What is the name of (1-a)*current + a*sample

Question

Say we take a sample every time unit, we have a calibrated value which needs tracking for drift, however every sample has a given variance causing jumping if we were to use it directly as the new calibrated value, therefore we use $(1-a)*\textrm{current} + a*\textrm{sample}$ with $a$ some suitable value between $0$ and $1$. What is this kind of method called?

Answer 1

It is called exponential smoothing. If sample is constant, then the value converges to this value exponentially.

If $u_n$ represents the 'raw' data, then $x_n$ is the 'exponentially smoothed' version in the following (with $\alpha \in (0,1]$):

$$x_{n+1} = (1-\alpha) x_n + \alpha u_n$$

The explicit solution is given by $$ x_n = (1-\alpha)^n x_0 + \sum_{k=0}^{n-1} (1-\alpha)^k \alpha u_{n-1-k}$$ If $u_n = \hat{u}$ for all $n$, then this gives $$x_n = (1-\alpha)^n x_0 + \hat{u} (1 - (1-\alpha)^n)$$ from which we see that $x_n \to \hat{u}$ with rate $(1-\alpha)$.