyfunc : Takes an arbitrary Python function and returns a ufunc Notes ----- The `vectorize` function is provided primarily for convenience, not for performance. The implementation is essentially a for loop. If `otypes` is not specified, then a call to the function with the first argument will be used to determine the number of outputs. The results of this call will be cached if `cache` is `True` to prevent calling the function twice. However, to implement the cache, the original function must be wrapped which will slow down subsequent calls, so only do this if your function is expensive. The new keyword argument interface and `excluded` argument support further degrades performance. References ---------- .. [1] NumPy Reference, section `Generalized Universal Function API <http://docs.scipy.org/doc/numpy/reference/c-api.generalized-ufuncs.html>`_. 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Covariance indicates the level to which two variables vary together. If we examine N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]^T`, then the covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i`. See the notes for an outline of the algorithm. Parameters ---------- m : array_like A 1-D or 2-D array containing multiple variables and observations. Each row of `m` represents a variable, and each column a single observation of all those variables. Also see `rowvar` below. y : array_like, optional An additional set of variables and observations. `y` has the same form as that of `m`. rowvar : bool, optional If `rowvar` is True (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : bool, optional Default normalization (False) is by ``(N - 1)``, where ``N`` is the number of observations given (unbiased estimate). If `bias` is True, then normalization is by ``N``. These values can be overridden by using the keyword ``ddof`` in numpy versions >= 1.5. ddof : int, optional If not ``None`` the default value implied by `bias` is overridden. Note that ``ddof=1`` will return the unbiased estimate, even if both `fweights` and `aweights` are specified, and ``ddof=0`` will return the simple average. See the notes for the details. The default value is ``None``. .. versionadded:: 1.5 fweights : array_like, int, optional 1-D array of integer freguency weights; the number of times each observation vector should be repeated. .. versionadded:: 1.10 aweights : array_like, optional 1-D array of observation vector weights. These relative weights are typically large for observations considered "important" and smaller for observations considered less "important". If ``ddof=0`` the array of weights can be used to assign probabilities to observation vectors. .. versionadded:: 1.10 Returns ------- out : ndarray The covariance matrix of the variables. See Also -------- corrcoef : Normalized covariance matrix Notes ----- Assume that the observations are in the columns of the observation array `m` and let ``f = fweights`` and ``a = aweights`` for brevity. The steps to compute the weighted covariance are as follows:: >>> w = f * a >>> v1 = np.sum(w) >>> v2 = np.sum(w * a) >>> m -= np.sum(m * w, axis=1, keepdims=True) / v1 >>> cov = np.dot(m * w, m.T) * v1 / (v1**2 - ddof * v2) Note that when ``a == 1``, the normalization factor ``v1 / (v1**2 - ddof * v2)`` goes over to ``1 / (np.sum(f) - ddof)`` as it should. Examples -------- Consider two variables, :math:`x_0` and :math:`x_1`, which correlate perfectly, but in opposite directions: >>> x = np.array([[0, 2], [1, 1], [2, 0]]).T >>> x array([[0, 1, 2], [2, 1, 0]]) Note how :math:`x_0` increases while :math:`x_1` decreases. The covariance matrix shows this clearly: >>> np.cov(x) array([[ 1., -1.], [-1., 1.]]) Note that element :math:`C_{0,1}`, which shows the correlation between :math:`x_0` and :math:`x_1`, is negative. Further, note how `x` and `y` are combined: >>> x = [-2.1, -1, 4.3] >>> y = [3, 1.1, 0.12] >>> X = np.vstack((x,y)) >>> print(np.cov(X)) [[ 11.71 -4.286 ] [ -4.286 2.14413333]] >>> print(np.cov(x, y)) [[ 11.71 -4.286 ] [ -4.286 2.14413333]] >>> print(np.cov(x)) 11.71 Nzddof must be integerri z m has more than 2 dimensionsz y has more than 2 dimensions)�ndminr� r r4 F)rD rw r� )r� zfweights must be integerz'cannot handle multidimensional fweightsz,incompatible numbers of samples and fweightszfweights cannot be negativez'cannot handle multidimensional aweightsz,incompatible numbers of samples and aweightszaweights cannot be negativeT)ry r� r� z!Degrees of freedom <= 0 for slice)r� g g �?)r� rm r r rn r� r r r� r� r� r� r� r� r r| r� r/ rS r0 r� r� �RuntimeWarningr �conjr� )rp r} �rowvar�bias�ddofZfweightsZaweightsr� �X�wr� Zw_sumr ZX_Tr� rs rs rt rV H s� r c C s� |t jk s|t jk r$tjdtdd� t| ||�}y t|�}W n tk rT || S X t|j �}||dd�df }||ddd�f }t j |j dd|j d� t � |�r�t j |j dd|j d� |S )a� Return Pearson product-moment correlation coefficients. Please refer to the documentation for `cov` for more detail. The relationship between the correlation coefficient matrix, `R`, and the covariance matrix, `C`, is .. math:: R_{ij} = \frac{ C_{ij} } { \sqrt{ C_{ii} * C_{jj} } } The values of `R` are between -1 and 1, inclusive. Parameters ---------- x : array_like A 1-D or 2-D array containing multiple variables and observations. Each row of `x` represents a variable, and each column a single observation of all those variables. Also see `rowvar` below. y : array_like, optional An additional set of variables and observations. `y` has the same shape as `x`. rowvar : bool, optional If `rowvar` is True (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : _NoValue, optional Has no effect, do not use. .. deprecated:: 1.10.0 ddof : _NoValue, optional Has no effect, do not use. .. deprecated:: 1.10.0 Returns ------- R : ndarray The correlation coefficient matrix of the variables. See Also -------- cov : Covariance matrix Notes ----- Due to floating point rounding the resulting array may not be Hermitian, the diagonal elements may not be 1, and the elements may not satisfy the inequality abs(a) <= 1. The real and imaginary parts are clipped to the interval [-1, 1] in an attempt to improve on that situation but is not much help in the complex case. This function accepts but discards arguments `bias` and `ddof`. This is for backwards compatibility with previous versions of this function. These arguments had no effect on the return values of the function and can be safely ignored in this and previous versions of numpy. z/bias and ddof have no effect and are deprecatedri )r� Nru r4 )r ) r �_NoValuer� r� r� rV r3 rm r% r� Zclipr r� )r� r} rz r{ r| r� r� �stddevrs rs rt rW s : c C sh | dk rt g �S | dkr"tdt�S td| �}ddtdt | | d � dtdt | | d � S )a� Return the Blackman window. The Blackman window is a taper formed by using the first three terms of a summation of cosines. It was designed to have close to the minimal leakage possible. It is close to optimal, only slightly worse than a Kaiser window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. Returns ------- out : ndarray The window, with the maximum value normalized to one (the value one appears only if the number of samples is odd). See Also -------- bartlett, hamming, hanning, kaiser Notes ----- The Blackman window is defined as .. math:: w(n) = 0.42 - 0.5 \cos(2\pi n/M) + 0.08 \cos(4\pi n/M) Most references to the Blackman window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. It is known as a "near optimal" tapering function, almost as good (by some measures) as the kaiser window. References ---------- Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471. Examples -------- >>> np.blackman(12) array([ -1.38777878e-17, 3.26064346e-02, 1.59903635e-01, 4.14397981e-01, 7.36045180e-01, 9.67046769e-01, 9.67046769e-01, 7.36045180e-01, 4.14397981e-01, 1.59903635e-01, 3.26064346e-02, -1.38777878e-17]) Plot the window and the frequency response: >>> from numpy.fft import fft, fftshift >>> window = np.blackman(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Blackman window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show() >>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Blackman window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show() r4 r g�z�G��?g �?g @g{�G�z�?g @)r r r� r r# r )r� r� rs rs rt r^ l s Z c C sb | dk rt g �S | dkr"tdt�S td| �}tt|| d d �d| | d dd| | d �S )a� Return the Bartlett window. The Bartlett window is very similar to a triangular window, except that the end points are at zero. It is often used in signal processing for tapering a signal, without generating too much ripple in the frequency domain. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. Returns ------- out : array The triangular window, with the maximum value normalized to one (the value one appears only if the number of samples is odd), with the first and last samples equal to zero. See Also -------- blackman, hamming, hanning, kaiser Notes ----- The Bartlett window is defined as .. math:: w(n) = \frac{2}{M-1} \left( \frac{M-1}{2} - \left|n - \frac{M-1}{2}\right| \right) Most references to the Bartlett window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. Note that convolution with this window produces linear interpolation. It is also known as an apodization (which means"removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. The fourier transform of the Bartlett is the product of two sinc functions. Note the excellent discussion in Kanasewich. References ---------- .. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", Biometrika 37, 1-16, 1950. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal Processing", Prentice-Hall, 1999, pp. 468-471. .. [4] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 429. Examples -------- >>> np.bartlett(12) array([ 0. , 0.18181818, 0.36363636, 0.54545455, 0.72727273, 0.90909091, 0.90909091, 0.72727273, 0.54545455, 0.36363636, 0.18181818, 0. ]) Plot the window and its frequency response (requires SciPy and matplotlib): >>> from numpy.fft import fft, fftshift >>> window = np.bartlett(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Bartlett window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show() >>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Bartlett window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show() r4 r g @)r r r� r r r$ )r� r� rs rs rt r] � s b c C sL | dk rt g �S | dkr"tdt�S td| �}ddtdt | | d � S )aa Return the Hanning window. The Hanning window is a taper formed by using a weighted cosine. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. Returns ------- out : ndarray, shape(M,) The window, with the maximum value normalized to one (the value one appears only if `M` is odd). See Also -------- bartlett, blackman, hamming, kaiser Notes ----- The Hanning window is defined as .. math:: w(n) = 0.5 - 0.5cos\left(\frac{2\pi{n}}{M-1}\right) \qquad 0 \leq n \leq M-1 The Hanning was named for Julius von Hann, an Austrian meteorologist. It is also known as the Cosine Bell. Some authors prefer that it be called a Hann window, to help avoid confusion with the very similar Hamming window. Most references to the Hanning window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 106-108. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 425. Examples -------- >>> np.hanning(12) array([ 0. , 0.07937323, 0.29229249, 0.57115742, 0.82743037, 0.97974649, 0.97974649, 0.82743037, 0.57115742, 0.29229249, 0.07937323, 0. ]) Plot the window and its frequency response: >>> from numpy.fft import fft, fftshift >>> window = np.hanning(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Hann window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show() >>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of the Hann window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show() r4 r g �?g @)r r r� r r# r )r� r� rs rs rt r\ 8 s \ c C sL | dk rt g �S | dkr"tdt�S td| �}ddtdt | | d � S )aQ Return the Hamming window. The Hamming window is a taper formed by using a weighted cosine. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. Returns ------- out : ndarray The window, with the maximum value normalized to one (the value one appears only if the number of samples is odd). See Also -------- bartlett, blackman, hanning, kaiser Notes ----- The Hamming window is defined as .. math:: w(n) = 0.54 - 0.46cos\left(\frac{2\pi{n}}{M-1}\right) \qquad 0 \leq n \leq M-1 The Hamming was named for R. W. Hamming, an associate of J. W. Tukey and is described in Blackman and Tukey. It was recommended for smoothing the truncated autocovariance function in the time domain. Most references to the Hamming window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 425. Examples -------- >>> np.hamming(12) array([ 0.08 , 0.15302337, 0.34890909, 0.60546483, 0.84123594, 0.98136677, 0.98136677, 0.84123594, 0.60546483, 0.34890909, 0.15302337, 0.08 ]) Plot the window and the frequency response: >>> from numpy.fft import fft, fftshift >>> window = np.hamming(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Hamming window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show() >>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Hamming window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show() r4 r gH�z�G�?gq= ףp�?g @)r r r� r r# r )r� r� rs rs rt r[ � s Z g��4!�\T�g��}�b3�<g�� r넱�g�^�����<g��"P �g'&&KF�5=g��bLa�g$ӛ�/��=g�j�z����g<t̾��=gV�����g4�T��&>g�0 ���K�g5dM�v;p>g�"�c쑾g��$��>g'd��o�ҾgY(��X?�>gZ �Y&+�g�|t�(?gR� ��B�g�u�Z?gI� ^�q�g����a��?g�!�N��g-��Ί>�?g�-4pK��g���w���?g��W��ӿg*�5�N��?g��T��`�g0�f�FV�g!����<g�A`��<g� ҫ `��g8��箸�g ��}��<g�攐�*�<g�be~���g2�hϙ]'�gE�_ �V=gs��k� [=g�&�GCi=gf�C��g�{~5���g%t9 Q��gO �$�=guo�� >g�["�d,->gm�ր�VX>gna����>g���+A�>gR��x�?gI�墌�k?g� ��b��?c C sL |d }d}x2t dt|��D ] }|}|}| | | || }q W d|| S )Nr g r4 g �?)r� rl )r� r� Zb0Zb1r� Zb2rs rs rt �_chbevl= s r� c C s t | �t| d d t� S )Ng @ri )r( r� �_i0A)r� rs rs rt �_i0_1I s r� c C s" t | �td| d t� t| � S )Ng @@g @)r( r� �_i0Br% )r� rs rs rt �_i0_2M s r� c C s` t | ��� } t| �}| dk }| | | |< | dk}t| | �||< | }t| | �||< |�� S )a0 Modified Bessel function of the first kind, order 0. Usually denoted :math:`I_0`. This function does broadcast, but will *not* "up-cast" int dtype arguments unless accompanied by at least one float or complex dtype argument (see Raises below). Parameters ---------- x : array_like, dtype float or complex Argument of the Bessel function. Returns ------- out : ndarray, shape = x.shape, dtype = x.dtype The modified Bessel function evaluated at each of the elements of `x`. Raises ------ TypeError: array cannot be safely cast to required type If argument consists exclusively of int dtypes. See Also -------- scipy.special.iv, scipy.special.ive Notes ----- We use the algorithm published by Clenshaw [1]_ and referenced by Abramowitz and Stegun [2]_, for which the function domain is partitioned into the two intervals [0,8] and (8,inf), and Chebyshev polynomial expansions are employed in each interval. Relative error on the domain [0,30] using IEEE arithmetic is documented [3]_ as having a peak of 5.8e-16 with an rms of 1.4e-16 (n = 30000). References ---------- .. [1] C. W. Clenshaw, "Chebyshev series for mathematical functions", in *National Physical Laboratory Mathematical Tables*, vol. 5, London: Her Majesty's Stationery Office, 1962. .. [2] M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions*, 10th printing, New York: Dover, 1964, pp. 379. http://www.math.sfu.ca/~cbm/aands/page_379.htm .. [3] http://kobesearch.cpan.org/htdocs/Math-Cephes/Math/Cephes.html Examples -------- >>> np.i0([0.]) array(1.0) >>> np.i0([0., 1. + 2j]) array([ 1.00000000+0.j , 0.18785373+0.64616944j]) r g @)r rD r r� r� r� )r� r} �indZind2rs rs rt ra Q s 6 c C sb ddl m} | dkr t�dg�S td| �}| d d }||td|| | d � �|t|�� S )a� Return the Kaiser window. The Kaiser window is a taper formed by using a Bessel function. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. beta : float Shape parameter for window. Returns ------- out : array The window, with the maximum value normalized to one (the value one appears only if the number of samples is odd). See Also -------- bartlett, blackman, hamming, hanning Notes ----- The Kaiser window is defined as .. math:: w(n) = I_0\left( \beta \sqrt{1-\frac{4n^2}{(M-1)^2}} \right)/I_0(\beta) with .. math:: \quad -\frac{M-1}{2} \leq n \leq \frac{M-1}{2}, where :math:`I_0` is the modified zeroth-order Bessel function. The Kaiser was named for Jim Kaiser, who discovered a simple approximation to the DPSS window based on Bessel functions. The Kaiser window is a very good approximation to the Digital Prolate Spheroidal Sequence, or Slepian window, which is the transform which maximizes the energy in the main lobe of the window relative to total energy. The Kaiser can approximate many other windows by varying the beta parameter. ==== ======================= beta Window shape ==== ======================= 0 Rectangular 5 Similar to a Hamming 6 Similar to a Hanning 8.6 Similar to a Blackman ==== ======================= A beta value of 14 is probably a good starting point. Note that as beta gets large, the window narrows, and so the number of samples needs to be large enough to sample the increasingly narrow spike, otherwise NaNs will get returned. Most references to the Kaiser window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. References ---------- .. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285. John Wiley and Sons, New York, (1966). .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 177-178. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function Examples -------- >>> np.kaiser(12, 14) array([ 7.72686684e-06, 3.46009194e-03, 4.65200189e-02, 2.29737120e-01, 5.99885316e-01, 9.45674898e-01, 9.45674898e-01, 5.99885316e-01, 2.29737120e-01, 4.65200189e-02, 3.46009194e-03, 7.72686684e-06]) Plot the window and the frequency response: >>> from numpy.fft import fft, fftshift >>> window = np.kaiser(51, 14) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Kaiser window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show() >>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Kaiser window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show() r )ra r4 g �?g @)Z numpy.dualra r r r r% r� )r� �betara r� �alphars rs rt r_ � s w c C s* t �| �} tt| dkd| � }t|�| S )a� Return the sinc function. The sinc function is :math:`\sin(\pi x)/(\pi x)`. Parameters ---------- x : ndarray Array (possibly multi-dimensional) of values for which to to calculate ``sinc(x)``. Returns ------- out : ndarray ``sinc(x)``, which has the same shape as the input. Notes ----- ``sinc(0)`` is the limit value 1. The name sinc is short for "sine cardinal" or "sinus cardinalis". The sinc function is used in various signal processing applications, including in anti-aliasing, in the construction of a Lanczos resampling filter, and in interpolation. For bandlimited interpolation of discrete-time signals, the ideal interpolation kernel is proportional to the sinc function. References ---------- .. [1] Weisstein, Eric W. "Sinc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SincFunction.html .. [2] Wikipedia, "Sinc function", http://en.wikipedia.org/wiki/Sinc_function Examples -------- >>> x = np.linspace(-4, 4, 41) >>> np.sinc(x) array([ -3.89804309e-17, -4.92362781e-02, -8.40918587e-02, -8.90384387e-02, -5.84680802e-02, 3.89804309e-17, 6.68206631e-02, 1.16434881e-01, 1.26137788e-01, 8.50444803e-02, -3.89804309e-17, -1.03943254e-01, -1.89206682e-01, -2.16236208e-01, -1.55914881e-01, 3.89804309e-17, 2.33872321e-01, 5.04551152e-01, 7.56826729e-01, 9.35489284e-01, 1.00000000e+00, 9.35489284e-01, 7.56826729e-01, 5.04551152e-01, 2.33872321e-01, 3.89804309e-17, -1.55914881e-01, -2.16236208e-01, -1.89206682e-01, -1.03943254e-01, -3.89804309e-17, 8.50444803e-02, 1.26137788e-01, 1.16434881e-01, 6.68206631e-02, 3.89804309e-17, -5.84680802e-02, -8.90384387e-02, -8.40918587e-02, -4.92362781e-02, -3.89804309e-17]) >>> plt.plot(x, np.sinc(x)) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Sinc Function") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("X") <matplotlib.text.Text object at 0x...> >>> plt.show() It works in 2-D as well: >>> x = np.linspace(-4, 4, 401) >>> xx = np.outer(x, x) >>> plt.imshow(np.sinc(xx)) <matplotlib.image.AxesImage object at 0x...> r g#B� ���;)r r r r r& )r� r} rs rs rt rZ s J c C s t | ddd�}|�d� |S )ak Return a copy of an array sorted along the first axis. Parameters ---------- a : array_like Array to be sorted. Returns ------- sorted_array : ndarray Array of the same type and shape as `a`. See Also -------- sort Notes ----- ``np.msort(a)`` is equivalent to ``np.sort(a, axis=0)``. T)rm rD r )r r, )r� r� rs rs rt rX b s c K s� t �| �} |�dd�}|dk r�t| j�}| j}t�||�}x|D ] }d||< q@W t|�dkrj|d |d< q�t t |��t |� }t|�}x$t t |��D ]\} } | � | | �} q�W | �| jd|� d �} d|d< n dg| j }|| f|�} | |fS )a0 Internal Function. Call `func` with `a` as first argument swapping the axes to use extended axis on functions that don't support it natively. Returns result and a.shape with axis dims set to 1. Parameters ---------- a : array_like Input array or object that can be converted to an array. func : callable Reduction function capable of receiving a single axis argument. It is is called with `a` as first argument followed by `kwargs`. kwargs : keyword arguments additional keyword arguments to pass to `func`. Returns ------- result : tuple Result of func(a, **kwargs) and a.shape with axis dims set to 1 which can be used to reshape the result to the same shape a ufunc with keepdims=True would produce. ry Nr4 r )ru ru )r r �getr� r� rn r r rl r) r� r r( r� r� ) r� r` r ry Zkeepdimr Zaxr� Znkeepr� r� �rrs rs rt �_ureduce~ s& r� c C s, t | t|||d�\}}|r$|�|�S |S dS )a Compute the median along the specified axis. Returns the median of the array elements. Parameters ---------- a : array_like Input array or object that can be converted to an array. axis : {int, sequence of int, None}, optional Axis