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Vehicle-Anti-Theft-Face-Rec.../venv/Lib/site-packages/numpy/lib/function_base.py

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Created starter files for the project.
2020-10-03 01:26:03 +00:00
import collections.abc
import functools
import re
import sys
import warnings
import numpy as np
import numpy.core.numeric as _nx
from numpy.core import transpose
from numpy.core.numeric import (
ones, zeros, arange, concatenate, array, asarray, asanyarray, empty,
ndarray, around, floor, ceil, 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.overrides import set_module
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 (
_insert, 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
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'
]
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.
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
"""
try:
iter(y)
except TypeError:
return False
return True
def _average_dispatcher(a, axis=None, weights=None, returned=None):
return (a, weights)
@array_function_dispatch(_average_dispatcher)
def average(a, axis=None, weights=None, returned=False):
"""
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.
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
"""
a = np.asanyarray(a)
if weights is None:
avg = a.mean(axis)
scl = avg.dtype.type(a.size/avg.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)
if np.any(scl == 0.0):
raise ZeroDivisionError(
"Weights sum to zero, can't be normalized")
avg = np.multiply(a, wgt, dtype=result_dtype).sum(axis)/scl
if returned:
if scl.shape != avg.shape:
scl = np.broadcast_to(scl, avg.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'}, optional
Whether to use row-major (C-style) or
column-major (Fortran-style) memory representation.
Defaults to 'C'.
Returns
-------
out : ndarray
Array interpretation of `a`. No copy is performed if the input
is already an ndarray. If `a` is a subclass of ndarray, a base
class ndarray is returned.
Raises
------
ValueError
Raises ValueError if `a` contains NaN (Not a Number) or Inf (Infinity).
See Also
--------
asarray : Create and array.
asanyarray : Similar function which passes through subclasses.
ascontiguousarray : Convert input to a contiguous array.
asfarray : Convert input to a floating point ndarray.
asfortranarray : Convert input to an ndarray with column-major
memory order.
fromiter : Create an array from an iterator.
fromfunction : Construct an array by executing a function on grid
positions.
Examples
--------
Convert a list into an array. If all elements are finite
``asarray_chkfinite`` is identical to ``asarray``.
>>> a = [1, 2]
>>> np.asarray_chkfinite(a, dtype=float)
array([1., 2.])
Raises ValueError if array_like contains Nans or Infs.
>>> a = [1, 2, np.inf]
>>> try:
... np.asarray_chkfinite(a)
... except ValueError:
... print('ValueError')
...
ValueError
"""
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 = array(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(x.shape, x.dtype)
for k in range(n):
item = funclist[k]
if not isinstance(item, collections.abc.Callable):
y[condlist[k]] = item
else:
vals = x[condlist[k]]
if vals.size > 0:
y[condlist[k]] = item(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(10)
>>> condlist = [x<3, x>5]
>>> choicelist = [x, x**2]
>>> np.select(condlist, choicelist)
array([ 0, 1, 2, ..., 49, 64, 81])
"""
# 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]
choicelist.append(np.asarray(default))
# need to get the result type before broadcasting for correct scalar
# behaviour
dtype = np.result_type(*choicelist)
# 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 in range(len(condlist)):
cond = condlist[i]
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 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 set of ndarrays (or a single ndarray if there is only one dimension)
corresponding to the derivatives of f with respect to each dimension.
Each derivative has the same shape as f.
Examples
--------
>>> f = np.array([1, 2, 4, 7, 11, 16], dtype=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]
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.
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`
Notes
-----
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):
return (p,)
@array_function_dispatch(_unwrap_dispatcher)
def unwrap(p, discont=pi, axis=-1):
"""
Unwrap by changing deltas between values to 2*pi complement.
Unwrap radian phase `p` by changing absolute jumps greater than
`discont` to their 2*pi complement along the given axis.
