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Average relative accuracy indices
In FoReco: Forecast Reconciliation
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
Let
$$
\hat{e}{i,j,t}^{[k],h} = y{i,t+h}^{[k]} - \hat{y}_{i,j,t}^{[k],h}, \quad
\begin{array}{l}
i=1,\ldots,n, \
j=0,\ldots,J,
\end{array} \; t=1,\ldots, q , \;
\begin{array}{l}
k \in {\cal K}, \
h=1,\ldots,h_k ,
\end{array}
$$
be the forecast error, where $y$ and $\hat{y}$ are the actual and the forecasted values, respectively, suffix $i$ denotes the variable of interest, $j$ is the forecasting technique, where $j=0$ is the benchmark forecasting procedure, $t$ is the forecast origin, ${\cal K}$ is the set of the time frequencies at which the series is observed, and $h$ is the forecast horizon, whose lead time depends on the time frequency $k$.
Denote by A${i,j}^{[k],h}$ the forecasting accuracy of the technique $j$, computed across $q$ forecast origins, for the $h$-step-ahead forecasts of the variable $i$ at the temporal aggregation level $k$. For example, A${i,j}^{[k],h} = MSE_{i,j}^{[k],h}$ (type="mse" in score_index() function, default), otherwise we might have A${i,j}^{[k],h} = MAE{i,j}^{[k],h}$ (type="mae" in score_index() function) or A${i,j}^{[k],h} = RMSE{i,j}^{[k],h}$ (type="rmse" in score_index() function), where
$$
\begin{array}{rcl}
MSE_{i,j}^{[k],h} & = & \displaystyle\frac{1}{q}\displaystyle\sum_{t=1}^{q} \left(\hat{e}{i,j}^{[k],h}\right)^2 \
MAE{i,j}^{[k],h} & = & \displaystyle\frac{1}{q}\displaystyle\sum_{t=1}^{q} \left|\hat{e}{i,j}^{[k],h}\right| \
RMSE{i,j}^{[k],h} & = & \sqrt{\displaystyle\frac{1}{q}\displaystyle\sum_{t=1}^{q} \left(\hat{e}_{i,j}^{[k],h}\right)^2}
\end{array}
$$
\noindent In any case, we consider the relative version of the accuracy index $A_{i,j}^{[k],h}$, given by:
$$
r_{i,j}^{[k],h} = \displaystyle\frac{A_{i,j}^{[k],h}}{A_{i,0}^{[k],h}} ,
\quad i=1,\ldots,n, \quad j=0,\ldots, J, \quad k \in {\cal K}, \quad h=1, \ldots, h_k,
$$
and use it to compute the Average relative accuracy index of the forecasting procedure $j$, for given $k$ and $h$, through the geometric mean:
$$
\text{AvgRelA}{j}^{[k],h} = \left(\displaystyle\prod{i=1}^{n} r_{i,j}^{[k],h} \right)^{\frac{1}{n}} , \quad j=0,\ldots,J .
$$
\noindent We may consider the following average relative accuracy indices for selected groups of variables/time frequencies and forecast horizons:
Average relative accuracy indices for a single variable at a given time frequency, for multiple forecast horizons
$$
\label{AvgRelAikh1h2}
\text{AvgRelA}{i,j}^{[k],q_1:q_2} = \left(\prod{h=q_1}^{q_2} r_{i,j}^{[k],h}\right)^{\frac{1}{q_2 - q_1 + 1}}, \;
\begin{array}{l}
i=1,\ldots,n, \
j=0,\ldots,J,
\end{array} \;
\begin{array}{l}
k \in {\cal K}, \
1 \le q_1 \le q_2 \le h_k
\end{array} .
%i=1,\ldots,n , \; j=0,\ldots, J, \; k \in {\cal K}, \; 1 \le q_1 \le q_2 \le h_k.
