관리-도구

편집 파일: function_base.py

import collections.abc import functools import re import sys import warnings from .._utils import set_module import numpy as np import numpy.core.numeric as _nx from numpy.core import transpose from numpy.core.numeric import ( ones, zeros_like, arange, concatenate, array, asarray, asanyarray, empty, ndarray, take, dot, where, intp, integer, isscalar, absolute ) from numpy.core.umath import ( pi, add, arctan2, frompyfunc, cos, less_equal, sqrt, sin, mod, exp, not_equal, subtract ) from numpy.core.fromnumeric import ( ravel, nonzero, partition, mean, any, sum ) from numpy.core.numerictypes import typecodes from numpy.core import overrides from numpy.core.function_base import add_newdoc from numpy.lib.twodim_base import diag from numpy.core.multiarray import ( _place, add_docstring, bincount, normalize_axis_index, _monotonicity, interp as compiled_interp, interp_complex as compiled_interp_complex ) from numpy.core.umath import _add_newdoc_ufunc as add_newdoc_ufunc import builtins # needed in this module for compatibility from numpy.lib.histograms import histogram, histogramdd # noqa: F401 array_function_dispatch = functools.partial( overrides.array_function_dispatch, module='numpy') __all__ = [ 'select', 'piecewise', 'trim_zeros', 'copy', 'iterable', 'percentile', 'diff', 'gradient', 'angle', 'unwrap', 'sort_complex', 'disp', 'flip', 'rot90', 'extract', 'place', 'vectorize', 'asarray_chkfinite', 'average', 'bincount', 'digitize', 'cov', 'corrcoef', 'msort', 'median', 'sinc', 'hamming', 'hanning', 'bartlett', 'blackman', 'kaiser', 'trapz', 'i0', 'add_newdoc', 'add_docstring', 'meshgrid', 'delete', 'insert', 'append', 'interp', 'add_newdoc_ufunc', 'quantile' ] # _QuantileMethods is a dictionary listing all the supported methods to # compute quantile/percentile. # # Below virtual_index refer to the index of the element where the percentile # would be found in the sorted sample. # When the sample contains exactly the percentile wanted, the virtual_index is # an integer to the index of this element. # When the percentile wanted is in between two elements, the virtual_index # is made of a integer part (a.k.a 'i' or 'left') and a fractional part # (a.k.a 'g' or 'gamma') # # Each method in _QuantileMethods has two properties # get_virtual_index : Callable # The function used to compute the virtual_index. # fix_gamma : Callable # A function used for discret methods to force the index to a specific value. _QuantileMethods = dict( # --- HYNDMAN and FAN METHODS # Discrete methods inverted_cdf=dict( get_virtual_index=lambda n, quantiles: _inverted_cdf(n, quantiles), fix_gamma=lambda gamma, _: gamma, # should never be called ), averaged_inverted_cdf=dict( get_virtual_index=lambda n, quantiles: (n * quantiles) - 1, fix_gamma=lambda gamma, _: _get_gamma_mask( shape=gamma.shape, default_value=1., conditioned_value=0.5, where=gamma == 0), ), closest_observation=dict( get_virtual_index=lambda n, quantiles: _closest_observation(n, quantiles), fix_gamma=lambda gamma, _: gamma, # should never be called ), # Continuous methods interpolated_inverted_cdf=dict( get_virtual_index=lambda n, quantiles: _compute_virtual_index(n, quantiles, 0, 1), fix_gamma=lambda gamma, _: gamma, ), hazen=dict( get_virtual_index=lambda n, quantiles: _compute_virtual_index(n, quantiles, 0.5, 0.5), fix_gamma=lambda gamma, _: gamma, ), weibull=dict( get_virtual_index=lambda n, quantiles: _compute_virtual_index(n, quantiles, 0, 0), fix_gamma=lambda gamma, _: gamma, ), # Default method. # To avoid some rounding issues, `(n-1) * quantiles` is preferred to # `_compute_virtual_index(n, quantiles, 1, 1)`. # They are mathematically equivalent. linear=dict( get_virtual_index=lambda n, quantiles: (n - 1) * quantiles, fix_gamma=lambda gamma, _: gamma, ), median_unbiased=dict( get_virtual_index=lambda n, quantiles: _compute_virtual_index(n, quantiles, 1 / 3.0, 1 / 3.0), fix_gamma=lambda gamma, _: gamma, ), normal_unbiased=dict( get_virtual_index=lambda n, quantiles: _compute_virtual_index(n, quantiles, 3 / 8.0, 3 / 8.0), fix_gamma=lambda gamma, _: gamma, ), # --- OTHER METHODS lower=dict( get_virtual_index=lambda n, quantiles: np.floor( (n - 1) * quantiles).astype(np.intp), fix_gamma=lambda gamma, _: gamma, # should never be called, index dtype is int ), higher=dict( get_virtual_index=lambda n, quantiles: np.ceil( (n - 1) * quantiles).astype(np.intp), fix_gamma=lambda gamma, _: gamma, # should never be called, index dtype is int ), midpoint=dict( get_virtual_index=lambda n, quantiles: 0.5 * ( np.floor((n - 1) * quantiles) + np.ceil((n - 1) * quantiles)), fix_gamma=lambda gamma, index: _get_gamma_mask( shape=gamma.shape, default_value=0.5, conditioned_value=0., where=index % 1 == 0), ), nearest=dict( get_virtual_index=lambda n, quantiles: np.around( (n - 1) * quantiles).astype(np.intp), fix_gamma=lambda gamma, _: gamma, # should never be called, index dtype is int )) def _rot90_dispatcher(m, k=None, axes=None): return (m,) @array_function_dispatch(_rot90_dispatcher) def rot90(m, k=1, axes=(0, 1)): """ Rotate an array by 90 degrees in the plane specified by axes. Rotation direction is from the first towards the second axis. This means for a 2D array with the default `k` and `axes`, the rotation will be counterclockwise. 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. .. versionadded:: 1.12.0 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]]]) """ axes = tuple(axes) if len(axes) != 2: raise ValueError("len(axes) must be 2.") m = asanyarray(m) if axes[0] == axes[1] or absolute(axes[0] - axes[1]) == m.ndim: raise ValueError("Axes must be different.") if (axes[0] >= m.ndim or axes[0] < -m.ndim or axes[1] >= m.ndim or axes[1] < -m.ndim): raise ValueError("Axes={} out of range for array of ndim={}." .format(axes, m.ndim)) k %= 4 if k == 0: return m[:] if k == 2: return flip(flip(m, axes[0]), axes[1]) axes_list = arange(0, m.ndim) (axes_list[axes[0]], axes_list[axes[1]]) = (axes_list[axes[1]], axes_list[axes[0]]) if k == 1: return transpose(flip(m, axes[1]), axes_list) else: # k == 3 return flip(transpose(m, axes_list), axes[1]) def _flip_dispatcher(m, axis=None): return (m,) @array_function_dispatch(_flip_dispatcher) def flip(m, axis=None): """ 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 : None or int or tuple of ints, optional Axis or axes along which to flip over. The default, axis=None, will flip over all of the axes of the input array. If axis is negative it counts from the last to the first axis. If axis is a tuple of ints, flipping is performed on all of the axes specified in the tuple. .. versionchanged:: 1.15.0 None and tuples of axes are supported 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. flip(m) corresponds to ``m[::-1,::-1,...,::-1]`` with ``::-1`` at all positions. flip(m, (0, 1)) corresponds to ``m[::-1,::-1,...]`` with ``::-1`` at position 0 and position 1. Examples -------- >>> A = np.arange(8).reshape((2,2,2)) >>> A array([[[0, 1], [2, 3]], [[4, 5], [6, 7]]]) >>> np.flip(A, 0) array([[[4, 5], [6, 7]], [[0, 1], [2, 3]]]) >>> np.flip(A, 1) array([[[2, 3], [0, 1]], [[6, 7], [4, 5]]]) >>> np.flip(A) array([[[7, 6], [5, 4]], [[3, 2], [1, 0]]]) >>> np.flip(A, (0, 2)) array([[[5, 4], [7, 6]], [[1, 0], [3, 2]]]) >>> A = np.random.randn(3,4,5) >>> np.all(np.flip(A,2) == A[:,:,::-1,...]) True """ if not hasattr(m, 'ndim'): m = asarray(m) if axis is None: indexer = (np.s_[::-1],) * m.ndim else: axis = _nx.normalize_axis_tuple(axis, m.ndim) indexer = [np.s_[:]] * m.ndim for ax in axis: indexer[ax] = np.s_[::-1] indexer = tuple(indexer) return m[indexer] @set_module('numpy') def iterable(y): """ 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 Notes ----- In most cases, the results of ``np.iterable(obj)`` are consistent with ``isinstance(obj, collections.abc.Iterable)``. One notable exception is the treatment of 0-dimensional arrays:: >>> from collections.abc import Iterable >>> a = np.array(1.0) # 0-dimensional numpy array >>> isinstance(a, Iterable) True >>> np.iterable(a) False """ try: iter(y) except TypeError: return False return True def _average_dispatcher(a, axis=None, weights=None, returned=None, *, keepdims=None): return (a, weights) @array_function_dispatch(_average_dispatcher) def average(a, axis=None, weights=None, returned=False, *, keepdims=np._NoValue): """ 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. The 1-D calculation is:: avg = sum(a * weights) / sum(weights) The only constraint on `weights` is that `sum(weights)` must not be 0. 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. 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 `a`. *Note:* `keepdims` will not work with instances of `numpy.matrix` or other classes whose methods do not support `keepdims`. .. versionadded:: 1.23.0 Returns ------- retval, [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. `sum_of_weights` is of the same type as `retval`. The result dtype follows a genereal pattern. If `weights` is None, the result dtype will be that of `a` , or ``float64`` if `a` is integral. Otherwise, if `weights` is not None and `a` is non- integral, the result type will be the type of lowest precision capable of representing values of both `a` and `weights`. If `a` happens to be integral, the previous rules still applies but the result dtype will at least be ``float64``. 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 numpy.result_type : Returns the type that results from applying the numpy type promotion rules to the arguments. Examples -------- >>> data = np.arange(1, 5) >>> data array([1, 2, 3, 4]) >>> np.average(data) 2.5 >>> np.average(np.arange(1, 11), weights=np.arange(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. >>> a = np.ones(5, dtype=np.float128) >>> w = np.ones(5, dtype=np.complex64) >>> avg = np.average(a, weights=w) >>> print(avg.dtype) complex256 With ``keepdims=True``, the following result has shape (3, 1). >>> np.average(data, axis=1, keepdims=True) array([[0.5], [2.5], [4.5]]) """ a = np.asanyarray(a) if keepdims is np._NoValue: # Don't pass on the keepdims argument if one wasn't given. keepdims_kw = {} else: keepdims_kw = {'keepdims': keepdims} if weights is None: avg = a.mean(axis, **keepdims_kw) avg_as_array = np.asanyarray(avg) scl = avg_as_array.dtype.type(a.size/avg_as_array.size) else: wgt = np.asanyarray(weights) if issubclass(a.dtype.type, (np.integer, np.bool_)): result_dtype = np.result_type(a.dtype, wgt.dtype, 'f8') else: result_dtype = np.result_type(a.dtype, wgt.dtype) # Sanity checks if a.shape != wgt.shape: if axis is None: raise TypeError( "Axis must be specified when shapes of a and weights " "differ.") if wgt.ndim != 1: raise TypeError( "1D weights expected when shapes of a and weights differ.") if wgt.shape[0] != a.shape[axis]: raise ValueError( "Length of weights not compatible with specified axis.") # setup wgt to broadcast along axis wgt = np.broadcast_to(wgt, (a.ndim-1)*(1,) + wgt.shape) wgt = wgt.swapaxes(-1, axis) scl = wgt.sum(axis=axis, dtype=result_dtype, **keepdims_kw) if np.any(scl == 0.0): raise ZeroDivisionError( "Weights sum to zero, can't be normalized") avg = avg_as_array = np.multiply(a, wgt, dtype=result_dtype).sum(axis, **keepdims_kw) / scl if returned: if scl.shape != avg_as_array.shape: scl = np.broadcast_to(scl, avg_as_array.shape).copy() return avg, scl else: return avg @set_module('numpy') def asarray_chkfinite(a, dtype=None, order=None): """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', 'A', 'K'}, optional Memory layout. 'A' and 'K' depend on the order of input array a. 'C' row-major (C-style), 'F' column-major (Fortran-style) memory representation. 