|
# Accessing collections abstact classes from collections # has been deprecated since Python 3.3 except ImportError: import collections as collections_abc
ones, zeros, arange, concatenate, array, asarray, asanyarray, empty, empty_like, ndarray, around, floor, ceil, take, dot, where, intp, integer, isscalar, absolute ) pi, add, arctan2, frompyfunc, cos, less_equal, sqrt, sin, mod, exp, not_equal, subtract ) ravel, nonzero, partition, mean, any, sum ) _insert, add_docstring, bincount, normalize_axis_index, _monotonicity, interp as compiled_interp, interp_complex as compiled_interp_complex )
# Force range to be a generator, for np.delete's usage. range = xrange import __builtin__ as builtins else:
overrides.array_function_dispatch, module='numpy')
# needed in this module for compatibility
'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' ]
return (m,)
""" 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])
return (m,)
""" 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]]]) >>> flip(A, 0) array([[[4, 5], [6, 7]], [[0, 1], [2, 3]]]) >>> 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(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]
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
"""
return (a, weights)
""" Compute the weighted average along the specified axis.
Parameters ---------- a : array_like Array containing data to be averaged. If `a` is not an array, a conversion is attempted. axis : None or int or tuple of ints, optional Axis or axes along which to average `a`. The default, axis=None, will average over all of the elements of the input array. If axis is negative it counts from the last to the first axis.
.. versionadded:: 1.7.0
If axis is a tuple of ints, averaging is performed on all of the axes specified in the tuple instead of a single axis or all the axes as before. weights : array_like, optional An array of weights associated with the values in `a`. Each value in `a` contributes to the average according to its associated weight. The weights array can either be 1-D (in which case its length must be the size of `a` along the given axis) or of the same shape as `a`. If `weights=None`, then all data in `a` are assumed to have a weight equal to one. returned : bool, optional Default is `False`. If `True`, the tuple (`average`, `sum_of_weights`) is returned, otherwise only the average is returned. If `weights=None`, `sum_of_weights` is equivalent to the number of elements over which the average is taken.
Returns ------- 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 = range(1,5) >>> data [1, 2, 3, 4] >>> np.average(data) 2.5 >>> np.average(range(1,11), weights=range(10,0,-1)) 4.0
>>> data = np.arange(6).reshape((3,2)) >>> data array([[0, 1], [2, 3], [4, 5]]) >>> np.average(data, axis=1, weights=[1./4, 3./4]) array([ 0.75, 2.75, 4.75]) >>> np.average(data, weights=[1./4, 3./4])
Traceback (most recent call last): ... TypeError: Axis must be specified when shapes of a and weights differ.
>>> a = np.ones(5, dtype=np.float128) >>> w = np.ones(5, dtype=np.complex64) >>> avg = np.average(a, weights=w) >>> print(avg.dtype) complex256 """
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
else: return avg
"""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
""" raise ValueError( "array must not contain infs or NaNs")
yield x # support the undocumented behavior of allowing scalars if np.iterable(condlist): for c in condlist: yield c
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
for c in condlist: yield c for c in choicelist: yield c
""" Return an array drawn from elements in choicelist, depending on conditions.
Parameters ---------- condlist : list of bool ndarrays The list of conditions which determine from which array in `choicelist` the output elements are taken. When multiple conditions are satisfied, the first one encountered in `condlist` is used. choicelist : list of ndarrays The list of arrays from which the output elements are taken. It has to be of the same length as `condlist`. default : scalar, optional The element inserted in `output` when all conditions evaluate to False.
Returns ------- output : ndarray The output at position m is the m-th element of the array in `choicelist` where the m-th element of the corresponding array in `condlist` is True.
See Also -------- where : Return elements from one of two arrays depending on condition. take, choose, compress, diag, diagonal
Examples -------- >>> x = np.arange(10) >>> condlist = [x<3, x>5] >>> choicelist = [x, x**2] >>> np.select(condlist, choicelist) array([ 0, 1, 2, 0, 0, 0, 36, 49, 64, 81])
""" # 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: # 2014-02-24, 1.9 warnings.warn("select with an empty condition list is not possible" "and will be deprecated", DeprecationWarning, stacklevel=2) return np.asarray(default)[()]
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. deprecated_ints = False for i in range(len(condlist)): cond = condlist[i] if cond.dtype.type is not np.bool_: if np.issubdtype(cond.dtype, np.integer): # A previous implementation accepted int ndarrays accidentally. # Supported here deliberately, but deprecated. condlist[i] = condlist[i].astype(bool) deprecated_ints = True else: raise ValueError( 'invalid entry {} in condlist: should be boolean ndarray'.format(i))
if deprecated_ints: # 2014-02-24, 1.9 msg = "select condlists containing integer ndarrays is deprecated " \ "and will be removed in the future. Use `.astype(bool)` to " \ "convert to bools." warnings.warn(msg, DeprecationWarning, stacklevel=2)
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
return (a,)
""" Return an array copy of the given object.
