Computational tools

Statistical functions

Percent change

Series and DataFrame have a method pct_change() to compute the percent change over a given number of periods (using fill_method to fill NA/null values before computing the percent change).

In [1]: ser = pd.Series(np.random.randn(8))

In [2]: ser.pct_change()
Out[2]: 
0         NaN
1   -1.602976
2    4.334938
3   -0.247456
4   -2.067345
5   -1.142903
6   -1.688214
7   -9.759729
dtype: float64

In [3]: df = pd.DataFrame(np.random.randn(10, 4))

In [4]: df.pct_change(periods=3)
Out[4]: 
          0         1         2         3
0       NaN       NaN       NaN       NaN
1       NaN       NaN       NaN       NaN
2       NaN       NaN       NaN       NaN
3 -0.218320 -1.054001  1.987147 -0.510183
4 -0.439121 -1.816454  0.649715 -4.822809
5 -0.127833 -3.042065 -5.866604 -1.776977
6 -2.596833 -1.959538 -2.111697 -3.798900
7 -0.117826 -2.169058  0.036094 -0.067696
8  2.492606 -1.357320 -1.205802 -1.558697
9 -1.012977  2.324558 -1.003744 -0.371806

Covariance

Series.cov() can be used to compute covariance between series (excluding missing values).

In [5]: s1 = pd.Series(np.random.randn(1000))

In [6]: s2 = pd.Series(np.random.randn(1000))

In [7]: s1.cov(s2)
Out[7]: 0.0006801088174310875

Analogously, DataFrame.cov() to compute pairwise covariances among the series in the DataFrame, also excluding NA/null values.

Note

Assuming the missing data are missing at random this results in an estimate for the covariance matrix which is unbiased. However, for many applications this estimate may not be acceptable because the estimated covariance matrix is not guaranteed to be positive semi-definite. This could lead to estimated correlations having absolute values which are greater than one, and/or a non-invertible covariance matrix. See Estimation of covariance matrices for more details.

In [8]: frame = pd.DataFrame(np.random.randn(1000, 5),
   ...:                      columns=['a', 'b', 'c', 'd', 'e'])
   ...: 

In [9]: frame.cov()
Out[9]: 
          a         b         c         d         e
a  1.000882 -0.003177 -0.002698 -0.006889  0.031912
b -0.003177  1.024721  0.000191  0.009212  0.000857
c -0.002698  0.000191  0.950735 -0.031743 -0.005087
d -0.006889  0.009212 -0.031743  1.002983 -0.047952
e  0.031912  0.000857 -0.005087 -0.047952  1.042487

DataFrame.cov also supports an optional min_periods keyword that specifies the required minimum number of observations for each column pair in order to have a valid result.

In [10]: frame = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c'])

In [11]: frame.loc[frame.index[:5], 'a'] = np.nan

In [12]: frame.loc[frame.index[5:10], 'b'] = np.nan

In [13]: frame.cov()
Out[13]: 
          a         b         c
a  1.123670 -0.412851  0.018169
b -0.412851  1.154141  0.305260
c  0.018169  0.305260  1.301149

In [14]: frame.cov(min_periods=12)
Out[14]: 
          a         b         c
a  1.123670       NaN  0.018169
b       NaN  1.154141  0.305260
c  0.018169  0.305260  1.301149

Correlation

Correlation may be computed using the corr() method. Using the method parameter, several methods for computing correlations are provided:

Method name	Description
pearson (default)	Standard correlation coefficient
kendall	Kendall Tau correlation coefficient
spearman	Spearman rank correlation coefficient

All of these are currently computed using pairwise complete observations. Wikipedia has articles covering the above correlation coefficients:

• Pearson correlation coefficient

• Kendall rank correlation coefficient

• Spearman’s rank correlation coefficient

Note

Please see the caveats associated with this method of calculating correlation matrices in the covariance section.

In [15]: frame = pd.DataFrame(np.random.randn(1000, 5),
   ....:                      columns=['a', 'b', 'c', 'd', 'e'])
   ....: 

In [16]: frame.iloc[::2] = np.nan

# Series with Series
In [17]: frame['a'].corr(frame['b'])
Out[17]: 0.013479040400098775

In [18]: frame['a'].corr(frame['b'], method='spearman')
---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
<ipython-input-18-3a09724e6474> in <module>
----> 1 frame['a'].corr(frame['b'], method='spearman')

~/scipy/pandas/pandas/core/series.py in corr(self, other, method, min_periods)
   2179         if method in ["pearson", "spearman", "kendall"] or callable(method):
   2180             return nanops.nancorr(
-> 2181                 this.values, other.values, method=method, min_periods=min_periods
   2182             )
   2183 

~/scipy/pandas/pandas/core/nanops.py in _f(*args, **kwargs)
     67             try:
     68                 with np.errstate(invalid="ignore"):
---> 69                     return f(*args, **kwargs)
     70             except ValueError as e:
     71                 # we want to transform an object array

~/scipy/pandas/pandas/core/nanops.py in nancorr(a, b, method, min_periods)
   1237         return np.nan
   1238 
-> 1239     f = get_corr_func(method)
   1240     return f(a, b)
   1241 

~/scipy/pandas/pandas/core/nanops.py in get_corr_func(method)
   1243 def get_corr_func(method):
   1244     if method in ["kendall", "spearman"]:
-> 1245         from scipy.stats import kendalltau, spearmanr
   1246     elif method in ["pearson"]:
   1247         pass

ModuleNotFoundError: No module named 'scipy'

# Pairwise correlation of DataFrame columns
In [19]: frame.corr()
Out[19]: 
          a         b         c         d         e
a  1.000000  0.013479 -0.049269 -0.042239 -0.028525
b  0.013479  1.000000 -0.020433 -0.011139  0.005654
c -0.049269 -0.020433  1.000000  0.018587 -0.054269
d -0.042239 -0.011139  0.018587  1.000000 -0.017060
e -0.028525  0.005654 -0.054269 -0.017060  1.000000

Note that non-numeric columns will be automatically excluded from the correlation calculation.

Like cov, corr also supports the optional min_periods keyword:

In [20]: frame = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c'])

In [21]: frame.loc[frame.index[:5], 'a'] = np.nan

In [22]: frame.loc[frame.index[5:10], 'b'] = np.nan

In [23]: frame.corr()
Out[23]: 
          a         b         c
a  1.000000 -0.121111  0.069544
b -0.121111  1.000000  0.051742
c  0.069544  0.051742  1.000000

In [24]: frame.corr(min_periods=12)
Out[24]: 
          a         b         c
a  1.000000       NaN  0.069544
b       NaN  1.000000  0.051742
c  0.069544  0.051742  1.000000

New in version 0.24.0.

