Windowing Operations

Overview

pandas supports 4 types of windowing operations:

1. Rolling window: Generic fixed or variable sliding window over the values.

2. Weighted window: Weighted, non-rectangular window supplied by the scipy.signal library.

3. Expanding window: Accumulating window over the values.

4. Exponentially Weighted window: Accumulating and exponentially weighted window over the values.

As noted above, some operations support specifying a window based on a time offset:

In [4]: s = pd.Series(range(5), index=pd.date_range('2020-01-01', periods=5, freq='1D'))

In [5]: s.rolling(window='2D').sum()
Out[5]: 
2020-01-01    0.0
2020-01-02    1.0
2020-01-03    3.0
2020-01-04    5.0
2020-01-05    7.0
Freq: D, dtype: float64

Additionally, some methods support chaining a groupby operation with a windowing operation which will first group the data by the specified keys and then perform a windowing operation per group.

In [6]: df = pd.DataFrame({'A': ['a', 'b', 'a', 'b', 'a'], 'B': range(5)})

In [7]: df.groupby('A').expanding().sum()
Out[7]: 
       B
A       
a 0  0.0
  2  2.0
  4  6.0
b 1  1.0
  3  4.0

Some windowing operations also support the method='table' option in the constructor which performs the windowing operation over an entire DataFrame instead of a single column or row at a time. This can provide a useful performance benefit for a DataFrame with many columns or rows (with the corresponding axis argument) or the ability to utilize other columns during the windowing operation. The method='table' option can only be used if engine='numba' is specified in the corresponding method call.

For example, a weighted mean calculation can be calculated with apply() by specifying a separate column of weights.

In [8]: def weighted_mean(x):
   ...:     arr = np.ones((1, x.shape[1]))
   ...:     arr[:, :2] = (x[:, :2] * x[:, 2]).sum(axis=0) / x[:, 2].sum()
   ...:     return arr
   ...: 

In [9]: df = pd.DataFrame([[1, 2, 0.6], [2, 3, 0.4], [3, 4, 0.2], [4, 5, 0.7]])

In [10]: df.rolling(2, method="table", min_periods=0).apply(weighted_mean, raw=True, engine="numba")  # noqa:E501
Out[10]: 
          0         1    2
0  1.000000  2.000000  1.0
1  1.800000  2.000000  1.0
2  3.333333  2.333333  1.0
3  1.555556  7.000000  1.0

Some windowing operations also support an online method after constructing a windowing object which returns a new object that supports passing in new DataFrame or Series objects to continue the windowing calculation with the new values (i.e. online calculations).

The methods on this new windowing objects must call the aggregation method first to “prime” the initial state of the online calculation. Then, new DataFrame or Series objects can be passed in the update argument to continue the windowing calculation.

In [11]: df = pd.DataFrame([[1, 2, 0.6], [2, 3, 0.4], [3, 4, 0.2], [4, 5, 0.7]])

In [12]: df.ewm(0.5).mean()
Out[12]: 
          0         1         2
0  1.000000  2.000000  0.600000
1  1.750000  2.750000  0.450000
2  2.615385  3.615385  0.276923
3  3.550000  4.550000  0.562500

In [13]: online_ewm = df.head(2).ewm(0.5).online()

In [14]: online_ewm.mean()
Out[14]: 
      0     1     2
0  1.00  2.00  0.60
1  1.75  2.75  0.45

In [15]: online_ewm.mean(update=df.tail(1))
Out[15]: 
          0         1         2
3  3.307692  4.307692  0.623077

All windowing operations support a min_periods argument that dictates the minimum amount of non-np.nan values a window must have; otherwise, the resulting value is np.nan. min_periods defaults to 1 for time-based windows and window for fixed windows

In [16]: s = pd.Series([np.nan, 1, 2, np.nan, np.nan, 3])

In [17]: s.rolling(window=3, min_periods=1).sum()
Out[17]: 
0    NaN
1    1.0
2    3.0
3    3.0
4    2.0
5    3.0
dtype: float64

In [18]: s.rolling(window=3, min_periods=2).sum()
Out[18]: 
0    NaN
1    NaN
2    3.0
3    3.0
4    NaN
5    NaN
dtype: float64

# Equivalent to min_periods=3
In [19]: s.rolling(window=3, min_periods=None).sum()
Out[19]: 
0   NaN
1   NaN
2   NaN
3   NaN
4   NaN
5   NaN
dtype: float64

Additionally, all windowing operations supports the aggregate method for returning a result of multiple aggregations applied to a window.

