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window.py
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"""
provide a generic structure to support window functions,
similar to how we have a Groupby object
"""
from __future__ import division
import warnings
import numpy as np
from collections import defaultdict
from datetime import timedelta
from pandas.core.dtypes.generic import (
ABCSeries,
ABCDataFrame,
ABCDatetimeIndex,
ABCTimedeltaIndex,
ABCPeriodIndex)
from pandas.core.dtypes.common import (
is_integer,
is_bool,
is_float_dtype,
is_integer_dtype,
needs_i8_conversion,
is_timedelta64_dtype,
is_list_like,
_ensure_float64,
is_scalar)
import pandas as pd
from pandas.core.base import (PandasObject, SelectionMixin,
GroupByMixin)
import pandas.core.common as com
import pandas._libs.window as _window
from pandas.tseries.offsets import DateOffset
from pandas import compat
from pandas.compat.numpy import function as nv
from pandas.util._decorators import (Substitution, Appender,
cache_readonly)
from pandas.core.generic import _shared_docs
from textwrap import dedent
_shared_docs = dict(**_shared_docs)
_doc_template = """
Returns
-------
same type as input
See also
--------
pandas.Series.%(name)s
pandas.DataFrame.%(name)s
"""
class _Window(PandasObject, SelectionMixin):
_attributes = ['window', 'min_periods', 'freq', 'center', 'win_type',
'axis', 'on', 'closed']
exclusions = set()
def __init__(self, obj, window=None, min_periods=None, freq=None,
center=False, win_type=None, axis=0, on=None, closed=None,
**kwargs):
if freq is not None:
warnings.warn("The freq kw is deprecated and will be removed in a "
"future version. You can resample prior to passing "
"to a window function", FutureWarning, stacklevel=3)
self.__dict__.update(kwargs)
self.blocks = []
self.obj = obj
self.on = on
self.closed = closed
self.window = window
self.min_periods = min_periods
self.freq = freq
self.center = center
self.win_type = win_type
self.win_freq = None
self.axis = obj._get_axis_number(axis) if axis is not None else None
self.validate()
@property
def _constructor(self):
return Window
@property
def is_datetimelike(self):
return None
@property
def _on(self):
return None
@property
def is_freq_type(self):
return self.win_type == 'freq'
def validate(self):
if self.center is not None and not is_bool(self.center):
raise ValueError("center must be a boolean")
if self.min_periods is not None and not \
is_integer(self.min_periods):
raise ValueError("min_periods must be an integer")
if self.closed is not None and self.closed not in \
['right', 'both', 'left', 'neither']:
raise ValueError("closed must be 'right', 'left', 'both' or "
"'neither'")
def _convert_freq(self, how=None):
""" resample according to the how, return a new object """
obj = self._selected_obj
index = None
if (self.freq is not None and
isinstance(obj, (ABCSeries, ABCDataFrame))):
if how is not None:
warnings.warn("The how kw argument is deprecated and removed "
"in a future version. You can resample prior "
"to passing to a window function", FutureWarning,
stacklevel=6)
obj = obj.resample(self.freq).aggregate(how or 'asfreq')
return obj, index
def _create_blocks(self, how):
""" split data into blocks & return conformed data """
obj, index = self._convert_freq(how)
if index is not None:
index = self._on
# filter out the on from the object
if self.on is not None:
if obj.ndim == 2:
obj = obj.reindex(columns=obj.columns.difference([self.on]),
copy=False)
blocks = obj.as_blocks(copy=False).values()
return blocks, obj, index
def _gotitem(self, key, ndim, subset=None):
"""
sub-classes to define
return a sliced object
Parameters
----------
key : string / list of selections
ndim : 1,2
requested ndim of result
subset : object, default None
subset to act on
"""
# create a new object to prevent aliasing
if subset is None:
subset = self.obj
self = self._shallow_copy(subset)
self._reset_cache()
if subset.ndim == 2:
if is_scalar(key) and key in subset or is_list_like(key):
self._selection = key
return self
def __getattr__(self, attr):
if attr in self._internal_names_set:
return object.__getattribute__(self, attr)
if attr in self.obj:
return self[attr]
raise AttributeError("%r object has no attribute %r" %
(type(self).__name__, attr))
