Definition:Probability Limit
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 $\hat z_t$ 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 $l$.
A probability limit, for a given probability $p$, is the deviation $\Delta_p$, either positive or negative, from $\map {\hat z_t} l$ such that the probability that the actual value lies within $\Delta_p$ of $\map {\hat z_t} l$ is greater than $p$.
That is, the actual value of the time series at $l$, when it occurs, will be within those probability limits within that stated probability $p$.
Typical values for $p$ are whatever may be convenient, for example $50 \%$ or $95 \%$.
Upper
Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$.
The upper probability limit of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l + \Delta_p$.
Lower
Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$.
The lower probability limit of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l - \Delta_p$.
Sources
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):
- $1$: Introduction:
- $1.1$ Four Important Practical Problems:
- $1.1.1$ Forecasting Time Series
- $1.1$ Four Important Practical Problems:
- $1$: Introduction: