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).

  1. In [1]: ser = pd.Series(np.random.randn(8))
  3. In [2]: ser.pct_change()
  4. Out[2]: 
  5. 0         NaN
  6. 1   -1.602976
  7. 2    4.334938
  8. 3   -0.247456
  9. 4   -2.067345
  10. 5   -1.142903
  11. 6   -1.688214
  12. 7   -9.759729
  13. dtype: float64
  1. In [3]: df = pd.DataFrame(np.random.randn(10, 4))
  3. In [4]: df.pct_change(periods=3)
  4. Out[4]: 
  5.           0         1         2         3
  6. 0       NaN       NaN       NaN       NaN
  7. 1       NaN       NaN       NaN       NaN
  8. 2       NaN       NaN       NaN       NaN
  9. 3 -0.218320 -1.054001  1.987147 -0.510183
  10. 4 -0.439121 -1.816454  0.649715 -4.822809
  11. 5 -0.127833 -3.042065 -5.866604 -1.776977
  12. 6 -2.596833 -1.959538 -2.111697 -3.798900
  13. 7 -0.117826 -2.169058  0.036094 -0.067696
  14. 8  2.492606 -1.357320 -1.205802 -1.558697
  15. 9 -1.012977  2.324558 -1.003744 -0.371806

Covariance

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

  1. In [5]: s1 = pd.Series(np.random.randn(1000))
  3. In [6]: s2 = pd.Series(np.random.randn(1000))
  5. In [7]: s1.cov(s2)
  6. Out[7]: 0.000680108817431082

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

::: tip 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.

:::

  1. In [8]: frame = pd.DataFrame(np.random.randn(1000, 5),
  2.    ...:                      columns=['a', 'b', 'c', 'd', 'e'])
  3.    ...: 
  5. In [9]: frame.cov()
  6. Out[9]: 
  7.           a         b         c         d         e
  8. a  1.000882 -0.003177 -0.002698 -0.006889  0.031912
  9. b -0.003177  1.024721  0.000191  0.009212  0.000857
  10. c -0.002698  0.000191  0.950735 -0.031743 -0.005087
  11. d -0.006889  0.009212 -0.031743  1.002983 -0.047952
  12. 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.

  1. In [10]: frame = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c'])
  3. In [11]: frame.loc[frame.index[:5], 'a'] = np.nan
  5. In [12]: frame.loc[frame.index[5:10], 'b'] = np.nan
  7. In [13]: frame.cov()
  8. Out[13]: 
  9.           a         b         c
  10. a  1.123670 -0.412851  0.018169
  11. b -0.412851  1.154141  0.305260
  12. c  0.018169  0.305260  1.301149
  14. In [14]: frame.cov(min_periods=12)
  15. Out[14]: 
  16.           a         b         c
  17. a  1.123670       NaN  0.018169
  18. b       NaN  1.154141  0.305260
  19. 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:

::: tip Note

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

:::

  1. In [15]: frame = pd.DataFrame(np.random.randn(1000, 5),
  2.    ....:                      columns=['a', 'b', 'c', 'd', 'e'])
  3.    ....: 
  5. In [16]: frame.iloc[::2] = np.nan
  7. # Series with Series
  8. In [17]: frame['a'].corr(frame['b'])
  9. Out[17]: 0.013479040400098794
  11. In [18]: frame['a'].corr(frame['b'], method='spearman')
  12. Out[18]: -0.007289885159540637
  14. # Pairwise correlation of DataFrame columns
  15. In [19]: frame.corr()
  16. Out[19]: 
  17.           a         b         c         d         e
  18. a  1.000000  0.013479 -0.049269 -0.042239 -0.028525
  19. b  0.013479  1.000000 -0.020433 -0.011139  0.005654
  20. c -0.049269 -0.020433  1.000000  0.018587 -0.054269
  21. d -0.042239 -0.011139  0.018587  1.000000 -0.017060
  22. 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:

  1. In [20]: frame = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c'])
  3. In [21]: frame.loc[frame.index[:5], 'a'] = np.nan
  5. In [22]: frame.loc[frame.index[5:10], 'b'] = np.nan
  7. In [23]: frame.corr()
  8. Out[23]: 
  9.           a         b         c
  10. a  1.000000 -0.121111  0.069544
  11. b -0.121111  1.000000  0.051742
  12. c  0.069544  0.051742  1.000000
  14. In [24]: frame.corr(min_periods=12)
  15. Out[24]: 
  16.           a         b         c
  17. a  1.000000       NaN  0.069544
  18. b       NaN  1.000000  0.051742
  19. 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:

  1. # histogram intersection
  2. In [25]: def histogram_intersection(a, b):
  3.    ....:     return np.minimum(np.true_divide(a, a.sum()),
  4.    ....:                       np.true_divide(b, b.sum())).sum()
  5.    ....: 
  7. In [26]: frame.corr(method=histogram_intersection)
  8. Out[26]: 
  9.           a          b          c
  10. a  1.000000  -6.404882  -2.058431
  11. b -6.404882   1.000000 -19.255743
  12. 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.

  1. In [27]: index = ['a', 'b', 'c', 'd', 'e']
  3. In [28]: columns = ['one', 'two', 'three', 'four']
  5. In [29]: df1 = pd.DataFrame(np.random.randn(5, 4), index=index, columns=columns)
  7. In [30]: df2 = pd.DataFrame(np.random.randn(4, 4), index=index[:4], columns=columns)
  9. In [31]: df1.corrwith(df2)
  10. Out[31]: 
  11. one     -0.125501
  12. two     -0.493244
  13. three    0.344056
  14. four     0.004183
  15. dtype: float64
  17. In [32]: df2.corrwith(df1, axis=1)
  18. Out[32]: 
  19. a   -0.675817
  20. b    0.458296
  21. c    0.190809
  22. d   -0.186275
  23. e         NaN
  24. 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:

  1. In [33]: s = pd.Series(np.random.np.random.randn(5), index=list('abcde'))
  3. In [34]: s['d'] = s['b']  # so there's a tie
  5. In [35]: s.rank()
  6. Out[35]: 
  7. a    5.0
  8. b    2.5
  9. c    1.0
  10. d    2.5
  11. e    4.0
  12. 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.

  1. In [36]: df = pd.DataFrame(np.random.np.random.randn(10, 6))
  3. In [37]: df[4] = df[2][:5]  # some ties
  5. In [38]: df
  6. Out[38]: 
  7.           0         1         2         3         4         5
  8. 0 -0.904948 -1.163537 -1.457187  0.135463 -1.457187  0.294650
  9. 1 -0.976288 -0.244652 -0.748406 -0.999601 -0.748406 -0.800809
  10. 2  0.401965  1.460840  1.256057  1.308127  1.256057  0.876004
  11. 3  0.205954  0.369552 -0.669304  0.038378 -0.669304  1.140296
  12. 4 -0.477586 -0.730705 -1.129149 -0.601463 -1.129149 -0.211196
  13. 5 -1.092970 -0.689246  0.908114  0.204848       NaN  0.463347
  14. 6  0.376892  0.959292  0.095572 -0.593740       NaN -0.069180
  15. 7 -1.002601  1.957794 -0.120708  0.094214       NaN -1.467422
  16. 8 -0.547231  0.664402 -0.519424 -0.073254       NaN -1.263544
  17. 9 -0.250277 -0.237428 -1.056443  0.419477       NaN  1.375064
  19. In [39]: df.rank(1)
  20. Out[39]: 
  21.      0    1    2    3    4    5
  22. 0  4.0  3.0  1.5  5.0  1.5  6.0
  23. 1  2.0  6.0  4.5  1.0  4.5  3.0
  24. 2  1.0  6.0  3.5  5.0  3.5  2.0
  25. 3  4.0  5.0  1.5  3.0  1.5  6.0
  26. 4  5.0  3.0  1.5  4.0  1.5  6.0
  27. 5  1.0  2.0  5.0  3.0  NaN  4.0
  28. 6  4.0  5.0  3.0  1.0  NaN  2.0
  29. 7  2.0  5.0  3.0  4.0  NaN  1.0
  30. 8  2.0  5.0  3.0  4.0  NaN  1.0
  31. 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.

::: tip 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.

  1. In [40]: s = pd.Series(np.random.randn(1000),
  2.    ....:               index=pd.date_range('1/1/2000', periods=1000))
  3.    ....: 
  5. In [41]: s = s.cumsum()
  7. In [42]: s
  8. Out[42]: 
  9. 2000-01-01    -0.268824
  10. 2000-01-02    -1.771855
  11. 2000-01-03    -0.818003
  12. 2000-01-04    -0.659244
  13. 2000-01-05    -1.942133
  14.                 ...    
  15. 2002-09-22   -67.457323
  16. 2002-09-23   -69.253182
  17. 2002-09-24   -70.296818
  18. 2002-09-25   -70.844674
  19. 2002-09-26   -72.475016
  20. Freq: D, Length: 1000, dtype: float64

These are created from methods on Series and DataFrame.