def _dir_additions(self):
return self.obj._dir_additions()
def _get_window(self, other=None):
return self.window
@property
def _window_type(self):
return self.__class__.__name__
def __unicode__(self):
""" provide a nice str repr of our rolling object """
attrs = ["{k}={v}".format(k=k, v=getattr(self, k))
for k in self._attributes
if getattr(self, k, None) is not None]
return "{klass} [{attrs}]".format(klass=self._window_type,
attrs=','.join(attrs))
def _get_index(self, index=None):
"""
Return index as ndarrays
Returns
-------
tuple of (index, index_as_ndarray)
"""
if self.is_freq_type:
if index is None:
index = self._on
return index, index.asi8
return index, index
def _prep_values(self, values=None, kill_inf=True, how=None):
if values is None:
values = getattr(self._selected_obj, 'values', self._selected_obj)
# GH #12373 : rolling functions error on float32 data
# make sure the data is coerced to float64
if is_float_dtype(values.dtype):
values = _ensure_float64(values)
elif is_integer_dtype(values.dtype):
values = _ensure_float64(values)
elif needs_i8_conversion(values.dtype):
raise NotImplementedError("ops for {action} for this "
"dtype {dtype} are not "
"implemented".format(
action=self._window_type,
dtype=values.dtype))
else:
try:
values = _ensure_float64(values)
except (ValueError, TypeError):
raise TypeError("cannot handle this type -> {0}"
"".format(values.dtype))
if kill_inf:
values = values.copy()
values[np.isinf(values)] = np.NaN
return values
def _wrap_result(self, result, block=None, obj=None):
""" wrap a single result """
if obj is None:
obj = self._selected_obj
index = obj.index
if isinstance(result, np.ndarray):
# coerce if necessary
if block is not None:
if is_timedelta64_dtype(block.values.dtype):
result = pd.to_timedelta(
result.ravel(), unit='ns').values.reshape(result.shape)
if result.ndim == 1:
from pandas import Series
return Series(result, index, name=obj.name)
return type(obj)(result, index=index, columns=block.columns)
return result
def _wrap_results(self, results, blocks, obj):
"""
wrap the results
Paramters
---------
results : list of ndarrays
blocks : list of blocks
obj : conformed data (may be resampled)
"""
from pandas import Series
from pandas.core.index import _ensure_index
final = []
for result, block in zip(results, blocks):
result = self._wrap_result(result, block=block, obj=obj)
if result.ndim == 1:
return result
final.append(result)
# if we have an 'on' column
# we want to put it back into the results
# in the same location
columns = self._selected_obj.columns
if self.on is not None \
and not self._on.equals(obj.index):
name = self._on.name
final.append(Series(self._on, index=obj.index, name=name))
if self._selection is not None:
selection = _ensure_index(self._selection)
# need to reorder to include original location of
# the on column (if its not already there)
if name not in selection:
columns = self.obj.columns
indexer = columns.get_indexer(selection.tolist() + [name])
columns = columns.take(sorted(indexer))
if not len(final):
return obj.astype('float64')
return pd.concat(final, axis=1).reindex(columns=columns,
copy=False)
def _center_window(self, result, window):
""" center the result in the window """
if self.axis > result.ndim - 1:
raise ValueError("Requested axis is larger then no. of argument "
"dimensions")
from pandas import Series, DataFrame
offset = _offset(window, True)
if offset > 0:
if isinstance(result, (Series, DataFrame)):
result = result.slice_shift(-offset, axis=self.axis)
else:
lead_indexer = [slice(None)] * result.ndim
lead_indexer[self.axis] = slice(offset, None)
result = np.copy(result[tuple(lead_indexer)])
return result
def aggregate(self, arg, *args, **kwargs):
result, how = self._aggregate(arg, *args, **kwargs)
if result is None:
return self.apply(arg, args=args, kwargs=kwargs)
return result
agg = aggregate
_shared_docs['sum'] = dedent("""
%(name)s sum
Parameters
----------
how : string, default None
.. deprecated:: 0.18.0
Method for down- or re-sampling""")
_shared_docs['mean'] = dedent("""
%(name)s mean
Parameters
----------
how : string, default None
.. deprecated:: 0.18.0
Method for down- or re-sampling""")
class Window(_Window):
"""
Provides rolling window calculations.