In [20]: df = pd.DataFrame({"A": range(5), "B": range(10, 15)})

In [21]: df.expanding().agg([np.sum, np.mean, np.std])
Out[21]: 
      A                    B                
    sum mean       std   sum  mean       std
0   0.0  0.0       NaN  10.0  10.0       NaN
1   1.0  0.5  0.707107  21.0  10.5  0.707107
2   3.0  1.0  1.000000  33.0  11.0  1.000000
3   6.0  1.5  1.290994  46.0  11.5  1.290994
4  10.0  2.0  1.581139  60.0  12.0  1.581139

Rolling window

Generic rolling windows support specifying windows as a fixed number of observations or variable number of observations based on an offset. If a time based offset is provided, the corresponding time based index must be monotonic.

In [22]: times = ['2020-01-01', '2020-01-03', '2020-01-04', '2020-01-05', '2020-01-29']

In [23]: s = pd.Series(range(5), index=pd.DatetimeIndex(times))

In [24]: s
Out[24]: 
2020-01-01    0
2020-01-03    1
2020-01-04    2
2020-01-05    3
2020-01-29    4
dtype: int64

# Window with 2 observations
In [25]: s.rolling(window=2).sum()
Out[25]: 
2020-01-01    NaN
2020-01-03    1.0
2020-01-04    3.0
2020-01-05    5.0
2020-01-29    7.0
dtype: float64

# Window with 2 days worth of observations
In [26]: s.rolling(window='2D').sum()
Out[26]: 
2020-01-01    0.0
2020-01-03    1.0
2020-01-04    3.0
2020-01-05    5.0
2020-01-29    4.0
dtype: float64

For all supported aggregation functions, see Rolling window functions.

In [27]: s = pd.Series(range(10))

In [28]: s.rolling(window=5).mean()
Out[28]: 
0    NaN
1    NaN
2    NaN
3    NaN
4    2.0
5    3.0
6    4.0
7    5.0
8    6.0
9    7.0
dtype: float64

In [29]: s.rolling(window=5, center=True).mean()
Out[29]: 
0    NaN
1    NaN
2    2.0
3    3.0
4    4.0
5    5.0
6    6.0
7    7.0
8    NaN
9    NaN
dtype: float64

In [30]: df = pd.DataFrame(
   ....:     {"A": [0, 1, 2, 3, 4]}, index=pd.date_range("2020", periods=5, freq="1D")
   ....: )
   ....: 

In [31]: df
Out[31]: 
            A
2020-01-01  0
2020-01-02  1
2020-01-03  2
2020-01-04  3
2020-01-05  4

In [32]: df.rolling("2D", center=False).mean()
Out[32]: 
              A
2020-01-01  0.0
2020-01-02  0.5
2020-01-03  1.5
2020-01-04  2.5
2020-01-05  3.5

In [33]: df.rolling("2D", center=True).mean()
Out[33]: 
              A
2020-01-01  0.5
2020-01-02  1.5
2020-01-03  2.5
2020-01-04  3.5
2020-01-05  4.0

In [34]: 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 [35]: df["right"] = df.rolling("2s", closed="right").x.sum()  # default

In [36]: df["both"] = df.rolling("2s", closed="both").x.sum()

In [37]: df["left"] = df.rolling("2s", closed="left").x.sum()

In [38]: df["neither"] = df.rolling("2s", closed="neither").x.sum()

In [39]: df
Out[39]: 
                     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

In [40]: use_expanding = [True, False, True, False, True]

In [41]: use_expanding
Out[41]: [True, False, True, False, True]

In [42]: df = pd.DataFrame({"values": range(5)})

In [43]: df
Out[43]: 
   values
0       0
1       1
2       2
3       3
4       4