  1. In [43]: r = s.rolling(window=60)
  3. In [44]: r
  4. Out[44]: Rolling [window=60,center=False,axis=0]

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

  1. In [14]: r.<TAB>                                          # noqa: E225, E999
  2. r.agg         r.apply       r.count       r.exclusions  r.max         r.median      r.name        r.skew        r.sum
  3. 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:

  1. In [45]: r.mean()
  2. Out[45]: 
  3. 2000-01-01          NaN
  4. 2000-01-02          NaN
  5. 2000-01-03          NaN
  6. 2000-01-04          NaN
  7. 2000-01-05          NaN
  8.                 ...    
  9. 2002-09-22   -62.914971
  10. 2002-09-23   -63.061867
  11. 2002-09-24   -63.213876
  12. 2002-09-25   -63.375074
  13. 2002-09-26   -63.539734
  14. Freq: D, Length: 1000, dtype: float64
  1. In [46]: s.plot(style='k--')
  2. Out[46]: <matplotlib.axes._subplots.AxesSubplot at 0x7f66048bdef0>
  4. In [47]: r.mean().plot(style='k')
  5. Out[47]: <matplotlib.axes._subplots.AxesSubplot at 0x7f66048bdef0>

rolling_mean_ex

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:

  1. In [48]: df = pd.DataFrame(np.random.randn(1000, 4),
  2.    ....:                   index=pd.date_range('1/1/2000', periods=1000),
  3.    ....:                   columns=['A', 'B', 'C', 'D'])
  4.    ....: 
  6. In [49]: df = df.cumsum()
  8. In [50]: df.rolling(window=60).sum().plot(subplots=True)
  9. Out[50]: 
  10. array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f66075a7f60>,
  11.        <matplotlib.axes._subplots.AxesSubplot object at 0x7f65f79e55c0>,
  12.        <matplotlib.axes._subplots.AxesSubplot object at 0x7f65f7998588>,
  13.        <matplotlib.axes._subplots.AxesSubplot object at 0x7f65f794b550>],
  14.       dtype=object)

rolling_mean_frame

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)

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:

  1. In [51]: def mad(x):
  2.    ....:     return np.fabs(x - x.mean()).mean()
  3.    ....: 
  5. In [52]: s.rolling(window=60).apply(mad, raw=True).plot(style='k')
  6. Out[52]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f795fac8>

rolling_apply_ex

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).
  1. In [53]: ser = pd.Series(np.random.randn(10),
  2.    ....:                 index=pd.date_range('1/1/2000', periods=10))
  3.    ....: 
  5. In [54]: ser.rolling(window=5, win_type='triang').mean()
  6. Out[54]: 
  7. 2000-01-01         NaN
  8. 2000-01-02         NaN
  9. 2000-01-03         NaN
  10. 2000-01-04         NaN
  11. 2000-01-05   -1.037870
  12. 2000-01-06   -0.767705
  13. 2000-01-07   -0.383197
  14. 2000-01-08   -0.395513
  15. 2000-01-09   -0.558440
  16. 2000-01-10   -0.672416
  17. Freq: D, dtype: float64

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

  1. In [55]: ser.rolling(window=5, win_type='boxcar').mean()
  2. Out[55]: 
  3. 2000-01-01         NaN
  4. 2000-01-02         NaN
  5. 2000-01-03         NaN
  6. 2000-01-04         NaN
  7. 2000-01-05   -0.841164
  8. 2000-01-06   -0.779948
  9. 2000-01-07   -0.565487
  10. 2000-01-08   -0.502815
  11. 2000-01-09   -0.553755
  12. 2000-01-10   -0.472211
  13. Freq: D, dtype: float64
  15. In [56]: ser.rolling(window=5).mean()
  16. Out[56]: 
  17. 2000-01-01         NaN
  18. 2000-01-02         NaN
  19. 2000-01-03         NaN
  20. 2000-01-04         NaN
  21. 2000-01-05   -0.841164
  22. 2000-01-06   -0.779948
  23. 2000-01-07   -0.565487
  24. 2000-01-08   -0.502815
  25. 2000-01-09   -0.553755
  26. 2000-01-10   -0.472211
  27. Freq: D, dtype: float64

