Suppose that a time series $s_t$ it is known to be obtained via exponential smoothing of an underlying signal $x_t$, that is

$$ s_{0}= x_0 $$


$$ s_{t} = (1-\alpha)\,x_t+\alpha\,s_{t-1}. $$

I am wondering wether there exist procedures/filters to re-construct or approximate the sequence of the $x_t$ having observed the sequence of the $s_t$ and having an estimate of the smoothing parameter $\alpha$.


If you know $\alpha$ and have $s_1,\dots,s_n$, if

$$ s_{t} = (1-\alpha)\,x_t+\alpha\,s_{t-1} $$

then by simple arithmetic

$$ x_t = \frac{s_{t} - \alpha\,s_{t-1}}{(1-\alpha)} $$

So you can reconstruct it exactly, not just approximately.

Below you can see the code example.


n <- 250
x <- runif(n)

alpha <- 0.6
s <- numeric(n)
s[1] <- x[1]
for (t in 2:n)
  s[t] <- (1-alpha)*x[t] + alpha*s[t-1]

z <- numeric(n)
z[1] <- x[1] # since s[1] <- x[1]
for (t in n:2)
  z[t] <- (s[t] - s[t-1]*alpha)/(1-alpha)

all.equal(z, x)
## [1] TRUE

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