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{{ header }}

Windowing Operations

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 :class:`Series` and :class:`DataFrame` call the windowing method with necessary parameters and then subsequently call the aggregation function.

.. ipython:: python

   s = pd.Series(range(5))
   s.rolling(window=2).sum()

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:

.. ipython:: python

   for window in s.rolling(window=2):
       print(window)

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.
Concept Method Returned Object Supports time-based windows Supports chained groupby Supports table method
Rolling window rolling Rolling Yes Yes Yes (as of version 1.3)
Weighted window rolling Window No No No
Expanding window expanding Expanding No Yes Yes (as of version 1.3)
Exponentially Weighted window ewm ExponentialMovingWindow No Yes (as of version 1.2) No

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

.. ipython:: python

   s = pd.Series(range(5), index=pd.date_range('2020-01-01', periods=5, freq='1D'))
   s.rolling(window='2D').sum()

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.

.. ipython:: python

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

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.

.. versionadded:: 1.3

Some windowing operations also support the method='table' option in the constructor which performs the windowing operaion over an entire :class:`DataFrame` instead of a single column or row at a time. This can provide a useful performance benefit for a :class:`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 :meth:`~Rolling.apply` by specifying a separate column of weights.

.. ipython:: python

   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

   df = pd.DataFrame([[1, 2, 0.6], [2, 3, 0.4], [3, 4, 0.2], [4, 5, 0.7]])
   df.rolling(2, method="table", min_periods=0).apply(weighted_mean, raw=True, engine="numba")  # noqa:E501

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_peridos defaults to 1 for time-based windows and window for fixed windows

.. ipython:: python

   s = pd.Series([np.nan, 1, 2, np.nan, np.nan, 3])
   s.rolling(window=3, min_periods=1).sum()
   s.rolling(window=3, min_periods=2).sum()
   # Equivalent to min_periods=3
   s.rolling(window=3, min_periods=None).sum()

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

.. ipython:: python

   df = pd.DataFrame({"A": range(5), "B": range(10, 15)})
   df.expanding().agg([np.sum, np.mean, np.std])

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.

.. ipython:: python

   times = ['2020-01-01', '2020-01-03', '2020-01-04', '2020-01-05', '2020-01-29']
   s = pd.Series(range(5), index=pd.DatetimeIndex(times))
   s
   # Window with 2 observations
   s.rolling(window=2).sum()
   # Window with 2 days worth of observations
   s.rolling(window='2D').sum()

For all supported aggregation functions, see :ref:`api.functions_rolling`.

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.

.. ipython:: python

   s = pd.Series(range(10))
   s.rolling(window=5).mean()
   s.rolling(window=5, center=True).mean()

Rolling window endpoints

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

Value Behavior
right' close right endpoint
'left' close left endpoint
'both' close both endpoints
'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.

.. ipython:: python

   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"),
       ],
   )

   df["right"] = df.rolling("2s", closed="right").x.sum()  # default
   df["both"] = df.rolling("2s", closed="both").x.sum()
   df["left"] = df.rolling("2s", closed="left").x.sum()
   df["neither"] = df.rolling("2s", closed="neither").x.sum()

   df

Custom window rolling

.. versionadded:: 1.0

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

For example, if we have the following :class:DataFrame:

.. ipython:: python

   use_expanding = [True, False, True, False, True]
   use_expanding
   df = pd.DataFrame({"values": range(5)})
   df

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

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

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

You can view other examples of BaseIndexer subclasses here

.. versionadded:: 1.1

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

.. ipython:: python

   from pandas.api.indexers import VariableOffsetWindowIndexer

   df = pd.DataFrame(range(10), index=pd.date_range("2020", periods=10))
   offset = pd.offsets.BDay(1)
   indexer = VariableOffsetWindowIndexer(index=df.index, offset=offset)
   df
   df.rolling(indexer).sum()

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. :func:`FixedForwardWindowIndexer <pandas.api.indexers.FixedForwardWindowIndexer>` class is available for this purpose. This :func:`BaseIndexer <pandas.api.indexers.BaseIndexer>` subclass implements a closed fixed-width forward-looking rolling window, and we can use it as follows:

.. ipython:: ipython

   from pandas.api.indexers import FixedForwardWindowIndexer
   indexer = FixedForwardWindowIndexer(window_size=2)
   df.rolling(indexer, min_periods=1).sum()

Rolling apply

The :meth:`~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. raw specifies whether the windows are cast as :class:`Series` objects (raw=False) or ndarray objects (raw=True).

.. ipython:: python

   def mad(x):
       return np.fabs(x - x.mean()).mean()

   s = pd.Series(range(10))
   s.rolling(window=4).apply(mad, raw=True)

Numba engine

.. versionadded:: 1.0

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

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

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

Note

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

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

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

In [3]: def f(x):
   ...:     return np.sum(x) + 5
# Run the first time, compilation time will affect performance
In [4]: %timeit -r 1 -n 1 roll.apply(f, engine='numba', raw=True)  # noqa: E225, E999
1.23 s ± 0 ns per loop (mean ± std. dev. of 1 run, 1 loop each)
# Function is cached and performance will improve
In [5]: %timeit roll.apply(f, engine='numba', raw=True)
188 ms ± 1.93 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

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

Binary window functions

:meth:`~Rolling.cov` and :meth:`~Rolling.corr` can compute moving window statistics about two :class:`Series` or any combination of :class:`DataFrame`/:class:`Series` or :class:`DataFrame`/:class:`DataFrame`. Here is the behavior in each case:

For example:

.. ipython:: python

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

   df2 = df[:4]
   df2.rolling(window=2).corr(df2["B"])

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

Note

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

.. ipython:: python

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

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.

.. ipython:: python

   s = pd.Series(range(10))
   s.rolling(window=5).mean()
   s.rolling(window=5, win_type="triang").mean()
   # Supplementary Scipy arguments passed in the aggregation function
   s.rolling(window=5, win_type="gaussian").mean(std=0.1)

For all supported aggregation functions, see :ref:`api.functions_window`.

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:

.. ipython:: python

   df = pd.DataFrame(range(5))
   df.rolling(window=len(df), min_periods=1).mean()
   df.expanding(min_periods=1).mean()

For all supported aggregation functions, see :ref:`api.functions_expanding`.

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 :ref:`api.functions_ewm`.

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

y_0 &= x_0 \\
y_t &= (1 - \alpha) y_{t-1} + \alpha x_t,

which is equivalent to using weights

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

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

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}

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:

\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}

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.
.. versionadded:: 1.1.0

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.

.. ipython:: python

    df = pd.DataFrame({"B": [0, 1, 2, np.nan, 4]})
    df
    times = ["2020-01-01", "2020-01-03", "2020-01-10", "2020-01-15", "2020-01-17"]
    df.ewm(halflife="4 days", times=pd.DatetimeIndex(times)).mean()

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

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

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