The method argument can also be a callable for a generic correlation calculation. In this case, it should be a single function that produces a single value from two ndarray inputs. Suppose we wanted to compute the correlation based on histogram intersection:

# histogram intersection
In [25]: def histogram_intersection(a, b):
   ....:     return np.minimum(np.true_divide(a, a.sum()),
   ....:                       np.true_divide(b, b.sum())).sum()
   ....: 

In [26]: frame.corr(method=histogram_intersection)
Out[26]: 
          a          b          c
a  1.000000  -6.404882  -2.058431
b -6.404882   1.000000 -19.255743
c -2.058431 -19.255743   1.000000

A related method corrwith() is implemented on DataFrame to compute the correlation between like-labeled Series contained in different DataFrame objects.

In [27]: index = ['a', 'b', 'c', 'd', 'e']

In [28]: columns = ['one', 'two', 'three', 'four']

In [29]: df1 = pd.DataFrame(np.random.randn(5, 4), index=index, columns=columns)

In [30]: df2 = pd.DataFrame(np.random.randn(4, 4), index=index[:4], columns=columns)

In [31]: df1.corrwith(df2)
Out[31]: 
one     -0.125501
two     -0.493244
three    0.344056
four     0.004183
dtype: float64

In [32]: df2.corrwith(df1, axis=1)
Out[32]: 
a   -0.675817
b    0.458296
c    0.190809
d   -0.186275
e         NaN
dtype: float64

Data ranking

The rank() method produces a data ranking with ties being assigned the mean of the ranks (by default) for the group:

In [33]: s = pd.Series(np.random.randn(5), index=list('abcde'))

In [34]: s['d'] = s['b']  # so there's a tie

In [35]: s.rank()
Out[35]: 
a    5.0
b    2.5
c    1.0
d    2.5
e    4.0
dtype: float64

rank() is also a DataFrame method and can rank either the rows (axis=0) or the columns (axis=1). NaN values are excluded from the ranking.

In [36]: df = pd.DataFrame(np.random.randn(10, 6))

In [37]: df[4] = df[2][:5]  # some ties

In [38]: df
Out[38]: 
          0         1         2         3         4         5
0 -0.904948 -1.163537 -1.457187  0.135463 -1.457187  0.294650
1 -0.976288 -0.244652 -0.748406 -0.999601 -0.748406 -0.800809
2  0.401965  1.460840  1.256057  1.308127  1.256057  0.876004
3  0.205954  0.369552 -0.669304  0.038378 -0.669304  1.140296
4 -0.477586 -0.730705 -1.129149 -0.601463 -1.129149 -0.211196
5 -1.092970 -0.689246  0.908114  0.204848       NaN  0.463347
6  0.376892  0.959292  0.095572 -0.593740       NaN -0.069180
7 -1.002601  1.957794 -0.120708  0.094214       NaN -1.467422
8 -0.547231  0.664402 -0.519424 -0.073254       NaN -1.263544
9 -0.250277 -0.237428 -1.056443  0.419477       NaN  1.375064

In [39]: df.rank(1)
Out[39]: 
     0    1    2    3    4    5
0  4.0  3.0  1.5  5.0  1.5  6.0
1  2.0  6.0  4.5  1.0  4.5  3.0
2  1.0  6.0  3.5  5.0  3.5  2.0
3  4.0  5.0  1.5  3.0  1.5  6.0
4  5.0  3.0  1.5  4.0  1.5  6.0
5  1.0  2.0  5.0  3.0  NaN  4.0
6  4.0  5.0  3.0  1.0  NaN  2.0
7  2.0  5.0  3.0  4.0  NaN  1.0
8  2.0  5.0  3.0  4.0  NaN  1.0
9  2.0  3.0  1.0  4.0  NaN  5.0

rank optionally takes a parameter ascending which by default is true; when false, data is reverse-ranked, with larger values assigned a smaller rank.

rank supports different tie-breaking methods, specified with the method parameter:

• average : average rank of tied group

• min : lowest rank in the group

• max : highest rank in the group

• first : ranks assigned in the order they appear in the array

Window Functions

For working with data, a number of window functions are provided for computing common window or rolling statistics. Among these are count, sum, mean, median, correlation, variance, covariance, standard deviation, skewness, and kurtosis.

The rolling() and expanding() functions can be used directly from DataFrameGroupBy objects, see the groupby docs.

Note

The API for window statistics is quite similar to the way one works with GroupBy objects, see the documentation here.

We work with rolling, expanding and exponentially weighted data through the corresponding objects, Rolling, Expanding and EWM.

In [40]: s = pd.Series(np.random.randn(1000),
   ....:               index=pd.date_range('1/1/2000', periods=1000))
   ....: 

In [41]: s = s.cumsum()

In [42]: s
Out[42]: 
2000-01-01    -0.268824
2000-01-02    -1.771855
2000-01-03    -0.818003
2000-01-04    -0.659244
2000-01-05    -1.942133
                ...    
2002-09-22   -67.457323
2002-09-23   -69.253182
2002-09-24   -70.296818
2002-09-25   -70.844674
2002-09-26   -72.475016
Freq: D, Length: 1000, dtype: float64

These are created from methods on Series and DataFrame.

In [43]: r = s.rolling(window=60)

In [44]: r
Out[44]: Rolling [window=60,center=False,axis=0]

These object provide tab-completion of the available methods and properties.

