i'm trying to understand time series and lag operators. However there is a lot of stuff about it on the internet (and too on stack overflow, but not what I'm looking for), I cannot understand, how to compute Lag (sometimes called Backshift) operator from a given time series. I know, how it works ($BX_{t}=X_{t-1}$). Imagine, that we have some time series T:
$$
T = (x_{1},x_{2},x_{3},...,x_{n-1},x_{n})
$$
in time $$(t_{1},t_{2},t_{3},...,t_{n-1},t_{n})$$
Now for example you have ARMA model, where you use equation
$$
\phi(B)X_{t}=θ_{0}+θ(B)a_{t}
$$
where $\phi(B)=1-\phi_{1}B-...-\phi_{p}B^{p}$ and $θ(B)=1-θ_{1}B-...-θ_{q}B^{q}$ are polynomials in B with degrees p and q.
Now what I want is to compute all B values (or better polynomials $\phi(B)$ and $θ(B)$) for equation above (for simplicity imagine, that we have ARMA(1,2)). I have seen some equations and computing using lag operator, but values were already given (see here page 4). I cannot find how to find out values for lag.
I was thinking, that I can somehow use linear regression, but I'm not sure. My second idea was to compute it as here:
If $BX_{t}=X_{t-1}$, than I can compute B as B = $X_{t-1}/X_{t}$, but it cannot be same for all times $T_{s}$ same right?
For example: $B_{1} = x_{1}/x_{2}$, $B_{2}=x_{2}/x_{3}$,... but $B_{1}\ne B_{2}$, so which one to use? Or should I use both?
I was thinking, that B can be a polynom itself, but how then compute polynomials $\phi(B)$ and $θ(B)$?
Maybe I all understand it wrong.
Thank you