'A' (any) means 'F' if `a` is Fortran contiguous, 'C' otherwise 'K' (keep) preserve input order 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 """ a = asarray(a, dtype=dtype, order=order) if a.dtype.char in typecodes['AllFloat'] and not np.isfinite(a).all(): raise ValueError( "array must not contain infs or NaNs") return a def _piecewise_dispatcher(x, condlist, funclist, *args, **kw): yield x # support the undocumented behavior of allowing scalars if np.iterable(condlist): yield from condlist @array_function_dispatch(_piecewise_dispatcher) def piecewise(x, condlist, funclist, *args, **kw): """ 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 a 1d array as input and give an 1d 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) """ x = asanyarray(x) n2 = len(funclist) # undocumented: single condition is promoted to a list of one condition if isscalar(condlist) or ( not isinstance(condlist[0], (list, ndarray)) and x.ndim != 0): condlist = [condlist] condlist = asarray(condlist, dtype=bool) n = len(condlist) if n == n2 - 1: # compute the "otherwise" condition. condelse = ~np.any(condlist, axis=0, keepdims=True) condlist = np.concatenate([condlist, condelse], axis=0) n += 1 elif n != n2: raise ValueError( "with {} condition(s), either {} or {} functions are expected" .format(n, n, n+1) ) y = zeros_like(x) for cond, func in zip(condlist, funclist): if not isinstance(func, collections.abc.Callable): y[cond] = func else: vals = x[cond] if vals.size > 0: y[cond] = func(vals, *args, **kw) return y def _select_dispatcher(condlist, choicelist, default=None): yield from condlist yield from choicelist @array_function_dispatch(_select_dispatcher) def select(condlist, choicelist, default=0): """ 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(6) >>> condlist = [x<3, x>3] >>> choicelist = [x, x**2] >>> np.select(condlist, choicelist, 42) array([ 0, 1, 2, 42, 16, 25]) >>> condlist = [x<=4, x>3] >>> choicelist = [x, x**2] >>> np.select(condlist, choicelist, 55) array([ 0, 1, 2, 3, 4, 25]) """ # Check the size of condlist and choicelist are the same, or abort. if len(condlist) != len(choicelist): raise ValueError( 'list of cases must be same length as list of conditions') # Now that the dtype is known, handle the deprecated select([], []) case if len(condlist) == 0: raise ValueError("select with an empty condition list is not possible") choicelist = [np.asarray(choice) for choice in choicelist] try: intermediate_dtype = np.result_type(*choicelist) except TypeError as e: msg = f'Choicelist elements do not have a common dtype: {e}' raise TypeError(msg) from None default_array = np.asarray(default) choicelist.append(default_array) # need to get the result type before broadcasting for correct scalar # behaviour try: dtype = np.result_type(intermediate_dtype, default_array) except TypeError as e: msg = f'Choicelists and default value do not have a common dtype: {e}' raise TypeError(msg) from None # Convert conditions to arrays and broadcast conditions and choices # as the shape is needed for the result. Doing it separately optimizes # for example when all choices are scalars. condlist = np.broadcast_arrays(*condlist) choicelist = np.broadcast_arrays(*choicelist) # If cond array is not an ndarray in boolean format or scalar bool, abort. for i, cond in enumerate(condlist): if cond.dtype.type is not np.bool_: raise TypeError( 'invalid entry {} in condlist: should be boolean ndarray'.format(i)) if choicelist[0].ndim == 0: # This may be common, so avoid the call. result_shape = condlist[0].shape else: result_shape = np.broadcast_arrays(condlist[0], choicelist[0])[0].shape result = np.full(result_shape, choicelist[-1], dtype) # Use np.copyto to burn each choicelist array onto result, using the # corresponding condlist as a boolean mask. This is done in reverse # order since the first choice should take precedence. choicelist = choicelist[-2::-1] condlist = condlist[::-1] for choice, cond in zip(choicelist, condlist): np.copyto(result, choice, where=cond) return result def _copy_dispatcher(a, order=None, subok=None): return (a,) @array_function_dispatch(_copy_dispatcher) def copy(a, order='K', subok=False): """ 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.) subok : bool, optional If True, then sub-classes will be passed-through, otherwise the returned array will be forced to be a base-class array (defaults to False). .. versionadded:: 1.19.0 Returns ------- arr : ndarray Array interpretation of `a`. See Also -------- ndarray.copy : Preferred method for creating an array copy 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 Note that, np.copy clears previously set WRITEABLE=False flag. >>> a = np.array([1, 2, 3]) >>> a.flags["WRITEABLE"] = False >>> b = np.copy(a) >>> b.flags["WRITEABLE"] True >>> b[0] = 3 >>> b array([3, 2, 3]) Note that np.copy is a shallow copy and will not copy object elements within arrays. This is mainly important for arrays containing Python objects. The new array will contain the same object which may lead to surprises if that object can be modified (is mutable): >>> a = np.array([1, 'm', [2, 3, 4]], dtype=object) >>> b = np.copy(a) >>> b[2][0] = 10 >>> a array([1, 'm', list([10, 3, 4])], dtype=object) To ensure all elements within an ``object`` array are copied, use `copy.deepcopy`: >>> import copy >>> a = np.array([1, 'm', [2, 3, 4]], dtype=object) >>> c = copy.deepcopy(a) >>> c[2][0] = 10 >>> c array([1, 'm', list([10, 3, 4])], dtype=object) >>> a array([1, 'm', list([2, 3, 4])], dtype=object) """ return array(a, order=order, subok=subok, copy=True) # Basic operations def _gradient_dispatcher(f, *varargs, axis=None, edge_order=None): yield f yield from varargs @array_function_dispatch(_gradient_dispatcher) def gradient(f, *varargs, axis=None, edge_order=1): """ 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 list 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=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=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=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=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=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 :math:`h_{*}` be a non-homogeneous stepsize, we minimize the "consistency error" :math:`\\eta_{i}` between the true gradient and its estimate from a linear combination of the neighboring grid-points: .. 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>`_. """ f = np.asanyarray(f) N = f.ndim # number of dimensions if axis is None: axes = tuple(range(N)) else: axes = _nx.normalize_axis_tuple(axis, N) len_axes = len(axes) n = len(varargs) if n == 0: # no spacing argument - use 1 in all axes dx = [1.0] * len_axes elif n == 1 and np.ndim(varargs[0]) == 0: # single scalar for all axes dx = varargs * len_axes elif n == len_axes: # scalar or 1d array for each axis dx = list(varargs) for i, distances in enumerate(dx): distances = np.asanyarray(distances) if distances.ndim == 0: continue elif distances.ndim != 1: raise ValueError("distances must be either scalars or 1d") if len(distances) != f.shape[axes[i]]: raise ValueError("when 1d, distances must match " "the length of the corresponding dimension") if np.issubdtype(distances.dtype, np.integer): # Convert numpy integer types to float64 to avoid modular # arithmetic in np.diff(distances). distances = distances.astype(np.float64) diffx = np.diff(distances) # if distances are constant reduce to the scalar case # since it brings a consistent speedup if (diffx == diffx[0]).all(): diffx = diffx[0] dx[i] = diffx else: raise TypeError("invalid number of arguments") if edge_order > 2: raise ValueError("'edge_order' greater than 2 not supported") # use central differences on interior and one-sided differences on the # endpoints. This preserves second order-accuracy over the full domain. outvals = [] # create slice objects --- initially all are [:, :, ..., :] slice1 = [slice(None)]*N slice2 = [slice(None)]*N slice3 = [slice(None)]*N slice4 = [slice(None)]*N otype = f.dtype if otype.type is np.datetime64: # the timedelta dtype with the same unit information otype = np.dtype(otype.name.replace('datetime', 'timedelta')) # view as timedelta to allow addition f = f.view(otype) elif otype.type is np.timedelta64: pass elif np.issubdtype(otype, np.inexact): pass else: # All other types convert to floating point. # First check if f is a numpy integer type; if so, convert f to float64 # to avoid modular arithmetic when computing the changes in f. if np.issubdtype(otype, np.integer): f = f.astype(np.float64) otype = np.float64 for axis, ax_dx in zip(axes, dx): if f.shape[axis] < edge_order + 1: raise ValueError( "Shape of array too small to calculate a numerical gradient, " "at least (edge_order + 1) elements are required.") # result allocation out = np.empty_like(f, dtype=otype) # spacing for the current axis uniform_spacing = np.ndim(ax_dx) == 0 # Numerical differentiation: 2nd order interior slice1[axis] = slice(1, -1) slice2[axis] = slice(None, -2) slice3[axis] = slice(1, -1) slice4[axis] = slice(2, None) if uniform_spacing: out[tuple(slice1)] = (f[tuple(slice4)] - f[tuple(slice2)]) / (2. * ax_dx) else: dx1 = ax_dx[0:-1] dx2 = ax_dx[1:] a = -(dx2)/(dx1 * (dx1 + dx2)) b = (dx2 - dx1) / (dx1 * dx2) c = dx1 / (dx2 * (dx1 + dx2)) # fix the shape for broadcasting shape = np.ones(N, dtype=int) shape[axis] = -1 a.shape = b.shape = c.shape = shape # 1D equivalent -- out[1:-1] = a * f[:-2] + b * f[1:-1] + c * f[2:] out[tuple(slice1)] = a * f[tuple(slice2)] + b * f[tuple(slice3)] + c * f[tuple(slice4)] # Numerical differentiation: 1st order edges if edge_order == 1: slice1[axis] = 0 slice2[axis] = 1 slice3[axis] = 0 dx_0 = ax_dx if uniform_spacing else ax_dx[0] # 1D equivalent -- out[0] = (f[1] - f[0]) / (x[1] - x[0]) out[tuple(slice1)] = (f[tuple(slice2)] - f[tuple(slice3)]) / dx_0 slice1[axis] = -1 slice2[axis] = -1 slice3[axis] = -2 dx_n = ax_dx if uniform_spacing else ax_dx[-1] # 1D equivalent -- out[-1] = (f[-1] - f[-2]) / (x[-1] - x[-2]) out[tuple(slice1)] = (f[tuple(slice2)] - f[tuple(slice3)]) / dx_n # Numerical differentiation: 2nd order edges else: slice1[axis] = 0 slice2[axis] = 0 slice3[axis] = 1 slice4[axis] = 2 if uniform_spacing: a = -1.5 / ax_dx b = 2. / ax_dx c = -0.5 / ax_dx else: dx1 = ax_dx[0] dx2 = ax_dx[1] a = -(2. * dx1 + dx2)/(dx1 * (dx1 + dx2)) b = (dx1 + dx2) / (dx1 * dx2) c = - dx1 / (dx2 * (dx1 + dx2)) # 1D equivalent -- out[0] = a * f[0] + b * f[1] + c * f[2] out[tuple(slice1)] = a * f[tuple(slice2)] + b * f[tuple(slice3)] + c * f[tuple(slice4)] slice1[axis] = -1 slice2[axis] = -3 slice3[axis] = -2 slice4[axis] = -1 if uniform_spacing: a = 0.5 / ax_dx b = -2. / ax_dx c = 1.5 / ax_dx else: dx1 = ax_dx[-2] dx2 = ax_dx[-1] a = (dx2) / (dx1 * (dx1 + dx2)) b = - (dx2 + dx1) / (dx1 * dx2) c = (2. * dx2 + dx1) / (dx2 * (dx1 + dx2)) # 1D equivalent -- out[-1] = a * f[-3] + b * f[-2] + c * f[-1] out[tuple(slice1)] = a * f[tuple(slice2)] + b * f[tuple(slice3)] + c * f[tuple(slice4)] outvals.append(out) # reset the slice object in this dimension to ":" slice1[axis] = slice(None) slice2[axis] = slice(None) slice3[axis] = slice(None) slice4[axis] = slice(None) if len_axes == 1: return outvals[0] elif np._using_numpy2_behavior(): return tuple(outvals) else: return outvals def _diff_dispatcher(a, n=None, axis=None, prepend=None, append=None): return (a, prepend, append) @array_function_dispatch(_diff_dispatcher) def diff(a, n=1, axis=-1, prepend=np._NoValue, append=np._NoValue): """ Calculate the n-th discrete difference along the given axis. The first difference is given by ``out[i] = a[i+1] - a[i]`` 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. If zero, the input is returned as-is. axis : int, optional The axis along which the difference is taken, default is the last axis. prepend, append : array_like, optional Values to prepend or append to `a` along axis prior to performing the difference. Scalar values are expanded to arrays with length 1 in the direction of axis and the shape of the input array in along all other axes. Otherwise the dimension and shape must match `a` except along axis. .. versionadded:: 1.16.0 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 the type of the difference between any two elements of `a`. This is the same as the type of `a` in most cases. A notable exception is `datetime64`, which results in a `timedelta64` output array. See Also -------- gradient, ediff1d, cumsum Notes ----- Type is preserved for boolean arrays, so 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,...] 255 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]]) >>> x = np.arange('1066-10-13', '1066-10-16', dtype=np.datetime64) >>> np.diff(x) array([1, 1], dtype='timedelta64[D]') """ if n == 0: return a if n < 0: raise ValueError( "order must be non-negative but got " + repr(n)) a = asanyarray(a) nd = a.ndim if nd == 0: raise ValueError("diff requires input that is at least one dimensional") axis = normalize_axis_index(axis, nd) combined = [] if prepend is not np._NoValue: prepend = np.asanyarray(prepend) if prepend.ndim == 0: shape = list(a.shape) shape[axis] = 1 prepend = np.broadcast_to(prepend, tuple(shape)) combined.append(prepend) combined.append(a) if append is not np._NoValue: append = np.asanyarray(append) if append.ndim == 0: shape = list(a.shape) shape[axis] = 1 append = np.broadcast_to(append, tuple(shape)) combined.append(append) if len(combined) > 1: a = np.concatenate(combined, axis) slice1 = [slice(None)] * nd slice2 = [slice(None)] * nd slice1[axis] = slice(1, None) slice2[axis] = slice(None, -1) slice1 = tuple(slice1) slice2 = tuple(slice2) op = not_equal if a.dtype == np.bool_ else subtract for _ in range(n): a = op(a[slice1], a[slice2]) return a def _interp_dispatcher(x, xp, fp, left=None, right=None, period=None): return (x, xp, fp) @array_function_dispatch(_interp_dispatcher) def interp(x, xp, fp, left=None, right=None, period=None): """ One-dimensional linear interpolation for monotonically increasing sample points. Returns the one-dimensional piecewise linear interpolant to a function with given discrete data points (`xp`, `fp`), evaluated at `x`. Parameters ---------- x : array_like The x-coordinates at which to evaluate 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` See Also -------- scipy.interpolate Warnings -------- The x-coordinate sequence is expected to be increasing, but this is not explicitly enforced. However, if the sequence `xp` is non-increasing, interpolation results are meaningless. Note that, since NaN is unsortable, `xp` also cannot contain NaNs. A simple check for `xp` being strictly 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]) """ fp = np.asarray(fp) if np.iscomplexobj(fp): interp_func = compiled_interp_complex input_dtype = np.complex128 else: interp_func = compiled_interp input_dtype = np.float64 if period is not None: if period == 0: raise ValueError("period must be a non-zero value") period = abs(period) left = None right = None x = np.asarray(x, dtype=np.float64) xp = np.asarray(xp, dtype=np.float64) fp = np.asarray(fp, dtype=input_dtype) if xp.ndim != 1 or fp.ndim != 1: raise ValueError("Data points must be 1-D sequences") if xp.shape[0] != fp.shape[0]: raise ValueError("fp and xp are not of the same length") # normalizing periodic boundaries x = x % period xp = xp % period asort_xp = np.argsort(xp) xp = xp[asort_xp] fp = fp[asort_xp] xp = np.concatenate((xp[-1:]-period, xp, xp[0:1]+period)) fp = np.concatenate((fp[-1:], fp, fp[0:1])) return interp_func(x, xp, fp, left, right) def _angle_dispatcher(z, deg=None): return (z,) @array_function_dispatch(_angle_dispatcher) def angle(z, deg=False): """ 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 in the range ``(-pi, pi]``, with dtype as numpy.float64. .. versionchanged:: 1.16.0 This function works on subclasses of ndarray like `ma.array`. See Also -------- arctan2 absolute Notes ----- Although the angle of the complex number 0 is undefined, ``numpy.angle(0)`` returns the value 0. Examples -------- >>> np.angle([1.0, 1.0j, 1+1j]) # in radians array([ 0. , 1.57079633, 0.78539816]) # may vary >>> np.angle(1+1j, deg=True) # in degrees 45.0 """ z = asanyarray(z) if issubclass(z.dtype.type, _nx.complexfloating): zimag = z.imag zreal = z.real else: zimag = 0 zreal = z a = arctan2(zimag, zreal) if deg: a *= 180/pi return a def _unwrap_dispatcher(p, discont=None, axis=None, *, period=None): return (p,) @array_function_dispatch(_unwrap_dispatcher) def unwrap(p, discont=None, axis=-1, *, period=2*pi): r""" Unwrap by taking the complement of large deltas with respect to the period. This unwraps a signal `p` by changing elements which have an absolute difference from their predecessor of more than ``max(discont, period/2)`` to their `period`-complementary values. For the default case where `period` is :math:`2\pi` and `discont` is :math:`\pi`, this unwraps a radian phase `p` such that adjacent differences are never greater than :math:`\pi` by adding :math:`2k\pi` for some integer :math:`k`. Parameters ---------- p : array_like Input array. discont : float, optional Maximum discontinuity between values, default is ``period/2``. Values below ``period/2`` are treated as if they were ``period/2``. To have an effect different from the default, `discont` should be larger than ``period/2``. axis : int, optional Axis along which unwrap will operate, default is the last axis. period : float, optional Size of the range over which the input wraps. By default, it is ``2 pi``. .. versionadded:: 1.21.0 Returns ------- out : ndarray Output array. See Also -------- rad2deg, deg2rad Notes ----- If the discontinuity in `p` is smaller than ``period/2``, but larger than `discont`, no unwrapping is done because taking the 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]) # may vary >>> np.unwrap(phase) array([ 0. , 0.78539816, 1.57079633, -0.78539816, 0. ]) # may vary >>> np.unwrap([0, 1, 2, -1, 0], period=4) array([0, 1, 2, 3, 4]) >>> np.unwrap([ 1, 2, 3, 4, 5, 6, 1, 2, 3], period=6) array([1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> np.unwrap([2, 3, 4, 5, 2, 3, 4, 5], period=4) array([2, 3, 4, 5, 6, 7, 8, 9]) >>> phase_deg = np.mod(np.linspace(0 ,720, 19), 360) - 180 >>> np.unwrap(phase_deg, period=360) array([-180., -140., -100., -60., -20., 20., 60., 100., 140., 180., 220., 260., 300., 340., 380., 420., 460., 500., 540.]) """ p = asarray(p) nd = p.ndim dd = diff(p, axis=axis) if discont is None: discont = period/2 slice1 = [slice(None, None)]*nd # full slices slice1[axis] = slice(1, None) slice1 = tuple(slice1) dtype = np.result_type(dd, period) if _nx.issubdtype(dtype, _nx.integer): interval_high, rem = divmod(period, 2) boundary_ambiguous = rem == 0 else: interval_high = period / 2 boundary_ambiguous = True interval_low = -interval_high ddmod = mod(dd - interval_low, period) + interval_low if boundary_ambiguous: # for `mask = (abs(dd) == period/2)`, the above line made # `ddmod[mask] == -period/2`. correct these such that # `ddmod[mask] == sign(dd[mask])*period/2`. _nx.copyto(ddmod, interval_high, where=(ddmod == interval_low) & (dd > 0)) ph_correct = ddmod - dd _nx.copyto(ph_correct, 0, where=abs(dd) < discont) up = array(p, copy=True, dtype=dtype) up[slice1] = p[slice1] + ph_correct.cumsum(axis) return up def _sort_complex(a): return (a,) @array_function_dispatch(_sort_complex) def sort_complex(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]) """ b = array(a, copy=True) b.sort() if not issubclass(b.dtype.type, _nx.complexfloating): if b.dtype.char in 'bhBH': return b.astype('F') elif b.dtype.char == 'g': return b.astype('G') else: return b.astype('D') else: return b def _trim_zeros(filt, trim=None): return (filt,) @array_function_dispatch(_trim_zeros) def trim_zeros(filt, trim='fb'): """ 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, ..., 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] """ first = 0 trim = trim.upper() if 'F' in trim: for i in filt: if i != 0.: break else: first = first + 1 last = len(filt) if 'B' in trim: for i in filt[::-1]: if i != 0.: break else: last = last - 1 return filt[first:last] def _extract_dispatcher(condition, arr): return (condition, arr) @array_function_dispatch(_extract_dispatcher) def extract(condition, arr): """ 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]]) >>> np.extract(condition, arr) array([0, 3, 6, 9]) If `condition` is boolean: >>> arr[condition] array([0, 3, 6, 9]) """ return _nx.take(ravel(arr), nonzero(ravel(condition))[0]) def _place_dispatcher(arr, mask, vals): return (arr, mask, vals) @array_function_dispatch(_place_dispatcher) def place(arr, mask, vals): """ 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]]) """ return _place(arr, mask, vals) def disp(mesg, device=None, linefeed=True): """ 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 io import StringIO >>> buf = StringIO() >>> np.disp(u'"Display" in a file', device=buf) >>> buf.getvalue() '"Display" in a file\\n' """ if device is None: device = sys.stdout if linefeed: device.write('%s\n' % mesg) else: device.write('%s' % mesg) device.flush() return # See https://docs.scipy.org/doc/numpy/reference/c-api.generalized-ufuncs.html _DIMENSION_NAME = r'\w+' _CORE_DIMENSION_LIST = '(?:{0:}(?:,{0:})*)?'.format(_DIMENSION_NAME) _ARGUMENT = r'\({}\)'.format(_CORE_DIMENSION_LIST) _ARGUMENT_LIST = '{0:}(?:,{0:})*'.format(_ARGUMENT) _SIGNATURE = '^{0:}->{0:}$'.format(_ARGUMENT_LIST) def _parse_gufunc_signature(signature): """ 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, ...]]. """ signature = re.sub(r'\s+', '', signature) if not re.match(_SIGNATURE, signature): raise ValueError( 'not a valid gufunc signature: {}'.format(signature)) return tuple([tuple(re.findall(_DIMENSION_NAME, arg)) for arg in re.findall(_ARGUMENT, arg_list)] for arg_list in signature.split('->')) def _update_dim_sizes(dim_sizes, arg, core_dims): """ 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. """ if not core_dims: return num_core_dims = len(core_dims) if arg.ndim < num_core_dims: raise ValueError( '%d-dimensional argument does not have enough ' 'dimensions for all core dimensions %r' % (arg.ndim, core_dims)) core_shape = arg.shape[-num_core_dims:] for dim, size in zip(core_dims, core_shape): if dim in dim_sizes: if size != dim_sizes[dim]: raise ValueError( 'inconsistent size for core dimension %r: %r vs %r' % (dim, size, dim_sizes[dim])) else: dim_sizes[dim] = size def _parse_input_dimensions(args, input_core_dims): """ 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. """ broadcast_args = [] dim_sizes = {} for arg, core_dims in zip(args, input_core_dims): _update_dim_sizes(dim_sizes, arg, core_dims) ndim = arg.ndim - len(core_dims) dummy_array = np.lib.stride_tricks.as_strided(0, arg.shape[:ndim]) broadcast_args.append(dummy_array) broadcast_shape = np.lib.stride_tricks._broadcast_shape(*broadcast_args) return broadcast_shape, dim_sizes def _calculate_shapes(broadcast_shape, dim_sizes, list_of_core_dims): """Helper for calculating broadcast shapes with core dimensions.""" return [broadcast_shape + tuple(dim_sizes[dim] for dim in core_dims) for core_dims in list_of_core_dims] def _create_arrays(broadcast_shape, dim_sizes, list_of_core_dims, dtypes, results=None): """Helper for creating output arrays in vectorize.""" shapes = _calculate_shapes(broadcast_shape, dim_sizes, list_of_core_dims) if dtypes is None: dtypes = [None] * len(shapes) if results is None: arrays = tuple(np.empty(shape=shape, dtype=dtype) for shape, dtype in zip(shapes, dtypes)) else: arrays = tuple(np.empty_like(result, shape=shape, dtype=dtype) for result, shape, dtype in zip(results, shapes, dtypes)) return arrays @set_module('numpy') class vectorize: """ vectorize(pyfunc=np._NoValue, otypes=None, doc=None, excluded=None, cache=False, signature=None) Returns an object that acts like pyfunc, but takes arrays as input. Define a vectorized function which takes a nested sequence of objects or numpy arrays as inputs and returns a single numpy array or a tuple of numpy arrays. 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, optional A python function or method. Can be omitted to produce a decorator with keyword arguments. 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 ------- out : callable A vectorized function if ``pyfunc`` was provided, a decorator otherwise. 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] :doc:`/reference/c-api/generalized-ufuncs` 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]) <class 'numpy.int64'> >>> vfunc = np.vectorize(myfunc, otypes=[float]) >>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) <class '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.]]) Decorator syntax is supported. The decorator can be called as a function to provide keyword arguments. >>>@np.vectorize ...def identity(x): ... return x ... >>>identity([0, 1, 2]) array([0, 1, 2]) >>>@np.vectorize(otypes=[float]) ...def as_float(x): ... return x ... >>>as_float([0, 1, 2]) array([0., 1., 2.]) """ def __init__(self, pyfunc=np._NoValue, otypes=None, doc=None, excluded=None, cache=False, signature=None): if (pyfunc != np._NoValue) and (not callable(pyfunc)): #Splitting the error message to keep #the length below 79 characters. part1 = "When used as a decorator, " part2 = "only accepts keyword arguments." raise TypeError(part1 + part2) self.pyfunc = pyfunc self.cache = cache self.signature = signature if pyfunc != np._NoValue and hasattr(pyfunc, '__name__'): self.__name__ = pyfunc.__name__ self._ufunc = {} # Caching to improve default performance self._doc = None self.__doc__ = doc if doc is None and hasattr(pyfunc, '__doc__'): self.__doc__ = pyfunc.__doc__ else: self._doc = doc if isinstance(otypes, str): for char in otypes: if char not in typecodes['All']: raise ValueError("Invalid otype specified: %s" % (char,)) elif iterable(otypes): otypes = ''.join([_nx.dtype(x).char for x in otypes]) elif otypes is not None: raise ValueError("Invalid otype specification") self.otypes = otypes # Excluded variable support if excluded is None: excluded = set() self.excluded = set(excluded) if signature is not None: self._in_and_out_core_dims = _parse_gufunc_signature(signature) else: self._in_and_out_core_dims = None def _init_stage_2(self, pyfunc, *args, **kwargs): self.__name__ = pyfunc.__name__ self.pyfunc = pyfunc if self._doc is None: self.__doc__ = pyfunc.__doc__ else: self.