Parameters
----------
p : array_like
Input array.
discont : float, optional
Maximum discontinuity between values, default is ``pi``.
axis : int, optional
Axis along which unwrap will operate, default is the last axis.
Returns
-------
out : ndarray
Output array.
See Also
--------
rad2deg, deg2rad
Notes
-----
If the discontinuity in `p` is smaller than ``pi``, but larger than
`discont`, no unwrapping is done because taking the 2*pi complement
would only make the discontinuity larger.
Examples
--------
>>> phase = np.linspace(0, np.pi, num=5)
>>> phase[3:] += np.pi
>>> phase
array([ 0. , 0.78539816, 1.57079633, 5.49778714, 6.28318531]) # may vary
>>> np.unwrap(phase)
array([ 0. , 0.78539816, 1.57079633, -0.78539816, 0. ]) # may vary
"""
p = asarray(p)
nd = p.ndim
dd = diff(p, axis=axis)
slice1 = [slice(None, None)]*nd # full slices
slice1[axis] = slice(1, None)
slice1 = tuple(slice1)
ddmod = mod(dd + pi, 2*pi) - pi
_nx.copyto(ddmod, pi, where=(ddmod == -pi) & (dd > 0))
ph_correct = ddmod - dd
_nx.copyto(ph_correct, 0, where=abs(dd) < discont)
up = array(p, copy=True, dtype='d')
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]])
"""
if not isinstance(arr, np.ndarray):
raise TypeError("argument 1 must be numpy.ndarray, "
"not {name}".format(name=type(arr).__name__))
return _insert(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, ...]].
"""
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):
"""Helper for creating output arrays in vectorize."""
shapes = _calculate_shapes(broadcast_shape, dim_sizes, list_of_core_dims)
arrays = tuple(np.empty(shape, dtype=dtype)
for shape, dtype in zip(shapes, dtypes))
return arrays
@set_module('numpy')
class vectorize:
"""
vectorize(pyfunc, otypes=None, doc=None, excluded=None, cache=False,
signature=None)
Generalized function class.
Define a vectorized function which takes a nested sequence of objects or
numpy arrays as inputs and returns 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
A python function or method.
otypes : str or list of dtypes, optional
The output data type. It must be specified as either a string of
typecode characters or a list of data type specifiers. There should
be one data type specifier for each output.
doc : str, optional
The docstring for the function. If None, the docstring will be the
``pyfunc.__doc__``.
excluded : set, optional
Set of strings or integers representing the positional or keyword
arguments for which the function will not be vectorized. These will be
passed directly to `pyfunc` unmodified.
.. versionadded:: 1.7.0
cache : bool, optional
If `True`, then cache the first function call that determines the number
of outputs if `otypes` is not provided.
.. versionadded:: 1.7.0
signature : string, optional
Generalized universal function signature, e.g., ``(m,n),(n)->(m)`` for
vectorized matrix-vector multiplication. If provided, ``pyfunc`` will
be called with (and expected to return) arrays with shapes given by the
size of corresponding core dimensions. By default, ``pyfunc`` is
assumed to take scalars as input and output.
.. versionadded:: 1.12.0
Returns
-------
vectorized : callable
Vectorized function.
See Also
--------
frompyfunc : Takes an arbitrary Python function and returns a ufunc
Notes
-----
The `vectorize` function is provided primarily for convenience, not for
performance. The implementation is essentially a for loop.
If `otypes` is not specified, then a call to the function with the
first argument will be used to determine the number of outputs. The
results of this call will be cached if `cache` is `True` to prevent
calling the function twice. However, to implement the cache, the
original function must be wrapped which will slow down subsequent
calls, so only do this if your function is expensive.
The new keyword argument interface and `excluded` argument support
further degrades performance.
References
----------
.. [1] NumPy Reference, section `Generalized Universal Function API
<https://docs.scipy.org/doc/numpy/reference/c-api.generalized-ufuncs.html>`_.