In [2]: from pandas.api.indexers import BaseIndexer

In [3]: class CustomIndexer(BaseIndexer):
   ...:     def get_window_bounds(self, num_values, min_periods, center, closed):
   ...:         start = np.empty(num_values, dtype=np.int64)
   ...:         end = np.empty(num_values, dtype=np.int64)
   ...:         for i in range(num_values):
   ...:             if self.use_expanding[i]:
   ...:                 start[i] = 0
   ...:                 end[i] = i + 1
   ...:             else:
   ...:                 start[i] = i
   ...:                 end[i] = i + self.window_size
   ...:         return start, end

In [4]: indexer = CustomIndexer(window_size=1, use_expanding=use_expanding)

In [5]: df.rolling(indexer).sum()
Out[5]:
    values
0     0.0
1     1.0
2     3.0
3     3.0
4    10.0

In [44]: from pandas.api.indexers import VariableOffsetWindowIndexer

In [45]: df = pd.DataFrame(range(10), index=pd.date_range("2020", periods=10))

In [46]: offset = pd.offsets.BDay(1)

In [47]: indexer = VariableOffsetWindowIndexer(index=df.index, offset=offset)

In [48]: df
Out[48]: 
            0
2020-01-01  0
2020-01-02  1
2020-01-03  2
2020-01-04  3
2020-01-05  4
2020-01-06  5
2020-01-07  6
2020-01-08  7
2020-01-09  8
2020-01-10  9

In [49]: df.rolling(indexer).sum()
Out[49]: 
               0
2020-01-01   0.0
2020-01-02   1.0
2020-01-03   2.0
2020-01-04   3.0
2020-01-05   7.0
2020-01-06  12.0
2020-01-07   6.0
2020-01-08   7.0
2020-01-09   8.0
2020-01-10   9.0

In [50]: from pandas.api.indexers import FixedForwardWindowIndexer

In [51]: indexer = FixedForwardWindowIndexer(window_size=2)

In [52]: df.rolling(indexer, min_periods=1).sum()
Out[52]: 
               0
2020-01-01   1.0
2020-01-02   3.0
2020-01-03   5.0
2020-01-04   7.0
2020-01-05   9.0
2020-01-06  11.0
2020-01-07  13.0
2020-01-08  15.0
2020-01-09  17.0
2020-01-10   9.0

In [53]: df = pd.DataFrame(
   ....:     data=[
   ....:         [pd.Timestamp("2018-01-01 00:00:00"), 100],
   ....:         [pd.Timestamp("2018-01-01 00:00:01"), 101],
   ....:         [pd.Timestamp("2018-01-01 00:00:03"), 103],
   ....:         [pd.Timestamp("2018-01-01 00:00:04"), 111],
   ....:     ],
   ....:     columns=["time", "value"],
   ....: ).set_index("time")
   ....: 

In [54]: df
Out[54]: 
                     value
time                      
2018-01-01 00:00:00    100
2018-01-01 00:00:01    101
2018-01-01 00:00:03    103
2018-01-01 00:00:04    111

In [55]: reversed_df = df[::-1].rolling("2s").sum()[::-1]

In [56]: reversed_df
Out[56]: 
                     value
time                      
2018-01-01 00:00:00  201.0
2018-01-01 00:00:01  101.0
2018-01-01 00:00:03  214.0
2018-01-01 00:00:04  111.0

In [57]: def mad(x):
   ....:     return np.fabs(x - x.mean()).mean()
   ....: 

In [58]: s = pd.Series(range(10))

In [59]: s.rolling(window=4).apply(mad, raw=True)
Out[59]: 
0    NaN
1    NaN
2    NaN
3    1.0
4    1.0
5    1.0
6    1.0
7    1.0
8    1.0
9    1.0
dtype: float64

In [60]: df = pd.DataFrame(
   ....:     np.random.randn(10, 4),
   ....:     index=pd.date_range("2020-01-01", periods=10),
   ....:     columns=["A", "B", "C", "D"],
   ....: )
   ....: 

In [61]: df = df.cumsum()

In [62]: df2 = df[:4]