In [14]: r.<TAB>                                          # noqa: E225, E999
r.agg         r.apply       r.count       r.exclusions  r.max         r.median      r.name        r.skew        r.sum
r.aggregate   r.corr        r.cov         r.kurt        r.mean        r.min         r.quantile    r.std         r.var

Generally these methods all have the same interface. They all accept the following arguments:

• window: size of moving window

• min_periods: threshold of non-null data points to require (otherwise result is NA)

• center: boolean, whether to set the labels at the center (default is False)

We can then call methods on these rolling objects. These return like-indexed objects:

In [45]: r.mean()
Out[45]: 
2000-01-01          NaN
2000-01-02          NaN
2000-01-03          NaN
2000-01-04          NaN
2000-01-05          NaN
                ...    
2002-09-22   -62.914971
2002-09-23   -63.061867
2002-09-24   -63.213876
2002-09-25   -63.375074
2002-09-26   -63.539734
Freq: D, Length: 1000, dtype: float64

In [46]: s.plot(style='k--')
Out[46]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65bd288828>

In [47]: r.mean().plot(style='k')
Out[47]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65bd288828>

They can also be applied to DataFrame objects. This is really just syntactic sugar for applying the moving window operator to all of the DataFrame’s columns:

In [48]: df = pd.DataFrame(np.random.randn(1000, 4),
   ....:                   index=pd.date_range('1/1/2000', periods=1000),
   ....:                   columns=['A', 'B', 'C', 'D'])
   ....: 

In [49]: df = df.cumsum()

In [50]: df.rolling(window=60).sum().plot(subplots=True)
Out[50]: 
array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f6589eae4e0>,
       <matplotlib.axes._subplots.AxesSubplot object at 0x7f6589e69a58>,
       <matplotlib.axes._subplots.AxesSubplot object at 0x7f6589e19a20>,
       <matplotlib.axes._subplots.AxesSubplot object at 0x7f6589dcb9e8>],
      dtype=object)

Method summary

We provide a number of common statistical functions:

Method	Description
count()	Number of non-null observations
sum()	Sum of values
mean()	Mean of values
median()	Arithmetic median of values
min()	Minimum
max()	Maximum
std()	Bessel-corrected sample standard deviation
var()	Unbiased variance
skew()	Sample skewness (3rd moment)
kurt()	Sample kurtosis (4th moment)
quantile()	Sample quantile (value at %)
apply()	Generic apply
cov()	Unbiased covariance (binary)
corr()	Correlation (binary)

Rolling Apply

The apply() function takes an extra func argument and performs generic rolling computations. The func argument should be a single function that produces a single value from an ndarray input. Suppose we wanted to compute the mean absolute deviation on a rolling basis:

In [51]: def mad(x):
   ....:     return np.fabs(x - x.mean()).mean()
   ....: 

In [52]: s.rolling(window=60).apply(mad, raw=True).plot(style='k')
Out[52]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6589c61048>

New in version 1.0.

Additionally, apply() can leverage Numba if installed as an optional dependency. The apply aggregation can be executed using Numba by specifying engine='numba' and engine_kwargs arguments (raw must also be set to True). Numba will be applied in potentially two routines:

1. If func is a standard Python function, the engine will JIT the passed function. func can also be a JITed function in which case the engine will not JIT the function again.

2. The engine will JIT the for loop where the apply function is applied to each window.

The engine_kwargs argument is a dictionary of keyword arguments that will be passed into the numba.jit decorator. These keyword arguments will be applied to both the passed function (if a standard Python function) and the apply for loop over each window. Currently only nogil, nopython, and parallel are supported, and their default values are set to False, True and False respectively.

Note

In terms of performance, the first time a function is run using the Numba engine will be slow as Numba will have some function compilation overhead. However, rolling objects will cache the function and subsequent calls will be fast. In general, the Numba engine is performant with a larger amount of data points (e.g. 1+ million).

In [1]: data = pd.Series(range(1_000_000))

In [2]: roll = data.rolling(10)

In [3]: def f(x):
   ...:     return np.sum(x) + 5
# Run the first time, compilation time will affect performance
In [4]: %timeit -r 1 -n 1 roll.apply(f, engine='numba', raw=True)  # noqa: E225
1.23 s ± 0 ns per loop (mean ± std. dev. of 1 run, 1 loop each)
# Function is cached and performance will improve
In [5]: %timeit roll.apply(f, engine='numba', raw=True)
188 ms ± 1.93 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

In [6]: %timeit roll.apply(f, engine='cython', raw=True)
3.92 s ± 59 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

Rolling windows

Passing win_type to .rolling generates a generic rolling window computation, that is weighted according the win_type. The following methods are available:

Method	Description
sum()	Sum of values
mean()	Mean of values

The weights used in the window are specified by the win_type keyword. The list of recognized types are the scipy.signal window functions:

• boxcar

• triang

• blackman

• hamming

• bartlett

• parzen

• bohman

• blackmanharris

• nuttall

• barthann

• kaiser (needs beta)

• gaussian (needs std)

• general_gaussian (needs power, width)

• slepian (needs width)

• exponential (needs tau).

In [53]: ser = pd.Series(np.random.randn(10),
   ....:                 index=pd.date_range('1/1/2000', periods=10))
   ....: 

In [54]: ser.rolling(window=5, win_type='triang').mean()
---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
<ipython-input-54-a42237a4feee> in <module>
----> 1 ser.rolling(window=5, win_type='triang').mean()

~/scipy/pandas/pandas/core/generic.py in rolling(self, window, min_periods, center, win_type, on, axis, closed)
  10102                     on=on,
  10103                     axis=axis,
> 10104                     closed=closed,
  10105                 )
  10106 

~/scipy/pandas/pandas/core/window/rolling.py in __init__(self, obj, window, min_periods, center, win_type, axis, on, closed, **kwargs)
     92         self.win_freq = None
     93         self.axis = obj._get_axis_number(axis) if axis is not None else None
---> 94         self.validate()
     95         self._numba_func_cache: Dict[Optional[str], Callable] = dict()
     96 

~/scipy/pandas/pandas/core/window/rolling.py in validate(self)
    949                 raise ValueError("window must be > 0 ")
    950             import_optional_dependency(
--> 951                 "scipy", extra="Scipy is required to generate window weight."
    952             )
    953             import scipy.signal as sig

~/scipy/pandas/pandas/compat/_optional.py in import_optional_dependency(name, extra, raise_on_missing, on_version)
     89     except ImportError:
     90         if raise_on_missing:
---> 91             raise ImportError(msg) from None
     92         else:
     93             return None

ImportError: Missing optional dependency 'scipy'. Scipy is required to generate window weight. Use pip or conda to install scipy.

Note that the boxcar window is equivalent to mean().