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.]])
"""
def __init__(self, pyfunc, otypes=None, doc=None, excluded=None,
cache=False, signature=None):
self.pyfunc = pyfunc
self.cache = cache
self.signature = signature
self._ufunc = {} # Caching to improve default performance
if doc is None:
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 __call__(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 _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 = [array(a, copy=False, subok=True, dtype=object)
for a in args]
outputs = ufunc(*inputs)
if ufunc.nout == 1:
res = array(outputs, copy=False, subok=True, dtype=otypes[0])
else:
res = tuple([array(x, copy=False, subok=True, 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)
if otypes is None:
otypes = [asarray(result).dtype for result in results]
outputs = _create_arrays(broadcast_shape, dim_sizes,
output_core_dims, otypes)
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):
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):
"""
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
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 None:
dtype = np.result_type(m, np.float64)
else:
y = np.asarray(y)
if y.ndim > 2:
raise ValueError("y has more than 2 dimensions")
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=3)
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):
return (x, y)
@array_function_dispatch(_corrcoef_dispatcher)
def corrcoef(x, y=None, rowvar=True, bias=np._NoValue, ddof=np._NoValue):
"""
Return Pearson product-moment correlation coefficients.
Please refer to the documentation for `cov` for more detail. The
relationship between the correlation coefficient matrix, `R`, and the
covariance matrix, `C`, is
.. math:: R_{ij} = \\frac{ C_{ij} } { \\sqrt{ C_{ii} * C_{jj} } }
The values of `R` are between -1 and 1, inclusive.
Parameters
----------
x : array_like
A 1-D or 2-D array containing multiple variables and observations.
Each row of `x` represents a variable, and each column a single
observation of all those variables. Also see `rowvar` below.
y : array_like, optional
An additional set of variables and observations. `y` has the same
shape as `x`.
rowvar : bool, optional
If `rowvar` is True (default), then each row represents a
variable, with observations in the columns. Otherwise, the relationship
is transposed: each column represents a variable, while the rows
contain observations.
bias : _NoValue, optional
Has no effect, do not use.
.. deprecated:: 1.10.0
ddof : _NoValue, optional
Has no effect, do not use.
.. deprecated:: 1.10.0
Returns
-------
R : ndarray
The correlation coefficient matrix of the variables.
See Also
--------
cov : Covariance matrix
Notes
-----
Due to floating point rounding the resulting array may not be Hermitian,
the diagonal elements may not be 1, and the elements may not satisfy the
inequality abs(a) <= 1. The real and imaginary parts are clipped to the
interval [-1, 1] in an attempt to improve on that situation but is not
much help in the complex case.
This function accepts but discards arguments `bias` and `ddof`. This is
for backwards compatibility with previous versions of this function. These
arguments had no effect on the return values of the function and can be
safely ignored in this and previous versions of numpy.
"""
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=3)
c = cov(x, y, rowvar)
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()
"""
if M < 1:
return array([])
if M == 1:
return ones(1, float)
n = arange(0, M)
return 0.42 - 0.5*cos(2.0*pi*n/(M-1)) + 0.08*cos(4.0*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 is the product
of two sinc functions.
Note the excellent discussion in Kanasewich.
References
----------
.. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra",
Biometrika 37, 1-16, 1950.
.. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics",
The University of Alberta Press, 1975, pp. 109-110.
.. [3] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal
Processing", Prentice-Hall, 1999, pp. 468-471.
.. [4] Wikipedia, "Window function",
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()
"""
if M < 1:
return array([])
if M == 1:
return ones(1, float)
n = arange(0, M)
return where(less_equal(n, (M-1)/2.0), 2.0*n/(M-1), 2.0 - 2.0*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 - 0.5cos\\left(\\frac{2\\pi{n}}{M-1}\\right)
\\qquad 0 \\leq n \\leq M-1
The Hanning was named for Julius von Hann, an Austrian meteorologist.
