Definition:Deviation

Definition

Let $S$ be a set of observations.

Let $x \in S$.

The deviation of $x$ is the difference between $x$ and some other value, whose nature depends upon the context.

Let $x \in S$.

The deviation of $x$ from the mean is the difference between $x$ and the arithmetic mean $\bar x$ of $S$:

$x - \bar x$

Let $T$ be a time series.

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

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

Let $\hat z_t$ be a forecast function on $L$.

Let $\map {\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$.

Let $z_{t + l}$ denote the actual value of the observation at the timestamp of $l$.

The deviation (from forecast) is the difference between $\map {\hat z_t} l$ and $z_{t + l}$:

$\Delta_l := z_{t + l} - \map {\hat z_t} l$