Parameters ---------- a : array_like Input data. order : {'C', 'F', 'A', 'K'}, optional Controls the memory layout of the copy. 'C' means C-order, 'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous, 'C' otherwise. 'K' means match the layout of `a` as closely as possible. (Note that this function and :meth:`ndarray.copy` are very similar, but have different default values for their order= arguments.)
Returns ------- arr : ndarray Array interpretation of `a`.
Notes ----- This is equivalent to:
>>> np.array(a, copy=True) #doctest: +SKIP
Examples -------- Create an array x, with a reference y and a copy z:
>>> x = np.array([1, 2, 3]) >>> y = x >>> z = np.copy(x)
Note that, when we modify x, y changes, but not z:
>>> x[0] = 10 >>> x[0] == y[0] True >>> x[0] == z[0] False
""" return array(a, order=order, copy=True)
# Basic operations
yield f for v in varargs: yield v
def gradient(f, *varargs, **kwargs): """ 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
axes = kwargs.pop('axis', None) if axes is None: axes = tuple(range(N)) else: axes = _nx.normalize_axis_tuple(axes, 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): if np.ndim(distances) == 0: continue elif np.ndim(distances) != 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") 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")
edge_order = kwargs.pop('edge_order', 1) if kwargs: raise TypeError('"{}" are not valid keyword arguments.'.format( '", "'.join(kwargs.keys()))) 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 otype = np.double
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
return (a, prepend, append)
""" Calculate the n-th discrete difference along the given axis.
The first difference is given by ``out[n] = a[n+1] - a[n]`` along the given axis, higher differences are calculated by using `diff` recursively.
Parameters ---------- a : array_like Input array n : int, optional The number of times values are differenced. 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.
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,...] array(255, np.uint8)
If this is not desirable, then the array should be cast to a larger integer type first:
>>> i16_arr = u8_arr.astype(np.int16) >>> np.diff(i16_arr) array([-1], dtype=int16)
Examples -------- >>> x = np.array([1, 2, 4, 7, 0]) >>> np.diff(x) array([ 1, 2, 3, -7]) >>> np.diff(x, n=2) array([ 1, 1, -10])
>>> x = np.array([[1, 3, 6, 10], [0, 5, 6, 8]]) >>> np.diff(x) array([[2, 3, 4], [5, 1, 2]]) >>> np.diff(x, axis=0) array([[-1, 2, 0, -2]])
>>> x = np.arange('1066-10-13', '1066-10-16', dtype=np.datetime64) >>> np.diff(x) array([1, 1], dtype='timedelta64[D]')
""" return a raise ValueError( "order must be non-negative but got " + repr(n))
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)
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)
a = np.concatenate(combined, axis)
return (x, xp, fp)
""" 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 ----- Does not check that the x-coordinate sequence `xp` is increasing. If `xp` is not increasing, the results are nonsense. A simple check for increasing is::
np.all(np.diff(xp) > 0)
Examples -------- >>> xp = [1, 2, 3] >>> fp = [3, 2, 0] >>> np.interp(2.5, xp, fp) 1.0 >>> np.interp([0, 1, 1.5, 2.72, 3.14], xp, fp) array([ 3. , 3. , 2.5 , 0.56, 0. ]) >>> UNDEF = -99.0 >>> np.interp(3.14, xp, fp, right=UNDEF) -99.0
Plot an interpolant to the sine function:
>>> x = np.linspace(0, 2*np.pi, 10) >>> y = np.sin(x) >>> xvals = np.linspace(0, 2*np.pi, 50) >>> yinterp = np.interp(xvals, x, y) >>> import matplotlib.pyplot as plt >>> plt.plot(x, y, 'o') [<matplotlib.lines.Line2D object at 0x...>] >>> plt.plot(xvals, yinterp, '-x') [<matplotlib.lines.Line2D object at 0x...>] >>> plt.show()
Interpolation with periodic x-coordinates:
>>> x = [-180, -170, -185, 185, -10, -5, 0, 365] >>> xp = [190, -190, 350, -350] >>> fp = [5, 10, 3, 4] >>> np.interp(x, xp, fp, period=360) array([7.5, 5., 8.75, 6.25, 3., 3.25, 3.5, 3.75])
Complex interpolation:
>>> x = [1.5, 4.0] >>> xp = [2,3,5] >>> fp = [1.0j, 0, 2+3j] >>> np.interp(x, xp, fp) array([ 0.+1.j , 1.+1.5j])
"""
interp_func = compiled_interp_complex input_dtype = np.complex128 else:
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 (z,)
""" Return the angle of the complex argument.
