Windowing operations — pandas 2.2.3 documentation (original) (raw)

pandas contains a compact set of APIs for performing windowing operations - an operation that performs an aggregation over a sliding partition of values. The API functions similarly to the groupby API in that Series and DataFrame call the windowing method with necessary parameters and then subsequently call the aggregation function.

In [1]: s = pd.Series(range(5))

In [2]: s.rolling(window=2).sum() Out[2]: 0 NaN 1 1.0 2 3.0 3 5.0 4 7.0 dtype: float64

The windows are comprised by looking back the length of the window from the current observation. The result above can be derived by taking the sum of the following windowed partitions of data:

In [3]: for window in s.rolling(window=2): ...: print(window) ...: 0 0 dtype: int64 0 0 1 1 dtype: int64 1 1 2 2 dtype: int64 2 2 3 3 dtype: int64 3 3 4 4 dtype: int64

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

Note

Windowing operations currently only support numeric data (integer and float) and will always return float64 values.

Warning

Some windowing aggregation, mean, sum, var and std methods may suffer from numerical imprecision due to the underlying windowing algorithms accumulating sums. When values differ with magnitude \(1/np.finfo(np.double).eps\) this results in truncation. It must be noted, that large values may have an impact on windows, which do not include these values. Kahan summation is used to compute the rolling sums to preserve accuracy as much as possible.

Added in version 1.3.0.

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

Added in version 1.3.

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(["sum", "mean", "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.

Centering windows#

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

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

This can also be applied to datetime-like indices.

Added in version 1.3.0.

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

Rolling window endpoints#

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

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

In [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

Custom window rolling#

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

For example, if we have the following DataFrame

In [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

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

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

In [45]: class CustomIndexer(BaseIndexer): ....: def get_window_bounds(self, num_values, min_periods, center, closed, step): ....: 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 [46]: indexer = CustomIndexer(window_size=1, use_expanding=use_expanding)

In [47]: df.rolling(indexer).sum() Out[47]: values 0 0.0 1 1.0 2 3.0 3 3.0 4 10.0

You can view other examples of BaseIndexer subclasses here

One subclass of note within those examples is the VariableOffsetWindowIndexer that allows rolling operations over a non-fixed offset like a BusinessDay.

In [48]: from pandas.api.indexers import VariableOffsetWindowIndexer

In [49]: df = pd.DataFrame(range(10), index=pd.date_range("2020", periods=10))

In [50]: offset = pd.offsets.BDay(1)

In [51]: indexer = VariableOffsetWindowIndexer(index=df.index, offset=offset)

In [52]: df Out[52]: 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 [53]: df.rolling(indexer).sum() Out[53]: 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

For some problems knowledge of the future is available for analysis. For example, this occurs when each data point is a full time series read from an experiment, and the task is to extract underlying conditions. In these cases it can be useful to perform forward-looking rolling window computations.FixedForwardWindowIndexer class is available for this purpose. This BaseIndexer subclass implements a closed fixed-width forward-looking rolling window, and we can use it as follows:

In [54]: from pandas.api.indexers import FixedForwardWindowIndexer

In [55]: indexer = FixedForwardWindowIndexer(window_size=2)

In [56]: df.rolling(indexer, min_periods=1).sum() Out[56]: 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

We can also achieve this by using slicing, applying rolling aggregation, and then flipping the result as shown in example below:

In [57]: 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 [58]: df Out[58]: 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 [59]: reversed_df = df[::-1].rolling("2s").sum()[::-1]

In [60]: reversed_df Out[60]: 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

Rolling apply#

The apply() function takes an extra func argument and performs generic rolling computations. The func argument should be a single function that produces a single value from an ndarray input. raw specifies whether the windows are cast as Series objects (raw=False) or ndarray objects (raw=True).

In [61]: def mad(x): ....: return np.fabs(x - x.mean()).mean() ....:

In [62]: s = pd.Series(range(10))

In [63]: s.rolling(window=4).apply(mad, raw=True) Out[63]: 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

Numba engine#

Additionally, apply() can leverage Numbaif installed as an optional dependency. The apply aggregation can be executed using Numba by specifyingengine='numba' and engine_kwargs arguments (raw must also be set to True). See enhancing performance with Numba for general usage of the arguments and performance considerations.

Numba will be applied in potentially two routines:

1. If func is a standard Python function, the engine will JIT the passed function. func can also be a JITed function in which case the engine will not JIT the function again.
2. The engine will JIT the for loop where the apply function is applied to each window.

The engine_kwargs argument is a dictionary of keyword arguments that will be passed into thenumba.jit decorator. These keyword arguments will be applied to both the passed function (if a standard Python function) and the apply for loop over each window.

Added in version 1.3.0.

mean, median, max, min, and sum also support the engine and engine_kwargs arguments.

Binary window functions#

cov() and corr() can compute moving window statistics about two Series or any combination of DataFrame/Series orDataFrame/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 DataFrame with aMultiIndex whose values are the dates in question (see the next section).

For example:

In [64]: df = pd.DataFrame( ....: np.random.randn(10, 4), ....: index=pd.date_range("2020-01-01", periods=10), ....: columns=["A", "B", "C", "D"], ....: ) ....:

In [65]: df = df.cumsum()

In [66]: df2 = df[:4]

In [67]: df2.rolling(window=2).corr(df2["B"]) Out[67]: 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

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 ofDataFrame inputs will yield a MultiIndexed DataFrame whose index are the dates in question. In the case of a single DataFrame argument the pairwise argument can even be omitted:

Note

Missing values are ignored and each entry is computed using the pairwise complete observations.

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 matricesfor more details.

In [68]: covs = ( ....: df[["B", "C", "D"]] ....: .rolling(window=4) ....: .cov(df[["A", "B", "C"]], pairwise=True) ....: ) ....:

In [69]: covs Out[69]: 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 [70]: s = pd.Series(range(10))

In [71]: s.rolling(window=5).mean() Out[71]: 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 [72]: s.rolling(window=5, win_type="triang").mean() Out[72]: 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 [73]: s.rolling(window=5, win_type="gaussian").mean(std=0.1) Out[73]: 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 [74]: df = pd.DataFrame(range(5))

In [75]: df.rolling(window=len(df), min_periods=1).mean() Out[75]: 0 0 0.0 1 0.5 2 1.0 3 1.5 4 2.0

In [76]: df.expanding(min_periods=1).mean() Out[76]: 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

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

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

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

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

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

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

which is equivalent to using weights

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

Note

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

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

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

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

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

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

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

One must have \(0 < \alpha \leq 1\), and while it is possible to pass\(\alpha\) directly, it’s often easier to think about either thespan, 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-lifeand 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 [77]: df = pd.DataFrame({"B": [0, 1, 2, np.nan, 4]})

In [78]: df Out[78]: B 0 0.0 1 1.0 2 2.0 3 NaN 4 4.0

In [79]: times = ["2020-01-01", "2020-01-03", "2020-01-10", "2020-01-15", "2020-01-17"]

In [80]: df.ewm(halflife="4 days", times=pd.DatetimeIndex(times)).mean() Out[80]: 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:

\[y_t = \frac{\sum_{i=0}^t 0.5^\frac{t_{t} - t_{i}}{\lambda} x_{t-i}}{\sum_{i=0}^t 0.5^\frac{t_{t} - t_{i}}{\lambda}},\]

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

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

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 asewmvar(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 Varianceon Wikipedia for further details.