NumPy version of “Exponential weighted moving average”, equivalent to pandas.ewm().mean()

Question

How do I get the exponential weighted moving average in NumPy just like the following in pandas?

import pandas as pd
import pandas_datareader as pdr
from dat         


        

Answer 1

Fastest EWMA 23x pandas

The question is strictly asking for a numpy solution, however, it seems that the OP was actually just after a pure numpy solution to speed up runtime.

I solved a similar problem but instead looked towards numba.jit which massively speeds the compute time

In [24]: a = np.random.random(10**7)
    ...: df = pd.Series(a)
In [25]: %timeit numpy_ewma(a, 10)               # /a/42915307/4013571
    ...: %timeit df.ewm(span=10).mean()          # pandas
    ...: %timeit numpy_ewma_vectorized_v2(a, 10) # best w/o numba: /a/42926270/4013571
    ...: %timeit _ewma(a, 10)                    # fastest accurate (below)
    ...: %timeit _ewma_infinite_hist(a, 10)      # fastest overall (below)
4.14 s ± 116 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
991 ms ± 52.2 ms per loop (mean ± std. dev. of 7 runs, 1 loop each) 
396 ms ± 8.39 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
181 ms ± 1.01 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)   
39.6 ms ± 979 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)

Scaling down to smaller arrays of a = np.random.random(100) (results in the same order)

41.6 µs ± 491 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
945 ms ± 12 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
16 µs ± 93.5 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
1.66 µs ± 13.7 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
1.14 µs ± 5.57 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)

It is also worth pointing out that my functions below are identically aligned to the pandas (see the examples in docstr), whereas a few of the answers here take various different approximations. For example,

In [57]: print(pd.DataFrame([1,2,3]).ewm(span=2).mean().values.ravel())
    ...: print(numpy_ewma_vectorized_v2(np.array([1,2,3]), 2))
    ...: print(numpy_ewma(np.array([1,2,3]), 2))
[1.         1.75       2.61538462]
[1.         1.66666667 2.55555556]
[1.         1.18181818 1.51239669]

The source code which I have documented for my own library

import numpy as np
from numba import jit
from numba import float64
from numba import int64


@jit((float64[:], int64), nopython=True, nogil=True)
def _ewma(arr_in, window):
    r"""Exponentialy weighted moving average specified by a decay ``window``
    to provide better adjustments for small windows via:

        y[t] = (x[t] + (1-a)*x[t-1] + (1-a)^2*x[t-2] + ... + (1-a)^n*x[t-n]) /
               (1 + (1-a) + (1-a)^2 + ... + (1-a)^n).

    Parameters
    ----------
    arr_in : np.ndarray, float64
        A single dimenisional numpy array
    window : int64
        The decay window, or 'span'

    Returns
    -------
    np.ndarray
        The EWMA vector, same length / shape as ``arr_in``

    Examples
    --------
    >>> import pandas as pd
    >>> a = np.arange(5, dtype=float)
    >>> exp = pd.DataFrame(a).ewm(span=10, adjust=True).mean()
    >>> np.array_equal(_ewma_infinite_hist(a, 10), exp.values.ravel())
    True
    """
    n = arr_in.shape[0]
    ewma = np.empty(n, dtype=float64)
    alpha = 2 / float(window + 1)
    w = 1
    ewma_old = arr_in[0]
    ewma[0] = ewma_old
    for i in range(1, n):
        w += (1-alpha)**i
        ewma_old = ewma_old*(1-alpha) + arr_in[i]
        ewma[i] = ewma_old / w
    return ewma


@jit((float64[:], int64), nopython=True, nogil=True)
def _ewma_infinite_hist(arr_in, window):
    r"""Exponentialy weighted moving average specified by a decay ``window``
    assuming infinite history via the recursive form:

        (2) (i)  y[0] = x[0]; and
            (ii) y[t] = a*x[t] + (1-a)*y[t-1] for t>0.

    This method is less accurate that ``_ewma`` but
    much faster:

        In [1]: import numpy as np, bars
           ...: arr = np.random.random(100000)
           ...: %timeit bars._ewma(arr, 10)
           ...: %timeit bars._ewma_infinite_hist(arr, 10)
        3.74 ms ± 60.2 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
        262 µs ± 1.54 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

    Parameters
    ----------
    arr_in : np.ndarray, float64
        A single dimenisional numpy array
    window : int64
        The decay window, or 'span'

    Returns
    -------
    np.ndarray
        The EWMA vector, same length / shape as ``arr_in``

    Examples
    --------
    >>> import pandas as pd
    >>> a = np.arange(5, dtype=float)
    >>> exp = pd.DataFrame(a).ewm(span=10, adjust=False).mean()
    >>> np.array_equal(_ewma_infinite_hist(a, 10), exp.values.ravel())
    True
    """
    n = arr_in.shape[0]
    ewma = np.empty(n, dtype=float64)
    alpha = 2 / float(window + 1)
    ewma[0] = arr_in[0]
    for i in range(1, n):
        ewma[i] = arr_in[i] * alpha + ewma[i-1] * (1 - alpha)
    return ewma