Parameters ---------- z : array_like A complex number or sequence of complex numbers. deg : bool, optional Return angle in degrees if True, radians if False (default).
Returns ------- angle : ndarray or scalar The counterclockwise angle from the positive real axis on the complex plane, with dtype as numpy.float64.
..versionchanged:: 1.16.0 This function works on subclasses of ndarray like `ma.array`.
See Also -------- arctan2 absolute
Examples -------- >>> np.angle([1.0, 1.0j, 1+1j]) # in radians array([ 0. , 1.57079633, 0.78539816]) >>> np.angle(1+1j, deg=True) # in degrees 45.0
""" 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
return (p,)
""" Unwrap by changing deltas between values to 2*pi complement.
Unwrap radian phase `p` by changing absolute jumps greater than `discont` to their 2*pi complement along the given axis.
Parameters ---------- p : array_like Input array. discont : float, optional Maximum discontinuity between values, default is ``pi``. axis : int, optional Axis along which unwrap will operate, default is the last axis.
Returns ------- out : ndarray Output array.
See Also -------- rad2deg, deg2rad
Notes ----- If the discontinuity in `p` is smaller than ``pi``, but larger than `discont`, no unwrapping is done because taking the 2*pi complement would only make the discontinuity larger.
Examples -------- >>> phase = np.linspace(0, np.pi, num=5) >>> phase[3:] += np.pi >>> phase array([ 0. , 0.78539816, 1.57079633, 5.49778714, 6.28318531]) >>> np.unwrap(phase) array([ 0. , 0.78539816, 1.57079633, -0.78539816, 0. ])
""" 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
return (a,)
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
return (filt,)
""" Trim the leading and/or trailing zeros from a 1-D array or sequence.
Parameters ---------- filt : 1-D array or sequence Input array. trim : str, optional A string with 'f' representing trim from front and 'b' to trim from back. Default is 'fb', trim zeros from both front and back of the array.
Returns ------- trimmed : 1-D array or sequence The result of trimming the input. The input data type is preserved.
Examples -------- >>> a = np.array((0, 0, 0, 1, 2, 3, 0, 2, 1, 0)) >>> np.trim_zeros(a) array([1, 2, 3, 0, 2, 1])
>>> np.trim_zeros(a, 'b') array([0, 0, 0, 1, 2, 3, 0, 2, 1])
The input data type is preserved, list/tuple in means list/tuple out.
>>> np.trim_zeros([0, 1, 2, 0]) [1, 2]
""" else: first = first + 1 for i in filt[::-1]: if i != 0.: break else: last = last - 1
return (condition, arr)
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 (arr, mask, vals)
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]])
""" raise TypeError("argument 1 must be numpy.ndarray, " "not {name}".format(name=type(arr).__name__))
""" 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
""" 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('->'))
""" 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
""" 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
"""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]
"""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
""" 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.
Examples -------- >>> def myfunc(a, b): ... "Return a-b if a>b, otherwise return a+b" ... if a > b: ... return a - b ... else: ... return a + b
>>> vfunc = np.vectorize(myfunc) >>> vfunc([1, 2, 3, 4], 2) array([3, 4, 1, 2])
The docstring is taken from the input function to `vectorize` unless it is specified:
>>> vfunc.__doc__ 'Return a-b if a>b, otherwise return a+b' >>> vfunc = np.vectorize(myfunc, doc='Vectorized `myfunc`') >>> vfunc.__doc__ 'Vectorized `myfunc`'
The output type is determined by evaluating the first element of the input, unless it is specified:
>>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) <type 'numpy.int32'> >>> vfunc = np.vectorize(myfunc, otypes=[float]) >>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) <type 'numpy.float64'>
The `excluded` argument can be used to prevent vectorizing over certain arguments. This can be useful for array-like arguments of a fixed length such as the coefficients for a polynomial as in `polyval`:
>>> def mypolyval(p, x): ... _p = list(p) ... res = _p.pop(0) ... while _p: ... res = res*x + _p.pop(0) ... return res >>> vpolyval = np.vectorize(mypolyval, excluded=['p']) >>> vpolyval(p=[1, 2, 3], x=[0, 1]) array([3, 6])
Positional arguments may also be excluded by specifying their position:
>>> vpolyval.excluded.add(0) >>> vpolyval([1, 2, 3], x=[0, 1]) array([3, 6])
The `signature` argument allows for vectorizing functions that act on non-scalar arrays of fixed length. For example, you can use it for a vectorized calculation of Pearson correlation coefficient and its p-value:
>>> import scipy.stats >>> pearsonr = np.vectorize(scipy.stats.pearsonr, ... signature='(n),(n)->(),()') >>> pearsonr([[0, 1, 2, 3]], [[1, 2, 3, 4], [4, 3, 2, 1]]) (array([ 1., -1.]), array([ 0., 0.]))