In [55]: ser.rolling(window=5, win_type='boxcar').mean()
---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
<ipython-input-55-d77bdc3f0969> in <module>
----> 1 ser.rolling(window=5, win_type='boxcar').mean()

~/scipy/pandas/pandas/core/generic.py in rolling(self, window, min_periods, center, win_type, on, axis, closed)
  10102                     on=on,
  10103                     axis=axis,
> 10104                     closed=closed,
  10105                 )
  10106 

~/scipy/pandas/pandas/core/window/rolling.py in __init__(self, obj, window, min_periods, center, win_type, axis, on, closed, **kwargs)
     92         self.win_freq = None
     93         self.axis = obj._get_axis_number(axis) if axis is not None else None
---> 94         self.validate()
     95         self._numba_func_cache: Dict[Optional[str], Callable] = dict()
     96 

~/scipy/pandas/pandas/core/window/rolling.py in validate(self)
    949                 raise ValueError("window must be > 0 ")
    950             import_optional_dependency(
--> 951                 "scipy", extra="Scipy is required to generate window weight."
    952             )
    953             import scipy.signal as sig

~/scipy/pandas/pandas/compat/_optional.py in import_optional_dependency(name, extra, raise_on_missing, on_version)
     89     except ImportError:
     90         if raise_on_missing:
---> 91             raise ImportError(msg) from None
     92         else:
     93             return None

ImportError: Missing optional dependency 'scipy'. Scipy is required to generate window weight. Use pip or conda to install scipy.

In [56]: ser.rolling(window=5).mean()
Out[56]: 
2000-01-01         NaN
2000-01-02         NaN
2000-01-03         NaN
2000-01-04         NaN
2000-01-05   -0.841164
2000-01-06   -0.779948
2000-01-07   -0.565487
2000-01-08   -0.502815
2000-01-09   -0.553755
2000-01-10   -0.472211
Freq: D, dtype: float64

For some windowing functions, additional parameters must be specified:

In [57]: ser.rolling(window=5, win_type='gaussian').mean(std=0.1)
---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
<ipython-input-57-dbbb08454bfa> in <module>
----> 1 ser.rolling(window=5, win_type='gaussian').mean(std=0.1)

~/scipy/pandas/pandas/core/generic.py in rolling(self, window, min_periods, center, win_type, on, axis, closed)
  10102                     on=on,
  10103                     axis=axis,
> 10104                     closed=closed,
  10105                 )
  10106 

~/scipy/pandas/pandas/core/window/rolling.py in __init__(self, obj, window, min_periods, center, win_type, axis, on, closed, **kwargs)
     92         self.win_freq = None
     93         self.axis = obj._get_axis_number(axis) if axis is not None else None
---> 94         self.validate()
     95         self._numba_func_cache: Dict[Optional[str], Callable] = dict()
     96 

~/scipy/pandas/pandas/core/window/rolling.py in validate(self)
    949                 raise ValueError("window must be > 0 ")
    950             import_optional_dependency(
--> 951                 "scipy", extra="Scipy is required to generate window weight."
    952             )
    953             import scipy.signal as sig

~/scipy/pandas/pandas/compat/_optional.py in import_optional_dependency(name, extra, raise_on_missing, on_version)
     89     except ImportError:
     90         if raise_on_missing:
---> 91             raise ImportError(msg) from None
     92         else:
     93             return None

ImportError: Missing optional dependency 'scipy'. Scipy is required to generate window weight. Use pip or conda to install scipy.

Note

For .sum() with a win_type, there is no normalization done to the weights for the window. Passing custom weights of [1, 1, 1] will yield a different result than passing weights of [2, 2, 2], for example. When passing a win_type instead of explicitly specifying the weights, the weights are already normalized so that the largest weight is 1.

In contrast, the nature of the .mean() calculation is such that the weights are normalized with respect to each other. Weights of [1, 1, 1] and [2, 2, 2] yield the same result.

Time-aware rolling

It is possible to pass an offset (or convertible) to a .rolling() method and have it produce variable sized windows based on the passed time window. For each time point, this includes all preceding values occurring within the indicated time delta.

This can be particularly useful for a non-regular time frequency index.

In [58]: dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
   ....:                    index=pd.date_range('20130101 09:00:00',
   ....:                                        periods=5,
   ....:                                        freq='s'))
   ....: 

In [59]: dft
Out[59]: 
                       B
2013-01-01 09:00:00  0.0
2013-01-01 09:00:01  1.0
2013-01-01 09:00:02  2.0
2013-01-01 09:00:03  NaN
2013-01-01 09:00:04  4.0

This is a regular frequency index. Using an integer window parameter works to roll along the window frequency.

In [60]: dft.rolling(2).sum()
Out[60]: 
                       B
2013-01-01 09:00:00  NaN
2013-01-01 09:00:01  1.0
2013-01-01 09:00:02  3.0
2013-01-01 09:00:03  NaN
2013-01-01 09:00:04  NaN

In [61]: dft.rolling(2, min_periods=1).sum()
Out[61]: 
                       B
2013-01-01 09:00:00  0.0
2013-01-01 09:00:01  1.0
2013-01-01 09:00:02  3.0
2013-01-01 09:00:03  2.0
2013-01-01 09:00:04  4.0

Specifying an offset allows a more intuitive specification of the rolling frequency.

In [62]: dft.rolling('2s').sum()
Out[62]: 
                       B
2013-01-01 09:00:00  0.0
2013-01-01 09:00:01  1.0
2013-01-01 09:00:02  3.0
2013-01-01 09:00:03  2.0
2013-01-01 09:00:04  4.0

Using a non-regular, but still monotonic index, rolling with an integer window does not impart any special calculation.

In [63]: dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
   ....:                    index=pd.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')],
   ....:                                   name='foo'))
   ....: 

In [64]: dft
Out[64]: 
                       B
foo                     
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

In [65]: dft.rolling(2).sum()
Out[65]: 
                       B
foo                     
2013-01-01 09:00:00  NaN
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  NaN

Using the time-specification generates variable windows for this sparse data.

In [66]: dft.rolling('2s').sum()
Out[66]: 
                       B
foo                     
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

Furthermore, we now allow an optional on parameter to specify a column (rather than the default of the index) in a DataFrame.

In [67]: dft = dft.reset_index()

In [68]: dft
Out[68]: 
                  foo    B
0 2013-01-01 09:00:00  0.0
1 2013-01-01 09:00:02  1.0
2 2013-01-01 09:00:03  2.0
3 2013-01-01 09:00:05  NaN
4 2013-01-01 09:00:06  4.0

In [69]: dft.rolling('2s', on='foo').sum()
Out[69]: 
                  foo    B
0 2013-01-01 09:00:00  0.0
1 2013-01-01 09:00:02  1.0
2 2013-01-01 09:00:03  3.0
3 2013-01-01 09:00:05  NaN
4 2013-01-01 09:00:06  4.0

Custom window rolling

New in version 1.0.