or axes along which the medians are computed. The default is to compute the median along a flattened version of the array. A sequence of axes is supported since version 1.9.0. out : ndarray, optional Alternative output array in which to place the result. It must have the same shape and buffer length as the expected output, but the type (of the output) will be cast if necessary. overwrite_input : bool, optional If True, then allow use of memory of input array `a` for calculations. The input array will be modified by the call to `median`. This will save memory when you do not need to preserve the contents of the input array. Treat the input as undefined, but it will probably be fully or partially sorted. Default is False. If `overwrite_input` is ``True`` and `a` is not already an `ndarray`, an error will be raised. keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original `arr`. .. versionadded:: 1.9.0 Returns ------- median : ndarray A new array holding the result. If the input contains integers or floats smaller than ``float64``, then the output data-type is ``np.float64``. Otherwise, the data-type of the output is the same as that of the input. If `out` is specified, that array is returned instead. See Also -------- mean, percentile Notes ----- Given a vector ``V`` of length ``N``, the median of ``V`` is the middle value of a sorted copy of ``V``, ``V_sorted`` - i e., ``V_sorted[(N-1)/2]``, when ``N`` is odd, and the average of the two middle values of ``V_sorted`` when ``N`` is even. Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.median(a) 3.5 >>> np.median(a, axis=0) array([ 6.5, 4.5, 2.5]) >>> np.median(a, axis=1) array([ 7., 2.]) >>> m = np.median(a, axis=0) >>> out = np.zeros_like(m) >>> np.median(a, axis=0, out=m) array([ 6.5, 4.5, 2.5]) >>> m array([ 6.5, 4.5, 2.5]) >>> b = a.copy() >>> np.median(b, axis=1, overwrite_input=True) array([ 7., 2.]) >>> assert not np.all(a==b) >>> b = a.copy() >>> np.median(b, axis=None, overwrite_input=True) 3.5 >>> assert not np.all(a==b) )r` ry r �overwrite_inputN)r� �_medianr� )r� ry r r� �keepdimsr� rq rs rs rt rY � s Q c C s� t �| �} |d kr| j}n | j| }|d dkrF|d }|d |g}n|d d g}t �| jt j�rn|�d� |r�|d kr�| �� }|� |� q�| j ||d� | }nt | ||d�}|jdkr�|� � S |d kr�d}t d �g|j }|j| d } |j| d dk�rt | | d �||< nt | d | d �||< t �| jt j��rl|dk�rlt || ||d�} t jj�|| ||�S t || ||d�S d S )Nri r r4 ru )ry rs )ry r )r r r� r� r� r� r� rf r* r- r� rw rn r. rF �utilsZ_median_nancheck) r� ry r r� �szZszhZkth�partrz rr Zroutrs rs rt r� s< r� �linearc C s` t |tjdd�}t| t|||||d�\}}|rX|jdkrB|�|�S |�t|�g| �S n|S dS )a� Compute the qth percentile of the data along the specified axis. Returns the qth percentile(s) of the array elements. Parameters ---------- a : array_like Input array or object that can be converted to an array. q : float in range of [0,100] (or sequence of floats) Percentile to compute, which must be between 0 and 100 inclusive. axis : {int, sequence of int, None}, optional Axis or axes along which the percentiles are computed. The default is to compute the percentile(s) along a flattened version of the array. A sequence of axes is supported since version 1.9.0. out : ndarray, optional Alternative output array in which to place the result. It must have the same shape and buffer length as the expected output, but the type (of the output) will be cast if necessary. overwrite_input : bool, optional If True, then allow use of memory of input array `a` calculations. The input array will be modified by the call to `percentile`. This will save memory when you do not need to preserve the contents of the input array. In this case you should not make any assumptions about the contents of the input `a` after this function completes -- treat it as undefined. Default is False. If `a` is not already an array, this parameter will have no effect as `a` will be converted to an array internally regardless of the value of this parameter. interpolation : {'linear', 'lower', 'higher', 'midpoint', 'nearest'} This optional parameter specifies the interpolation method to use when the desired quantile lies between two data points ``i < j``: * linear: ``i + (j - i) * fraction``, where ``fraction`` is the fractional part of the index surrounded by ``i`` and ``j``. * lower: ``i``. * higher: ``j``. * nearest: ``i`` or ``j``, whichever is nearest. * midpoint: ``(i + j) / 2``. .. versionadded:: 1.9.0 keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array `a`. .. versionadded:: 1.9.0 Returns ------- percentile : scalar or ndarray If `q` is a single percentile and `axis=None`, then the result is a scalar. If multiple percentiles are given, first axis of the result corresponds to the percentiles. The other axes are the axes that remain after the reduction of `a`. If the input contains integers or floats smaller than ``float64``, the output data-type is ``float64``. Otherwise, the output data-type is the same as that of the input. If `out` is specified, that array is returned instead. See Also -------- mean, median, nanpercentile Notes ----- Given a vector ``V`` of length ``N``, the ``q``-th percentile of ``V`` is the value ``q/100`` of the way from the minimum to the maximum in a sorted copy of ``V``. The values and distances of the two nearest neighbors as well as the `interpolation` parameter will determine the percentile if the normalized ranking does not match the location of ``q`` exactly. This function is the same as the median if ``q=50``, the same as the minimum if ``q=0`` and the same as the maximum if ``q=100``. Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.percentile(a, 50) 3.5 >>> np.percentile(a, 50, axis=0) array([[ 6.5, 4.5, 2.5]]) >>> np.percentile(a, 50, axis=1) array([ 7., 2.]) >>> np.percentile(a, 50, axis=1, keepdims=True) array([[ 7.], [ 2.]]) >>> m = np.percentile(a, 50, axis=0) >>> out = np.zeros_like(m) >>> np.percentile(a, 50, axis=0, out=out) array([[ 6.5, 4.5, 2.5]]) >>> m array([[ 6.5, 4.5, 2.5]]) >>> b = a.copy() >>> np.percentile(b, 50, axis=1, overwrite_input=True) array([ 7., 2.]) >>> assert not np.all(a == b) T)r� rD )r` �qry r r� � interpolationr N)r r r r� � _percentilern r� rl ) r� r� ry r r� r� r� r� rq rs rs rt rF C s l c C sd t | �} |jdkr d}|d }nd}|jdk rtxpt|j�D ]4}|| dk sV|| dkr^td��|| d < q:W n,t�|dk �s�t�|dk�r�td��|d }|r�|d kr�| �� } q�| } n|d kr�| �� } n| � � } |d kr�d}| j | } || d } |d k�rt | �� t �} nh|d k�r,t| �� t �} nN|d k�rLd t | �t| � } n.|d k�rft| �� t �} n|dk�rrntd��tjdtd�} | jt k�r*t�| jtj��r�t| dgf�} | j| |d� t�| |d�} d}t�| jtj��r| d d� } t�| dd �df �} |�r| d } t| | ||d�} �ntt | �� t �}|d }| d ||| d k< t�| jtj��rtt|dgf�}| | }d| }dg| j }t| �||< ||_ ||_ | jt||f�|d� t�| |d�} t�||d�}t�||d�}d}t�| jtj��r |d d� }t�| dd �df �} t| ||d�| }t| ||d�| }t�||d�}t�||d�}|�rz|�d�}|�d�}|d k �r�t |||d�} n t ||�} t� | ��r`t j dt dd� |�r | jdk�r�|d k �r�| j�!tj"�|d< |} n| j�!tj"�} n | j�!tj"�| d| �d�f< nD| jdk�r@| j�!tj"�| d d �< n | j�!tj"�| d| �#|jd�f< | S )Nr TFr� g g Y@z(Percentiles must be in the range [0,100]r4 �lowerZhigherZmidpointg �?Znearestr� zNinterpolation can only be 'linear', 'lower' 'higher', 'midpoint', or 'nearest')r� ru )ry .)ry r g �?)r z'Invalid value encountered in percentiler@ )r� )$r rn r� r� rm r Z count_nonzeror* r' rD r� r r� r r r r r� r� r� r� r r- �rollaxis�isnanr rl r� r r/ r� r� rx r� �nan�repeat)r� r� ry r r� r� r� r� r� ZapZNxr� r� r� Z indices_belowZ indices_aboveZ weights_aboveZ weights_belowZ weights_shape�x1Zx2rs rs rt r� � s� r� � �?c C s t | �} |dkr|}nLt |�}|jdkrVt|�}dg| j }|jd ||< |�|�}n t||d�}| j}td�g| }td�g| }tdd�||< tdd�||< y"|| | | | d �|�} W nJ tk �r t� |�}t� | �} t � || | | | d |�} Y nX | S )af Integrate along the given axis using the composite trapezoidal rule. Integrate `y` (`x`) along given axis. Parameters ---------- y : array_like Input array to integrate. x : array_like, optional The sample points corresponding to the `y` values. If `x` is None, the sample points are assumed to be evenly spaced `dx` apart. The default is None. dx : scalar, optional The spacing between sample points when `x` is None. The default is 1. axis : int, optional The axis along which to integrate. Returns ------- trapz : float Definite integral as approximated by trapezoidal rule. See Also -------- sum, cumsum Notes ----- Image [2]_ illustrates trapezoidal rule -- y-axis locations of points will be taken from `y` array, by default x-axis distances between points will be 1.0, alternatively they can be provided with `x` array or with `dx` scalar. Return value will be equal to combined area under the red lines. References ---------- .. [1] Wikipedia page: http://en.wikipedia.org/wiki/Trapezoidal_rule .. [2] Illustration image: http://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png Examples -------- >>> np.trapz([1,2,3]) 4.0 >>> np.trapz([1,2,3], x=[4,6,8]) 8.0 >>> np.trapz([1,2,3], dx=2) 8.0 >>> a = np.arange(6).reshape(2, 3) >>> a array([[0, 1, 2], [3, 4, 5]]) >>> np.trapz(a, axis=0) array([ 1.5, 2.5, 3.5]) >>> np.trapz(a, axis=1) array([ 2., 8.]) Nr4 r )ry ru g @) r rn rG r� r� rw r0 rm r r r r� ) r} r� r ry r� r� r r r �retrs rs rt r` L s, > " &c C s� y�t t| t� i |g�|�}t|t�r4t||�� � n\t|t�r\tt ||d �|d �� � n4t|t�r�x(|D ] }tt ||d �|d �� � qlW W n Y nX dS )a� Adds documentation to obj which is in module place. If doc is a string add it to obj as a docstring If doc is a tuple, then the first element is interpreted as an attribute of obj and the second as the docstring (method, docstring) If doc is a list, then each element of the list should be a sequence of length two --> [(method1, docstring1), (method2, docstring2), ...] This routine never raises an error. This routine cannot modify read-only docstrings, as appear in new-style classes or built-in functions. Because this routine never raises an error the caller must check manually that the docstrings were changed. r r4 N) �getattr� __import__�globalsr� rR r7 �striprk r� )rP �objrW �new�valrs rs rt rb � s $c s� t | �}|�dd�}|�dd�}|�dd�}|rFtdt|�d f ��|d krVtd ��d | � � fd d �t| �D �}|dkr�|dkr�d� dd� |d _d� dd� |d _|s�tj|ddi�}|r�dd � |D �}|S )a� Return coordinate matrices from coordinate vectors. Make N-D coordinate arrays for vectorized evaluations of N-D scalar/vector fields over N-D grids, given one-dimensional coordinate arrays x1, x2,..., xn. .. versionchanged:: 1.9 1-D and 0-D cases are allowed. Parameters ---------- x1, x2,..., xn : array_like 1-D arrays representing the coordinates of a grid. indexing : {'xy', 'ij'}, optional Cartesian ('xy', default) or matrix ('ij') indexing of output. See Notes for more details. .. versionadded:: 1.7.0 sparse : bool, optional If True a sparse grid is returned in order to conserve memory. Default is False. .. versionadded:: 1.7.0 copy : bool, optional If False, a view into the original arrays are returned in order to conserve memory. Default is True. Please note that ``sparse=False, copy=False`` will likely return non-contiguous arrays. Furthermore, more than one element of a broadcast