3.5 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 thru the .ewm method to receive an EWM object. A number of expanding EW (exponentially weighted) methods are provided:

Function	Description
`mean()`	EW moving average
`var()`	EW moving variance
`std()`	EW moving standard deviation
`corr()`	EW moving correlation
`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 and $y_t$ is the result.

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:

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

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 shows the equivalence of the above two variants for infinite series. When adjust=True we have $y_0 = x_0$ and from the last representation above we have $y_t = \alpha x_t + (1 - \alpha) y_{t-1}$ , 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:

In [1]: s.plot(style='k--')
Out[1]: <matplotlib.axes._subplots.AxesSubplot at 0x2b35b9f3e150>

In [2]: s.ewm(span=20).mean().plot(style='k')
Out[2]: <matplotlib.axes._subplots.AxesSubplot at 0x2b35b9f3e150>

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. (This is a change from versions prior to 0.15.0, in which the min_periods argument affected only the min_periods consecutive entries starting at the first non-null value.)

EWM also has an ignore_na argument, which deterines 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 (which reproduces the behavior in versions prior to 0.15.0), 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 for further details.