.. versionadded:: 0.18.0
Parameters
----------
window : int, or offset
Size of the moving window. This is the number of observations used for
calculating the statistic. Each window will be a fixed size.
If its an offset then this will be the time period of each window. Each
window will be a variable sized based on the observations included in
the time-period. This is only valid for datetimelike indexes. This is
new in 0.19.0
min_periods : int, default None
Minimum number of observations in window required to have a value
(otherwise result is NA). For a window that is specified by an offset,
this will default to 1.
freq : string or DateOffset object, optional (default None)
.. deprecated:: 0.18.0
Frequency to conform the data to before computing the statistic.
Specified as a frequency string or DateOffset object.
center : boolean, default False
Set the labels at the center of the window.
win_type : string, default None
Provide a window type. See the notes below.
on : string, optional
For a DataFrame, column on which to calculate
the rolling window, rather than the index
closed : string, default None
Make the interval closed on the 'right', 'left', 'both' or
'neither' endpoints.
For offset-based windows, it defaults to 'right'.
For fixed windows, defaults to 'both'. Remaining cases not implemented
for fixed windows.
.. versionadded:: 0.20.0
axis : int or string, default 0
Returns
-------
a Window or Rolling sub-classed for the particular operation
Examples
--------
>>> df = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]})
>>> df
B
0 0.0
1 1.0
2 2.0
3 NaN
4 4.0
Rolling sum with a window length of 2, using the 'triang'
window type.
>>> df.rolling(2, win_type='triang').sum()
B
0 NaN
1 1.0
2 2.5
3 NaN
4 NaN
Rolling sum with a window length of 2, min_periods defaults
to the window length.
>>> df.rolling(2).sum()
B
0 NaN
1 1.0
2 3.0
3 NaN
4 NaN
Same as above, but explicity set the min_periods
>>> df.rolling(2, min_periods=1).sum()
B
0 0.0
1 1.0
2 3.0
3 2.0
4 4.0
A ragged (meaning not-a-regular frequency), time-indexed DataFrame
>>> df = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
....: index = [pd.Timestamp('20130101 09:00:00'),
....: pd.Timestamp('20130101 09:00:02'),
....: pd.Timestamp('20130101 09:00:03'),
....: pd.Timestamp('20130101 09:00:05'),
....: pd.Timestamp('20130101 09:00:06')])
>>> df
B
2013-01-01 09:00:00 0.0
2013-01-01 09:00:02 1.0
2013-01-01 09:00:03 2.0
2013-01-01 09:00:05 NaN
2013-01-01 09:00:06 4.0
Contrasting to an integer rolling window, this will roll a variable
length window corresponding to the time period.
The default for min_periods is 1.
>>> df.rolling('2s').sum()
B
2013-01-01 09:00:00 0.0
2013-01-01 09:00:02 1.0
2013-01-01 09:00:03 3.0
2013-01-01 09:00:05 NaN
2013-01-01 09:00:06 4.0
Notes
-----
By default, the result is set to the right edge of the window. This can be
changed to the center of the window by setting ``center=True``.
The `freq` keyword is used to conform time series data to a specified
frequency by resampling the data. This is done with the default parameters
of :meth:`~pandas.Series.resample` (i.e. using the `mean`).
To learn more about the offsets & frequency strings, please see `this link
<http://pandas.pydata.org/pandas-docs/stable/timeseries.html#offset-aliases>`__.
The recognized win_types are:
* ``boxcar``
* ``triang``
* ``blackman``
* ``hamming``
* ``bartlett``
* ``parzen``
* ``bohman``
* ``blackmanharris``
* ``nuttall``
* ``barthann``
* ``kaiser`` (needs beta)
* ``gaussian`` (needs std)
* ``general_gaussian`` (needs power, width)
* ``slepian`` (needs width).