array may refer to a single memory location. If you need to write to the arrays, make copies first. .. versionadded:: 1.7.0 Returns ------- X1, X2,..., XN : ndarray For vectors `x1`, `x2`,..., 'xn' with lengths ``Ni=len(xi)`` , return ``(N1, N2, N3,...Nn)`` shaped arrays if indexing='ij' or ``(N2, N1, N3,...Nn)`` shaped arrays if indexing='xy' with the elements of `xi` repeated to fill the matrix along the first dimension for `x1`, the second for `x2` and so on. Notes ----- This function supports both indexing conventions through the indexing keyword argument. Giving the string 'ij' returns a meshgrid with matrix indexing, while 'xy' returns a meshgrid with Cartesian indexing. In the 2-D case with inputs of length M and N, the outputs are of shape (N, M) for 'xy' indexing and (M, N) for 'ij' indexing. In the 3-D case with inputs of length M, N and P, outputs are of shape (N, M, P) for 'xy' indexing and (M, N, P) for 'ij' indexing. The difference is illustrated by the following code snippet:: xv, yv = np.meshgrid(x, y, sparse=False, indexing='ij') for i in range(nx): for j in range(ny): # treat xv[i,j], yv[i,j] xv, yv = np.meshgrid(x, y, sparse=False, indexing='xy') for i in range(nx): for j in range(ny): # treat xv[j,i], yv[j,i] In the 1-D and 0-D case, the indexing and sparse keywords have no effect. See Also -------- index_tricks.mgrid : Construct a multi-dimensional "meshgrid" using indexing notation. index_tricks.ogrid : Construct an open multi-dimensional "meshgrid" using indexing notation. Examples -------- >>> nx, ny = (3, 2) >>> x = np.linspace(0, 1, nx) >>> y = np.linspace(0, 1, ny) >>> xv, yv = np.meshgrid(x, y) >>> xv array([[ 0. , 0.5, 1. ], [ 0. , 0.5, 1. ]]) >>> yv array([[ 0., 0., 0.], [ 1., 1., 1.]]) >>> xv, yv = np.meshgrid(x, y, sparse=True) # make sparse output arrays >>> xv array([[ 0. , 0.5, 1. ]]) >>> yv array([[ 0.], [ 1.]]) `meshgrid` is very useful to evaluate functions on a grid. >>> x = np.arange(-5, 5, 0.1) >>> y = np.arange(-5, 5, 0.1) >>> xx, yy = np.meshgrid(x, y, sparse=True) >>> z = np.sin(xx**2 + yy**2) / (xx**2 + yy**2) >>> h = plt.contourf(x,y,z) rD T�sparseF�indexingr� z2meshgrid() got an unexpected keyword argument '%s'r )r� Zijz.Valid values for `indexing` are 'xy' and 'ij'.)r4 c s< g | ]4\}}t �|��� d |� d � |d d � ��qS )N)ru r4 )r r r� )r� r� r� )�s0rs rt r� ? s z meshgrid.<locals>.<listcomp>r4 )r4 ru ri N)ru r4 rm c S s g | ] }|� � �qS rs )rD )r� r� rs rs rt r� L s ) rl r r| r� rm r r� r r� )�xir rn Zcopy_r� r� rt rs )r� rt rc � s* f c C sD d}t | �tk r0y | j}W n tk r. Y nX t| �} | j}| jjrJdnd}|dkrp|dkrf| �� } | j}d}|dkr�t j dt dd � |r�|| �S | j |d �S t ||�}td�g| }| j| }t| j�}t|t��r`|�|�\} } } t| | | �} t| �} | dk�r(|�r || j |d ��S | j |d �S | dk �rL| } | d } | d d } || | 8 < t|| j|�}| dk�rvntd| �||< | | ||< | |k�r�n:t| | d�||< td�g| }t| d�||< | | ||< | dk�r�nlt| | td �}d |d| | | �< t| | | �||< td�g| }t| | �||< | | } |||< | | ||< |�r\||�S |S |}t�|�}|jtk�r�t j d tdd � |�t �}t|t t t f��r\|� � }|| k �s�||k�r�t!d|||f ��|dk �r�||7 }|| d8 < t|| j|�}td|�||< | | ||< t|d�||< td�g| }t|d d�||< | | ||< n�|j"dk�r�t|tj��s�|�t �}t�#|t d��s�t j dt dd � |�t �}t|td �}||k || k@ }|�$� �s�t j dt dd � || }|dk}|�$� �st j dtdd � || }d ||f< |||< | | }|�r<||�S |S dS )a Return a new array with sub-arrays along an axis deleted. For a one dimensional array, this returns those entries not returned by `arr[obj]`. Parameters ---------- arr : array_like Input array. obj : slice, int or array of ints Indicate which sub-arrays to remove. axis : int, optional The axis along which to delete the subarray defined by `obj`. If `axis` is None, `obj` is applied to the flattened array. Returns ------- out : ndarray A copy of `arr` with the elements specified by `obj` removed. Note that `delete` does not occur in-place. If `axis` is None, `out` is a flattened array. See Also -------- insert : Insert elements into an array. append : Append elements at the end of an array. Notes ----- Often it is preferable to use a boolean mask. For example: >>> mask = np.ones(len(arr), dtype=bool) >>> mask[[0,2,4]] = False >>> result = arr[mask,...] Is equivalent to `np.delete(arr, [0,2,4], axis=0)`, but allows further use of `mask`. Examples -------- >>> arr = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]]) >>> arr array([[ 1, 2, 3, 4], [ 5, 6, 7, 8], [ 9, 10, 11, 12]]) >>> np.delete(arr, 1, 0) array([[ 1, 2, 3, 4], [ 9, 10, 11, 12]]) >>> np.delete(arr, np.s_[::2], 1) array([[ 2, 4], [ 6, 8], [10, 12]]) >>> np.delete(arr, [1,3,5], None) array([ 1, 3, 5, 7, 8, 9, 10, 11, 12]) Nr� �Cr4 ru r z\in the future the special handling of scalars will be removed from delete and raise an errorri )r� )r� )r� Fzpin the future insert will treat boolean arrays and array-likes as boolean index instead of casting it to integerz2index %i is out of bounds for axis %i with size %i� same_kindzPusing a non-integer array as obj in delete will result in an error in the futurezcin the future out of bounds indices will raise an error instead of being ignored by `numpy.delete`.zEin the future negative indices will not be ignored by `numpy.delete`.)%r� r �__array_wrap__r� r rn �flags�fncr* r� r� r� rD r: rw r� r� r� r� r� rl r r� r r� r � FutureWarningr� r r� r> r r� rx r� r� r� )r/ r� ry �wraprn �arrorder�slobjr� �newshape�start�stop�stepZxrZnumtodelr� �slobj2r� �_objZ inside_boundsZpositive_indicesrs rs rt rd Q s� : c C sx d}t | �tk r0y | j}W n tk r. Y nX t| �} | j}| jjrJdnd}|dkrv|dkrf| �� } | j}|d }nH|dkr�t j dt dd� | j |d �} || d < |r�|| �S | S n t ||�}td�g| }| j| }t| j�} t|t��rt|�|�d ti�} nFt�|�} | jtk�r4t j d tdd� | �t�} n| jdk�rHtd ��| j dk�rv| � � } | | k �sr| |k�r�t d|||f ��| dk �r�| |7 } t|d| j| jd�}| jdk�r�t� |d||j d �}|j| } | | | 7 < t | | j|�} td| �||< | | | |< t| | | �||< || |< t| | d�||< td�g| }t| d�||< | | | |< |�rr|| �S | S | j dk�r�t|tj��s�| �t�} t�!