In [63]: df2.rolling(window=2).corr(df2["B"])
Out[63]: 
              A    B    C    D
2020-01-01  NaN  NaN  NaN  NaN
2020-01-02 -1.0  1.0 -1.0  1.0
2020-01-03  1.0  1.0  1.0 -1.0
2020-01-04 -1.0  1.0  1.0 -1.0

In [64]: covs = (
   ....:     df[["B", "C", "D"]]
   ....:     .rolling(window=4)
   ....:     .cov(df[["A", "B", "C"]], pairwise=True)
   ....: )
   ....: 

In [65]: covs
Out[65]: 
                     B         C         D
2020-01-01 A       NaN       NaN       NaN
           B       NaN       NaN       NaN
           C       NaN       NaN       NaN
2020-01-02 A       NaN       NaN       NaN
           B       NaN       NaN       NaN
...                ...       ...       ...
2020-01-09 B  0.342006  0.230190  0.052849
           C  0.230190  1.575251  0.082901
2020-01-10 A -0.333945  0.006871 -0.655514
           B  0.649711  0.430860  0.469271
           C  0.430860  0.829721  0.055300

[30 rows x 3 columns]

Weighted window

The win_type argument in .rolling generates a weighted windows that are commonly used in filtering and spectral estimation. win_type must be string that corresponds to a scipy.signal window function. Scipy must be installed in order to use these windows, and supplementary arguments that the Scipy window methods take must be specified in the aggregation function.

In [66]: s = pd.Series(range(10))

In [67]: s.rolling(window=5).mean()
Out[67]: 
0    NaN
1    NaN
2    NaN
3    NaN
4    2.0
5    3.0
6    4.0
7    5.0
8    6.0
9    7.0
dtype: float64

In [68]: s.rolling(window=5, win_type="triang").mean()
Out[68]: 
0    NaN
1    NaN
2    NaN
3    NaN
4    2.0
5    3.0
6    4.0
7    5.0
8    6.0
9    7.0
dtype: float64

# Supplementary Scipy arguments passed in the aggregation function
In [69]: s.rolling(window=5, win_type="gaussian").mean(std=0.1)
Out[69]: 
0    NaN
1    NaN
2    NaN
3    NaN
4    2.0
5    3.0
6    4.0
7    5.0
8    6.0
9    7.0
dtype: float64

For all supported aggregation functions, see Weighted window functions.

Expanding window

An expanding window yields the value of an aggregation statistic with all the data available up to that point in time. Since these calculations are a special case of rolling statistics, they are implemented in pandas such that the following two calls are equivalent:

In [70]: df = pd.DataFrame(range(5))

In [71]: df.rolling(window=len(df), min_periods=1).mean()
Out[71]: 
     0
0  0.0
1  0.5
2  1.0
3  1.5
4  2.0

In [72]: df.expanding(min_periods=1).mean()
Out[72]: 
     0
0  0.0
1  0.5
2  1.0
3  1.5
4  2.0

For all supported aggregation functions, see Expanding window functions.

Exponentially Weighted window

An exponentially weighted window is similar to an expanding window but with each prior point being exponentially weighted down relative to the current point.

In general, a weighted moving average is calculated as

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

For all supported aggregation functions, see Exponentially-weighted window functions.

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

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

which is equivalent to using weights

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:

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

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

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

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.

You can also specify halflife in terms of a timedelta convertible unit to specify the amount of time it takes for an observation to decay to half its value when also specifying a sequence of times.

In [73]: df = pd.DataFrame({"B": [0, 1, 2, np.nan, 4]})

In [74]: df
Out[74]: 
     B
0  0.0
1  1.0
2  2.0
3  NaN
4  4.0

In [75]: times = ["2020-01-01", "2020-01-03", "2020-01-10", "2020-01-15", "2020-01-17"]

In [76]: df.ewm(halflife="4 days", times=pd.DatetimeIndex(times)).mean()
Out[76]: 
          B
0  0.000000
1  0.585786
2  1.523889
3  1.523889
4  3.233686

The following formula is used to compute exponentially weighted mean with an input vector of times:

ExponentialMovingWindow 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

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

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

(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.