"""
def validate(self):
super(Window, self).validate()
window = self.window
if isinstance(window, (list, tuple, np.ndarray)):
pass
elif is_integer(window):
if window < 0:
raise ValueError("window must be non-negative")
try:
import scipy.signal as sig
except ImportError:
raise ImportError('Please install scipy to generate window '
'weight')
if not isinstance(self.win_type, compat.string_types):
raise ValueError('Invalid win_type {0}'.format(self.win_type))
if getattr(sig, self.win_type, None) is None:
raise ValueError('Invalid win_type {0}'.format(self.win_type))
else:
raise ValueError('Invalid window {0}'.format(window))
def _prep_window(self, **kwargs):
"""
provide validation for our window type, return the window
we have already been validated
"""
window = self._get_window()
if isinstance(window, (list, tuple, np.ndarray)):
return com._asarray_tuplesafe(window).astype(float)
elif is_integer(window):
import scipy.signal as sig
# the below may pop from kwargs
def _validate_win_type(win_type, kwargs):
arg_map = {'kaiser': ['beta'],
'gaussian': ['std'],
'general_gaussian': ['power', 'width'],
'slepian': ['width']}
if win_type in arg_map:
return tuple([win_type] + _pop_args(win_type,
arg_map[win_type],
kwargs))
return win_type
def _pop_args(win_type, arg_names, kwargs):
msg = '%s window requires %%s' % win_type
all_args = []
for n in arg_names:
if n not in kwargs:
raise ValueError(msg % n)
all_args.append(kwargs.pop(n))
return all_args
win_type = _validate_win_type(self.win_type, kwargs)
# GH #15662. `False` makes symmetric window, rather than periodic.
return sig.get_window(win_type, window, False).astype(float)
def _apply_window(self, mean=True, how=None, **kwargs):
"""
Applies a moving window of type ``window_type`` on the data.
Parameters
----------
mean : boolean, default True
If True computes weighted mean, else weighted sum
how : string, default to None
.. deprecated:: 0.18.0
how to resample
Returns
-------
y : type of input argument
"""
window = self._prep_window(**kwargs)
center = self.center
blocks, obj, index = self._create_blocks(how=how)
results = []
for b in blocks:
try:
values = self._prep_values(b.values)
except TypeError:
results.append(b.values.copy())
continue
if values.size == 0:
results.append(values.copy())
continue
offset = _offset(window, center)
additional_nans = np.array([np.NaN] * offset)
def f(arg, *args, **kwargs):
minp = _use_window(self.min_periods, len(window))
return _window.roll_window(np.concatenate((arg,
additional_nans))
if center else arg, window, minp,
avg=mean)
result = np.apply_along_axis(f, self.axis, values)
if center:
result = self._center_window(result, window)
results.append(result)
return self._wrap_results(results, blocks, obj)
_agg_doc = dedent("""
Examples
--------
>>> df = pd.DataFrame(np.random.randn(10, 3), columns=['A', 'B', 'C'])
>>> df
A B C
0 -2.385977 -0.102758 0.438822
1 -1.004295 0.905829 -0.954544
2 0.735167 -0.165272 -1.619346
3 -0.702657 -1.340923 -0.706334
4 -0.246845 0.211596 -0.901819
5 2.463718 3.157577 -1.380906
6 -1.142255 2.340594 -0.039875
7 1.396598 -1.647453 1.677227
8 -0.543425 1.761277 -0.220481
9 -0.640505 0.289374 -1.550670
>>> df.rolling(3, win_type='boxcar').agg('mean')
A B C
0 NaN NaN NaN
1 NaN NaN NaN
2 -0.885035 0.212600 -0.711689
3 -0.323928 -0.200122 -1.093408
4 -0.071445 -0.431533 -1.075833
5 0.504739 0.676083 -0.996353
6 0.358206 1.903256 -0.774200
7 0.906020 1.283573 0.085482
8 -0.096361 0.818139 0.472290
9 0.070889 0.134399 -0.031308
See also
--------
pandas.DataFrame.rolling.aggregate
pandas.DataFrame.aggregate
""")
@Appender(_agg_doc)
@Appender(_shared_docs['aggregate'] % dict(
versionadded='',
klass='Series/DataFrame'))
def aggregate(self, arg, *args, **kwargs):
result, how = self._aggregate(arg, *args, **kwargs)
if result is None:
# these must apply directly
result = arg(self)
return result
agg = aggregate
@Substitution(name='window')
@Appender(_doc_template)
@Appender(_shared_docs['sum'])
def sum(self, *args, **kwargs):
nv.validate_window_func('sum', args, kwargs)
return self._apply_window(mean=False, **kwargs)
@Substitution(name='window')
