= Linear innovation process
{title2=$\varepsilon_t=X_t-\operatorname{proj}_{\mathcal H_{t-1}}X_t$}
= Linear innovation
{synonym}
For a square-integrable <time series>, let $\mathcal H_{t-1}$ be the closed linear span of its past. The innovation $\varepsilon_t=X_t-\operatorname{proj}_{\mathcal H_{t-1}}X_t$ is orthogonal to every past linear observation. A causal invertible <autoregressive moving-average model> is driven by these innovations, which are <weak white noise> for a <weakly stationary process>.
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