__doc__ = self._doc def _call_as_normal(self, *args, **kwargs): """ Return arrays with the results of `pyfunc` broadcast (vectorized) over `args` and `kwargs` not in `excluded`. """ excluded = self.excluded if not kwargs and not excluded: func = self.pyfunc vargs = args else: # The wrapper accepts only positional arguments: we use `names` and # `inds` to mutate `the_args` and `kwargs` to pass to the original # function. nargs = len(args) names = [_n for _n in kwargs if _n not in excluded] inds = [_i for _i in range(nargs) if _i not in excluded] the_args = list(args) def func(*vargs): for _n, _i in enumerate(inds): the_args[_i] = vargs[_n] kwargs.update(zip(names, vargs[len(inds):])) return self.pyfunc(*the_args, **kwargs) vargs = [args[_i] for _i in inds] vargs.extend([kwargs[_n] for _n in names]) return self._vectorize_call(func=func, args=vargs) def __call__(self, *args, **kwargs): if self.pyfunc is np._NoValue: self._init_stage_2(*args, **kwargs) return self return self._call_as_normal(*args, **kwargs) def _get_ufunc_and_otypes(self, func, args): """Return (ufunc, otypes).""" # frompyfunc will fail if args is empty if not args: raise ValueError('args can not be empty') if self.otypes is not None: otypes = self.otypes # self._ufunc is a dictionary whose keys are the number of # arguments (i.e. len(args)) and whose values are ufuncs created # by frompyfunc. len(args) can be different for different calls if # self.pyfunc has parameters with default values. We only use the # cache when func is self.pyfunc, which occurs when the call uses # only positional arguments and no arguments are excluded. nin = len(args) nout = len(self.otypes) if func is not self.pyfunc or nin not in self._ufunc: ufunc = frompyfunc(func, nin, nout) else: ufunc = None # We'll get it from self._ufunc if func is self.pyfunc: ufunc = self._ufunc.setdefault(nin, ufunc) else: # Get number of outputs and output types by calling the function on # the first entries of args. We also cache the result to prevent # the subsequent call when the ufunc is evaluated. # Assumes that ufunc first evaluates the 0th elements in the input # arrays (the input values are not checked to ensure this) args = [asarray(arg) for arg in args] if builtins.any(arg.size == 0 for arg in args): raise ValueError('cannot call `vectorize` on size 0 inputs ' 'unless `otypes` is set') inputs = [arg.flat[0] for arg in args] outputs = func(*inputs) # Performance note: profiling indicates that -- for simple # functions at least -- this wrapping can almost double the # execution time. # Hence we make it optional. if self.cache: _cache = [outputs] def _func(*vargs): if _cache: return _cache.pop() else: return func(*vargs) else: _func = func if isinstance(outputs, tuple): nout = len(outputs) else: nout = 1 outputs = (outputs,) otypes = ''.join([asarray(outputs[_k]).dtype.char for _k in range(nout)]) # Performance note: profiling indicates that creating the ufunc is # not a significant cost compared with wrapping so it seems not # worth trying to cache this. ufunc = frompyfunc(_func, len(args), nout) return ufunc, otypes def _vectorize_call(self, func, args): """Vectorized call to `func` over positional `args`.""" if self.signature is not None: res = self._vectorize_call_with_signature(func, args) elif not args: res = func() else: ufunc, otypes = self._get_ufunc_and_otypes(func=func, args=args) # Convert args to object arrays first inputs = [asanyarray(a, dtype=object) for a in args] outputs = ufunc(*inputs) if ufunc.nout == 1: res = asanyarray(outputs, dtype=otypes[0]) else: res = tuple([asanyarray(x, dtype=t) for x, t in zip(outputs, otypes)]) return res def _vectorize_call_with_signature(self, func, args): """Vectorized call over positional arguments with a signature.""" input_core_dims, output_core_dims = self._in_and_out_core_dims if len(args) != len(input_core_dims): raise TypeError('wrong number of positional arguments: ' 'expected %r, got %r' % (len(input_core_dims), len(args))) args = tuple(asanyarray(arg) for arg in args) broadcast_shape, dim_sizes = _parse_input_dimensions( args, input_core_dims) input_shapes = _calculate_shapes(broadcast_shape, dim_sizes, input_core_dims) args = [np.broadcast_to(arg, shape, subok=True) for arg, shape in zip(args, input_shapes)] outputs = None otypes = self.otypes nout = len(output_core_dims) for index in np.ndindex(*broadcast_shape): results = func(*(arg[index] for arg in args)) n_results = len(results) if isinstance(results, tuple) else 1 if nout != n_results: raise ValueError( 'wrong number of outputs from pyfunc: expected %r, got %r' % (nout, n_results)) if nout == 1: results = (results,) if outputs is None: for result, core_dims in zip(results, output_core_dims): _update_dim_sizes(dim_sizes, result, core_dims) outputs = _create_arrays(broadcast_shape, dim_sizes, output_core_dims, otypes, results) for output, result in zip(outputs, results): output[index] = result if outputs is None: # did not call the function even once if otypes is None: raise ValueError('cannot call `vectorize` on size 0 inputs ' 'unless `otypes` is set') if builtins.any(dim not in dim_sizes for dims in output_core_dims for dim in dims): raise ValueError('cannot call `vectorize` with a signature ' 'including new output dimensions on size 0 ' 'inputs') outputs = _create_arrays(broadcast_shape, dim_sizes, output_core_dims, otypes) return outputs[0] if nout == 1 else outputs def _cov_dispatcher(m, y=None, rowvar=None, bias=None, ddof=None, fweights=None, aweights=None, *, dtype=None): return (m, y, fweights, aweights) @array_function_dispatch(_cov_dispatcher) def cov(m, y=None, rowvar=True, bias=False, ddof=None, fweights=None, aweights=None, *, dtype=None): """ Estimate a covariance matrix, given data and weights. 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 frequency 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 dtype : data-type, optional Data-type of the result. By default, the return data-type will have at least `numpy.float64` precision. .. versionadded:: 1.20 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:: >>> m = np.arange(10, dtype=np.float64) >>> f = np.arange(10) * 2 >>> a = np.arange(10) ** 2. >>> ddof = 1 >>> w = f * a >>> v1 = np.sum(w) >>> v2 = np.sum(w * a) >>> m -= np.sum(m * w, axis=None, 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.stack((x, y), axis=0) >>> np.cov(X) array([[11.71 , -4.286 ], # may vary [-4.286 , 2.144133]]) >>> np.cov(x, y) array([[11.71 , -4.286 ], # may vary [-4.286 , 2.144133]]) >>> np.cov(x) array(11.71) """ # Check inputs if ddof is not None and ddof != int(ddof): raise ValueError( "ddof must be integer") # Handles complex arrays too m = np.asarray(m) if m.ndim > 2: raise ValueError("m has more than 2 dimensions") if y is not None: y = np.asarray(y) if y.ndim > 2: raise ValueError("y has more than 2 dimensions") if dtype is None: if y is None: dtype = np.result_type(m, np.float64) else: dtype = np.result_type(m, y, np.float64) X = array(m, ndmin=2, dtype=dtype) if not rowvar and X.shape[0] != 1: X = X.T if X.shape[0] == 0: return np.array([]).reshape(0, 0) if y is not None: y = array(y, copy=False, ndmin=2, dtype=dtype) if not rowvar and y.shape[0] != 1: y = y.T X = np.concatenate((X, y), axis=0) if ddof is None: if bias == 0: ddof = 1 else: ddof = 0 # Get the product of frequencies and weights w = None if fweights is not None: fweights = np.asarray(fweights, dtype=float) if not np.all(fweights == np.around(fweights)): raise TypeError( "fweights must be integer") if fweights.ndim > 1: raise RuntimeError( "cannot handle multidimensional fweights") if fweights.shape[0] != X.shape[1]: raise RuntimeError( "incompatible numbers of samples and fweights") if any(fweights < 0): raise ValueError( "fweights cannot be negative") w = fweights if aweights is not None: aweights = np.asarray(aweights, dtype=float) if aweights.ndim > 1: raise RuntimeError( "cannot handle multidimensional aweights") if aweights.shape[0] != X.shape[1]: raise RuntimeError( "incompatible numbers of samples and aweights") if any(aweights < 0): raise ValueError( "aweights cannot be negative") if w is None: w = aweights else: w *= aweights avg, w_sum = average(X, axis=1, weights=w, returned=True) w_sum = w_sum[0] # Determine the normalization if w is None: fact = X.shape[1] - ddof elif ddof == 0: fact = w_sum elif aweights is None: fact = w_sum - ddof else: fact = w_sum - ddof*sum(w*aweights)/w_sum if fact <= 0: warnings.warn("Degrees of freedom <= 0 for slice", RuntimeWarning, stacklevel=2) fact = 0.0 X -= avg[:, None] if w is None: X_T = X.T else: X_T = (X*w).T c = dot(X, X_T.conj()) c *= np.true_divide(1, fact) return c.squeeze() def _corrcoef_dispatcher(x, y=None, rowvar=None, bias=None, ddof=None, *, dtype=None): return (x, y) @array_function_dispatch(_corrcoef_dispatcher) def corrcoef(x, y=None, rowvar=True, bias=np._NoValue, ddof=np._NoValue, *, dtype=None): """ 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 dtype : data-type, optional Data-type of the result. By default, the return data-type will have at least `numpy.float64` precision. .. versionadded:: 1.20 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. Examples -------- In this example we generate two random arrays, ``xarr`` and ``yarr``, and compute the row-wise and column-wise Pearson correlation coefficients, ``R``. Since ``rowvar`` is true by default, we first find the row-wise Pearson correlation coefficients between the variables of ``xarr``. >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> xarr = rng.random((3, 3)) >>> xarr array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) >>> R1 = np.corrcoef(xarr) >>> R1 array([[ 1. , 0.99256089, -0.68080986], [ 0.99256089, 1. , -0.76492172], [-0.68080986, -0.76492172, 1. ]]) If we add another set of variables and observations ``yarr``, we can compute the row-wise Pearson correlation coefficients between the variables in ``xarr`` and ``yarr``. >>> yarr = rng.random((3, 3)) >>> yarr array([[0.45038594, 0.37079802, 0.92676499], [0.64386512, 0.82276161, 0.4434142 ], [0.22723872, 0.55458479, 0.06381726]]) >>> R2 = np.corrcoef(xarr, yarr) >>> R2 array([[ 1. , 0.99256089, -0.68080986, 0.75008178, -0.934284 , -0.99004057], [ 0.99256089, 1. , -0.76492172, 0.82502011, -0.97074098, -0.99981569], [-0.68080986, -0.76492172, 1. , -0.99507202, 0.89721355, 0.77714685], [ 0.75008178, 0.82502011, -0.99507202, 1. , -0.93657855, -0.83571711], [-0.934284 , -0.97074098, 0.89721355, -0.93657855, 1. , 0.97517215], [-0.99004057, -0.99981569, 0.77714685, -0.83571711, 0.97517215, 1. ]]) Finally if we use the option ``rowvar=False``, the columns are now being treated as the variables and we will find the column-wise Pearson correlation coefficients between variables in ``xarr`` and ``yarr``. >>> R3 = np.corrcoef(xarr, yarr, rowvar=False) >>> R3 array([[ 1. , 0.77598074, -0.47458546, -0.75078643, -0.9665554 , 0.22423734], [ 0.77598074, 1. , -0.92346708, -0.99923895, -0.58826587, -0.44069024], [-0.47458546, -0.92346708, 1. , 0.93773029, 0.23297648, 0.75137473], [-0.75078643, -0.99923895, 0.93773029, 1. , 0.55627469, 0.47536961], [-0.9665554 , -0.58826587, 0.23297648, 0.55627469, 1. , -0.46666491], [ 0.22423734, -0.44069024, 0.75137473, 0.47536961, -0.46666491, 1. ]]) """ if bias is not np._NoValue or ddof is not np._NoValue: # 2015-03-15, 1.10 warnings.warn('bias and ddof have no effect and are deprecated', DeprecationWarning, stacklevel=2) c = cov(x, y, rowvar, dtype=dtype) try: d = diag(c) except ValueError: # scalar covariance # nan if incorrect value (nan, inf, 0), 1 otherwise return c / c stddev = sqrt(d.real) c /= stddev[:, None] c /= stddev[None, :] # Clip real and imaginary parts to [-1, 1]. This does not guarantee # abs(a[i,j]) <= 1 for complex arrays, but is the best we can do without # excessive work. np.clip(c.real, -1, 1, out=c.real) if np.iscomplexobj(c): np.clip(c.imag, -1, 1, out=c.imag) return c @set_module('numpy') def blackman(M): """ 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 -------- >>> import matplotlib.pyplot as plt >>> np.blackman(12) array([-1.38777878e-17, 3.26064346e-02, 1.59903635e-01, # may vary 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") Text(0.5, 1.0, 'Blackman window') >>> plt.ylabel("Amplitude") Text(0, 0.5, 'Amplitude') >>> plt.xlabel("Sample") Text(0.5, 0, 'Sample') >>> plt.show() >>> plt.figure() <Figure size 640x480 with 0 Axes> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> with np.errstate(divide='ignore', invalid='ignore'): ... 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") Text(0.5, 1.0, 'Frequency response of Blackman window') >>> plt.ylabel("Magnitude [dB]") Text(0, 0.5, 'Magnitude [dB]') >>> plt.xlabel("Normalized frequency [cycles per sample]") Text(0.5, 0, 'Normalized frequency [cycles per sample]') >>> _ = plt.axis('tight') >>> plt.show() """ # Ensures at least float64 via 0.0. M should be an integer, but conversion # to double is safe for a range. values = np.array([0.0, M]) M = values[1] if M < 1: return array([], dtype=values.dtype) if M == 1: return ones(1, dtype=values.dtype) n = arange(1-M, M, 2) return 0.42 + 0.5*cos(pi*n/(M-1)) + 0.08*cos(2.0*pi*n/(M-1)) @set_module('numpy') def bartlett(M): """ 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 window is the product of two sinc functions. Note the excellent discussion in Kanasewich [2]_. 