Or for a vectorized convolution:
>>> convolve = np.vectorize(np.convolve, signature='(n),(m)->(k)') >>> convolve(np.eye(4), [1, 2, 1]) array([[ 1., 2., 1., 0., 0., 0.], [ 0., 1., 2., 1., 0., 0.], [ 0., 0., 1., 2., 1., 0.], [ 0., 0., 0., 1., 2., 1.]])
See Also -------- frompyfunc : Takes an arbitrary Python function and returns a ufunc
Notes ----- The `vectorize` function is provided primarily for convenience, not for performance. The implementation is essentially a for loop.
If `otypes` is not specified, then a call to the function with the first argument will be used to determine the number of outputs. The results of this call will be cached if `cache` is `True` to prevent calling the function twice. However, to implement the cache, the original function must be wrapped which will slow down subsequent calls, so only do this if your function is expensive.
The new keyword argument interface and `excluded` argument support further degrades performance.
References ---------- .. [1] NumPy Reference, section `Generalized Universal Function API <https://docs.scipy.org/doc/numpy/reference/c-api.generalized-ufuncs.html>`_. """
cache=False, signature=None):
else: self.__doc__ = doc
raise ValueError("Invalid otype specified: %s" % (char,)) otypes = ''.join([_nx.dtype(x).char for x in otypes]) raise ValueError("Invalid otype specification")
# Excluded variable support
self._in_and_out_core_dims = _parse_gufunc_signature(signature) else:
""" 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)
"""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 nout = len(otypes)
# Note logic here: We only *use* self._ufunc if func is self.pyfunc # even though we set self._ufunc regardless. if func is self.pyfunc and self._ufunc is not None: ufunc = self._ufunc else: ufunc = self._ufunc = frompyfunc(func, len(args), nout) 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
"""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
"""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
fweights=None, aweights=None): return (m, y, fweights, aweights)
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::
>>> w = f * a >>> v1 = np.sum(w) >>> v2 = np.sum(w * a) >>> m -= np.sum(m * w, axis=1, keepdims=True) / v1 >>> cov = np.dot(m * w, m.T) * v1 / (v1**2 - ddof * v2)
Note that when ``a == 1``, the normalization factor ``v1 / (v1**2 - ddof * v2)`` goes over to ``1 / (np.sum(f) - ddof)`` as it should.
Examples -------- Consider two variables, :math:`x_0` and :math:`x_1`, which correlate perfectly, but in opposite directions:
>>> x = np.array([[0, 2], [1, 1], [2, 0]]).T >>> x array([[0, 1, 2], [2, 1, 0]])
Note how :math:`x_0` increases while :math:`x_1` decreases. The covariance matrix shows this clearly:
>>> np.cov(x) array([[ 1., -1.], [-1., 1.]])
Note that element :math:`C_{0,1}`, which shows the correlation between :math:`x_0` and :math:`x_1`, is negative.
