I am writing a computer program where the user has the choice of one of two sequences, $a_k$ or $b_k$. These sequences are defined as follows:
- $a_k = c k + \varepsilon_k$;
- $b_k = b_{k-1} + c + \varepsilon_k$
in both cases, $\varepsilon_k$ are i.i.d. zero-mean random variables and c is a fixed constant. That is, $a_k$ is an arithmetic series with noise superimposed, whereas $b_k$ is an AR(1) autoregressive process.
Is there a good adjective that distinguishes between the two cases, i.e. something to take the place of ????? in
make_series(growth_rate = $c$, noise_dist = $\mathcal{N}(0, 1)$, ????? = true)
It is not an ajective. They are instances of completely different classes of models:
$a_k$ is a linear trend with zero intercept.
$b_k = b_{k-1} + c + \epsilon_k = b_{k-2} + 2c + \epsilon_k + \epsilon_{k-1} = \\ =b_{k-3} + 3c + \sum_{j=0}^2\epsilon_{k-j} = b_{k-T} + Tc + \sum_{j=0}^{T-1}\epsilon_{k-j} \hspace{.2cm} \forall T \in \mathbb{N}$
I.e. $b_k$ is a random walk with drift c.