In addition to accepting an integer or offset as a window argument, rolling also accepts a BaseIndexer subclass that allows a user to define a custom method for calculating window bounds. The BaseIndexer subclass will need to define a get_window_bounds method that returns a tuple of two arrays, the first being the starting indices of the windows and second being the ending indices of the windows. Additionally, num_values, min_periods, center, closed and will automatically be passed to get_window_bounds and the defined method must always accept these arguments.

For example, if we have the following DataFrame:

In [70]: use_expanding = [True, False, True, False, True]

In [71]: use_expanding
Out[71]: [True, False, True, False, True]

In [72]: df = pd.DataFrame({'values': range(5)})

In [73]: df
Out[73]: 
   values
0       0
1       1
2       2
3       3
4       4

and we want to use an expanding window where use_expanding is True otherwise a window of size 1, we can create the following BaseIndexer:

In [2]: from pandas.api.indexers import BaseIndexer
...:
...: class CustomIndexer(BaseIndexer):
...:
...:    def get_window_bounds(self, num_values, min_periods, center, closed):
...:        start = np.empty(num_values, dtype=np.int64)
...:        end = np.empty(num_values, dtype=np.int64)
...:        for i in range(num_values):
...:            if self.use_expanding[i]:
...:                start[i] = 0
...:                end[i] = i + 1
...:            else:
...:                start[i] = i
...:                end[i] = i + self.window_size
...:        return start, end
...:

In [3]: indexer = CustomIndexer(window_size=1, use_expanding=use_expanding)

In [4]: df.rolling(indexer).sum()
Out[4]:
    values
0     0.0
1     1.0
2     3.0
3     3.0
4    10.0

Rolling window endpoints

The inclusion of the interval endpoints in rolling window calculations can be specified with the closed parameter:

closed	Description	Default for
right	close right endpoint	time-based windows
left	close left endpoint
both	close both endpoints	fixed windows
neither	open endpoints

For example, having the right endpoint open is useful in many problems that require that there is no contamination from present information back to past information. This allows the rolling window to compute statistics “up to that point in time”, but not including that point in time.

In [74]: df = pd.DataFrame({'x': 1},
   ....:                   index=[pd.Timestamp('20130101 09:00:01'),
   ....:                          pd.Timestamp('20130101 09:00:02'),
   ....:                          pd.Timestamp('20130101 09:00:03'),
   ....:                          pd.Timestamp('20130101 09:00:04'),
   ....:                          pd.Timestamp('20130101 09:00:06')])
   ....: 

In [75]: df["right"] = df.rolling('2s', closed='right').x.sum()  # default

In [76]: df["both"] = df.rolling('2s', closed='both').x.sum()

In [77]: df["left"] = df.rolling('2s', closed='left').x.sum()

In [78]: df["neither"] = df.rolling('2s', closed='neither').x.sum()

In [79]: df
Out[79]: 
                     x  right  both  left  neither
2013-01-01 09:00:01  1    1.0   1.0   NaN      NaN
2013-01-01 09:00:02  1    2.0   2.0   1.0      1.0
2013-01-01 09:00:03  1    2.0   3.0   2.0      1.0
2013-01-01 09:00:04  1    2.0   3.0   2.0      1.0
2013-01-01 09:00:06  1    1.0   2.0   1.0      NaN

Currently, this feature is only implemented for time-based windows. For fixed windows, the closed parameter cannot be set and the rolling window will always have both endpoints closed.

Time-aware rolling vs. resampling

Using .rolling() with a time-based index is quite similar to resampling. They both operate and perform reductive operations on time-indexed pandas objects.

When using .rolling() with an offset. The offset is a time-delta. Take a backwards-in-time looking window, and aggregate all of the values in that window (including the end-point, but not the start-point). This is the new value at that point in the result. These are variable sized windows in time-space for each point of the input. You will get a same sized result as the input.

When using .resample() with an offset. Construct a new index that is the frequency of the offset. For each frequency bin, aggregate points from the input within a backwards-in-time looking window that fall in that bin. The result of this aggregation is the output for that frequency point. The windows are fixed size in the frequency space. Your result will have the shape of a regular frequency between the min and the max of the original input object.

To summarize, .rolling() is a time-based window operation, while .resample() is a frequency-based window operation.

Centering windows

By default the labels are set to the right edge of the window, but a center keyword is available so the labels can be set at the center.

In [80]: ser.rolling(window=5).mean()
Out[80]: 
2000-01-01         NaN
2000-01-02         NaN
2000-01-03         NaN
2000-01-04         NaN
2000-01-05   -0.841164
2000-01-06   -0.779948
2000-01-07   -0.565487
2000-01-08   -0.502815
2000-01-09   -0.553755
2000-01-10   -0.472211
Freq: D, dtype: float64

In [81]: ser.rolling(window=5, center=True).mean()
Out[81]: 
2000-01-01         NaN
2000-01-02         NaN
2000-01-03   -0.841164
2000-01-04   -0.779948
2000-01-05   -0.565487
2000-01-06   -0.502815
2000-01-07   -0.553755
2000-01-08   -0.472211
2000-01-09         NaN
2000-01-10         NaN
Freq: D, dtype: float64

Binary window functions

cov() and corr() can compute moving window statistics about two Series or any combination of DataFrame/Series or DataFrame/DataFrame. Here is the behavior in each case:

• two Series: compute the statistic for the pairing.

• DataFrame/Series: compute the statistics for each column of the DataFrame with the passed Series, thus returning a DataFrame.

• DataFrame/DataFrame: by default compute the statistic for matching column names, returning a DataFrame. If the keyword argument pairwise=True is passed then computes the statistic for each pair of columns, returning a MultiIndexed DataFrame whose index are the dates in question (see the next section).

