One-step and multi-step residuals

These functions can be used to arrange residuals to reconcile temporal or cross-temporal forecasts.

res2matrix takes as input a set of temporal and cross-temporal residuals and re-organizes them into a matrix where the rows correspond to different forecast horizons, capturing the temporal dimension. Meanwhile, the columns are ordered based on the specific arrangement as described in Di Fonzo and Girolimetto (2023).

arrange_hres takes as input a list of multi-step residuals and is designed to organize them in accordance with their time order (Girolimetto et al. 2023). When applied, this function ensures that the sequence of multi-step residuals aligns with the chronological order in which they occurred.

Usage

res2matrix(res, agg_order)

arrange_hres(list_res)

Arguments

res: A ($n \times N(k^\ast+m)$) numeric matrix (cross-temporal framework) or an ($N(k^\ast+m) \times 1$) numeric vector (temporal framework) representing the in-sample residuals at all the temporal frequencies ordered from the lowest frequency to the highest frequency (columns) for each variable (rows).
agg_order: Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation, $m$), or a vector representing a subset of $p$ factors of $m$.
list_res: A list of $H$ multi-step residuals. Each element in the list can be either a ($T \times 1$) vector (temporal framework) or a ($T \times n$) matrix (cross-temporal framework).

Value

res2matrix returns a ($N \times n(k^\ast + m)$) matrix, where $n = 1$

for the temporal framework.

arrange_hres returns a ($N(k^\ast+m) \times 1$) vector (temporal framework) or a ($n \times N(k^\ast+m)$) matrix (cross-temporal framework) of multi-step residuals.

Details

Let $Z_t$, $t=1,\dots,T$, be a univariate time series. We can define the multi-step residuals such us $$\widehat{\varepsilon}_{h,t} = Z_{t+h} - \widehat{Z}_{t+h|t} \qquad h \le t \le T-h$$ where $\widehat{Z}_{t+h|t}$ is the $h$-step fitted value, calculated as the $h$-step ahead forecast condition to the information up to time $t$. Given the list of errors at different steps $$\left([\widehat{\varepsilon}_{1,1}, \; \dots, \; \widehat{\varepsilon}_{1,T}], \dots, [\widehat{\varepsilon}_{H,1}, \; \dots, \; \widehat{\varepsilon}_{H,T}]\right),$$ arrange_hres returns a $T$-vector with the residuals, organized in the following way: $$[\varepsilon_{1,1} \; \varepsilon_{2,2} \; \dots \; \varepsilon_{H,H} \; \varepsilon_{1,H+1} \; \dots \; \varepsilon_{H,T-H}]'$$ A similar organisation can be apply to a multivariate time series.

References

Di Fonzo, T. and Girolimetto, D. (2023), Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives, International Journal of Forecasting, 39, 1, 39-57. doi:10.1016/j.ijforecast.2021.08.004

Girolimetto, D., Athanasopoulos, G., Di Fonzo, T. and Hyndman, R.J. (2024), Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues. International Journal of Forecasting, 40, 3, 1134-1151. doi:10.1016/j.ijforecast.2023.10.003

Usage

Arguments

Value

Details

References

See also