In coatless/ITS: A Tour of Time Series Analysis with R

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Definition: Autoregressive Process of Order p = 1

This process is generally denoted as AR(1) and is defined as: ${y_t} = {\phi 1}{y{t - 1}} + {w_t},$

where ${w_t}\mathop \sim \limits^{iid} WN\left( {0,\sigma _w^2} \right)$

If $\phi _1 = 1$, then the process is equivalent to a random walk.

The process can be simplified using backsubstitution to being:

white noise can actually be generalize

The process name of white noise has meaning in the notion of colors of noise. Specifically, the white noise is a process that mirrors white light's flat frequency spectrum. So, the process has equal frequencies in any interval of time.

Definition: White Noise

$w_t$ or $\varepsilon _t$ is a white noise process if $w_t$ are uncorrelated identically distributed random variables with $E\left[w_t\right] = 0$ and $Var\left[w_t\right] = \sigma ^2$, for all $t$. We can represent this algebraically as: $$y_t = w_t,$$ where ${w_t}\mathop \sim \limits^{id} WN\left( {0,\sigma _w^2} \right)$

Now, if the $w_t$ are Normally (Gaussian) distributed, then the process is known as a Gaussian White Noise e.g. ${w_t}\mathop \sim \limits^{iid} N\left( {0,{\sigma ^2}} \right)$

coatless/ITS documentation built on May 13, 2019, 8:45 p.m.