# Computational tools ## Statistical functions ### Percent change ``Series`` and ``DataFrame`` have a method [``pct_change()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.pct_change.html#pandas.DataFrame.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). ``` python 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 ``` ``` python 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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.Series.cov.html#pandas.Series.cov) can be used to compute covariance between series (excluding missing values). ``` python 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]: 0.000680108817431082 ``` Analogously, [``DataFrame.cov()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cov.html#pandas.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](http://en.wikipedia.org/w/index.php?title=Estimation_of_covariance_matrices) for more details. ::: ``` python 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. ``` python 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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.corr.html#pandas.DataFrame.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](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient) - [Kendall rank correlation coefficient](https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient) - [Spearman’s rank correlation coefficient](https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient) ::: tip Note Please see the [caveats](#computation-covariance-caveats) associated with this method of calculating correlation matrices in the [covariance section](#computation-covariance). ::: ``` python 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]: 0.013479040400098794 In [18]: frame['a'].corr(frame['b'], method='spearman') Out[18]: -0.007289885159540637 # 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: ``` python 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: ``` python # 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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.corrwith.html#pandas.DataFrame.corrwith) is implemented on DataFrame to compute the correlation between like-labeled Series contained in different DataFrame objects. ``` python 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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.Series.rank.html#pandas.Series.rank) method produces a data ranking with ties being assigned the mean of the ranks (by default) for the group: ``` python In [33]: s = pd.Series(np.random.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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.rank.html#pandas.DataFrame.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. ``` python In [36]: df = pd.DataFrame(np.random.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](groupby.html#groupby-transform-window-resample). ::: tip Note The API for window statistics is quite similar to the way one works with ``GroupBy`` objects, see the documentation [here](groupby.html#groupby). ::: We work with ``rolling``, ``expanding`` and ``exponentially weighted`` data through the corresponding objects, ``Rolling``, ``Expanding`` and ``EWM``. ``` python 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``. ``` python 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. ``` python In [14]: r. # 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: ``` python 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 ``` ``` python In [46]: s.plot(style='k--') Out[46]: In [47]: r.mean().plot(style='k') Out[47]: ``` ![rolling_mean_ex](https://static.pypandas.cn/public/static/images/rolling_mean_ex.png) 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: ``` python 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([, , , ], dtype=object) ``` ![rolling_mean_frame](https://static.pypandas.cn/public/static/images/rolling_mean_frame.png) ### Method summary We provide a number of common statistical functions: Method | Description ---|--- [count()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.count.html#pandas.core.window.Rolling.count) | Number of non-null observations [sum()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.sum.html#pandas.core.window.Rolling.sum) | Sum of values [mean()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.mean.html#pandas.core.window.Rolling.mean) | Mean of values [median()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.median.html#pandas.core.window.Rolling.median) | Arithmetic median of values [min()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.min.html#pandas.core.window.Rolling.min) | Minimum [max()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.max.html#pandas.core.window.Rolling.max) | Maximum [std()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.std.html#pandas.core.window.Rolling.std) | Bessel-corrected sample standard deviation [var()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.var.html#pandas.core.window.Rolling.var) | Unbiased variance [skew()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.skew.html#pandas.core.window.Rolling.skew) | Sample skewness (3rd moment) [kurt()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.kurt.html#pandas.core.window.Rolling.kurt) | Sample kurtosis (4th moment) [quantile()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.quantile.html#pandas.core.window.Rolling.quantile) | Sample quantile (value at %) [apply()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.apply.html#pandas.core.window.Rolling.apply) | Generic apply [cov()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.cov.html#pandas.core.window.Rolling.cov) | Unbiased covariance (binary) [corr()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.corr.html#pandas.core.window.Rolling.corr) | Correlation (binary) The [``apply()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.apply.html#pandas.core.window.Rolling.