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", https://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 -------- >>> import matplotlib.pyplot as plt >>> np.bartlett(12) array([ 0. , 0.18181818, 0.36363636, 0.54545455, 0.72727273, # may vary 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") Text(0.5, 1.0, 'Bartlett window') >>> plt.ylabel("Amplitude") Text(0, 0.5, 'Amplitude') >>> plt.xlabel("Sample") Text(0.5, 0, 'Sample') >>> plt.show() >>> plt.figure() <Figure size 640x480 with 0 Axes> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> with np.errstate(divide='ignore', invalid='ignore'): ... 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") Text(0.5, 1.0, 'Frequency response of Bartlett window') >>> plt.ylabel("Magnitude [dB]") Text(0, 0.5, 'Magnitude [dB]') >>> plt.xlabel("Normalized frequency [cycles per sample]") Text(0.5, 0, 'Normalized frequency [cycles per sample]') >>> _ = plt.axis('tight') >>> plt.show() """ # Ensures at least float64 via 0.0. M should be an integer, but conversion # to double is safe for a range. values = np.array([0.0, M]) M = values[1] if M < 1: return array([], dtype=values.dtype) if M == 1: return ones(1, dtype=values.dtype) n = arange(1-M, M, 2) return where(less_equal(n, 0), 1 + n/(M-1), 1 - n/(M-1)) @set_module('numpy') def hanning(M): """ 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 - 0.5\\cos\\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", https://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: >>> import matplotlib.pyplot as plt >>> from numpy.fft import fft, fftshift >>> window = np.hanning(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Hann window") Text(0.5, 1.0, 'Hann window') >>> plt.ylabel("Amplitude") Text(0, 0.5, 'Amplitude') >>> plt.xlabel("Sample") Text(0.5, 0, 'Sample') >>> plt.show() >>> plt.figure() <Figure size 640x480 with 0 Axes> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> with np.errstate(divide='ignore', invalid='ignore'): ... 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") Text(0.5, 1.0, 'Frequency response of the Hann window') >>> plt.ylabel("Magnitude [dB]") Text(0, 0.5, 'Magnitude [dB]') >>> plt.xlabel("Normalized frequency [cycles per sample]") Text(0.5, 0, 'Normalized frequency [cycles per sample]') >>> plt.axis('tight') ... >>> plt.show() """ # Ensures at least float64 via 0.0. M should be an integer, but conversion # to double is safe for a range. values = np.array([0.0, M]) M = values[1] if M < 1: return array([], dtype=values.dtype) if M == 1: return ones(1, dtype=values.dtype) n = arange(1-M, M, 2) return 0.5 + 0.5*cos(pi*n/(M-1)) @set_module('numpy') def hamming(M): """ 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 - 0.46\\cos\\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", https://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, # may vary 0.98136677, 0.98136677, 0.84123594, 0.60546483, 0.34890909, 0.15302337, 0.08 ]) Plot the window and the frequency response: >>> import matplotlib.pyplot as plt >>> from numpy.fft import fft, fftshift >>> window = np.hamming(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Hamming window") Text(0.5, 1.0, 'Hamming window') >>> plt.ylabel("Amplitude") Text(0, 0.5, 'Amplitude') >>> plt.xlabel("Sample") Text(0.5, 0, 'Sample') >>> plt.show() >>> plt.figure() <Figure size 640x480 with 0 Axes> >>> 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") Text(0.5, 1.0, 'Frequency response of Hamming window') >>> plt.ylabel("Magnitude [dB]") Text(0, 0.5, 'Magnitude [dB]') >>> plt.xlabel("Normalized frequency [cycles per sample]") Text(0.5, 0, 'Normalized frequency [cycles per sample]') >>> plt.axis('tight') ... >>> plt.show() """ # Ensures at least float64 via 0.0. M should be an integer, but conversion # to double is safe for a range. values = np.array([0.0, M]) M = values[1] if M < 1: return array([], dtype=values.dtype) if M == 1: return ones(1, dtype=values.dtype) n = arange(1-M, M, 2) return 0.54 + 0.46*cos(pi*n/(M-1)) ## Code from cephes for i0 _i0A = [ -4.41534164647933937950E-18, 3.33079451882223809783E-17, -2.43127984654795469359E-16, 1.71539128555513303061E-15, -1.16853328779934516808E-14, 7.67618549860493561688E-14, -4.85644678311192946090E-13, 2.95505266312963983461E-12, -1.72682629144155570723E-11, 9.67580903537323691224E-11, -5.18979560163526290666E-10, 2.65982372468238665035E-9, -1.30002500998624804212E-8, 6.04699502254191894932E-8, -2.67079385394061173391E-7, 1.11738753912010371815E-6, -4.41673835845875056359E-6, 1.64484480707288970893E-5, -5.75419501008210370398E-5, 1.88502885095841655729E-4, -5.76375574538582365885E-4, 1.63947561694133579842E-3, -4.32430999505057594430E-3, 1.05464603945949983183E-2, -2.37374148058994688156E-2, 4.93052842396707084878E-2, -9.49010970480476444210E-2, 1.71620901522208775349E-1, -3.04682672343198398683E-1, 6.76795274409476084995E-1 ] _i0B = [ -7.23318048787475395456E-18, -4.83050448594418207126E-18, 4.46562142029675999901E-17, 3.46122286769746109310E-17, -2.82762398051658348494E-16, -3.42548561967721913462E-16, 1.77256013305652638360E-15, 3.81168066935262242075E-15, -9.55484669882830764870E-15, -4.15056934728722208663E-14, 1.54008621752140982691E-14, 3.85277838274214270114E-13, 7.18012445138366623367E-13, -1.79417853150680611778E-12, -1.32158118404477131188E-11, -3.14991652796324136454E-11, 1.18891471078464383424E-11, 4.94060238822496958910E-10, 3.39623202570838634515E-9, 2.26666899049817806459E-8, 2.04891858946906374183E-7, 2.89137052083475648297E-6, 6.88975834691682398426E-5, 3.36911647825569408990E-3, 8.04490411014108831608E-1 ] def _chbevl(x, vals): b0 = vals[0] b1 = 0.0 for i in range(1, len(vals)): b2 = b1 b1 = b0 b0 = x*b1 - b2 + vals[i] return 0.5*(b0 - b2) def _i0_1(x): return exp(x) * _chbevl(x/2.0-2, _i0A) def _i0_2(x): return exp(x) * _chbevl(32.0/x - 2.0, _i0B) / sqrt(x) def _i0_dispatcher(x): return (x,) @array_function_dispatch(_i0_dispatcher) def i0(x): """ Modified Bessel function of the first kind, order 0. Usually denoted :math:`I_0`. Parameters ---------- x : array_like of float Argument of the Bessel function. Returns ------- out : ndarray, shape = x.shape, dtype = float The modified Bessel function evaluated at each of the elements of `x`. See Also -------- scipy.special.i0, scipy.special.iv, scipy.special.ive Notes ----- The scipy implementation is recommended over this function: it is a proper ufunc written in C, and more than an order of magnitude faster. 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. https://personal.math.ubc.ca/~cbm/aands/page_379.htm .. [3] https://metacpan.org/pod/distribution/Math-Cephes/lib/Math/Cephes.pod#i0:-Modified-Bessel-function-of-order-zero Examples -------- >>> np.i0(0.) array(1.0) >>> np.i0([0, 1, 2, 3]) array([1. , 1.26606588, 2.2795853 , 4.88079259]) """ x = np.asanyarray(x) if x.dtype.kind == 'c': raise TypeError("i0 not supported for complex values") if x.dtype.kind != 'f': x = x.astype(float) x = np.abs(x) return piecewise(x, [x <= 8.0], [_i0_1, _i0_2]) ## End of cephes code for i0 @set_module('numpy') def kaiser(M, beta): """ 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", https://en.wikipedia.org/wiki/Window_function Examples -------- >>> import matplotlib.pyplot as plt >>> np.kaiser(12, 14) array([7.72686684e-06, 3.46009194e-03, 4.65200189e-02, # may vary 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") Text(0.5, 1.0, 'Kaiser window') >>> plt.ylabel("Amplitude") Text(0, 0.5, 'Amplitude') >>> plt.xlabel("Sample") Text(0.5, 0, 'Sample') >>> plt.show() >>> plt.figure() <Figure size 640x480 with 0 Axes> >>> 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") Text(0.5, 1.0, 'Frequency response of Kaiser window') >>> plt.ylabel("Magnitude [dB]") Text(0, 0.5, 'Magnitude [dB]') >>> plt.xlabel("Normalized frequency [cycles per sample]") Text(0.5, 0, 'Normalized frequency [cycles per sample]') >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) # may vary >>> plt.show() """ # Ensures at least float64 via 0.0. M should be an integer, but conversion # to double is safe for a range. (Simplified result_type with 0.0 # strongly typed. result-type is not/less order sensitive, but that mainly # matters for integers anyway.) values = np.array([0.0, M, beta]) M = values[1] beta = values[2] if M == 1: return np.ones(1, dtype=values.dtype) n = arange(0, M) alpha = (M-1)/2.0 return i0(beta * sqrt(1-((n-alpha)/alpha)**2.0))/i0(beta) def _sinc_dispatcher(x): return (x,) @array_function_dispatch(_sinc_dispatcher) def sinc(x): r""" Return the normalized sinc function. The sinc function is equal to :math:`\sin(\pi x)/(\pi x)` for any argument :math:`x\ne 0`. ``sinc(0)`` takes the limit value 1, making ``sinc`` not only everywhere continuous but also infinitely differentiable. .. note:: Note the normalization factor of ``pi`` used in the definition. This is the most commonly used definition in signal processing. Use ``sinc(x / np.pi)`` to obtain the unnormalized sinc function :math:`\sin(x)/x` that is more common in mathematics. Parameters ---------- x : ndarray Array (possibly multi-dimensional) of values for which to calculate ``sinc(x)``. Returns ------- out : ndarray ``sinc(x)``, which has the same shape as the input. Notes ----- 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", https://en.wikipedia.org/wiki/Sinc_function Examples -------- >>> import matplotlib.pyplot as plt >>> x = np.linspace(-4, 4, 41) >>> np.sinc(x) array([-3.89804309e-17, -4.92362781e-02, -8.40918587e-02, # may vary -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") Text(0.5, 1.0, 'Sinc Function') >>> plt.ylabel("Amplitude") Text(0, 0.5, 'Amplitude') >>> plt.xlabel("X") Text(0.5, 0, 'X') >>> plt.show() """ x = np.asanyarray(x) y = pi * where(x == 0, 1.0e-20, x) return sin(y)/y def _msort_dispatcher(a): return (a,) @array_function_dispatch(_msort_dispatcher) def msort(a): """ Return a copy of an array sorted along the first axis. .. deprecated:: 1.24 msort is deprecated, use ``np.sort(a, axis=0)`` instead. 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)``. Examples -------- >>> a = np.array([[1, 4], [3, 1]]) >>> np.msort(a) # sort along the first axis array([[1, 1], [3, 4]]) """ # 2022-10-20 1.24 warnings.warn( "msort is deprecated, use np.sort(a, axis=0) instead", DeprecationWarning, stacklevel=2, ) b = array(a, subok=True, copy=True) b.sort(0) return b def _ureduce(a, func, keepdims=False, **kwargs): """ 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 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. """ a = np.asanyarray(a) axis = kwargs.get('axis', None) out = kwargs.get('out', None) if keepdims is np._NoValue: keepdims = False nd = a.ndim if axis is not None: axis = _nx.normalize_axis_tuple(axis, nd) if keepdims: if out is not None: index_out = tuple( 0 if i in axis else slice(None) for i in range(nd)) kwargs['out'] = out[(Ellipsis, ) + index_out] if len(axis) == 1: kwargs['axis'] = axis[0] else: keep = set(range(nd)) - set(axis) nkeep = len(keep) # swap axis that should not be reduced to front for i, s in enumerate(sorted(keep)): a = a.swapaxes(i, s) # merge reduced axis a = a.reshape(a.shape[:nkeep] + (-1,)) kwargs['axis'] = -1 else: if keepdims: if out is not None: index_out = (0, ) * nd kwargs['out'] = out[(Ellipsis, ) + index_out] r = func(a, **kwargs) if out is not None: return out if keepdims: if axis is None: index_r = (np.newaxis, ) * nd else: index_r = tuple( np.newaxis if i in axis else slice(None) for i in range(nd)) r = r[(Ellipsis, ) + index_r] return r def _median_dispatcher( a, axis=None, out=None, overwrite_input=None, keepdims=None): return (a, out) @array_function_dispatch(_median_dispatcher) def median(a, axis=None, out=None, overwrite_input=False, keepdims=False): """ 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) """ return _ureduce(a, func=_median, keepdims=keepdims, axis=axis, out=out, overwrite_input=overwrite_input) def _median(a, axis=None, out=None, overwrite_input=False): # can't be reasonably be implemented in terms of percentile as we have to # call mean to not break astropy a = np.asanyarray(a) # Set the partition indexes if axis is None: sz = a.size else: sz = a.shape[axis] if sz % 2 == 0: szh = sz // 2 kth = [szh - 1, szh] else: kth = [(sz - 1) // 2] # We have to check for NaNs (as of writing 'M' doesn't actually work). supports_nans = np.issubdtype(a.dtype, np.inexact) or a.dtype.kind in 'Mm' if supports_nans: kth.append(-1) if overwrite_input: if axis is None: part = a.ravel() part.partition(kth) else: a.partition(kth, axis=axis) part = a else: part = partition(a, kth, axis=axis) if part.shape == (): # make 0-D arrays work return part.item() if axis is None: axis = 0 indexer = [slice(None)] * part.ndim index = part.shape[axis] // 2 if part.shape[axis] % 2 == 1: # index with slice to allow mean (below) to work indexer[axis] = slice(index, index+1) else: indexer[axis] = slice(index-1, index+1) indexer = tuple(indexer) # Use mean in both odd and even case to coerce data type, # using out array if needed. rout = mean(part[indexer], axis=axis, out=out) if supports_nans and sz > 0: # If nans are possible, warn and replace by nans like mean would. rout = np.lib.utils._median_nancheck(part, rout, axis) return rout def _percentile_dispatcher(a, q, axis=None, out=None, overwrite_input=None, method=None, keepdims=None, *, interpolation=None): return (a, q, out) @array_function_dispatch(_percentile_dispatcher) def percentile(a, q, axis=None, out=None, overwrite_input=False, method="linear", keepdims=False, *, interpolation=None): """ Compute the q-th percentile of the data along the specified axis. Returns the q-th percentile(s) of the array elements. Parameters ---------- a : array_like of real numbers Input array or object that can be converted to an array. q : array_like of float Percentage or sequence of percentages for the percentiles to compute. Values must be between 0 and 100 inclusive. axis : {int, tuple 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. .. versionchanged:: 1.9.0 A tuple of axes is supported 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 the input array `a` to be modified by intermediate calculations, to save memory. In this case, the contents of the input `a` after this function completes is undefined. method : str, optional This parameter specifies the method to use for estimating the percentile. There are many different methods, some unique to NumPy. See the notes for explanation. The options sorted by their R type as summarized in the H&F paper [1]_ are: 1. 