For some windowing functions, additional parameters must be specified:

  1. In [57]: ser.rolling(window=5, win_type='gaussian').mean(std=0.1)
  2. Out[57]: 
  3. 2000-01-01         NaN
  4. 2000-01-02         NaN
  5. 2000-01-03         NaN
  6. 2000-01-04         NaN
  7. 2000-01-05   -1.309989
  8. 2000-01-06   -1.153000
  9. 2000-01-07    0.606382
  10. 2000-01-08   -0.681101
  11. 2000-01-09   -0.289724
  12. 2000-01-10   -0.996632
  13. Freq: D, dtype: float64

::: tip 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

New in version 0.19.0.

New in version 0.19.0 are the ability 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.

  1. In [58]: dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
  2.    ....:                    index=pd.date_range('20130101 09:00:00',
  3.    ....:                                        periods=5,
  4.    ....:                                        freq='s'))
  5.    ....: 
  7. In [59]: dft
  8. Out[59]: 
  9.                        B
  10. 2013-01-01 09:00:00  0.0
  11. 2013-01-01 09:00:01  1.0
  12. 2013-01-01 09:00:02  2.0
  13. 2013-01-01 09:00:03  NaN
  14. 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.

  1. In [60]: dft.rolling(2).sum()
  2. Out[60]: 
  3.                        B
  4. 2013-01-01 09:00:00  NaN
  5. 2013-01-01 09:00:01  1.0
  6. 2013-01-01 09:00:02  3.0
  7. 2013-01-01 09:00:03  NaN
  8. 2013-01-01 09:00:04  NaN
  10. In [61]: dft.rolling(2, min_periods=1).sum()
  11. Out[61]: 
  12.                        B
  13. 2013-01-01 09:00:00  0.0
  14. 2013-01-01 09:00:01  1.0
  15. 2013-01-01 09:00:02  3.0
  16. 2013-01-01 09:00:03  2.0
  17. 2013-01-01 09:00:04  4.0

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

  1. In [62]: dft.rolling('2s').sum()
  2. Out[62]: 
  3.                        B
  4. 2013-01-01 09:00:00  0.0
  5. 2013-01-01 09:00:01  1.0
  6. 2013-01-01 09:00:02  3.0
  7. 2013-01-01 09:00:03  2.0
  8. 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.

  1. In [63]: dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
  2.    ....:                    index=pd.Index([pd.Timestamp('20130101 09:00:00'),
  3.    ....:                                    pd.Timestamp('20130101 09:00:02'),
  4.    ....:                                    pd.Timestamp('20130101 09:00:03'),
  5.    ....:                                    pd.Timestamp('20130101 09:00:05'),
  6.    ....:                                    pd.Timestamp('20130101 09:00:06')],
  7.    ....:                                   name='foo'))
  8.    ....: 
  10. In [64]: dft
  11. Out[64]: 
  12.                        B
  13. foo                     
  14. 2013-01-01 09:00:00  0.0
  15. 2013-01-01 09:00:02  1.0
  16. 2013-01-01 09:00:03  2.0
  17. 2013-01-01 09:00:05  NaN
  18. 2013-01-01 09:00:06  4.0
  20. In [65]: dft.rolling(2).sum()
  21. Out[65]: 
  22.                        B
  23. foo                     
  24. 2013-01-01 09:00:00  NaN
  25. 2013-01-01 09:00:02  1.0
  26. 2013-01-01 09:00:03  3.0
  27. 2013-01-01 09:00:05  NaN
  28. 2013-01-01 09:00:06  NaN

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

  1. In [66]: dft.rolling('2s').sum()
  2. Out[66]: 
  3.                        B
  4. foo                     
  5. 2013-01-01 09:00:00  0.0
  6. 2013-01-01 09:00:02  1.0
  7. 2013-01-01 09:00:03  3.0
  8. 2013-01-01 09:00:05  NaN
  9. 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.