| td��s�t j dt dd� | �t�} | | dk |7 < t"| �} | j#dd�}| | t�| �7 < | | | 7 < t$| | td�}d|| < t | | j|�} td�g| }| ||< |||< || |< | | |< |�rt|| �S | S )a Insert values along the given axis before the given indices. Parameters ---------- arr : array_like Input array. obj : int, slice or sequence of ints Object that defines the index or indices before which `values` is inserted. .. versionadded:: 1.8.0 Support for multiple insertions when `obj` is a single scalar or a sequence with one element (similar to calling insert multiple times). values : array_like Values to insert into `arr`. If the type of `values` is different from that of `arr`, `values` is converted to the type of `arr`. `values` should be shaped so that ``arr[...,obj,...] = values`` is legal. axis : int, optional Axis along which to insert `values`. If `axis` is None then `arr` is flattened first. Returns ------- out : ndarray A copy of `arr` with `values` inserted. Note that `insert` does not occur in-place: a new array is returned. If `axis` is None, `out` is a flattened array. See Also -------- append : Append elements at the end of an array. concatenate : Join a sequence of arrays along an existing axis. delete : Delete elements from an array. Notes ----- Note that for higher dimensional inserts `obj=0` behaves very different from `obj=[0]` just like `arr[:,0,:] = values` is different from `arr[:,[0],:] = values`. Examples -------- >>> a = np.array([[1, 1], [2, 2], [3, 3]]) >>> a array([[1, 1], [2, 2], [3, 3]]) >>> np.insert(a, 1, 5) array([1, 5, 1, 2, 2, 3, 3]) >>> np.insert(a, 1, 5, axis=1) array([[1, 5, 1], [2, 5, 2], [3, 5, 3]]) Difference between sequence and scalars: >>> np.insert(a, [1], [[1],[2],[3]], axis=1) array([[1, 1, 1], [2, 2, 2], [3, 3, 3]]) >>> np.array_equal(np.insert(a, 1, [1, 2, 3], axis=1), ... np.insert(a, [1], [[1],[2],[3]], axis=1)) True >>> b = a.flatten() >>> b array([1, 1, 2, 2, 3, 3]) >>> np.insert(b, [2, 2], [5, 6]) array([1, 1, 5, 6, 2, 2, 3, 3]) >>> np.insert(b, slice(2, 4), [5, 6]) array([1, 1, 5, 2, 6, 2, 3, 3]) >>> np.insert(b, [2, 2], [7.13, False]) # type casting array([1, 1, 7, 0, 2, 2, 3, 3]) >>> x = np.arange(8).reshape(2, 4) >>> idx = (1, 3) >>> np.insert(x, idx, 999, axis=1) array([[ 0, 999, 1, 2, 999, 3], [ 4, 999, 5, 6, 999, 7]]) Nr� r� r4 r z\in the future the special handling of scalars will be removed from insert and raise an errorri )r� )r� .r� zrin the future insert will treat boolean arrays and array-likes as a boolean index instead of casting it to integerzDindex array argument obj to insert must be one dimensional or scalarz2index %i is out of bounds for axis %i with size %iF)rD rw r� r� zPusing a non-integer array as obj in insert will result in an error in the futureZ mergesort)r� )r� )%r� r r� r� r rn r� r� r* r� r� r� rD r: rw r� r� r� r r� r r r r� r� r� r� rm r� r� rx r� r r� rl r� r )r/ r� �valuesry r� rn r� r� r� r� r� rr Znumnewr� r� r� Zold_maskrs rs rt re s� X c C sD t | �} |dkr4| jdkr"| �� } t|�}| jd }t| |f|d�S )a� Append values to the end of an array. Parameters ---------- arr : array_like Values are appended to a copy of this array. values : array_like These values are appended to a copy of `arr`. It must be of the correct shape (the same shape as `arr`, excluding `axis`). If `axis` is not specified, `values` can be any shape and will be flattened before use. axis : int, optional The axis along which `values` are appended. If `axis` is not given, both `arr` and `values` are flattened before use. Returns ------- append : ndarray A copy of `arr` with `values` appended to `axis`. Note that `append` does not occur in-place: a new array is allocated and filled. If `axis` is None, `out` is a flattened array. See Also -------- insert : Insert elements into an array. delete : Delete elements from an array. Examples -------- >>> np.append([1, 2, 3], [[4, 5, 6], [7, 8, 9]]) array([1, 2, 3, 4, 5, 6, 7, 8, 9]) When `axis` is specified, `values` must have the correct shape. >>> np.append([[1, 2, 3], [4, 5, 6]], [[7, 8, 9]], axis=0) array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> np.append([[1, 2, 3], [4, 5, 6]], [7, 8, 9], axis=0) Traceback (most recent call last): ... ValueError: arrays must have same number of dimensions Nr4 )ry )r rn r* r )r/ r� ry rs rs rt rf � s . )r4 rh )r� NFNN)r� NFN)NNF)NN)r )r� )r4 ru )NNN)r )r" )NT)NTFNNN)NNFF)NNF)NNFr� F)NNFr� F)Nr� ru )N)N)N)�� __future__r r r r� �operatorr6 r2 r� �numpyr Znumpy.core.numericr� �numericr Z numpy.corer r r r r r r r r r r r r r r r r r r r r r r r r Znumpy.core.umathr r r r! r" r# r$ r% r& r' r( r) Znumpy.core.fromnumericr* r+ r, r- r. r/ r0 Znumpy.core.numerictypesr1 r2 Znumpy.lib.twodim_baser3 r� r5 Znumpy.core.multiarrayr6 r7 r8 r9 r: r; r r< r r= rg � numpy.compatr> Znumpy.compat.py3kr? � version_info�xranger� � __builtin__rh �__all__rN rM rE r� r� r� r� r� r� r� r� rT rU rS rR rB rA rD rH rG rI rJ rK rC r- rO rP rL r8 ro Z_CORE_DIMENSION_LISTr: Z_ARGUMENT_LISTr> rA rE rI rK rL rn rQ rV r rW r^ r] r\ r[ r� r� r� r� r� ra r_ rZ rX r� rY r� rF r� r` rb rc rd re rf rs rs rs rt �<module> sZ \8$ $ WH " F ~ C g 1 \ V .7$ 3 3+ + " k QSbjdd CO 6 X 7 x [# N O