For example:

In [82]: df = pd.DataFrame(np.random.randn(1000, 4),
   ....:                   index=pd.date_range('1/1/2000', periods=1000),
   ....:                   columns=['A', 'B', 'C', 'D'])
   ....: 

In [83]: df = df.cumsum()

In [84]: df2 = df[:20]

In [85]: df2.rolling(window=5).corr(df2['B'])
Out[85]: 
                   A    B         C         D
2000-01-01       NaN  NaN       NaN       NaN
2000-01-02       NaN  NaN       NaN       NaN
2000-01-03       NaN  NaN       NaN       NaN
2000-01-04       NaN  NaN       NaN       NaN
2000-01-05  0.768775  1.0 -0.977990  0.800252
...              ...  ...       ...       ...
2000-01-16  0.691078  1.0  0.807450 -0.939302
2000-01-17  0.274506  1.0  0.582601 -0.902954
2000-01-18  0.330459  1.0  0.515707 -0.545268
2000-01-19  0.046756  1.0 -0.104334 -0.419799
2000-01-20 -0.328241  1.0 -0.650974 -0.777777

[20 rows x 4 columns]

Computing rolling pairwise covariances and correlations

In financial data analysis and other fields it’s common to compute covariance and correlation matrices for a collection of time series. Often one is also interested in moving-window covariance and correlation matrices. This can be done by passing the pairwise keyword argument, which in the case of DataFrame inputs will yield a MultiIndexed DataFrame whose index are the dates in question. In the case of a single DataFrame argument the pairwise argument can even be omitted:

Note

Missing values are ignored and each entry is computed using the pairwise complete observations. Please see the covariance section for caveats associated with this method of calculating covariance and correlation matrices.

In [86]: covs = (df[['B', 'C', 'D']].rolling(window=50)
   ....:                            .cov(df[['A', 'B', 'C']], pairwise=True))
   ....: 

In [87]: covs.loc['2002-09-22':]
Out[87]: 
                     B         C         D
2002-09-22 A  1.367467  8.676734 -8.047366
           B  3.067315  0.865946 -1.052533
           C  0.865946  7.739761 -4.943924
2002-09-23 A  0.910343  8.669065 -8.443062
           B  2.625456  0.565152 -0.907654
           C  0.565152  7.825521 -5.367526
2002-09-24 A  0.463332  8.514509 -8.776514
           B  2.306695  0.267746 -0.732186
           C  0.267746  7.771425 -5.696962
2002-09-25 A  0.467976  8.198236 -9.162599
           B  2.307129  0.267287 -0.754080
           C  0.267287  7.466559 -5.822650
2002-09-26 A  0.545781  7.899084 -9.326238
           B  2.311058  0.322295 -0.844451
           C  0.322295  7.038237 -5.684445

In [88]: correls = df.rolling(window=50).corr()

In [89]: correls.loc['2002-09-22':]
Out[89]: 
                     A         B         C         D
2002-09-22 A  1.000000  0.186397  0.744551 -0.769767
           B  0.186397  1.000000  0.177725 -0.240802
           C  0.744551  0.177725  1.000000 -0.712051
           D -0.769767 -0.240802 -0.712051  1.000000
2002-09-23 A  1.000000  0.134723  0.743113 -0.758758
...                ...       ...       ...       ...
2002-09-25 D -0.739160 -0.164179 -0.704686  1.000000
2002-09-26 A  1.000000  0.087756  0.727792 -0.736562
           B  0.087756  1.000000  0.079913 -0.179477
           C  0.727792  0.079913  1.000000 -0.692303
           D -0.736562 -0.179477 -0.692303  1.000000

[20 rows x 4 columns]

You can efficiently retrieve the time series of correlations between two columns by reshaping and indexing:

In [90]: correls.unstack(1)[('A', 'C')].plot()
Out[90]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65bd171048>

Aggregation

Once the Rolling, Expanding or EWM objects have been created, several methods are available to perform multiple computations on the data. These operations are similar to the aggregating API, groupby API, and resample API.

In [91]: dfa = pd.DataFrame(np.random.randn(1000, 3),
   ....:                    index=pd.date_range('1/1/2000', periods=1000),
   ....:                    columns=['A', 'B', 'C'])
   ....: 

In [92]: r = dfa.rolling(window=60, min_periods=1)

In [93]: r
Out[93]: Rolling [window=60,min_periods=1,center=False,axis=0]

We can aggregate by passing a function to the entire DataFrame, or select a Series (or multiple Series) via standard __getitem__.

In [94]: r.aggregate(np.sum)
Out[94]: 
                   A          B         C
2000-01-01 -0.289838  -0.370545 -1.284206
2000-01-02 -0.216612  -1.675528 -1.169415
2000-01-03  1.154661  -1.634017 -1.566620
2000-01-04  2.969393  -4.003274 -1.816179
2000-01-05  4.690630  -4.682017 -2.717209
...              ...        ...       ...
2002-09-22  2.860036  -9.270337  6.415245
2002-09-23  3.510163  -8.151439  5.177219
2002-09-24  6.524983 -10.168078  5.792639
2002-09-25  6.409626  -9.956226  5.704050
2002-09-26  5.093787  -7.074515  6.905823

[1000 rows x 3 columns]

In [95]: r['A'].aggregate(np.sum)
Out[95]: 
2000-01-01   -0.289838
2000-01-02   -0.216612
2000-01-03    1.154661
2000-01-04    2.969393
2000-01-05    4.690630
                ...   
2002-09-22    2.860036
2002-09-23    3.510163
2002-09-24    6.524983
2002-09-25    6.409626
2002-09-26    5.093787
Freq: D, Name: A, Length: 1000, dtype: float64

In [96]: r[['A', 'B']].aggregate(np.sum)
Out[96]: 
                   A          B
2000-01-01 -0.289838  -0.370545
2000-01-02 -0.216612  -1.675528
2000-01-03  1.154661  -1.634017
2000-01-04  2.969393  -4.003274
2000-01-05  4.690630  -4.682017
...              ...        ...
2002-09-22  2.860036  -9.270337
2002-09-23  3.510163  -8.151439
2002-09-24  6.524983 -10.168078
2002-09-25  6.409626  -9.956226
2002-09-26  5.093787  -7.074515

[1000 rows x 2 columns]

As you can see, the result of the aggregation will have the selected columns, or all columns if none are selected.

