Exponential weighted moving skewness/kurtosis

Question

There are well-known on-line formulas for computing exponentially weighted moving averages and standard deviations of a process $(x_n)_{n=0,1,2,\dots}$. For the mean,

$\mu_n = (1-\alpha) \mu_{n-1} + \alpha x_n$

and for the variance

$\sigma_n^2 = (1-\alpha) \sigma_{n-1}^2 + \alpha(x_n - \mu_{n-1})(x_n - \mu_n)$

from which you can compute the standard deviation.

Are there similar formulas for on-line computation of exponential weighted third- and fourth-central moments? My intuition is that they should take the form

$M_{3,n} = (1-\alpha) M_{3,n-1} + \alpha f(x_n,\mu_n,\mu_{n-1},S_n,S_{n-1})$

and

$M_{4,n} = (1-\alpha) M_{4,n-1} + \alpha f(x_n,\mu_n,\mu_{n-1},S_n,S_{n-1},M_{3,n},M_{3,n-1})$

from which you could compute the skewness $\gamma_n = M_{3,n} / \sigma_n^3$ and the kurtosis $k_n = M_{4,n}/\sigma_n^4$ but I've not been able to find simple, closed-form expression for the functions $f$ and $g$.

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Edit: Some more information. The updating formula for moving variance is a special case of the formula for the exponential weighted moving covariance, which can be computed via

$C_n(x,y) = (1-\alpha) C_{n-1}(x,y) + \alpha (x_n - \bar{x}_n) (y_n - \bar{y}_{n-1})$

where $\bar{x}_n$ and $\bar{y}_n$ are the exponential moving means of $x$ and $y$. The asymmetry between $x$ and $y$ is illusory, and disappears when you notice that $y-\bar{y}_n = (1-\alpha) (y-\bar{y}_{n-1})$.

Formulas like this can be computed by writing the central moment as an expectation $E_n(\cdot)$, where weights in the expectation are understood to be exponential, and using the fact that for any function $f(x)$ we have

$E_n(f(x)) = \alpha f(x_n) + (1-\alpha) E_{n-1}(f(x))$

It's easy to derive the updating formulas for the mean and variance using this relation, but it's proving to be more tricky for the third and fourth central moments.

Answer 1

The formulas are straightforward but they are not as simple as intimated in the question.

Let $Y$ be the previous EWMA and let $X = x_n$, which is presumed independent of $Y$. By definition, the new weighted average is $Z = \alpha X + (1 - \alpha)Y$ for a constant value $\alpha$. For notational convenience, set $\beta = 1-\alpha$. Let $F$ denote the CDF of a random variable and $\phi$ denote its moment generating function, so that

$$\phi_X(t) = \mathbb{E}_F[\exp(t X)] = \int_\mathbb{R}{\exp(t x) dF_X(x)}.$$

With Kendall and Stuart, let $\mu_k^{'}(Z)$ denote the non-central moment of order $k$ for the random variable $Z$; that is, $\mu_k^{'}(Z) = \mathbb{E}[Z^k]$. The skewness and kurtosis are expressible in terms of the $\mu_k^{'}$ for $k = 1,2,3,4$; for example, the skewness is defined as $\mu_3 / \mu_2^{3/2}$ where

$$\mu_3 = \mu_3^{'} - 3 \mu_2^{'}\mu_1^{'} + 2{\mu_1^{'}}^3 \text{ and }\mu_2 = \mu_2^{'} - {\mu_1^{'}}^2$$

are the third and second central moments, respectively.

By standard elementary results,

$$\eqalign{ &1 + \mu_1^{'}(Z) t + \frac{1}{2!} \mu_2^{'}(Z) t^2 + \frac{1}{3!} \mu_3^{'}(Z) t^3 + \frac{1}{4!} \mu_4^{'}(Z) t^4 +O(t^5) \cr &= \phi_Z(t) \cr &= \phi_{\alpha X}(t) \phi_{\beta Y}(t) \cr &= \phi_X(\alpha t) \phi_Y(\beta t) \cr &= (1 + \mu_1^{'}(X) \alpha t + \frac{1}{2!} \mu_2^{'}(X) \alpha^2 t^2 + \cdots) (1 + \mu_1^{'}(Y) \beta t + \frac{1}{2!} \mu_2^{'}(Y) \beta^2 t^2 + \cdots). } $$

To obtain the desired non-central moments, multiply the latter power series through fourth order in $t$ and equate the result term-by-term with the terms in $\phi_Z(t)$.

Answer 2

I think that the following updating formula works for the third moment, although I'd be glad to have someone check it:

$M_{3,n} = (1-\alpha)M_{3,n-1} + \alpha \Big[ x_n(x_n-\mu_n)(x_n-2\mu_n) - x_n\mu_{n-1}(\mu_{n-1}-2\mu_n) - \dots$ $\dots - \mu_{n-1}(\mu_n-\mu_{n-1})^2 - 3(x_n-\mu_n) \sigma_{n-1}^2 \Big]$

Updating formula for the kurtosis still open...