I'm a newbie to Signal Processing - and so I'm providing a "background" and a "question" section separately.
Background:
Suppose I have a linear filter (w.r.t. given as
$\theta_{t+1} = \theta_t + \lambda (x_t - \theta_t)$
In Financial Trading, such a filter is often called an exponential moving average, often applied to both prices (to get smoothened prices) and to square of price returns (to get "EWMA" volatility). By conventional definition:
$\lambda = \frac{2}{n+1}$, where $n$ is periods (example $20$-day EWMA volatility).
Usually, a lot of folks in Financial Trading define "lag" as the expected value of the filter, when $x_u$ is replace by $t-u$. For instance, the above filter can also be written as
$\theta_{t+1} = \lambda (x_t+(1-\lambda) x_{t-1}+ (1-\lambda)^2 x_{t-2}+ ...)$
To get the "lag" of this filter in Financial Trading, we write
$L(\theta) = \lambda (0+(1-\lambda) 1+ (1-\lambda)^2 2+ ...)$
which gives
$L(\theta) = (1-\lambda)/\lambda = \frac{n-1}{2}$.
Question:
Now suppose I have a filter that is non-linear in $\theta$. Example, suppose it is
$\theta_{t+1} = \theta_t + \lambda \frac{1}{2}(x_{t}^2 e^{2\theta_t}-1)$
I have no idea what would be the equivalent of a lag here. Some folks told me that the definition of lag in Financial Trading is equivalent to the group delay (or may be phase delay) in Signal Processing.
(1) I looked up the definition of group delay and phase delay (e.g. on Wiki), but I couldn't exactly relate them to the definition of lag in Financial Trading. For the linear example:
$\theta_{t+1} = \theta_t + \lambda (x_t - \theta_t)$
what would be the group delay (or may be phase delay if that's more equivalent)?
(2) For the non-linear example:
$\theta_{t+1} = \theta_t + \lambda \frac{1}{2}(x_{t}^2 e^{2\theta_t}-1)$
what would be the group delay / phase delay or the equivalent of lag as applied to the linear filters in Financial Trading?