It is also known as the Cosine Bell. Some authors prefer that it be
called a Hann window, to help avoid confusion with the very similar
Hamming window.
Most references to the Hanning window come from the signal processing
literature, where it is used as one of many windowing functions for
smoothing values. It is also known as an apodization (which means
"removing the foot", i.e. smoothing discontinuities at the beginning
and end of the sampled signal) or tapering function.
References
----------
.. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power
spectra, Dover Publications, New York.
.. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics",
The University of Alberta Press, 1975, pp. 106-108.
.. [3] Wikipedia, "Window function",
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()
"""
if M < 1:
return array([])
if M == 1:
return ones(1, float)
n = arange(0, M)
return 0.5 - 0.5*cos(2.0*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 - 0.46cos\\left(\\frac{2\\pi{n}}{M-1}\\right)
\\qquad 0 \\leq n \\leq M-1
The Hamming was named for R. W. Hamming, an associate of J. W. Tukey
and is described in Blackman and Tukey. It was recommended for
smoothing the truncated autocovariance function in the time domain.
Most references to the Hamming window come from the signal processing
literature, where it is used as one of many windowing functions for
smoothing values. It is also known as an apodization (which means
"removing the foot", i.e. smoothing discontinuities at the beginning
and end of the sampled signal) or tapering function.
References
----------
.. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power
spectra, Dover Publications, New York.
.. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The
University of Alberta Press, 1975, pp. 109-110.
.. [3] Wikipedia, "Window function",
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()
"""
if M < 1:
return array([])
if M == 1:
return ones(1, float)
n = arange(0, M)
return 0.54 - 0.46*cos(2.0*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`. This function does broadcast, but will *not*
"up-cast" int dtype arguments unless accompanied by at least one float or
complex dtype argument (see Raises below).
Parameters
----------
x : array_like, dtype float or complex
Argument of the Bessel function.
Returns
-------
out : ndarray, shape = x.shape, dtype = x.dtype
The modified Bessel function evaluated at each of the elements of `x`.
Raises
------
TypeError: array cannot be safely cast to required type
If argument consists exclusively of int dtypes.
See Also
--------
scipy.special.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.
http://www.math.sfu.ca/~cbm/aands/page_379.htm
.. [3] http://kobesearch.cpan.org/htdocs/Math-Cephes/Math/Cephes.html
Examples
--------
>>> np.i0(0.)
array(1.0) # may vary
>>> np.i0([0., 1. + 2j])
array([ 1.00000000+0.j , 0.18785373+0.64616944j]) # may vary
"""
x = np.asanyarray(x)
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()
"""
from numpy.dual import i0
if M == 1:
return np.array([1.])
n = arange(0, M)
alpha = (M-1)/2.0
return i0(beta * sqrt(1-((n-alpha)/alpha)**2.0))/i0(float(beta))
def _sinc_dispatcher(x):
return (x,)
@array_function_dispatch(_sinc_dispatcher)
def sinc(x):
"""
Return the sinc function.
The sinc function is :math:`\\sin(\\pi x)/(\\pi x)`.
Parameters
----------
x : ndarray
Array (possibly multi-dimensional) of values for which to to
calculate ``sinc(x)``.
Returns
-------
out : ndarray
``sinc(x)``, which has the same shape as the input.
Notes
-----
``sinc(0)`` is the limit value 1.
The name sinc is short for "sine cardinal" or "sinus cardinalis".
The sinc function is used in various signal processing applications,
including in anti-aliasing, in the construction of a Lanczos resampling
filter, and in interpolation.
For bandlimited interpolation of discrete-time signals, the ideal
interpolation kernel is proportional to the sinc function.
References
----------
.. [1] Weisstein, Eric W. "Sinc Function." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/SincFunction.html
.. [2] Wikipedia, "Sinc function",
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.
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)``.