Applying multiple functions

With windowed Series you can also pass a list of functions to do aggregation with, outputting a DataFrame:

In [97]: r['A'].agg([np.sum, np.mean, np.std])
Out[97]: 
                 sum      mean       std
2000-01-01 -0.289838 -0.289838       NaN
2000-01-02 -0.216612 -0.108306  0.256725
2000-01-03  1.154661  0.384887  0.873311
2000-01-04  2.969393  0.742348  1.009734
2000-01-05  4.690630  0.938126  0.977914
...              ...       ...       ...
2002-09-22  2.860036  0.047667  1.132051
2002-09-23  3.510163  0.058503  1.134296
2002-09-24  6.524983  0.108750  1.144204
2002-09-25  6.409626  0.106827  1.142913
2002-09-26  5.093787  0.084896  1.151416

[1000 rows x 3 columns]

On a windowed DataFrame, you can pass a list of functions to apply to each column, which produces an aggregated result with a hierarchical index:

In [98]: r.agg([np.sum, np.mean])
Out[98]: 
                   A                    B                   C          
                 sum      mean        sum      mean       sum      mean
2000-01-01 -0.289838 -0.289838  -0.370545 -0.370545 -1.284206 -1.284206
2000-01-02 -0.216612 -0.108306  -1.675528 -0.837764 -1.169415 -0.584708
2000-01-03  1.154661  0.384887  -1.634017 -0.544672 -1.566620 -0.522207
2000-01-04  2.969393  0.742348  -4.003274 -1.000819 -1.816179 -0.454045
2000-01-05  4.690630  0.938126  -4.682017 -0.936403 -2.717209 -0.543442
...              ...       ...        ...       ...       ...       ...
2002-09-22  2.860036  0.047667  -9.270337 -0.154506  6.415245  0.106921
2002-09-23  3.510163  0.058503  -8.151439 -0.135857  5.177219  0.086287
2002-09-24  6.524983  0.108750 -10.168078 -0.169468  5.792639  0.096544
2002-09-25  6.409626  0.106827  -9.956226 -0.165937  5.704050  0.095068
2002-09-26  5.093787  0.084896  -7.074515 -0.117909  6.905823  0.115097

[1000 rows x 6 columns]

Passing a dict of functions has different behavior by default, see the next section.

Applying different functions to DataFrame columns

By passing a dict to aggregate you can apply a different aggregation to the columns of a DataFrame:

In [99]: r.agg({'A': np.sum, 'B': lambda x: np.std(x, ddof=1)})
Out[99]: 
                   A         B
2000-01-01 -0.289838       NaN
2000-01-02 -0.216612  0.660747
2000-01-03  1.154661  0.689929
2000-01-04  2.969393  1.072199
2000-01-05  4.690630  0.939657
...              ...       ...
2002-09-22  2.860036  1.113208
2002-09-23  3.510163  1.132381
2002-09-24  6.524983  1.080963
2002-09-25  6.409626  1.082911
2002-09-26  5.093787  1.136199

[1000 rows x 2 columns]

The function names can also be strings. In order for a string to be valid it must be implemented on the windowed object

In [100]: r.agg({'A': 'sum', 'B': 'std'})
Out[100]: 
                   A         B
2000-01-01 -0.289838       NaN
2000-01-02 -0.216612  0.660747
2000-01-03  1.154661  0.689929
2000-01-04  2.969393  1.072199
2000-01-05  4.690630  0.939657
...              ...       ...
2002-09-22  2.860036  1.113208
2002-09-23  3.510163  1.132381
2002-09-24  6.524983  1.080963
2002-09-25  6.409626  1.082911
2002-09-26  5.093787  1.136199

[1000 rows x 2 columns]

Furthermore you can pass a nested dict to indicate different aggregations on different columns.

In [101]: r.agg({'A': ['sum', 'std'], 'B': ['mean', 'std']})
Out[101]: 
                   A                   B          
                 sum       std      mean       std
2000-01-01 -0.289838       NaN -0.370545       NaN
2000-01-02 -0.216612  0.256725 -0.837764  0.660747
2000-01-03  1.154661  0.873311 -0.544672  0.689929
2000-01-04  2.969393  1.009734 -1.000819  1.072199
2000-01-05  4.690630  0.977914 -0.936403  0.939657
...              ...       ...       ...       ...
2002-09-22  2.860036  1.132051 -0.154506  1.113208
2002-09-23  3.510163  1.134296 -0.135857  1.132381
2002-09-24  6.524983  1.144204 -0.169468  1.080963
2002-09-25  6.409626  1.142913 -0.165937  1.082911
2002-09-26  5.093787  1.151416 -0.117909  1.136199

[1000 rows x 4 columns]

Expanding windows

A common alternative to rolling statistics is to use an expanding window, which yields the value of the statistic with all the data available up to that point in time.

These follow a similar interface to .rolling, with the .expanding method returning an Expanding object.

As these calculations are a special case of rolling statistics, they are implemented in pandas such that the following two calls are equivalent:

In [102]: df.rolling(window=len(df), min_periods=1).mean()[:5]
Out[102]: 
                   A         B         C         D
2000-01-01  0.314226 -0.001675  0.071823  0.892566
2000-01-02  0.654522 -0.171495  0.179278  0.853361
2000-01-03  0.708733 -0.064489 -0.238271  1.371111
2000-01-04  0.987613  0.163472 -0.919693  1.566485
2000-01-05  1.426971  0.288267 -1.358877  1.808650

In [103]: df.expanding(min_periods=1).mean()[:5]
Out[103]: 
                   A         B         C         D
2000-01-01  0.314226 -0.001675  0.071823  0.892566
2000-01-02  0.654522 -0.171495  0.179278  0.853361
2000-01-03  0.708733 -0.064489 -0.238271  1.371111
2000-01-04  0.987613  0.163472 -0.919693  1.566485
2000-01-05  1.426971  0.288267 -1.358877  1.808650

These have a similar set of methods to .rolling methods.

Method summary

Function	Description
count()	Number of non-null observations
sum()	Sum of values
mean()	Mean of values
median()	Arithmetic median of values
min()	Minimum
max()	Maximum
std()	Unbiased standard deviation
var()	Unbiased variance
skew()	Unbiased skewness (3rd moment)
kurt()	Unbiased kurtosis (4th moment)
quantile()	Sample quantile (value at %)
apply()	Generic apply
cov()	Unbiased covariance (binary)
corr()	Correlation (binary)

Aside from not having a window parameter, these functions have the same interfaces as their .rolling counterparts. Like above, the parameters they all accept are:

• min_periods: threshold of non-null data points to require. Defaults to minimum needed to compute statistic. No NaNs will be output once min_periods non-null data points have been seen.

• center: boolean, whether to set the labels at the center (default is False).