Further, note how `x` and `y` are combined:
>>> x = [-2.1, -1, 4.3] >>> y = [3, 1.1, 0.12] >>> X = np.stack((x, y), axis=0) >>> print(np.cov(X)) [[ 11.71 -4.286 ] [ -4.286 2.14413333]] >>> print(np.cov(x, y)) [[ 11.71 -4.286 ] [ -4.286 2.14413333]] >>> print(np.cov(x)) 11.71
""" # Check inputs raise ValueError( "ddof must be integer")
# Handles complex arrays too raise ValueError("m has more than 2 dimensions")
else: raise ValueError("y has more than 2 dimensions")
X = X.T return np.array([]).reshape(0, 0) y = y.T
else:
# Get the product of frequencies and weights 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 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
# Determine the normalization elif ddof == 0: fact = w_sum elif aweights is None: fact = w_sum - ddof else: fact = w_sum - ddof*sum(w*aweights)/w_sum
warnings.warn("Degrees of freedom <= 0 for slice", RuntimeWarning, stacklevel=2) fact = 0.0
else: X_T = (X*w).T
return (x, y)
""" 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=2) 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
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, 4.14397981e-01, 7.36045180e-01, 9.67046769e-01, 9.67046769e-01, 7.36045180e-01, 4.14397981e-01, 1.59903635e-01, 3.26064346e-02, -1.38777878e-17])
Plot the window and the frequency response:
>>> from numpy.fft import fft, fftshift >>> window = np.blackman(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Blackman window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show()
>>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Blackman window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show()
""" 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))
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 -------- >>> np.bartlett(12) array([ 0. , 0.18181818, 0.36363636, 0.54545455, 0.72727273, 0.90909091, 0.90909091, 0.72727273, 0.54545455, 0.36363636, 0.18181818, 0. ])
Plot the window and its frequency response (requires SciPy and matplotlib):
>>> from numpy.fft import fft, fftshift >>> window = np.bartlett(51) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Bartlett window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show()
>>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Bartlett window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show()
""" 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))
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") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show()
>>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of the Hann window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show()
""" return array([]) return ones(1, float)
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, 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") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show()
>>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Hamming window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show()
""" 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
-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 ]
-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 ]
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)
return exp(x) * _chbevl(x/2.0-2, _i0A)
return exp(x) * _chbevl(32.0/x - 2.0, _i0B) / sqrt(x)
return (x,)
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.iv, scipy.special.ive
Notes ----- We use the algorithm published by Clenshaw [1]_ and referenced by Abramowitz and Stegun [2]_, for which the function domain is partitioned into the two intervals [0,8] and (8,inf), and Chebyshev polynomial expansions are employed in each interval. Relative error on the domain [0,30] using IEEE arithmetic is documented [3]_ as having a peak of 5.8e-16 with an rms of 1.4e-16 (n = 30000).
References ---------- .. [1] C. W. Clenshaw, "Chebyshev series for mathematical functions", in *National Physical Laboratory Mathematical Tables*, vol. 5, London: Her Majesty's Stationery Office, 1962. .. [2] M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions*, 10th printing, New York: Dover, 1964, pp. 379. http://www.math.sfu.ca/~cbm/aands/page_379.htm .. [3] http://kobesearch.cpan.org/htdocs/Math-Cephes/Math/Cephes.html
Examples -------- >>> np.i0([0.]) array(1.0) >>> np.i0([0., 1. + 2j]) array([ 1.00000000+0.j , 0.18785373+0.64616944j])
""" x = atleast_1d(x).copy() y = empty_like(x) ind = (x < 0) x[ind] = -x[ind] ind = (x <= 8.0) y[ind] = _i0_1(x[ind]) ind2 = ~ind y[ind2] = _i0_2(x[ind2]) return y.squeeze()
## End of cephes code for i0
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, 2.29737120e-01, 5.99885316e-01, 9.45674898e-01, 9.45674898e-01, 5.99885316e-01, 2.29737120e-01, 4.65200189e-02, 3.46009194e-03, 7.72686684e-06])
Plot the window and the frequency response:
>>> from numpy.fft import fft, fftshift >>> window = np.kaiser(51, 14) >>> plt.plot(window) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Kaiser window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Sample") <matplotlib.text.Text object at 0x...> >>> plt.show()
>>> plt.figure() <matplotlib.figure.Figure object at 0x...> >>> A = fft(window, 2048) / 25.5 >>> mag = np.abs(fftshift(A)) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(mag) >>> response = np.clip(response, -100, 100) >>> plt.plot(freq, response) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Frequency response of Kaiser window") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Magnitude [dB]") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("Normalized frequency [cycles per sample]") <matplotlib.text.Text object at 0x...> >>> plt.axis('tight') (-0.5, 0.5, -100.0, ...) >>> plt.show()
""" 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))
return (x,)
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, -8.90384387e-02, -5.84680802e-02, 3.89804309e-17, 6.68206631e-02, 1.16434881e-01, 1.26137788e-01, 8.50444803e-02, -3.89804309e-17, -1.03943254e-01, -1.89206682e-01, -2.16236208e-01, -1.55914881e-01, 3.89804309e-17, 2.33872321e-01, 5.04551152e-01, 7.56826729e-01, 9.35489284e-01, 1.00000000e+00, 9.35489284e-01, 7.56826729e-01, 5.04551152e-01, 2.33872321e-01, 3.89804309e-17, -1.55914881e-01, -2.16236208e-01, -1.89206682e-01, -1.03943254e-01, -3.89804309e-17, 8.50444803e-02, 1.26137788e-01, 1.16434881e-01, 6.68206631e-02, 3.89804309e-17, -5.84680802e-02, -8.90384387e-02, -8.40918587e-02, -4.92362781e-02, -3.89804309e-17])
>>> plt.plot(x, np.sinc(x)) [<matplotlib.lines.Line2D object at 0x...>] >>> plt.title("Sinc Function") <matplotlib.text.Text object at 0x...> >>> plt.ylabel("Amplitude") <matplotlib.text.Text object at 0x...> >>> plt.xlabel("X") <matplotlib.text.Text object at 0x...> >>> plt.show()
It works in 2-D as well:
>>> x = np.linspace(-4, 4, 401) >>> xx = np.outer(x, x) >>> plt.imshow(np.sinc(xx)) <matplotlib.image.AxesImage object at 0x...>
"""
return (a,)
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
""" 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.