"""
b = array(a, subok=True, copy=True)
b.sort(0)
return b
def _ureduce(a, func, **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)
if axis is not None:
keepdim = list(a.shape)
nd = a.ndim
axis = _nx.normalize_axis_tuple(axis, nd)
for ax in axis:
keepdim[ax] = 1
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
keepdim = tuple(keepdim)
else:
keepdim = (1,) * a.ndim
r = func(a, **kwargs)
return r, keepdim
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)
"""
r, k = _ureduce(a, func=_median, axis=axis, out=out,
overwrite_input=overwrite_input)
if keepdims:
return r.reshape(k)
else:
return r
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]
# Check if the array contains any nan's
if np.issubdtype(a.dtype, np.inexact):
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)
# Check if the array contains any nan's
if np.issubdtype(a.dtype, np.inexact) and sz > 0:
# warn and return nans like mean would
rout = mean(part[indexer], axis=axis, out=out)
return np.lib.utils._median_nancheck(part, rout, axis, out)
else:
# if there are no nans
# Use mean in odd and even case to coerce data type
# and check, use out array.
return mean(part[indexer], axis=axis, out=out)
def _percentile_dispatcher(a, q, axis=None, out=None, overwrite_input=None,
interpolation=None, keepdims=None):
return (a, q, out)
@array_function_dispatch(_percentile_dispatcher)
def percentile(a, q, axis=None, out=None,
overwrite_input=False, interpolation='linear', keepdims=False):
"""
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
Input array or object that can be converted to an array.
q : array_like of float
Percentile or sequence of percentiles to compute, which 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.
interpolation : {'linear', 'lower', 'higher', 'midpoint', 'nearest'}
This optional parameter specifies the interpolation method to
use when the desired percentile lies between two data points
``i < j``:
* 'linear': ``i + (j - i) * fraction``, where ``fraction``
is the fractional part of the index surrounded by ``i``
and ``j``.
* 'lower': ``i``.
* 'higher': ``j``.
* 'nearest': ``i`` or ``j``, whichever is nearest.
* 'midpoint': ``(i + j) / 2``.
.. versionadded:: 1.9.0
keepdims : bool, optional
If this is set to True, the axes which are reduced are left in
the result as dimensions with size one. With this option, the
result will broadcast correctly against the original array `a`.
.. versionadded:: 1.9.0
Returns
-------
percentile : scalar or ndarray
If `q` is a single percentile and `axis=None`, then the result
is a scalar. If multiple percentiles are given, first axis of
the result corresponds to the percentiles. The other axes are
the axes that remain after the reduction of `a`. If the input
contains integers or floats smaller than ``float64``, the output
data-type is ``float64``. Otherwise, the output data-type is the
same as that of the input. If `out` is specified, that array is
returned instead.
See Also
--------
mean
median : equivalent to ``percentile(..., 50)``
nanpercentile
quantile : equivalent to percentile, except with 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 `interpolation` parameter
will determine the percentile if the normalized ranking does not
match the location of ``q`` exactly. This function is the same as
the median if ``q=50``, the same as the minimum if ``q=0`` and the
same as the maximum if ``q=100``.
Examples
--------
>>> a = np.array([[10, 7, 4], [3, 2, 1]])
>>> a
array([[10, 7, 4],
[ 3, 2, 1]])
>>> np.percentile(a, 50)
3.5
>>> np.percentile(a, 50, axis=0)
array([6.5, 4.5, 2.5])
>>> np.percentile(a, 50, axis=1)
array([7., 2.])
>>> np.percentile(a, 50, axis=1, keepdims=True)
array([[7.],
[2.]])
>>> m = np.percentile(a, 50, axis=0)
>>> out = np.zeros_like(m)
>>> np.percentile(a, 50, axis=0, out=out)
array([6.5, 4.5, 2.5])
>>> m
array([6.5, 4.5, 2.5])
>>> b = a.copy()
>>> np.percentile(b, 50, axis=1, overwrite_input=True)
array([7., 2.])