'inverted_cdf' 2. 'averaged_inverted_cdf' 3. 'closest_observation' 4. 'interpolated_inverted_cdf' 5. 'hazen' 6. 'weibull' 7. 'linear' (default) 8. 'median_unbiased' 9. 'normal_unbiased' The first three methods are discontinuous. NumPy further defines the following discontinuous variations of the default 'linear' (7.) option: * 'lower' * 'higher', * 'midpoint' * 'nearest' .. versionchanged:: 1.22.0 This argument was previously called "interpolation" and only offered the "linear" default and last four options. 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 interpolation : str, optional Deprecated name for the method keyword argument. .. deprecated:: 1.22.0 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 : equivalent to ``percentile(..., 50)`` nanpercentile quantile : equivalent to percentile, except q in the range [0, 1]. 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 `method` 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``. The optional `method` parameter specifies the method to use when the desired percentile lies between two indexes ``i`` and ``j = i + 1``. In that case, we first determine ``i + g``, a virtual index that lies between ``i`` and ``j``, where ``i`` is the floor and ``g`` is the fractional part of the index. The final result is, then, an interpolation of ``a[i]`` and ``a[j]`` based on ``g``. During the computation of ``g``, ``i`` and ``j`` are modified using correction constants ``alpha`` and ``beta`` whose choices depend on the ``method`` used. Finally, note that since Python uses 0-based indexing, the code subtracts another 1 from the index internally. The following formula determines the virtual index ``i + g``, the location of the percentile in the sorted sample: .. math:: i + g = (q / 100) * ( n - alpha - beta + 1 ) + alpha The different methods then work as follows inverted_cdf: method 1 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 ; then take i averaged_inverted_cdf: method 2 of H&F [1]_. This method give discontinuous results: * if g > 0 ; then take j * if g = 0 ; then average between bounds closest_observation: method 3 of H&F [1]_. This method give discontinuous results: * if g > 0 ; then take j * if g = 0 and index is odd ; then take j * if g = 0 and index is even ; then take i interpolated_inverted_cdf: method 4 of H&F [1]_. This method give continuous results using: * alpha = 0 * beta = 1 hazen: method 5 of H&F [1]_. This method give continuous results using: * alpha = 1/2 * beta = 1/2 weibull: method 6 of H&F [1]_. This method give continuous results using: * alpha = 0 * beta = 0 linear: method 7 of H&F [1]_. This method give continuous results using: * alpha = 1 * beta = 1 median_unbiased: method 8 of H&F [1]_. This method is probably the best method if the sample distribution function is unknown (see reference). This method give continuous results using: * alpha = 1/3 * beta = 1/3 normal_unbiased: method 9 of H&F [1]_. This method is probably the best method if the sample distribution function is known to be normal. This method give continuous results using: * alpha = 3/8 * beta = 3/8 lower: NumPy method kept for backwards compatibility. Takes ``i`` as the interpolation point. higher: NumPy method kept for backwards compatibility. Takes ``j`` as the interpolation point. nearest: NumPy method kept for backwards compatibility. Takes ``i`` or ``j``, whichever is nearest. midpoint: NumPy method kept for backwards compatibility. Uses ``(i + j) / 2``. 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) The different methods can be visualized graphically: .. plot:: import matplotlib.pyplot as plt a = np.arange(4) p = np.linspace(0, 100, 6001) ax = plt.gca() lines = [ ('linear', '-', 'C0'), ('inverted_cdf', ':', 'C1'), # Almost the same as `inverted_cdf`: ('averaged_inverted_cdf', '-.', 'C1'), ('closest_observation', ':', 'C2'), ('interpolated_inverted_cdf', '--', 'C1'), ('hazen', '--', 'C3'), ('weibull', '-.', 'C4'), ('median_unbiased', '--', 'C5'), ('normal_unbiased', '-.', 'C6'), ] for method, style, color in lines: ax.plot( p, np.percentile(a, p, method=method), label=method, linestyle=style, color=color) ax.set( title='Percentiles for different methods and data: ' + str(a), xlabel='Percentile', ylabel='Estimated percentile value', yticks=a) ax.legend(bbox_to_anchor=(1.03, 1)) plt.tight_layout() plt.show() References ---------- .. [1] R. J. Hyndman and Y. Fan, "Sample quantiles in statistical packages," The American Statistician, 50(4), pp. 361-365, 1996 """ if interpolation is not None: method = _check_interpolation_as_method( method, interpolation, "percentile") a = np.asanyarray(a) if a.dtype.kind == "c": raise TypeError("a must be an array of real numbers") q = np.true_divide(q, 100) q = asanyarray(q) # undo any decay that the ufunc performed (see gh-13105) if not _quantile_is_valid(q): raise ValueError("Percentiles must be in the range [0, 100]") return _quantile_unchecked( a, q, axis, out, overwrite_input, method, keepdims) def _quantile_dispatcher(a, q, axis=None, out=None, overwrite_input=None, method=None, keepdims=None, *, interpolation=None): return (a, q, out) @array_function_dispatch(_quantile_dispatcher) def quantile(a, q, axis=None, out=None, overwrite_input=False, method="linear", keepdims=False, *, interpolation=None): """ Compute the q-th quantile of the data along the specified axis. .. versionadded:: 1.15.0 Parameters ---------- a : array_like of real numbers Input array or object that can be converted to an array. q : array_like of float Probability or sequence of probabilities for the quantiles to compute. Values must be between 0 and 1 inclusive. axis : {int, tuple of int, None}, optional Axis or axes along which the quantiles are computed. The default is to compute the quantile(s) along a flattened version of the array. 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 the input array `a` to be modified by intermediate calculations, to save memory. In this case, the contents of the input `a` after this function completes is undefined. method : str, optional This parameter specifies the method to use for estimating the quantile. There are many different methods, some unique to NumPy. See the notes for explanation. The options sorted by their R type as summarized in the H&F paper [1]_ are: 1. 'inverted_cdf' 2. 'averaged_inverted_cdf' 3. 'closest_observation' 4. 'interpolated_inverted_cdf' 5. 'hazen' 6. 'weibull' 7. 'linear' (default) 8. 'median_unbiased' 9. 'normal_unbiased' The first three methods are discontinuous. NumPy further defines the following discontinuous variations of the default 'linear' (7.) option: * 'lower' * 'higher', * 'midpoint' * 'nearest' .. versionchanged:: 1.22.0 This argument was previously called "interpolation" and only offered the "linear" default and last four options. 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`. interpolation : str, optional Deprecated name for the method keyword argument. .. deprecated:: 1.22.0 Returns ------- quantile : scalar or ndarray If `q` is a single probability and `axis=None`, then the result is a scalar. If multiple probabilies levels are given, first axis of the result corresponds to the quantiles. 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 percentile : equivalent to quantile, but with q in the range [0, 100]. median : equivalent to ``quantile(..., 0.5)`` nanquantile Notes ----- Given a vector ``V`` of length ``n``, the q-th quantile of ``V`` is the value ``q`` 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 `method` parameter will determine the quantile if the normalized ranking does not match the location of ``q`` exactly. This function is the same as the median if ``q=0.5``, the same as the minimum if ``q=0.0`` and the same as the maximum if ``q=1.0``. The optional `method` parameter specifies the method to use when the desired quantile lies between two indexes ``i`` and ``j = i + 1``. In that case, we first determine ``i + g``, a virtual index that lies between ``i`` and ``j``, where ``i`` is the floor and ``g`` is the fractional part of the index. The final result is, then, an interpolation of ``a[i]`` and ``a[j]`` based on ``g``. During the computation of ``g``, ``i`` and ``j`` are modified using correction constants ``alpha`` and ``beta`` whose choices depend on the ``method`` used. Finally, note that since Python uses 0-based indexing, the code subtracts another 1 from the index internally. The following formula determines the virtual index ``i + g``, the location of the quantile in the sorted sample: .. math:: i + g = q * ( n - alpha - beta + 1 ) + alpha The different methods then work as follows inverted_cdf: method 1 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 ; then take i averaged_inverted_cdf: method 2 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 ; then average between bounds closest_observation: method 3 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 and index is odd ; then take j * if g = 0 and index is even ; then take i interpolated_inverted_cdf: method 4 of H&F [1]_. This method gives continuous results using: * alpha = 0 * beta = 1 hazen: method 5 of H&F [1]_. This method gives continuous results using: * alpha = 1/2 * beta = 1/2 weibull: method 6 of H&F [1]_. This method gives continuous results using: * alpha = 0 * beta = 0 linear: method 7 of H&F [1]_. This method gives continuous results using: * alpha = 1 * beta = 1 median_unbiased: method 8 of H&F [1]_. This method is probably the best method if the sample distribution function is unknown (see reference). This method gives continuous results using: * alpha = 1/3 * beta = 1/3 normal_unbiased: method 9 of H&F [1]_. This method is probably the best method if the sample distribution function is known to be normal. This method gives continuous results using: * alpha = 3/8 * beta = 3/8 lower: NumPy method kept for backwards compatibility. Takes ``i`` as the interpolation point. higher: NumPy method kept for backwards compatibility. Takes ``j`` as the interpolation point. nearest: NumPy method kept for backwards compatibility. Takes ``i`` or ``j``, whichever is nearest. midpoint: NumPy method kept for backwards compatibility. Uses ``(i + j) / 2``. Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.quantile(a, 0.5) 3.5 >>> np.quantile(a, 0.5, axis=0) array([6.5, 4.5, 2.5]) >>> np.quantile(a, 0.5, axis=1) array([7., 2.]) >>> np.quantile(a, 0.5, axis=1, keepdims=True) array([[7.], [2.]]) >>> m = np.quantile(a, 0.5, axis=0) >>> out = np.zeros_like(m) >>> np.quantile(a, 0.5, axis=0, out=out) array([6.5, 4.5, 2.5]) >>> m array([6.5, 4.5, 2.5]) >>> b = a.copy() >>> np.quantile(b, 0.5, axis=1, overwrite_input=True) array([7., 2.]) >>> assert not np.all(a == b) See also `numpy.percentile` for a visualization of most methods. References ---------- .. [1] R. J. Hyndman and Y. Fan, "Sample quantiles in statistical packages," The American Statistician, 50(4), pp. 361-365, 1996 """ if interpolation is not None: method = _check_interpolation_as_method( method, interpolation, "quantile") a = np.asanyarray(a) if a.dtype.kind == "c": raise TypeError("a must be an array of real numbers") q = np.asanyarray(q) if not _quantile_is_valid(q): raise ValueError("Quantiles must be in the range [0, 1]") return _quantile_unchecked( a, q, axis, out, overwrite_input, method, keepdims) def _quantile_unchecked(a, q, axis=None, out=None, overwrite_input=False, method="linear", keepdims=False): """Assumes that q is in [0, 1], and is an ndarray""" return _ureduce(a, func=_quantile_ureduce_func, q=q, keepdims=keepdims, axis=axis, out=out, overwrite_input=overwrite_input, method=method) def _quantile_is_valid(q): # avoid expensive reductions, relevant for arrays with < O(1000) elements if q.ndim == 1 and q.size < 10: for i in range(q.size): if not (0.0 <= q[i] <= 1.0): return False else: if not (np.all(0 <= q) and np.all(q <= 1)): return False return True def _check_interpolation_as_method(method, interpolation, fname): # Deprecated NumPy 1.22, 2021-11-08 warnings.warn( f"the `interpolation=` argument to {fname} was renamed to " "`method=`, which has additional options.\n" "Users of the modes 'nearest', 'lower', 'higher', or " "'midpoint' are encouraged to review the method they used. " "(Deprecated NumPy 1.22)", DeprecationWarning, stacklevel=4) if method != "linear": # sanity check, we assume this basically never happens raise TypeError( "You shall not pass both `method` and `interpolation`!\n" "(`interpolation` is Deprecated in favor of `method`)") return interpolation def _compute_virtual_index(n, quantiles, alpha: float, beta: float): """ Compute the floating point indexes of an array for the linear interpolation of quantiles. n : array_like The sample sizes. quantiles : array_like The quantiles values. alpha : float A constant used to correct the index computed. beta : float A constant used to correct the index computed. alpha and beta values depend on the chosen method (see quantile documentation) Reference: Hyndman&Fan paper "Sample Quantiles in Statistical Packages", DOI: 10.1080/00031305.1996.10473566 """ return n * quantiles + ( alpha + quantiles * (1 - alpha - beta) ) - 1 def _get_gamma(virtual_indexes, previous_indexes, method): """ Compute gamma (a.k.a 'm' or 'weight') for the linear interpolation of quantiles. virtual_indexes : array_like The indexes where the percentile is supposed to be found in the sorted sample. previous_indexes : array_like The floor values of virtual_indexes. interpolation : dict The interpolation method chosen, which may have a specific rule modifying gamma. gamma is usually the fractional part of virtual_indexes but can be modified by the interpolation method. """ gamma = np.asanyarray(virtual_indexes - previous_indexes) gamma = method["fix_gamma"](gamma, virtual_indexes) return np.asanyarray(gamma) def _lerp(a, b, t, out=None): """ Compute the linear interpolation weighted by gamma on each point of two same shape array. a : array_like Left bound. b : array_like Right bound. t : array_like The interpolation weight. out : array_like Output array. """ diff_b_a = subtract(b, a) # asanyarray is a stop-gap until gh-13105 lerp_interpolation = asanyarray(add(a, diff_b_a * t, out=out)) subtract(b, diff_b_a * (1 - t), out=lerp_interpolation, where=t >= 0.5) if lerp_interpolation.ndim == 0 and out is None: lerp_interpolation = lerp_interpolation[()] # unpack 0d arrays return lerp_interpolation def _get_gamma_mask(shape, default_value, conditioned_value, where): out = np.full(shape, default_value) np.copyto(out, conditioned_value, where=where, casting="unsafe") return out def _discret_interpolation_to_boundaries(index, gamma_condition_fun): previous = np.floor(index) next = previous + 1 gamma = index - previous res = _get_gamma_mask(shape=index.shape, default_value=next, conditioned_value=previous, where=gamma_condition_fun(gamma, index) ).astype(np.intp) # Some methods can lead to out-of-bound integers, clip them: res[res < 0] = 0 return res def _closest_observation(n, quantiles): gamma_fun = lambda gamma, index: (gamma == 0) & (np.floor(index) % 2 == 0) return _discret_interpolation_to_boundaries((n * quantiles) - 1 - 0.5, gamma_fun) def _inverted_cdf(n, quantiles): gamma_fun = lambda gamma, _: (gamma == 0) return _discret_interpolation_to_boundaries((n * quantiles) - 1, gamma_fun) def _quantile_ureduce_func( a: np.array, q: np.array, axis: int = None, out=None, overwrite_input: bool = False, method="linear", ) -> np.array: if q.ndim > 2: # The code below works fine for nd, but it might not have useful # semantics. For now, keep the supported dimensions the same as it was # before. raise ValueError("q must be a scalar or 1d") if overwrite_input: if axis is None: axis = 0 arr = a.ravel() else: arr = a else: if axis is None: axis = 0 arr = a.flatten() else: arr = a.copy() result = _quantile(arr, quantiles=q, axis=axis, method=method, out=out) return result def _get_indexes(arr, virtual_indexes, valid_values_count): """ Get the valid indexes of arr neighbouring virtual_indexes. Note This is a companion function to linear interpolation of Quantiles Returns ------- (previous_indexes, next_indexes): Tuple A Tuple of virtual_indexes neighbouring indexes """ previous_indexes = np.asanyarray(np.floor(virtual_indexes)) next_indexes = np.asanyarray(previous_indexes + 1) indexes_above_bounds = virtual_indexes >= valid_values_count - 1 # When indexes is above max index, take the max value of the array if indexes_above_bounds.any(): previous_indexes[indexes_above_bounds] = -1 next_indexes[indexes_above_bounds] = -1 # When indexes is below min index, take the min value of the array indexes_below_bounds = virtual_indexes < 0 if indexes_below_bounds.any(): previous_indexes[indexes_below_bounds] = 0 next_indexes[indexes_below_bounds] = 0 if np.issubdtype(arr.dtype, np.inexact): # After the sort, slices having NaNs will have for last element a NaN virtual_indexes_nans = np.isnan(virtual_indexes) if virtual_indexes_nans.any(): previous_indexes[virtual_indexes_nans] = -1 next_indexes[virtual_indexes_nans] = -1 previous_indexes = previous_indexes.astype(np.intp) next_indexes = next_indexes.astype(np.intp) return previous_indexes, next_indexes def _quantile( arr: np.array, quantiles: np.array, axis: int = -1, method="linear", out=None, ): """ Private function that doesn't support extended axis or keepdims. These methods are extended to this function using _ureduce See nanpercentile for parameter usage It computes the quantiles of the array for the given axis. A linear interpolation is performed based on the `interpolation`. By default, the method is "linear" where alpha == beta == 1 which performs the 7th method of Hyndman&Fan. With "median_unbiased" we get alpha == beta == 1/3 thus the 8th method of Hyndman&Fan. """ # --- Setup arr = np.asanyarray(arr) values_count = arr.shape[axis] # The dimensions of `q` are prepended to the output shape, so we need the # axis being sampled from `arr` to be last. if axis != 0: # But moveaxis is slow, so only call it if necessary. arr = np.moveaxis(arr, axis, destination=0) # --- Computation of indexes # Index where to find the value in the sorted array. # Virtual because it is a floating point value, not an valid index. # The nearest neighbours are used for interpolation try: method = _QuantileMethods[method] except KeyError: raise ValueError( f"{method!r} is not a valid method. Use one of: " f"{_QuantileMethods.keys()}") from None virtual_indexes = method["get_virtual_index"](values_count, quantiles) virtual_indexes = np.asanyarray(virtual_indexes) supports_nans = ( np.issubdtype(arr.dtype, np.inexact) or arr.dtype.kind in 'Mm') if np.issubdtype(virtual_indexes.dtype, np.integer): # No interpolation needed, take the points along axis if supports_nans: # may contain nan, which would sort to the end arr.partition(concatenate((virtual_indexes.ravel(), [-1])), axis=0) slices_having_nans = np.isnan(arr[-1, ...]) else: # cannot contain nan arr.partition(virtual_indexes.ravel(), axis=0) slices_having_nans = np.array(False, dtype=bool) result = take(arr, virtual_indexes, axis=0, out=out) else: previous_indexes, next_indexes = _get_indexes(arr, virtual_indexes, values_count) # --- Sorting arr.partition( np.unique(np.concatenate(([0, -1], previous_indexes.ravel(), next_indexes.ravel(), ))), axis=0) if supports_nans: slices_having_nans = np.isnan(arr[-1, ...]) else: slices_having_nans = None # --- Get values from indexes previous = arr[previous_indexes] next = arr[next_indexes] # --- Linear interpolation gamma = _get_gamma(virtual_indexes, previous_indexes, method) result_shape = virtual_indexes.shape + (1,) * (arr.ndim - 1) gamma = gamma.reshape(result_shape) result = _lerp(previous, next, gamma, out=out) if np.any(slices_having_nans): if result.ndim == 0 and out is None: # can't write to a scalar, but indexing will be correct result = arr[-1] else: np.copyto(result, arr[-1, ...], where=slices_having_nans) return result def _trapz_dispatcher(y, x=None, dx=None, axis=None): return (y, x) @array_function_dispatch(_trapz_dispatcher) def trapz(y, x=None, dx=1.0, axis=-1): r""" Integrate along the given axis using the composite trapezoidal rule. If `x` is provided, the integration happens in sequence along its elements - they are not sorted. Integrate `y` (`x`) along each 1d slice on the given axis, compute :math:`\int y(x) dx`. When `x` is specified, this integrates along the parametric curve, computing :math:`\int_t y(t) dt = \int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt`. 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 or ndarray Definite integral of `y` = n-dimensional array as approximated along a single axis by the trapezoidal rule. If `y` is a 1-dimensional array, then the result is a float. If `n` is greater than 1, then the result is an `n`-1 dimensional array. 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: https://en.wikipedia.org/wiki/Trapezoidal_rule .. [2] Illustration image: https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png Examples -------- Use the trapezoidal rule on evenly spaced points: >>> np.trapz([1, 2, 3]) 4.0 The spacing between sample points can be selected by either the ``x`` or ``dx`` arguments: >>> np.trapz([1, 2, 3], x=[4, 6, 8]) 8.0 >>> np.trapz([1, 2, 3], dx=2) 8.0 Using a decreasing ``x`` corresponds to integrating in reverse: >>> np.trapz([1, 2, 3], x=[8, 6, 4]) -8.0 More generally ``x`` is used to integrate along a parametric curve. We can estimate the integral :math:`\int_0^1 x^2 = 1/3` using: >>> x = np.linspace(0, 1, num=50) >>> y = x**2 >>> np.trapz(y, x) 0.33340274885464394 Or estimate the area of a circle, noting we repeat the sample which closes the curve: >>> theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True) >>> np.trapz(np.cos(theta), x=np.sin(theta)) 3.141571941375841 ``np.trapz`` can be applied along a specified axis to do multiple computations in one call: >>> 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.]) """ y = asanyarray(y) if x is None: d = dx else: x = asanyarray(x) if x.ndim == 1: d = diff(x) # reshape to correct shape shape = [1]*y.ndim shape[axis] = d.shape[0] d = d.reshape(shape) else: d = diff(x, axis=axis) nd = y.ndim slice1 = [slice(None)]*nd slice2 = [slice(None)]*nd slice1[axis] = slice(1, None) slice2[axis] = slice(None, -1) try: ret = (d * (y[tuple(slice1)] + y[tuple(slice2)]) / 2.0).sum(axis) except ValueError: # Operations didn't work, cast to ndarray d = np.asarray(d) y = np.asarray(y) ret = add.reduce(d * (y[tuple(slice1)]+y[tuple(slice2)])/2.0, axis) return ret # __array_function__ has no __code__ or other attributes normal Python funcs we # wrap everything into a C callable. SciPy however, tries to "clone" `trapz` # into a new Python function which requires `__code__` and a few other # attributes. So we create a dummy clone and copy over its attributes allowing # SciPy <= 1.10 to work: https://github.com/scipy/scipy/issues/17811 assert not hasattr(trapz, "__code__") def _fake_trapz(y, x=None, dx=1.0, axis=-1): return trapz(y, x=x, dx=dx, axis=axis) trapz.__code__ = _fake_trapz.__code__ trapz.__globals__ = _fake_trapz.__globals__ trapz.__defaults__ = _fake_trapz.__defaults__ trapz.__closure__ = _fake_trapz.__closure__ trapz.__kwdefaults__ = _fake_trapz.__kwdefaults__ def _meshgrid_dispatcher(*xi, copy=None, sparse=None, indexing=None): return xi # Based on scitools meshgrid @array_function_dispatch(_meshgrid_dispatcher) def meshgrid(*xi, copy=True, sparse=False, indexing='xy'): """ Return a list of 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 the shape of the returned coordinate array for dimension *i* is reduced from ``(N1, ..., Ni, ... Nn)`` to ``(1, ..., 1, Ni, 1, ..., 1)``. These sparse coordinate grids are intended to be use with :ref:`basics.broadcasting`. When all coordinates are used in an expression, broadcasting still leads to a fully-dimensonal result array. 