What does mean of a mean of a function represent?

Question

Let $y_k = f(x,t_k)$ be a random process which describes the evolution of $y_k$ as a function of random variable $x_k$ and time $t_k$. I tried to fit the following low pass filter to estimate $y_k$: \begin{equation} y_k = \alpha y_{k-1} + (1-\alpha)z_k \end{equation} where, $\alpha = \exp(-t_k/\tau)$ and $\tau$ is a time constant. However, I tried many values of $\alpha$ and my estimate is still noisy. So, I did the following: \begin{equation} y1_n = \frac{1}{n}\sum\limits_{k=1}^ny_k \end{equation} for $n = 1,2,\cdots, N$. Then I assume that $y1_n$ is an approximate of $y_k$. However, in some cases, even $y1_n$ is noisy. So, I did another averaging like this: \begin{equation} y2_m = \frac{1}{m}\sum\limits_{n=1}^my1_n \end{equation} $y2_m$ is smoother for most cases. But I don't know how to interpret this new average i.e. $y2_m$. Can anyone tell what happens when one continues to take average like this for long i.e. compute $y3$ from $y2$, compute $y4$ from $y3$ and so on. And what do these averages represent?

Answer 1

$\alpha$ should be a constant. You have made it a function of $t_k$. So, as $t_k$ increases, $\alpha\to0$, and $y_k\approx z_k$. So as time goes on, the filtered sequence $y_k$ gets closer to the input sequence $z_k$.

$y1$ is the cumulative mean. So it is $$a,\frac{a+b}2,\frac{a+b+c}3,\ldots$$ $y2$ is weighted towards the initial values.
$$a, \frac{a+\frac{a+b}2}2=\frac{3a+b}4,\frac{a+\frac{a+b}2+\frac{a+b+c}3}3=\frac{11a+5b+2c}{18},\ldots$$ Further $y3,y4$ are weighted more and more towards the initial value, and lose the evolution of $y$.