Note

The output of the .rolling and .expanding methods do not return a NaN if there are at least min_periods non-null values in the current window. For example:

In [104]: sn = pd.Series([1, 2, np.nan, 3, np.nan, 4])

In [105]: sn
Out[105]: 
0    1.0
1    2.0
2    NaN
3    3.0
4    NaN
5    4.0
dtype: float64

In [106]: sn.rolling(2).max()
Out[106]: 
0    NaN
1    2.0
2    NaN
3    NaN
4    NaN
5    NaN
dtype: float64

In [107]: sn.rolling(2, min_periods=1).max()
Out[107]: 
0    1.0
1    2.0
2    2.0
3    3.0
4    3.0
5    4.0
dtype: float64

In case of expanding functions, this differs from cumsum(), cumprod(), cummax(), and cummin(), which return NaN in the output wherever a NaN is encountered in the input. In order to match the output of cumsum with expanding, use fillna():

In [108]: sn.expanding().sum()
Out[108]: 
0     1.0
1     3.0
2     3.0
3     6.0
4     6.0
5    10.0
dtype: float64

In [109]: sn.cumsum()
Out[109]: 
0     1.0
1     3.0
2     NaN
3     6.0
4     NaN
5    10.0
dtype: float64

In [110]: sn.cumsum().fillna(method='ffill')
Out[110]: 
0     1.0
1     3.0
2     3.0
3     6.0
4     6.0
5    10.0
dtype: float64

An expanding window statistic will be more stable (and less responsive) than its rolling window counterpart as the increasing window size decreases the relative impact of an individual data point. As an example, here is the mean() output for the previous time series dataset:

In [111]: s.plot(style='k--')
Out[111]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6589c770b8>

In [112]: s.expanding().mean().plot(style='k')
Out[112]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6589c770b8>

Exponentially weighted windows

A related set of functions are exponentially weighted versions of several of the above statistics. A similar interface to .rolling and .expanding is accessed through the .ewm method to receive an EWM object. A number of expanding EW (exponentially weighted) methods are provided:

Function	Description
mean()	EW moving average
var()	EW moving variance
std()	EW moving standard deviation
corr()	EW moving correlation
cov()	EW moving covariance

In general, a weighted moving average is calculated as

\[y_t = \frac{\sum_{i=0}^t w_i x_{t-i}}{\sum_{i=0}^t w_i},\]

where \(x_t\) is the input, \(y_t\) is the result and the \(w_i\) are the weights.

The EW functions support two variants of exponential weights. The default, adjust=True, uses the weights \(w_i = (1 - \alpha)^i\) which gives

\[y_t = \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ... + (1 - \alpha)^t x_{0}}{1 + (1 - \alpha) + (1 - \alpha)^2 + ... + (1 - \alpha)^t}\]

When adjust=False is specified, moving averages are calculated as

\[\begin{split}y_0 &= x_0 \\ y_t &= (1 - \alpha) y_{t-1} + \alpha x_t,\end{split}\]

which is equivalent to using weights

\[\begin{split}w_i = \begin{cases} \alpha (1 - \alpha)^i & \text{if } i < t \\ (1 - \alpha)^i & \text{if } i = t. \end{cases}\end{split}\]

Note

These equations are sometimes written in terms of \(\alpha' = 1 - \alpha\), e.g.

\[y_t = \alpha' y_{t-1} + (1 - \alpha') x_t.\]

The difference between the above two variants arises because we are dealing with series which have finite history. Consider a series of infinite history, with adjust=True:

\[y_t = \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...} {1 + (1 - \alpha) + (1 - \alpha)^2 + ...}\]

Noting that the denominator is a geometric series with initial term equal to 1 and a ratio of \(1 - \alpha\) we have

\[\begin{split}y_t &= \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...} {\frac{1}{1 - (1 - \alpha)}}\\ &= [x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...] \alpha \\ &= \alpha x_t + [(1-\alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...]\alpha \\ &= \alpha x_t + (1 - \alpha)[x_{t-1} + (1 - \alpha) x_{t-2} + ...]\alpha\\ &= \alpha x_t + (1 - \alpha) y_{t-1}\end{split}\]

which is the same expression as adjust=False above and therefore shows the equivalence of the two variants for infinite series. When adjust=False, we have \(y_0 = x_0\) and \(y_t = \alpha x_t + (1 - \alpha) y_{t-1}\). Therefore, there is an assumption that \(x_0\) is not an ordinary value but rather an exponentially weighted moment of the infinite series up to that point.

One must have \(0 < \alpha \leq 1\), and while it is possible to pass \(\alpha\) directly, it’s often easier to think about either the span, center of mass (com) or half-life of an EW moment:

\[\begin{split}\alpha = \begin{cases} \frac{2}{s + 1}, & \text{for span}\ s \geq 1\\ \frac{1}{1 + c}, & \text{for center of mass}\ c \geq 0\\ 1 - \exp^{\frac{\log 0.5}{h}}, & \text{for half-life}\ h > 0 \end{cases}\end{split}\]

One must specify precisely one of span, center of mass, half-life and alpha to the EW functions:

• Span corresponds to what is commonly called an “N-day EW moving average”.

• Center of mass has a more physical interpretation and can be thought of in terms of span: \(c = (s - 1) / 2\).

• Half-life is the period of time for the exponential weight to reduce to one half.

• Alpha specifies the smoothing factor directly.

Here is an example for a univariate time series:

In [113]: s.plot(style='k--')
Out[113]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6589ac9668>

In [114]: s.ewm(span=20).mean().plot(style='k')
Out[114]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6589ac9668>

EWM has a min_periods argument, which has the same meaning it does for all the .expanding and .rolling methods: no output values will be set until at least min_periods non-null values are encountered in the (expanding) window.

EWM also has an ignore_na argument, which determines how intermediate null values affect the calculation of the weights. When ignore_na=False (the default), weights are calculated based on absolute positions, so that intermediate null values affect the result. When ignore_na=True, weights are calculated by ignoring intermediate null values. For example, assuming adjust=True, if ignore_na=False, the weighted average of 3, NaN, 5 would be calculated as

\[\frac{(1-\alpha)^2 \cdot 3 + 1 \cdot 5}{(1-\alpha)^2 + 1}.\]

Whereas if ignore_na=True, the weighted average would be calculated as

\[\frac{(1-\alpha) \cdot 3 + 1 \cdot 5}{(1-\alpha) + 1}.\]

The var(), std(), and cov() functions have a bias argument, specifying whether the result should contain biased or unbiased statistics. For example, if bias=True, ewmvar(x) is calculated as ewmvar(x) = ewma(x**2) - ewma(x)**2; whereas if bias=False (the default), the biased variance statistics are scaled by debiasing factors

\[\frac{\left(\sum_{i=0}^t w_i\right)^2}{\left(\sum_{i=0}^t w_i\right)^2 - \sum_{i=0}^t w_i^2}.\]

(For \(w_i = 1\), this reduces to the usual \(N / (N - 1)\) factor, with \(N = t + 1\).) See Weighted Sample Variance on Wikipedia for further details.

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