"""
else: keep = set(range(nd)) - set(axis) nkeep = len(keep) # swap axis that should not be reduced to front for i, s in enumerate(sorted(keep)): a = a.swapaxes(i, s) # merge reduced axis a = a.reshape(a.shape[:nkeep] + (-1,)) kwargs['axis'] = -1 else:
a, axis=None, out=None, overwrite_input=None, keepdims=None): return (a, out)
""" 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)
""" overwrite_input=overwrite_input) return r.reshape(k) else:
# can't be reasonably be implemented in terms of percentile as we have to # call mean to not break astropy
# Set the partition indexes else: else: # Check if the array contains any nan's
else: a.partition(kth, axis=axis) part = a else:
# make 0-D arrays work return part.item()
# index with slice to allow mean (below) to work else:
# Check if the array contains any nan's # warn and return nans like mean would 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)
interpolation=None, keepdims=None): return (a, q, out)
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()
""" raise ValueError("Percentiles must be in the range [0, 100]") a, q, axis, out, overwrite_input, interpolation, keepdims)
interpolation=None, keepdims=None): return (a, q, out)
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) """ raise ValueError("Quantiles must be in the range [0, 1]") a, q, axis, out, overwrite_input, interpolation, keepdims)
interpolation='linear', keepdims=False): """Assumes that q is in [0, 1], and is an ndarray""" overwrite_input=overwrite_input, interpolation=interpolation) return r.reshape(q.shape + k) else:
# avoid expensive reductions, relevant for arrays with < O(1000) elements return False else: # faster than any() return False
interpolation='linear', keepdims=False): # Do not allow 0-d arrays because following code fails for scalar else:
# prepare a for partitioning if axis is None: ap = a.ravel() else: ap = a else: else: ap = a.copy()
# round fractional indices according to interpolation method indices = floor(indices).astype(intp) indices = ceil(indices).astype(intp) indices = 0.5 * (floor(indices) + ceil(indices)) indices = around(indices).astype(intp) else: raise ValueError( "interpolation can only be 'linear', 'lower' 'higher', " "'midpoint', or 'nearest'")
# 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
# Check if the array contains any nan's
# ensure axis with q-th is first
# Check if the array contains any nan's
# ensure axis with q-th is first
r = add(x1, x2, out=out) else:
warnings.warn("Invalid value encountered in percentile", RuntimeWarning, stacklevel=3) 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 (y, x)
""" 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
return xi
# Based on scitools meshgrid def meshgrid(*xi, **kwargs): """ 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()
"""
raise TypeError("meshgrid() got an unexpected keyword argument '%s'" % (list(kwargs)[0],))
raise ValueError( "Valid values for `indexing` are 'xy' and 'ij'.")
for i, x in enumerate(xi)]
# switch first and second axis
# Return the full N-D matrix (not only the 1-D vector)
return (arr, obj)
""" Return a new array with sub-arrays along an axis deleted. For a one dimensional array, this returns those entries not returned by `arr[obj]`.
Parameters ---------- arr : array_like Input array. obj : slice, int or array of ints Indicate which sub-arrays to remove. axis : int, optional The axis along which to delete the subarray defined by `obj`. If `axis` is None, `obj` is applied to the flattened array.
Returns ------- out : ndarray A copy of `arr` with the elements specified by `obj` removed. Note that `delete` does not occur in-place. If `axis` is None, `out` is a flattened array.