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: ``` python 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]: ``` ![rolling_apply_ex](https://static.pypandas.cn/public/static/images/rolling_apply_ex.png) ### 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()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Window.sum.html#pandas.core.window.Window.sum) | Sum of values [mean()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Window.mean.html#pandas.core.window.Window.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](https://docs.scipy.org/doc/scipy/reference/signal.html#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). ``` python 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() Out[54]: 2000-01-01 NaN 2000-01-02 NaN 2000-01-03 NaN 2000-01-04 NaN 2000-01-05 -1.037870 2000-01-06 -0.767705 2000-01-07 -0.383197 2000-01-08 -0.395513 2000-01-09 -0.558440 2000-01-10 -0.672416 Freq: D, dtype: float64 ``` Note that the ``boxcar`` window is equivalent to [``mean()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.mean.html#pandas.core.window.Rolling.mean). ``` python In [55]: ser.rolling(window=5, win_type='boxcar').mean() Out[55]: 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 [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: ``` python In [57]: ser.rolling(window=5, win_type='gaussian').mean(std=0.1) Out[57]: 2000-01-01 NaN 2000-01-02 NaN 2000-01-03 NaN 2000-01-04 NaN 2000-01-05 -1.309989 2000-01-06 -1.153000 2000-01-07 0.606382 2000-01-08 -0.681101 2000-01-09 -0.289724 2000-01-10 -0.996632 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. ``` python 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. ``` python 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. ``` python 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. ``` python 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. ``` python 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. ``` python 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 ``` ### 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. ``` python In [70]: 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 [71]: df["right"] = df.rolling('2s', closed='right').x.sum() # default In [72]: df["both"] = df.rolling('2s', closed='both').x.sum() In [73]: df["left"] = df.rolling('2s', closed='left').x.sum() In [74]: df["neither"] = df.rolling('2s', closed='neither').x.sum() In [75]: df Out[75]: 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](timeseries.html#timeseries-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. ``` python In [76]: ser.rolling(window=5).mean() Out[76]: 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 [77]: ser.rolling(window=5, center=True).mean() Out[77]: 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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.cov.html#pandas.core.window.Rolling.cov) and [``corr()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.corr.html#pandas.core.window.Rolling.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](#stats-moments-corr-pairwise)). For example: ``` python In [78]: df = pd.DataFrame(np.random.randn(1000, 4), ....: index=pd.date_range('1/1/2000', periods=1000), ....: columns=['A', 'B', 'C', 'D']) ....: In [79]: df = df.cumsum() In [80]: df2 = df[:20] In [81]: df2.rolling(window=5).corr(df2['B']) Out[81]: 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: ::: tip Note Missing values are ignored and each entry is computed using the pairwise complete observations. Please see the [covariance section](#computation-covariance) for [caveats](#computation-covariance-caveats) associated with this method of calculating covariance and correlation matrices. ::: ``` python In [82]: covs = (df[['B', 'C', 'D']].rolling(window=50) ....: .cov(df[['A', 'B', 'C']], pairwise=True)) ....: In [83]: covs.loc['2002-09-22':] Out[83]: 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 ``` ``` python In [84]: correls = df.rolling(window=50).corr() In [85]: correls.loc['2002-09-22':] Out[85]: 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: ``` python In [86]: correls.unstack(1)[('A', 'C')].plot() Out[86]: ``` ![rolling_corr_pairwise_ex](https://static.pypandas.cn/public/static/images/rolling_corr_pairwise_ex.png) ## 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](https://pandas.pydata.org/pandas-docs/stable/getting_started/basics.html#basics-aggregate), [groupby API](groupby.html#groupby-aggregate), and [resample API](timeseries.html#timeseries-aggregate). ``` python In [87]: dfa = pd.DataFrame(np.random.randn(1000, 3), ....: index=pd.date_range('1/1/2000', periods=1000), ....: columns=['A', 'B', 'C']) ....: In [88]: r = dfa.rolling(window=60, min_periods=1) In [89]: r 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__``. ``` python In [90]: r.aggregate(np.sum) Out[90]: 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 [91]: r['A'].aggregate(np.sum) Out[91]: 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 [92]: r[['A', 'B']].aggregate(np.sum) Out[92]: 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: ``` python In [93]: r['A'].agg([np.sum, np.mean, np.std]) Out[93]: 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: ``` python In [94]: r.agg([np.sum, np.mean]) Out[94]: 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``: ``` python In [95]: r.agg({'A': np.sum, 'B': lambda x: np.std(x, ddof=1)}) Out[95]: 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 ``` python In [96]: r.agg({'A': 'sum', 'B': 'std'}) Out[96]: 