Definition:Forecast Function

Definition
Let $T$ be a time series.

Let $S$ denote the range of $T$.

Let $L$ denote the set of lead times of $T$.

Let $t$ denote the origin of $T$.

A forecast function $\hat z_t : L \to S$ is a function which provides forecasts of the future values of $T$ at the lead times $L$.

The objective is to make the mean squares of the deviations $z_{t + l} - \map {\hat z_t} l$ as small as possible for each lead time $l$.

In the above:
 * $\map {\hat z_t} l$ denotes the forecast value of the observation at the timestamp of lead time $l$
 * $z_{t + l}$ denotes the actual value of the observation at the timestamp of $l$.