See Also -------- insert : Insert elements into an array. append : Append elements at the end of an array.
Notes ----- Often it is preferable to use a boolean mask. For example:
>>> mask = np.ones(len(arr), dtype=bool) >>> mask[[0,2,4]] = False >>> result = arr[mask,...]
Is equivalent to `np.delete(arr, [0,2,4], axis=0)`, but allows further use of `mask`.
Examples -------- >>> arr = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]]) >>> arr array([[ 1, 2, 3, 4], [ 5, 6, 7, 8], [ 9, 10, 11, 12]]) >>> np.delete(arr, 1, 0) array([[ 1, 2, 3, 4], [ 9, 10, 11, 12]])
>>> np.delete(arr, np.s_[::2], 1) array([[ 2, 4], [ 6, 8], [10, 12]]) >>> np.delete(arr, [1,3,5], None) array([ 1, 3, 5, 7, 8, 9, 10, 11, 12])
""" 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() ndim = arr.ndim axis = -1
if ndim == 0: # 2013-09-24, 1.9 warnings.warn( "in the future the special handling of scalars will be removed " "from delete and raise an error", DeprecationWarning, stacklevel=2) if wrap: return wrap(arr) else: return arr.copy(order=arrorder)
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 chunck 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
_obj = obj obj = np.asarray(obj) # After removing the special handling of booleans and out of # bounds values, the conversion to the array can be removed. if obj.dtype == bool: warnings.warn("in the future insert will treat boolean arrays and " "array-likes as boolean index instead of casting it " "to integer", FutureWarning, stacklevel=2) obj = obj.astype(intp) if isinstance(_obj, (int, long, integer)): # optimization for a single value obj = obj.item() if (obj < -N or obj >= N): raise IndexError( "index %i is out of bounds for axis %i with " "size %i" % (obj, axis, N)) if (obj < 0): obj += N newshape[axis] -= 1 new = empty(newshape, arr.dtype, arrorder) slobj[axis] = slice(None, obj) new[tuple(slobj)] = arr[tuple(slobj)] slobj[axis] = slice(obj, None) slobj2 = [slice(None)]*ndim slobj2[axis] = slice(obj+1, None) new[tuple(slobj)] = arr[tuple(slobj2)] else: if obj.size == 0 and not isinstance(_obj, np.ndarray): obj = obj.astype(intp) if not np.can_cast(obj, intp, 'same_kind'): # obj.size = 1 special case always failed and would just # give superfluous warnings. # 2013-09-24, 1.9 warnings.warn( "using a non-integer array as obj in delete will result in an " "error in the future", DeprecationWarning, stacklevel=2) obj = obj.astype(intp) keep = ones(N, dtype=bool)
# Test if there are out of bound indices, this is deprecated inside_bounds = (obj < N) & (obj >= -N) if not inside_bounds.all(): # 2013-09-24, 1.9 warnings.warn( "in the future out of bounds indices will raise an error " "instead of being ignored by `numpy.delete`.", DeprecationWarning, stacklevel=2) obj = obj[inside_bounds] positive_indices = obj >= 0 if not positive_indices.all(): warnings.warn( "in the future negative indices will not be ignored by " "`numpy.delete`.", FutureWarning, stacklevel=2) obj = obj[positive_indices]
keep[obj, ] = False slobj[axis] = keep new = arr[tuple(slobj)]
if wrap: return wrap(new) else: return new
return (arr, obj, values)
""" Insert values along the given axis before the given indices.
Parameters ---------- arr : array_like Input array. obj : int, slice or sequence of ints Object that defines the index or indices before which `values` is inserted.
.. versionadded:: 1.8.0
Support for multiple insertions when `obj` is a single scalar or a sequence with one element (similar to calling insert multiple times). values : array_like Values to insert into `arr`. If the type of `values` is different from that of `arr`, `values` is converted to the type of `arr`. `values` should be shaped so that ``arr[...,obj,...] = values`` is legal. axis : int, optional Axis along which to insert `values`. If `axis` is None then `arr` is flattened first.
Returns ------- out : ndarray A copy of `arr` with `values` inserted. Note that `insert` does not occur in-place: a new array is returned. If `axis` is None, `out` is a flattened array.
See Also -------- append : Append elements at the end of an array. concatenate : Join a sequence of arrays along an existing axis. delete : Delete elements from an array.
Notes ----- Note that for higher dimensional inserts `obj=0` behaves very different from `obj=[0]` just like `arr[:,0,:] = values` is different from `arr[:,[0],:] = values`.
