The Time Series Forecasting Problem

Forecasting Problem

A time series forecasting problem can be formulated as the following.

Given a dataset $\mathcal D$, with

1. $y^{(i)}_t$, the sequential variable to be forecasted,
2. $x^{(i)}_t$, exogenous data for the time series data,
3. $u^{(i)}_t$, some features that can be obtained or planned in advance,

where ${}^{(i)}$ indicates the $i$th variable, ${}_ t$ denotes time. In a forecasting task, we use $y^{(i)} _ {t-K:t}$, $x^{(i) _ {t-K:t}}$, and $u^{(i)} _ {t-K:t+H}$, to forecast the future $y^{(i)} _ {t+1:t+H}$.

A model $f$ will use $x^{(i)} _ {t-K:t}$ and $u^{(i)} _ {t-K:t+H}$ to forecast $y^{(i)} _ {t+1:t+H}$.