>>> assert not np.all(a == b)
The different types of interpolation 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', None),
('higher', '--'),
('lower', '--'),
('nearest', '-.'),
('midpoint', '-.'),
]
for interpolation, style in lines:
ax.plot(
p, np.percentile(a, p, interpolation=interpolation),
label=interpolation, linestyle=style)
ax.set(
title='Interpolation methods for list: ' + str(a),
xlabel='Percentile',
ylabel='List item returned',
yticks=a)
ax.legend()
plt.show()
"""
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, interpolation, keepdims)
def _quantile_dispatcher(a, q, axis=None, out=None, overwrite_input=None,
interpolation=None, keepdims=None):
return (a, q, out)
@array_function_dispatch(_quantile_dispatcher)
def quantile(a, q, axis=None, out=None,
overwrite_input=False, interpolation='linear', keepdims=False):
"""
Compute the q-th quantile of the data along the specified axis.
.. versionadded:: 1.15.0
Parameters
----------
a : array_like
Input array or object that can be converted to an array.
q : array_like of float
Quantile or sequence of quantiles to compute, which 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.
interpolation : {'linear', 'lower', 'higher', 'midpoint', 'nearest'}
This optional parameter specifies the interpolation method to
use when the desired quantile lies between two data points
``i < j``:
* linear: ``i + (j - i) * fraction``, where ``fraction``
is the fractional part of the index surrounded by ``i``
and ``j``.
* lower: ``i``.
* higher: ``j``.
* nearest: ``i`` or ``j``, whichever is nearest.
* midpoint: ``(i + j) / 2``.
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`.
Returns
-------
quantile : scalar or ndarray
If `q` is a single quantile and `axis=None`, then the result
is a scalar. If multiple quantiles 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 `interpolation` 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``.
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)
"""
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, interpolation, keepdims)
def _quantile_unchecked(a, q, axis=None, out=None, overwrite_input=False,
interpolation='linear', keepdims=False):
"""Assumes that q is in [0, 1], and is an ndarray"""
r, k = _ureduce(a, func=_quantile_ureduce_func, q=q, axis=axis, out=out,
overwrite_input=overwrite_input,
interpolation=interpolation)
if keepdims:
return r.reshape(q.shape + k)
else:
return r
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 q[i] < 0.0 or q[i] > 1.0:
return False
else:
# faster than any()
if np.count_nonzero(q < 0.0) or np.count_nonzero(q > 1.0):
return False
return True
def _quantile_ureduce_func(a, q, axis=None, out=None, overwrite_input=False,
interpolation='linear', keepdims=False):
a = asarray(a)
if q.ndim == 0:
# Do not allow 0-d arrays because following code fails for scalar
zerod = True
q = q[None]
else:
zerod = False
# prepare a for partitioning
if overwrite_input:
if axis is None:
ap = a.ravel()
else:
ap = a
else:
if axis is None:
ap = a.flatten()
else:
ap = a.copy()
if axis is None:
axis = 0
Nx = ap.shape[axis]
indices = q * (Nx - 1)
# round fractional indices according to interpolation method
if interpolation == 'lower':
indices = floor(indices).astype(intp)
elif interpolation == 'higher':
indices = ceil(indices).astype(intp)
elif interpolation == 'midpoint':
indices = 0.5 * (floor(indices) + ceil(indices))
elif interpolation == 'nearest':
indices = around(indices).astype(intp)
elif interpolation == 'linear':
pass # keep index as fraction and interpolate
else:
raise ValueError(
"interpolation can only be 'linear', 'lower' 'higher', "
"'midpoint', or 'nearest'")
n = np.array(False, dtype=bool) # check for nan's flag
if np.issubdtype(indices.dtype, np.integer): # take the points along axis
# Check if the array contains any nan's
if np.issubdtype(a.dtype, np.inexact):
indices = concatenate((indices, [-1]))
ap.partition(indices, axis=axis)
# ensure axis with q-th is first
ap = np.moveaxis(ap, axis, 0)
axis = 0
# Check if the array contains any nan's
if np.issubdtype(a.dtype, np.inexact):
indices = indices[:-1]
n = np.isnan(ap[-1:, ...])