Examples -------- >>> a = np.array([[1, 1], [2, 2], [3, 3]]) >>> a array([[1, 1], [2, 2], [3, 3]]) >>> np.insert(a, 1, 5) array([1, 5, 1, 2, 2, 3, 3]) >>> np.insert(a, 1, 5, axis=1) array([[1, 5, 1], [2, 5, 2], [3, 5, 3]])
Difference between sequence and scalars:
>>> np.insert(a, [1], [[1],[2],[3]], axis=1) array([[1, 1, 1], [2, 2, 2], [3, 3, 3]]) >>> np.array_equal(np.insert(a, 1, [1, 2, 3], axis=1), ... np.insert(a, [1], [[1],[2],[3]], axis=1)) True
>>> b = a.flatten() >>> b array([1, 1, 2, 2, 3, 3]) >>> np.insert(b, [2, 2], [5, 6]) array([1, 1, 5, 6, 2, 2, 3, 3])
>>> np.insert(b, slice(2, 4), [5, 6]) array([1, 1, 5, 2, 6, 2, 3, 3])
>>> np.insert(b, [2, 2], [7.13, False]) # type casting array([1, 1, 7, 0, 2, 2, 3, 3])
>>> x = np.arange(8).reshape(2, 4) >>> idx = (1, 3) >>> np.insert(x, idx, 999, axis=1) array([[ 0, 999, 1, 2, 999, 3], [ 4, 999, 5, 6, 999, 7]])
""" try: wrap = arr.__array_wrap__ except AttributeError: pass
arr = arr.ravel() elif ndim == 0: # 2013-09-24, 1.9 warnings.warn( "in the future the special handling of scalars will be removed " "from insert and raise an error", DeprecationWarning, stacklevel=2) arr = arr.copy(order=arrorder) arr[...] = values if wrap: return wrap(arr) else: return arr else: axis = normalize_axis_index(axis, ndim)
# 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 # See also delete warnings.warn( "in the future insert will treat boolean arrays and " "array-likes as a boolean index instead of casting it to " "integer", FutureWarning, stacklevel=2) indices = indices.astype(intp) # Code after warning period: #if obj.ndim != 1: # raise ValueError('boolean array argument obj to insert ' # 'must be one dimensional') #indices = np.flatnonzero(obj) raise ValueError( "index array argument obj to insert must be one dimensional " "or scalar") raise IndexError( "index %i is out of bounds for axis %i with " "size %i" % (obj, axis, N)) index += N
# There are some object array corner cases here, but we cannot avoid # that: # 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,:]) return wrap(new) elif indices.size == 0 and not isinstance(obj, np.ndarray): # Can safely cast the empty list to intp indices = indices.astype(intp)
if not np.can_cast(indices, intp, 'same_kind'): # 2013-09-24, 1.9 warnings.warn( "using a non-integer array as obj in insert will result in an " "error in the future", DeprecationWarning, stacklevel=2) 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
return (arr, values)
""" Append values to the end of an array.
Parameters ---------- arr : array_like Values are appended to a copy of this array. values : array_like These values are appended to a copy of `arr`. It must be of the correct shape (the same shape as `arr`, excluding `axis`). If `axis` is not specified, `values` can be any shape and will be flattened before use. axis : int, optional The axis along which `values` are appended. If `axis` is not given, both `arr` and `values` are flattened before use.
Returns ------- append : ndarray A copy of `arr` with `values` appended to `axis`. Note that `append` does not occur in-place: a new array is allocated and filled. If `axis` is None, `out` is a flattened array.
See Also -------- insert : Insert elements into an array. delete : Delete elements from an array.
Examples -------- >>> np.append([1, 2, 3], [[4, 5, 6], [7, 8, 9]]) array([1, 2, 3, 4, 5, 6, 7, 8, 9])
When `axis` is specified, `values` must have the correct shape.
>>> np.append([[1, 2, 3], [4, 5, 6]], [[7, 8, 9]], axis=0) array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> np.append([[1, 2, 3], [4, 5, 6]], [7, 8, 9], axis=0) Traceback (most recent call last): ... ValueError: arrays must have same number of dimensions
""" 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)
return (x, bins)
""" 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]) """
# here for compatibility, searchsorted below is happy to take this raise TypeError("x may not be complex")
raise ValueError("bins must be monotonically increasing or decreasing")
# this is backwards because the arguments below are swapped # reverse the bins, and invert the results return len(bins) - _nx.searchsorted(bins[::-1], x, side=side) else: |