  1. In [67]: dft = dft.reset_index()
  3. In [68]: dft
  4. Out[68]: 
  5.                   foo    B
  6. 0 2013-01-01 09:00:00  0.0
  7. 1 2013-01-01 09:00:02  1.0
  8. 2 2013-01-01 09:00:03  2.0
  9. 3 2013-01-01 09:00:05  NaN
  10. 4 2013-01-01 09:00:06  4.0
  12. In [69]: dft.rolling('2s', on='foo').sum()
  13. Out[69]: 
  14.                   foo    B
  15. 0 2013-01-01 09:00:00  0.0
  16. 1 2013-01-01 09:00:02  1.0
  17. 2 2013-01-01 09:00:03  3.0
  18. 3 2013-01-01 09:00:05  NaN
  19. 4 2013-01-01 09:00:06  4.0

Rolling window endpoints

New in version 0.20.0.

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.

  1. In [70]: df = pd.DataFrame({'x': 1},
  2.    ....:                   index=[pd.Timestamp('20130101 09:00:01'),
  3.    ....:                          pd.Timestamp('20130101 09:00:02'),
  4.    ....:                          pd.Timestamp('20130101 09:00:03'),
  5.    ....:                          pd.Timestamp('20130101 09:00:04'),
  6.    ....:                          pd.Timestamp('20130101 09:00:06')])
  7.    ....: 
  9. In [71]: df["right"] = df.rolling('2s', closed='right').x.sum()  # default
  11. In [72]: df["both"] = df.rolling('2s', closed='both').x.sum()
  13. In [73]: df["left"] = df.rolling('2s', closed='left').x.sum()
  15. In [74]: df["neither"] = df.rolling('2s', closed='neither').x.sum()
  17. In [75]: df
  18. Out[75]: 
  19.                      x  right  both  left  neither
  20. 2013-01-01 09:00:01  1    1.0   1.0   NaN      NaN
  21. 2013-01-01 09:00:02  1    2.0   2.0   1.0      1.0
  22. 2013-01-01 09:00:03  1    2.0   3.0   2.0      1.0
  23. 2013-01-01 09:00:04  1    2.0   3.0   2.0      1.0
  24. 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.

  1. In [76]: ser.rolling(window=5).mean()
  2. Out[76]: 
  3. 2000-01-01         NaN
  4. 2000-01-02         NaN
  5. 2000-01-03         NaN
  6. 2000-01-04         NaN
  7. 2000-01-05   -0.841164
  8. 2000-01-06   -0.779948
  9. 2000-01-07   -0.565487
  10. 2000-01-08   -0.502815
  11. 2000-01-09   -0.553755
  12. 2000-01-10   -0.472211
  13. Freq: D, dtype: float64
  15. In [77]: ser.rolling(window=5, center=True).mean()
  16. Out[77]: 
  17. 2000-01-01         NaN
  18. 2000-01-02         NaN
  19. 2000-01-03   -0.841164
  20. 2000-01-04   -0.779948
  21. 2000-01-05   -0.565487
  22. 2000-01-06   -0.502815
  23. 2000-01-07   -0.553755
  24. 2000-01-08   -0.472211
  25. 2000-01-09         NaN
  26. 2000-01-10         NaN
  27. 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:

  1. In [78]: df = pd.DataFrame(np.random.randn(1000, 4),
  2.    ....:                   index=pd.date_range('1/1/2000', periods=1000),
  3.    ....:                   columns=['A', 'B', 'C', 'D'])
  4.    ....: 
  6. In [79]: df = df.cumsum()
  8. In [80]: df2 = df[:20]
  10. In [81]: df2.rolling(window=5).corr(df2['B'])
  11. Out[81]: 
  12.                    A    B         C         D
  13. 2000-01-01       NaN  NaN       NaN       NaN
  14. 2000-01-02       NaN  NaN       NaN       NaN
  15. 2000-01-03       NaN  NaN       NaN       NaN
  16. 2000-01-04       NaN  NaN       NaN       NaN
  17. 2000-01-05  0.768775  1.0 -0.977990  0.800252
  18. ...              ...  ...       ...       ...
  19. 2000-01-16  0.691078  1.0  0.807450 -0.939302
  20. 2000-01-17  0.274506  1.0  0.582601 -0.902954
  21. 2000-01-18  0.330459  1.0  0.515707 -0.545268
  22. 2000-01-19  0.046756  1.0 -0.104334 -0.419799
  23. 2000-01-20 -0.328241  1.0 -0.650974 -0.777777
  25. [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:

::: tip 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.

