Existen fórmulas en línea bien conocidas para calcular promedios móviles ponderados exponencialmente y desviaciones estándar de un proceso . Por la media,
y para la varianza
desde el cual puede calcular la desviación estándar.
¿Existen fórmulas similares para el cálculo en línea de momentos exponenciales ponderados del tercer y cuarto centro? Mi intuición es que deberían tomar la forma
y
from which you could compute the skewness and the kurtosis but I've not been able to find simple, closed-form expression for the functions and .
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
where and are the exponential moving means of and . The asymmetry between and is illusory, and disappears when you notice that .
Formulas like this can be computed by writing the central moment as an expectation , where weights in the expectation are understood to be exponential, and using the fact that for any function we have
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.
I think that the following updating formula works for the third moment, although I'd be glad to have someone check it:
Updating formula for the kurtosis still open...
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