@Appender(_doc_template)
@Appender(_shared_docs['mean'])
def mean(self, *args, **kwargs):
nv.validate_window_func('mean', args, kwargs)
return self._apply_window(mean=True, **kwargs)
class _GroupByMixin(GroupByMixin):
""" provide the groupby facilities """
def __init__(self, obj, *args, **kwargs):
parent = kwargs.pop('parent', None) # noqa
groupby = kwargs.pop('groupby', None)
if groupby is None:
groupby, obj = obj, obj.obj
self._groupby = groupby
self._groupby.mutated = True
self._groupby.grouper.mutated = True
super(GroupByMixin, self).__init__(obj, *args, **kwargs)
count = GroupByMixin._dispatch('count')
corr = GroupByMixin._dispatch('corr', other=None, pairwise=None)
cov = GroupByMixin._dispatch('cov', other=None, pairwise=None)
def _apply(self, func, name, window=None, center=None,
check_minp=None, how=None, **kwargs):
"""
dispatch to apply; we are stripping all of the _apply kwargs and
performing the original function call on the grouped object
"""
def f(x, name=name, *args):
x = self._shallow_copy(x)
if isinstance(name, compat.string_types):
return getattr(x, name)(*args, **kwargs)
return x.apply(name, *args, **kwargs)
return self._groupby.apply(f)
class _Rolling(_Window):
@property
def _constructor(self):
return Rolling
def _apply(self, func, name=None, window=None, center=None,
check_minp=None, how=None, **kwargs):
"""
Rolling statistical measure using supplied function. Designed to be
used with passed-in Cython array-based functions.
Parameters
----------
func : string/callable to apply
name : string, optional
name of this function
window : int/array, default to _get_window()
center : boolean, default to self.center
check_minp : function, default to _use_window
how : string, default to None
.. deprecated:: 0.18.0
how to resample
Returns
-------
y : type of input
"""
if center is None:
center = self.center
if window is None:
window = self._get_window()
if check_minp is None:
check_minp = _use_window
blocks, obj, index = self._create_blocks(how=how)
index, indexi = self._get_index(index=index)
results = []
for b in blocks:
try:
values = self._prep_values(b.values)
except TypeError:
results.append(b.values.copy())
continue
if values.size == 0:
results.append(values.copy())
continue
# if we have a string function name, wrap it
if isinstance(func, compat.string_types):
cfunc = getattr(_window, func, None)
if cfunc is None:
raise ValueError("we do not support this function "
"in _window.{0}".format(func))
def func(arg, window, min_periods=None, closed=None):
minp = check_minp(min_periods, window)
# ensure we are only rolling on floats
arg = _ensure_float64(arg)
return cfunc(arg,
window, minp, indexi, closed, **kwargs)
# calculation function
if center:
offset = _offset(window, center)
additional_nans = np.array([np.NaN] * offset)
def calc(x):
return func(np.concatenate((x, additional_nans)),
window, min_periods=self.min_periods,
closed=self.closed)
else:
def calc(x):
return func(x, window, min_periods=self.min_periods,
closed=self.closed)
with np.errstate(all='ignore'):
if values.ndim > 1:
result = np.apply_along_axis(calc, self.axis, values)
else:
result = calc(values)
if center:
result = self._center_window(result, window)
results.append(result)
return self._wrap_results(results, blocks, obj)
class _Rolling_and_Expanding(_Rolling):
_shared_docs['count'] = """%(name)s count of number of non-NaN
observations inside provided window."""
def count(self):
blocks, obj, index = self._create_blocks(how=None)
index, indexi = self._get_index(index=index)
window = self._get_window()
window = min(window, len(obj)) if not self.center else window
results = []
for b in blocks:
result = b.notnull().astype(int)
result = self._constructor(result, window=window, min_periods=0,
center=self.center,
closed=self.closed).sum()
results.append(result)
return self._wrap_results(results, blocks, obj)
_shared_docs['apply'] = dedent("""
%(name)s function apply
Parameters
----------
func : function
Must produce a single value from an ndarray input
\*args and \*\*kwargs are passed to the function""")
def apply(self, func, args=(), kwargs={}):