:::

  1. In [82]: covs = (df[['B', 'C', 'D']].rolling(window=50)
  2.    ....:                            .cov(df[['A', 'B', 'C']], pairwise=True))
  3.    ....: 
  5. In [83]: covs.loc['2002-09-22':]
  6. Out[83]: 
  7.                      B         C         D
  8. 2002-09-22 A  1.367467  8.676734 -8.047366
  9.            B  3.067315  0.865946 -1.052533
  10.            C  0.865946  7.739761 -4.943924
  11. 2002-09-23 A  0.910343  8.669065 -8.443062
  12.            B  2.625456  0.565152 -0.907654
  13.            C  0.565152  7.825521 -5.367526
  14. 2002-09-24 A  0.463332  8.514509 -8.776514
  15.            B  2.306695  0.267746 -0.732186
  16.            C  0.267746  7.771425 -5.696962
  17. 2002-09-25 A  0.467976  8.198236 -9.162599
  18.            B  2.307129  0.267287 -0.754080
  19.            C  0.267287  7.466559 -5.822650
  20. 2002-09-26 A  0.545781  7.899084 -9.326238
  21.            B  2.311058  0.322295 -0.844451
  22.            C  0.322295  7.038237 -5.684445
  1. In [84]: correls = df.rolling(window=50).corr()
  3. In [85]: correls.loc['2002-09-22':]
  4. Out[85]: 
  5.                      A         B         C         D
  6. 2002-09-22 A  1.000000  0.186397  0.744551 -0.769767
  7.            B  0.186397  1.000000  0.177725 -0.240802
  8.            C  0.744551  0.177725  1.000000 -0.712051
  9.            D -0.769767 -0.240802 -0.712051  1.000000
  10. 2002-09-23 A  1.000000  0.134723  0.743113 -0.758758
  11. ...                ...       ...       ...       ...
  12. 2002-09-25 D -0.739160 -0.164179 -0.704686  1.000000
  13. 2002-09-26 A  1.000000  0.087756  0.727792 -0.736562
  14.            B  0.087756  1.000000  0.079913 -0.179477
  15.            C  0.727792  0.079913  1.000000 -0.692303
  16.            D -0.736562 -0.179477 -0.692303  1.000000
  18. [20 rows x 4 columns]

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

  1. In [86]: correls.unstack(1)[('A', 'C')].plot()
  2. Out[86]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f8dc79e8>

rolling_corr_pairwise_ex

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.

  1. In [87]: dfa = pd.DataFrame(np.random.randn(1000, 3),
  2.    ....:                    index=pd.date_range('1/1/2000', periods=1000),
  3.    ....:                    columns=['A', 'B', 'C'])
  4.    ....: 
  6. In [88]: r = dfa.rolling(window=60, min_periods=1)
  8. In [89]: r
  9. Out[89]: 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__.

  1. In [90]: r.aggregate(np.sum)
  2. Out[90]: 
  3.                    A          B         C
  4. 2000-01-01 -0.289838  -0.370545 -1.284206
  5. 2000-01-02 -0.216612  -1.675528 -1.169415
  6. 2000-01-03  1.154661  -1.634017 -1.566620
  7. 2000-01-04  2.969393  -4.003274 -1.816179
  8. 2000-01-05  4.690630  -4.682017 -2.717209
  9. ...              ...        ...       ...
  10. 2002-09-22  2.860036  -9.270337  6.415245
  11. 2002-09-23  3.510163  -8.151439  5.177219
  12. 2002-09-24  6.524983 -10.168078  5.792639
  13. 2002-09-25  6.409626  -9.956226  5.704050
  14. 2002-09-26  5.093787  -7.074515  6.905823
  16. [1000 rows x 3 columns]
  18. In [91]: r['A'].aggregate(np.sum)
  19. Out[91]: 
  20. 2000-01-01   -0.289838
  21. 2000-01-02   -0.216612
  22. 2000-01-03    1.154661
  23. 2000-01-04    2.969393
  24. 2000-01-05    4.690630
  25.                 ...   
  26. 2002-09-22    2.860036
  27. 2002-09-23    3.510163
  28. 2002-09-24    6.524983
  29. 2002-09-25    6.409626
  30. 2002-09-26    5.093787
  31. Freq: D, Name: A, Length: 1000, dtype: float64
  33. In [92]: r[['A', 'B']].aggregate(np.sum)
  34. Out[92]: 
  35.                    A          B
  36. 2000-01-01 -0.289838  -0.370545
  37. 2000-01-02 -0.216612  -1.675528
  38. 2000-01-03  1.154661  -1.634017
  39. 2000-01-04  2.969393  -4.003274
  40. 2000-01-05  4.690630  -4.682017
  41. ...              ...        ...
  42. 2002-09-22  2.860036  -9.270337
  43. 2002-09-23  3.510163  -8.151439
  44. 2002-09-24  6.524983 -10.168078
  45. 2002-09-25  6.409626  -9.956226
  46. 2002-09-26  5.093787  -7.074515
  48. [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:

  1. In [93]: r['A'].agg([np.sum, np.mean, np.std])
  2. Out[93]: 
  3.                  sum      mean       std
  4. 2000-01-01 -0.289838 -0.289838       NaN
  5. 2000-01-02 -0.216612 -0.108306  0.256725
  6. 2000-01-03  1.154661  0.384887  0.873311
  7. 2000-01-04  2.969393  0.742348  1.009734
  8. 2000-01-05  4.690630  0.938126  0.977914
  9. ...              ...       ...       ...
  10. 2002-09-22  2.860036  0.047667  1.132051
  11. 2002-09-23  3.510163  0.058503  1.134296
  12. 2002-09-24  6.524983  0.108750  1.144204
  13. 2002-09-25  6.409626  0.106827  1.142913
  14. 2002-09-26  5.093787  0.084896  1.151416
  16. [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:

  1. In [94]: r.agg([np.sum, np.mean])
  2. Out[94]: 
  3.                    A                    B                   C          
  4.                  sum      mean        sum      mean       sum      mean
  5. 2000-01-01 -0.289838 -0.289838  -0.370545 -0.370545 -1.284206 -1.284206
  6. 2000-01-02 -0.216612 -0.108306  -1.675528 -0.837764 -1.169415 -0.584708
  7. 2000-01-03  1.154661  0.384887  -1.634017 -0.544672 -1.566620 -0.522207
  8. 2000-01-04  2.969393  0.742348  -4.003274 -1.000819 -1.816179 -0.454045
  9. 2000-01-05  4.690630  0.938126  -4.682017 -0.936403 -2.717209 -0.543442
  10. ...              ...       ...        ...       ...       ...       ...
  11. 2002-09-22  2.860036  0.047667  -9.270337 -0.154506  6.415245  0.106921
  12. 2002-09-23  3.510163  0.058503  -8.151439 -0.135857  5.177219  0.086287
  13. 2002-09-24  6.524983  0.108750 -10.168078 -0.169468  5.792639  0.096544
  14. 2002-09-25  6.409626  0.106827  -9.956226 -0.165937  5.704050  0.095068
  15. 2002-09-26  5.093787  0.084896  -7.074515 -0.117909  6.905823  0.115097
  17. [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:

  1. In [95]: r.agg({'A': np.sum, 'B': lambda x: np.std(x, ddof=1)})
  2. Out[95]: 
  3.                    A         B
  4. 2000-01-01 -0.289838       NaN
  5. 2000-01-02 -0.216612  0.660747
  6. 2000-01-03  1.154661  0.689929
  7. 2000-01-04  2.969393  1.072199
  8. 2000-01-05  4.690630  0.939657
  9. ...              ...       ...
  10. 2002-09-22  2.860036  1.113208
  11. 2002-09-23  3.510163  1.132381
  12. 2002-09-24  6.524983  1.080963
  13. 2002-09-25  6.409626  1.082911
  14. 2002-09-26  5.093787  1.136199
  16. [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

  1. In [96]: r.agg({'A': 'sum', 'B': 'std'})
  2. Out[96]: 
  3.                    A         B
  4. 2000-01-01 -0.289838       NaN
  5. 2000-01-02 -0.216612  0.660747
  6. 2000-01-03  1.154661  0.689929
  7. 2000-01-04  2.969393  1.072199
  8. 2000-01-05  4.690630  0.939657
  9. ...              ...       ...
  10. 2002-09-22  2.860036  1.113208
  11. 2002-09-23  3.510163  1.132381
  12. 2002-09-24  6.524983  1.080963
  13. 2002-09-25  6.409626  1.082911
  14. 2002-09-26  5.093787  1.136199
  16. [1000 rows x 2 columns]