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 : list of ndarrays For vectors `x1`, `x2`,..., `xn` with lengths ``Ni=len(xi)``, returns ``(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, 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, 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 -------- mgrid : Construct a multi-dimensional "meshgrid" using indexing notation. ogrid : Construct an open multi-dimensional "meshgrid" using indexing notation. how-to-index 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.]]) The result of `meshgrid` is a coordinate grid: >>> import matplotlib.pyplot as plt >>> plt.plot(xv, yv, marker='o', color='k', linestyle='none') >>> plt.show() You can create sparse output arrays to save memory and computation time. >>> xv, yv = np.meshgrid(x, y, sparse=True) >>> xv array([[0. , 0.5, 1. ]]) >>> yv array([[0.], [1.]]) `meshgrid` is very useful to evaluate functions on a grid. If the function depends on all coordinates, both dense and sparse outputs can be used. >>> x = np.linspace(-5, 5, 101) >>> y = np.linspace(-5, 5, 101) >>> # full coordinate arrays >>> xx, yy = np.meshgrid(x, y) >>> zz = np.sqrt(xx**2 + yy**2) >>> xx.shape, yy.shape, zz.shape ((101, 101), (101, 101), (101, 101)) >>> # sparse coordinate arrays >>> xs, ys = np.meshgrid(x, y, sparse=True) >>> zs = np.sqrt(xs**2 + ys**2) >>> xs.shape, ys.shape, zs.shape ((1, 101), (101, 1), (101, 101)) >>> np.array_equal(zz, zs) True >>> h = plt.contourf(x, y, zs) >>> plt.axis('scaled') >>> plt.colorbar() >>> plt.show() """ ndim = len(xi) if indexing not in ['xy', 'ij']: raise ValueError( "Valid values for `indexing` are 'xy' and 'ij'.") s0 = (1,) * ndim output = [np.asanyarray(x).reshape(s0[:i] + (-1,) + s0[i + 1:]) for i, x in enumerate(xi)] if indexing == 'xy' and ndim > 1: # switch first and second axis output[0].shape = (1, -1) + s0[2:] output[1].shape = (-1, 1) + s0[2:] if not sparse: # Return the full N-D matrix (not only the 1-D vector) output = np.broadcast_arrays(*output, subok=True) if copy: output = [x.copy() for x in output] return output def _delete_dispatcher(arr, obj, axis=None): return (arr, obj) @array_function_dispatch(_delete_dispatcher) def delete(arr, obj, axis=None): """ 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 indices of sub-arrays to remove along the specified axis. .. versionchanged:: 1.19.0 Boolean indices are now treated as a mask of elements to remove, rather than being cast to the integers 0 and 1. 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: >>> arr = np.arange(12) + 1 >>> 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]) """ wrap = None if type(arr) is not ndarray: try: wrap = arr.__array_wrap__ except AttributeError: pass arr = asarray(arr) ndim = arr.ndim arrorder = 'F' if arr.flags.fnc else 'C' if axis is None: if ndim != 1: arr = arr.ravel() # needed for np.matrix, which is still not 1d after being ravelled ndim = arr.ndim axis = ndim - 1 else: axis = normalize_axis_index(axis, ndim) slobj = [slice(None)]*ndim N = arr.shape[axis] newshape = list(arr.shape) if isinstance(obj, slice): start, stop, step = obj.indices(N) xr = range(start, stop, step) numtodel = len(xr) if numtodel <= 0: if wrap: return wrap(arr.copy(order=arrorder)) else: return arr.copy(order=arrorder) # Invert if step is negative: if step < 0: step = -step start = xr[-1] stop = xr[0] + 1 newshape[axis] -= numtodel new = empty(newshape, arr.dtype, arrorder) # copy initial chunk if start == 0: pass else: slobj[axis] = slice(None, start) new[tuple(slobj)] = arr[tuple(slobj)] # copy end chunk if stop == N: pass else: slobj[axis] = slice(stop-numtodel, None) slobj2 = [slice(None)]*ndim slobj2[axis] = slice(stop, None) new[tuple(slobj)] = arr[tuple(slobj2)] # copy middle pieces if step == 1: pass else: # use array indexing. keep = ones(stop-start, dtype=bool) keep[:stop-start:step] = False slobj[axis] = slice(start, stop-numtodel) slobj2 = [slice(None)]*ndim slobj2[axis] = slice(start, stop) arr = arr[tuple(slobj2)] slobj2[axis] = keep new[tuple(slobj)] = arr[tuple(slobj2)] if wrap: return wrap(new) else: return new if isinstance(obj, (int, integer)) and not isinstance(obj, bool): single_value = True else: single_value = False _obj = obj obj = np.asarray(obj) # `size == 0` to allow empty lists similar to indexing, but (as there) # is really too generic: if obj.size == 0 and not isinstance(_obj, np.ndarray): obj = obj.astype(intp) elif obj.size == 1 and obj.dtype.kind in "ui": # For a size 1 integer array we can use the single-value path # (most dtypes, except boolean, should just fail later). obj = obj.item() single_value = True if single_value: # optimization for a single value if (obj < -N or obj >= N): raise IndexError( "index %i is out of bounds for axis %i with " "size %i" % (obj, axis, N)) if (obj < 0): obj += N newshape[axis] -= 1 new = empty(newshape, arr.dtype, arrorder) slobj[axis] = slice(None, obj) new[tuple(slobj)] = arr[tuple(slobj)] slobj[axis] = slice(obj, None) slobj2 = [slice(None)]*ndim slobj2[axis] = slice(obj+1, None) new[tuple(slobj)] = arr[tuple(slobj2)] else: if obj.dtype == bool: if obj.shape != (N,): raise ValueError('boolean array argument obj to delete ' 'must be one dimensional and match the axis ' 'length of {}'.format(N)) # optimization, the other branch is slower keep = ~obj else: keep = ones(N, dtype=bool) keep[obj,] = False slobj[axis] = keep new = arr[tuple(slobj)] if wrap: return wrap(new) else: return new def _insert_dispatcher(arr, obj, values, axis=None): return (arr, obj, values) @array_function_dispatch(_insert_dispatcher) def insert(arr, obj, values, axis=None): """ 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, 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, ..., 2, 3, 3]) >>> np.insert(b, slice(2, 4), [5, 6]) array([1, 1, 5, ..., 2, 3, 3]) >>> np.insert(b, [2, 2], [7.13, False]) # type casting array([1, 1, 7, ..., 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]]) """ wrap = None if type(arr) is not ndarray: try: wrap = arr.__array_wrap__ except AttributeError: pass arr = asarray(arr) ndim = arr.ndim arrorder = 'F' if arr.flags.fnc else 'C' if axis is None: if ndim != 1: arr = arr.ravel() # needed for np.matrix, which is still not 1d after being ravelled ndim = arr.ndim axis = ndim - 1 else: axis = normalize_axis_index(axis, ndim) slobj = [slice(None)]*ndim N = arr.shape[axis] newshape = list(arr.shape) if isinstance(obj, slice): # turn it into a range object indices = arange(*obj.indices(N), dtype=intp) else: # need to copy obj, because indices will be changed in-place indices = np.array(obj) if indices.dtype == bool: # See also delete # 2012-10-11, NumPy 1.8 warnings.warn( "in the future insert will treat boolean arrays and " "array-likes as a boolean index instead of casting it to " "integer", FutureWarning, stacklevel=2) indices = indices.astype(intp) # Code after warning period: #if obj.ndim != 1: # raise ValueError('boolean array argument obj to insert ' # 'must be one dimensional') #indices = np.flatnonzero(obj) elif indices.ndim > 1: raise ValueError( "index array argument obj to insert must be one dimensional " "or scalar") if indices.size == 1: index = indices.item() if index < -N or index > N: raise IndexError(f"index {obj} is out of bounds for axis {axis} " f"with size {N}") if (index < 0): index += N # There are some object array corner cases here, but we cannot avoid # that: values = array(values, copy=False, ndmin=arr.ndim, dtype=arr.dtype) if indices.ndim == 0: # broadcasting is very different here, since a[:,0,:] = ... behaves # very different from a[:,[0],:] = ...! This changes values so that # it works likes the second case. (here a[:,0:1,:]) values = np.moveaxis(values, 0, axis) numnew = values.shape[axis] newshape[axis] += numnew new = empty(newshape, arr.dtype, arrorder) slobj[axis] = slice(None, index) new[tuple(slobj)] = arr[tuple(slobj)] slobj[axis] = slice(index, index+numnew) new[tuple(slobj)] = values slobj[axis] = slice(index+numnew, None) slobj2 = [slice(None)] * ndim slobj2[axis] = slice(index, None) new[tuple(slobj)] = arr[tuple(slobj2)] if wrap: return wrap(new) return new elif indices.size == 0 and not isinstance(obj, np.ndarray): # Can safely cast the empty list to intp indices = indices.astype(intp) indices[indices < 0] += N numnew = len(indices) order = indices.argsort(kind='mergesort') # stable sort indices[order] += np.arange(numnew) newshape[axis] += numnew old_mask = ones(newshape[axis], dtype=bool) old_mask[indices] = False new = empty(newshape, arr.dtype, arrorder) slobj2 = [slice(None)]*ndim slobj[axis] = indices slobj2[axis] = old_mask new[tuple(slobj)] = values new[tuple(slobj2)] = arr if wrap: return wrap(new) return new def _append_dispatcher(arr, values, axis=None): return (arr, values) @array_function_dispatch(_append_dispatcher) def append(arr, values, axis=None): """ 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, ..., 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: all the input arrays must have same number of dimensions, but the array at index 0 has 2 dimension(s) and the array at index 1 has 1 dimension(s) """ arr = asanyarray(arr) if axis is None: if arr.ndim != 1: arr = arr.ravel() values = ravel(values) axis = arr.ndim-1 return concatenate((arr, values), axis=axis) def _digitize_dispatcher(x, bins, right=None): return (x, bins) @array_function_dispatch(_digitize_dispatcher) def digitize(x, bins, right=False): """ Return the indices of the bins to which each value in input array belongs. ========= ============= ============================ `right` order of bins returned index `i` satisfies ========= ============= ============================ ``False`` increasing ``bins[i-1] <= x < bins[i]`` ``True`` increasing ``bins[i-1] < x <= bins[i]`` ``False`` decreasing ``bins[i-1] > x >= bins[i]`` ``True`` decreasing ``bins[i-1] >= x > bins[i]`` ========= ============= ============================ If values in `x` are beyond the bounds of `bins`, 0 or ``len(bins)`` is returned as appropriate. Parameters ---------- x : array_like Input array to be binned. Prior to NumPy 1.10.0, this array had to be 1-dimensional, but can now have any shape. bins : array_like Array of bins. It has to be 1-dimensional and monotonic. right : bool, optional Indicating whether the intervals include the right or the left bin edge. Default behavior is (right==False) indicating that the interval does not include the right edge. The left bin end is open in this case, i.e., bins[i-1] <= x < bins[i] is the default behavior for monotonically increasing bins. Returns ------- indices : ndarray of ints Output array of indices, of same shape as `x`. Raises ------ ValueError If `bins` is not monotonic. TypeError If the type of the input is complex. See Also -------- bincount, histogram, unique, searchsorted Notes ----- If values in `x` are such that they fall outside the bin range, attempting to index `bins` with the indices that `digitize` returns will result in an IndexError. .. versionadded:: 1.10.0 `np.digitize` is implemented in terms of `np.searchsorted`. This means that a binary search is used to bin the values, which scales much better for larger number of bins than the previous linear search. It also removes the requirement for the input array to be 1-dimensional. For monotonically _increasing_ `bins`, the following are equivalent:: np.digitize(x, bins, right=True) np.searchsorted(bins, x, side='left') Note that as the order of the arguments are reversed, the side must be too. The `searchsorted` call is marginally faster, as it does not do any monotonicity checks. Perhaps more importantly, it supports all dtypes. Examples -------- >>> x = np.array([0.2, 6.4, 3.0, 1.6]) >>> bins = np.array([0.0, 1.0, 2.5, 4.0, 10.0]) >>> inds = np.digitize(x, bins) >>> inds array([1, 4, 3, 2]) >>> for n in range(x.size): ... print(bins[inds[n]-1], "<=", x[n], "<", bins[inds[n]]) ... 0.0 <= 0.2 < 1.0 4.0 <= 6.4 < 10.0 2.5 <= 3.0 < 4.0 1.0 <= 1.6 < 2.5 >>> x = np.array([1.2, 10.0, 12.4, 15.5, 20.]) >>> bins = np.array([0, 5, 10, 15, 20]) >>> np.digitize(x,bins,right=True) array([1, 2, 3, 4, 4]) >>> np.digitize(x,bins,right=False) array([1, 3, 3, 4, 5]) """ x = _nx.asarray(x) bins = _nx.asarray(bins) # here for compatibility, searchsorted below is happy to take this if np.issubdtype(x.dtype, _nx.complexfloating): raise TypeError("x may not be complex") mono = _monotonicity(bins) if mono == 0: raise ValueError("bins must be monotonically increasing or decreasing") # this is backwards because the arguments below are swapped side = 'left' if right else 'right' if mono == -1: # reverse the bins, and invert the results return len(bins) - _nx.searchsorted(bins[::-1], x, side=side) else: return _nx.searchsorted(bins, x, side=side)