if zerod:
indices = indices[0]
r = take(ap, indices, axis=axis, out=out)
else: # weight the points above and below the indices
indices_below = floor(indices).astype(intp)
indices_above = indices_below + 1
indices_above[indices_above > Nx - 1] = Nx - 1
# Check if the array contains any nan's
if np.issubdtype(a.dtype, np.inexact):
indices_above = concatenate((indices_above, [-1]))
weights_above = indices - indices_below
weights_below = 1 - weights_above
weights_shape = [1, ] * ap.ndim
weights_shape[axis] = len(indices)
weights_below.shape = weights_shape
weights_above.shape = weights_shape
ap.partition(concatenate((indices_below, indices_above)), axis=axis)
# ensure axis with q-th is first
ap = np.moveaxis(ap, axis, 0)
weights_below = np.moveaxis(weights_below, axis, 0)
weights_above = np.moveaxis(weights_above, axis, 0)
axis = 0
# Check if the array contains any nan's
if np.issubdtype(a.dtype, np.inexact):
indices_above = indices_above[:-1]
n = np.isnan(ap[-1:, ...])
x1 = take(ap, indices_below, axis=axis) * weights_below
x2 = take(ap, indices_above, axis=axis) * weights_above
# ensure axis with q-th is first
x1 = np.moveaxis(x1, axis, 0)
x2 = np.moveaxis(x2, axis, 0)
if zerod:
x1 = x1.squeeze(0)
x2 = x2.squeeze(0)
if out is not None:
r = add(x1, x2, out=out)
else:
r = add(x1, x2)
if np.any(n):
if zerod:
if ap.ndim == 1:
if out is not None:
out[...] = a.dtype.type(np.nan)
r = out
else:
r = a.dtype.type(np.nan)
else:
r[..., n.squeeze(0)] = a.dtype.type(np.nan)
else:
if r.ndim == 1:
r[:] = a.dtype.type(np.nan)
else:
r[..., n.repeat(q.size, 0)] = a.dtype.type(np.nan)
return r
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):
"""
Integrate along the given axis using the composite trapezoidal rule.
Integrate `y` (`x`) along given axis.
Parameters
----------
y : array_like
Input array to integrate.
x : array_like, optional
The sample points corresponding to the `y` values. If `x` is None,
the sample points are assumed to be evenly spaced `dx` apart. The
default is None.
dx : scalar, optional
The spacing between sample points when `x` is None. The default is 1.
axis : int, optional
The axis along which to integrate.
Returns
-------
trapz : float
Definite integral as approximated by trapezoidal rule.
See Also
--------
sum, cumsum
Notes
-----
Image [2]_ illustrates trapezoidal rule -- y-axis locations of points
will be taken from `y` array, by default x-axis distances between
points will be 1.0, alternatively they can be provided with `x` array
or with `dx` scalar. Return value will be equal to combined area under
the red lines.
References
----------
.. [1] Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule
.. [2] Illustration image:
https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png
Examples
--------
>>> np.trapz([1,2,3])
4.0
>>> np.trapz([1,2,3], x=[4,6,8])
8.0
>>> np.trapz([1,2,3], dx=2)
8.0
>>> a = np.arange(6).reshape(2, 3)
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> np.trapz(a, axis=0)
array([1.5, 2.5, 3.5])
>>> np.trapz(a, axis=1)
array([2., 8.])
"""
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
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 coordinate matrices from coordinate vectors.