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

  1. In [97]: r.agg({'A': ['sum', 'std'], 'B': ['mean', 'std']})
  2. Out[97]: 
  3.                    A                   B          
  4.                  sum       std      mean       std
  5. 2000-01-01 -0.289838       NaN -0.370545       NaN
  6. 2000-01-02 -0.216612  0.256725 -0.837764  0.660747
  7. 2000-01-03  1.154661  0.873311 -0.544672  0.689929
  8. 2000-01-04  2.969393  1.009734 -1.000819  1.072199
  9. 2000-01-05  4.690630  0.977914 -0.936403  0.939657
  10. ...              ...       ...       ...       ...
  11. 2002-09-22  2.860036  1.132051 -0.154506  1.113208
  12. 2002-09-23  3.510163  1.134296 -0.135857  1.132381
  13. 2002-09-24  6.524983  1.144204 -0.169468  1.080963
  14. 2002-09-25  6.409626  1.142913 -0.165937  1.082911
  15. 2002-09-26  5.093787  1.151416 -0.117909  1.136199
  17. [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:

  1. In [98]: df.rolling(window=len(df), min_periods=1).mean()[:5]
  2. Out[98]: 
  3.                    A         B         C         D
  4. 2000-01-01  0.314226 -0.001675  0.071823  0.892566
  5. 2000-01-02  0.654522 -0.171495  0.179278  0.853361
  6. 2000-01-03  0.708733 -0.064489 -0.238271  1.371111
  7. 2000-01-04  0.987613  0.163472 -0.919693  1.566485
  8. 2000-01-05  1.426971  0.288267 -1.358877  1.808650
  10. In [99]: df.expanding(min_periods=1).mean()[:5]
  11. Out[99]: 
  12.                    A         B         C         D
  13. 2000-01-01  0.314226 -0.001675  0.071823  0.892566
  14. 2000-01-02  0.654522 -0.171495  0.179278  0.853361
  15. 2000-01-03  0.708733 -0.064489 -0.238271  1.371111
  16. 2000-01-04  0.987613  0.163472 -0.919693  1.566485
  17. 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).

::: tip 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:

  1. In [100]: sn = pd.Series([1, 2, np.nan, 3, np.nan, 4])
  3. In [101]: sn
  4. Out[101]: 
  5. 0    1.0
  6. 1    2.0
  7. 2    NaN
  8. 3    3.0
  9. 4    NaN
  10. 5    4.0
  11. dtype: float64
  13. In [102]: sn.rolling(2).max()
  14. Out[102]: 
  15. 0    NaN
  16. 1    2.0
  17. 2    NaN
  18. 3    NaN
  19. 4    NaN
  20. 5    NaN
  21. dtype: float64
  23. In [103]: sn.rolling(2, min_periods=1).max()
  24. Out[103]: 
  25. 0    1.0
  26. 1    2.0
  27. 2    2.0
  28. 3    3.0
  29. 4    3.0
  30. 5    4.0
  31. 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():

  1. In [104]: sn.expanding().sum()
  2. Out[104]: 
  3. 0     1.0
  4. 1     3.0
  5. 2     3.0
  6. 3     6.0
  7. 4     6.0
  8. 5    10.0
  9. dtype: float64
  11. In [105]: sn.cumsum()
  12. Out[105]: 
  13. 0     1.0
  14. 1     3.0
  15. 2     NaN
  16. 3     6.0
  17. 4     NaN
  18. 5    10.0
  19. dtype: float64
  21. In [106]: sn.cumsum().fillna(method='ffill')
  22. Out[106]: 
  23. 0     1.0
  24. 1     3.0
  25. 2     3.0
  26. 3     6.0
  27. 4     6.0
  28. 5    10.0
  29. 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:

  1. In [107]: s.plot(style='k--')
  2. Out[107]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6607b14400>
  4. In [108]: s.expanding().mean().plot(style='k')
  5. Out[108]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6607b14400>

expanding_mean_frame

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

[yt = \frac{\sum{i=0}^t wi 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

[yt = \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}y0 &= 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}]

::: tip Note

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

[yt = \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:

[yt = \alpha’ y{t-1} + (1 - \alpha’) x_t.]

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

[\begin{split}yt &= \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 \(y0 = x_0\) 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 since version 0.18.0 it has been 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:

  1. In [109]: s.plot(style='k--')
  2. Out[109]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f7537278>
  4. In [110]: s.ewm(span=20).mean().plot(style='k')
  5. Out[110]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f7537278>

ewma_ex

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 wi\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.