# TODO: _level is unused?
_level = kwargs.pop('_level', None) # noqa
window = self._get_window()
offset = _offset(window, self.center)
index, indexi = self._get_index()
def f(arg, window, min_periods, closed):
minp = _use_window(min_periods, window)
return _window.roll_generic(arg, window, minp, indexi, closed,
offset, func, args, kwargs)
return self._apply(f, func, args=args, kwargs=kwargs,
center=False)
def sum(self, *args, **kwargs):
nv.validate_window_func('sum', args, kwargs)
return self._apply('roll_sum', 'sum', **kwargs)
_shared_docs['max'] = dedent("""
%(name)s maximum
Parameters
----------
how : string, default 'max'
.. deprecated:: 0.18.0
Method for down- or re-sampling""")
def max(self, how=None, *args, **kwargs):
nv.validate_window_func('max', args, kwargs)
if self.freq is not None and how is None:
how = 'max'
return self._apply('roll_max', 'max', how=how, **kwargs)
_shared_docs['min'] = dedent("""
%(name)s minimum
Parameters
----------
how : string, default 'min'
.. deprecated:: 0.18.0
Method for down- or re-sampling""")
def min(self, how=None, *args, **kwargs):
nv.validate_window_func('min', args, kwargs)
if self.freq is not None and how is None:
how = 'min'
return self._apply('roll_min', 'min', how=how, **kwargs)
def mean(self, *args, **kwargs):
nv.validate_window_func('mean', args, kwargs)
return self._apply('roll_mean', 'mean', **kwargs)
_shared_docs['median'] = dedent("""
%(name)s median
Parameters
----------
how : string, default 'median'
.. deprecated:: 0.18.0
Method for down- or re-sampling""")
def median(self, how=None, **kwargs):
if self.freq is not None and how is None:
how = 'median'
return self._apply('roll_median_c', 'median', how=how, **kwargs)
_shared_docs['std'] = dedent("""
%(name)s standard deviation
Parameters
----------
ddof : int, default 1
Delta Degrees of Freedom. The divisor used in calculations
is ``N - ddof``, where ``N`` represents the number of elements.""")
def std(self, ddof=1, *args, **kwargs):
nv.validate_window_func('std', args, kwargs)
window = self._get_window()
index, indexi = self._get_index()
def f(arg, *args, **kwargs):
minp = _require_min_periods(1)(self.min_periods, window)
return _zsqrt(_window.roll_var(arg, window, minp, indexi,
self.closed, ddof))
return self._apply(f, 'std', check_minp=_require_min_periods(1),
ddof=ddof, **kwargs)
_shared_docs['var'] = dedent("""
%(name)s variance
Parameters
----------
ddof : int, default 1
Delta Degrees of Freedom. The divisor used in calculations
is ``N - ddof``, where ``N`` represents the number of elements.""")
def var(self, ddof=1, *args, **kwargs):
nv.validate_window_func('var', args, kwargs)
return self._apply('roll_var', 'var',
check_minp=_require_min_periods(1), ddof=ddof,
**kwargs)
_shared_docs['skew'] = """Unbiased %(name)s skewness"""
def skew(self, **kwargs):
return self._apply('roll_skew', 'skew',
check_minp=_require_min_periods(3), **kwargs)
_shared_docs['kurt'] = """Unbiased %(name)s kurtosis"""
def kurt(self, **kwargs):
return self._apply('roll_kurt', 'kurt',
check_minp=_require_min_periods(4), **kwargs)
_shared_docs['quantile'] = dedent("""
%(name)s quantile
Parameters
----------
quantile : float
0 <= quantile <= 1""")
def quantile(self, quantile, **kwargs):
window = self._get_window()
index, indexi = self._get_index()
def f(arg, *args, **kwargs):
minp = _use_window(self.min_periods, window)
return _window.roll_quantile(arg, window, minp, indexi,
self.closed, quantile)
return self._apply(f, 'quantile', quantile=quantile,
**kwargs)
_shared_docs['cov'] = dedent("""
%(name)s sample covariance
Parameters
----------
other : Series, DataFrame, or ndarray, optional
if not supplied then will default to self and produce pairwise output
pairwise : bool, default None
If False then only matching columns between self and other will be used
and the output will be a DataFrame.
If True then all pairwise combinations will be calculated and the
output will be a MultiIndexed DataFrame in the case of DataFrame
inputs. In the case of missing elements, only complete pairwise
observations will be used.
ddof : int, default 1
Delta Degrees of Freedom. The divisor used in calculations
is ``N - ddof``, where ``N`` represents the number of elements.""")