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. ``` python In [97]: r.agg({'A': ['sum', 'std'], 'B': ['mean', 'std']}) Out[97]: 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: ``` python In [98]: df.rolling(window=len(df), min_periods=1).mean()[:5] Out[98]: 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 [99]: df.expanding(min_periods=1).mean()[:5] Out[99]: 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()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.count.html#pandas.core.window.Expanding.count) | Number of non-null observations [sum()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.sum.html#pandas.core.window.Expanding.sum) | Sum of values [mean()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.mean.html#pandas.core.window.Expanding.mean) | Mean of values [median()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.median.html#pandas.core.window.Expanding.median) | Arithmetic median of values [min()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.min.html#pandas.core.window.Expanding.min) | Minimum [max()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.max.html#pandas.core.window.Expanding.max) | Maximum [std()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.std.html#pandas.core.window.Expanding.std) | Unbiased standard deviation [var()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.var.html#pandas.core.window.Expanding.var) | Unbiased variance [skew()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.skew.html#pandas.core.window.Expanding.skew) | Unbiased skewness (3rd moment) [kurt()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.kurt.html#pandas.core.window.Expanding.kurt) | Unbiased kurtosis (4th moment) [quantile()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.quantile.html#pandas.core.window.Expanding.quantile) | Sample quantile (value at %) [apply()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.apply.html#pandas.core.window.Expanding.apply) | Generic apply [cov()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.cov.html#pandas.core.window.Expanding.cov) | Unbiased covariance (binary) [corr()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.corr.html#pandas.core.window.Expanding.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: ``` python In [100]: sn = pd.Series([1, 2, np.nan, 3, np.nan, 4]) In [101]: sn Out[101]: 0 1.0 1 2.0 2 NaN 3 3.0 4 NaN 5 4.0 dtype: float64 In [102]: sn.rolling(2).max() Out[102]: 0 NaN 1 2.0 2 NaN 3 NaN 4 NaN 5 NaN dtype: float64 In [103]: sn.rolling(2, min_periods=1).max() Out[103]: 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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cumsum.html#pandas.DataFrame.cumsum), [``cumprod()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cumprod.html#pandas.DataFrame.cumprod), [``cummax()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cummax.html#pandas.DataFrame.cummax), and [``cummin()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cummin.html#pandas.DataFrame.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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.fillna.html#pandas.DataFrame.fillna): ``` python In [104]: sn.expanding().sum() Out[104]: 0 1.0 1 3.0 2 3.0 3 6.0 4 6.0 5 10.0 dtype: float64 In [105]: sn.cumsum() Out[105]: 0 1.0 1 3.0 2 NaN 3 6.0 4 NaN 5 10.0 dtype: float64 In [106]: sn.cumsum().fillna(method='ffill') Out[106]: 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()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.mean.html#pandas.core.window.Expanding.mean) output for the previous time series dataset: ``` python In [107]: s.plot(style='k--') Out[107]: In [108]: s.expanding().mean().plot(style='k') Out[108]: ``` ![expanding_mean_frame](https://static.pypandas.cn/public/static/images/expanding_mean_frame.png) ## 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()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.mean.html#pandas.core.window.EWM.mean) | EW moving average [var()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.var.html#pandas.core.window.EWM.var) | EW moving variance [std()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.std.html#pandas.core.window.EWM.std) | EW moving standard deviation [corr()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.corr.html#pandas.core.window.EWM.corr) | EW moving correlation [cov()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.cov.html#pandas.core.window.EWM.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
\[\begin{split}w_i = \begin{cases} \alpha (1 - \alpha)^i & \text{if } i < t \\ (1 - \alpha)^i & \text{if } i = t. \end{cases}\end{split}\]
::: tip 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``:
\[y_t = \alpha' y_{t-1} + (1 - \alpha') x_t.\]
Noting that the denominator is a geometric series with initial term equal to 1 and a ratio of \\(1 - \alpha\\) we have
\[\begin{split}y_t &= \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...} {\frac{1}{1 - (1 - \alpha)}}\\ &= [x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...] \alpha \\ &= \alpha x_t + [(1-\alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...]\alpha \\ &= \alpha x_t + (1 - \alpha)[x_{t-1} + (1 - \alpha) x_{t-2} + ...]\alpha\\ &= \alpha x_t + (1 - \alpha) y_{t-1}\end{split}\]
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 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: ``` python In [109]: s.plot(style='k--') Out[109]: In [110]: s.ewm(span=20).mean().plot(style='k') Out[110]: ``` ![ewma_ex](https://static.pypandas.cn/public/static/images/ewma_ex.png) 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](http://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Weighted_sample_variance) on Wikipedia for further details.