Make N-D coordinate arrays for vectorized evaluations of
N-D scalar/vector fields over N-D grids, given
one-dimensional coordinate arrays x1, x2,..., xn.
.. versionchanged:: 1.9
1-D and 0-D cases are allowed.
Parameters
----------
x1, x2,..., xn : array_like
1-D arrays representing the coordinates of a grid.
indexing : {'xy', 'ij'}, optional
Cartesian ('xy', default) or matrix ('ij') indexing of output.
See Notes for more details.
.. versionadded:: 1.7.0
sparse : bool, optional
If True a sparse grid is returned in order to conserve memory.
Default is False.
.. versionadded:: 1.7.0
copy : bool, optional
If False, a view into the original arrays are returned in order to
conserve memory. Default is True. Please note that
``sparse=False, copy=False`` will likely return non-contiguous
arrays. Furthermore, more than one element of a broadcast array
may refer to a single memory location. If you need to write to the
arrays, make copies first.
.. versionadded:: 1.7.0
Returns
-------
X1, X2,..., XN : ndarray
For vectors `x1`, `x2`,..., 'xn' with lengths ``Ni=len(xi)`` ,
return ``(N1, N2, N3,...Nn)`` shaped arrays if indexing='ij'
or ``(N2, N1, N3,...Nn)`` shaped arrays if indexing='xy'
with the elements of `xi` repeated to fill the matrix along
the first dimension for `x1`, the second for `x2` and so on.
Notes
-----
This function supports both indexing conventions through the indexing
keyword argument. Giving the string 'ij' returns a meshgrid with
matrix indexing, while 'xy' returns a meshgrid with Cartesian indexing.
In the 2-D case with inputs of length M and N, the outputs are of shape
(N, M) for 'xy' indexing and (M, N) for 'ij' indexing. In the 3-D case
with inputs of length M, N and P, outputs are of shape (N, M, P) for
'xy' indexing and (M, N, P) for 'ij' indexing. The difference is
illustrated by the following code snippet::
xv, yv = np.meshgrid(x, y, sparse=False, indexing='ij')
for i in range(nx):
for j in range(ny):
# treat xv[i,j], yv[i,j]
xv, yv = np.meshgrid(x, y, sparse=False, indexing='xy')
for i in range(nx):
for j in range(ny):
# treat xv[j,i], yv[j,i]
In the 1-D and 0-D case, the indexing and sparse keywords have no effect.
See Also
--------
index_tricks.mgrid : Construct a multi-dimensional "meshgrid"
using indexing notation.
index_tricks.ogrid : Construct an open multi-dimensional "meshgrid"
using indexing notation.
Examples
--------
>>> nx, ny = (3, 2)
>>> x = np.linspace(0, 1, nx)
>>> y = np.linspace(0, 1, ny)
>>> xv, yv = np.meshgrid(x, y)
>>> xv
array([[0. , 0.5, 1. ],
[0. , 0.5, 1. ]])
>>> yv
array([[0., 0., 0.],
[1., 1., 1.]])
>>> xv, yv = np.meshgrid(x, y, sparse=True) # make sparse output arrays
>>> xv
array([[0. , 0.5, 1. ]])
>>> yv
array([[0.],
[1.]])
`meshgrid` is very useful to evaluate functions on a grid.
>>> import matplotlib.pyplot as plt
>>> x = np.arange(-5, 5, 0.1)
>>> y = np.arange(-5, 5, 0.1)
>>> xx, yy = np.meshgrid(x, y, sparse=True)
>>> z = np.sin(xx**2 + yy**2) / (xx**2 + yy**2)
>>> h = plt.contourf(x,y,z)
>>> 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):
# 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:
_obj = obj
obj = np.asarray(obj)
if obj.size == 0 and not isinstance(_obj, np.ndarray):
obj = obj.astype(intp)
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=3)
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(
"index %i is out of bounds for axis %i with "
"size %i" % (obj, axis, 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)
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