$$
Average relative accuracy indices for a group of variables (either all, or selected groups, e.g. a: uts, b: bts) at a given time frequency, either for a single forecast horizon or across them
$$
\begin{array}{rcll}
\text{AvgRelA}^{[k],h}j & = &
\left(\displaystyle\prod{i=1}^{n} r_{i,j}^{[k],h}\right)^{\frac{1}{n}}, &
j=0,\ldots,J , \; k \in {\cal K}, \; h=1,\ldots,h_k \[.5cm]
\text{AvgRelA}^{[k],h}{a,j} & = & \left(\displaystyle\prod{i=1}^{n_a} r_{i,j}^{[k],h}\right)^{\frac{1}{n_a}}, &
j=0,\ldots, J, \; k \in {\cal K} \[.15cm]
\text{AvgRelA}^{[k],h}{b,j} & = & \left(\displaystyle\prod{i=n_a+1}^{n} r_{i,j}^{[k],h}\right)^{\frac{1}{n_b}}, &
j=0,\ldots, J, \; k \in {\cal K} \[.15cm]
\text{AvgRelA}^{[k]}j & = & \left(\displaystyle\prod{i=1}^{n} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n h_k}}, &
\; j=0,\ldots,J, \; k \in {\cal K} \[.15cm]
\text{AvgRelA}^{[k]}{a,j} & = &\left(\displaystyle\prod{i=1}^{n_a} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_a h_k}}, &
j=0,\ldots, J, \; k \in {\cal K} \[.15cm]
\text{AvgRelA}^{[k]}{b,j} & = &\left(\displaystyle\prod{i=n_a+1}^{n} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_b h_k}}, &
j=0,\ldots, J, \; k \in {\cal K}
\end{array}
$$
Average relative accuracy indices for a single variable or for a group of variables (all, a: uts, b: bts), across all time frequencies and forecast horizons
$$
\begin{array}{rcll}
\text{AvgRelA}{i,j} & = & \left(\displaystyle\prod{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{k^+m}}, &
\begin{array}{l}
i=1,\ldots,n \ j=0,\ldots,J
\end{array} \[.15cm]
\text{AvgRelA}j & = & \left(\displaystyle\prod{i=1}^{n} \prod_{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n(k^+m)}} , & j=0,\ldots,J \[.15cm]
\text{AvgRelA}{a,j} & = & \left(\displaystyle\prod{i=1}^{n_a} \prod_{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_a(k^+m)}} , & j=0,\ldots,J \[.15cm]
\text{AvgRelA}{b,j} & = & \left(\displaystyle\prod{i=n_a+1}^{n} \prod_{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_b(k^+m)}} , & j=0,\ldots,J
\end{array}
$$
The score_index() function outputs
The score_index() function returns a summary table called Avg$_$mat (if compact option is TRUE, default), otherwise it returns a list of four tables. Fixed method $j$, we have:
- Avg$_$mat (default output)
df <- cbind(all = c("$\\textbf{all}$","$\\text{AvgRelA}^{[m]}_j$","$\\vdots$","$\\text{AvgRelA}^{[k]}_j$","$\\vdots$",
"$\\text{AvgRelA}^{[1]}_j$","$\\vdots$","$\\text{AvgRelA}_j$"),
uts = c("$\\textbf{uts}$","$\\text{AvgRelA}^{[m]}_{a,j}$","$\\vdots$","$\\text{AvgRelA}^{[k]}_{a,j}$","$\\vdots$",
"$\\text{AvgRelA}^{[1]}_{a,j}$","$\\vdots$","$\\text{AvgRelA}_{a,j}$"),
bts = c("$\\textbf{bts}$","$\\text{AvgRelA}^{[m]}_{b,j}$","$\\vdots$","$\\text{AvgRelA}^{[k]}_{b,j}$","$\\vdots$",
"$\\text{AvgRelA}^{[1]}_{b,j}$","$\\vdots$","$\\text{AvgRelA}_{b,j}$"))
rownames(df) <- c("","$\\textbf{m}$","$\\vdots$","$\\textbf{k}$","$\\vdots$","$\\textbf{1}$","$\\vdots$","$\\textbf{all}$")
knitr::kable(df,align='cccc',escape = F, col.names = rep("",3))
- Rel$_$mat (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1}$","$\\text{RelA}^{[m],1}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[m],1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[m],1}_{n,j}$"),
col2 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col3 = c("", "$\\textbf{1}$","$\\text{RelA}^{[k],1}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[k],1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],1}_{n,j}$"),
col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col5 = c("", "$\\mathbf{h_k}$","$\\text{RelA}^{[k],h_k}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[k],h_k}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],h_k}_{n,j}$"),
col6 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col7 = c("", "$\\textbf{1}$","$\\text{RelA}^{[1],1}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[1],1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],1}_{n,j}$"),
col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col9 = c("", "$\\mathbf{m}$","$\\text{RelA}^{[1],m}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[1],m}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],m}_{n,j}$"))
rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{1}$","$\\vdots$","$\\textbf{i}$","$\\vdots$","$\\textbf{n}$")
knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
- Rel$_$mat$_$cum (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1:1}$","$\\text{RelA}^{[m],1:1}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[m],1:1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[m],1:1}_{n,j}$"),
col2 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col3 = c("", "$\\textbf{1:1}$","$\\text{RelA}^{[k],1:1}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[k],1:1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],1:1}_{n,j}$"),
col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col5 = c("", "$\\mathbf{1:h_k}$","$\\text{RelA}^{[k],1:h_k}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[k],1:h_k}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],1:h_k}_{n,j}$"),
col6 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col7 = c("", "$\\textbf{1:1}$","$\\text{RelA}^{[1],1:1}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[1],1:1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],1:1}_{n,j}$"),
col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col9 = c("", "$\\mathbf{1:m}$","$\\text{RelA}^{[1],1:m}_{1,j}$","$\\vdots$",
"$\\text{RelA}^{[1],1:m}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],1:m}_{n,j}$"))
rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{1}$","$\\vdots$","$\\textbf{i}$","$\\vdots$","$\\textbf{n}$")
knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
- AvgRelIND$_$ik (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\text{AvgRelA}^{[m]}_{1,j}$","$\\vdots$",
"$\\text{AvgRelA}^{[m]}_{i,j}$","$\\vdots$","$\\text{AvgRelA}^{[m]}_{n,j}$"),
col2 = c("$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col3 = c("$\\textbf{k}$", "$\\text{AvgRelA}^{[k]}_{1,j}$","$\\vdots$",
"$\\text{AvgRelA}^{[k]}_{i,j}$","$\\vdots$","$\\text{AvgRelA}^{[k]}_{n,j}$"),
col4 = c("$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col5 = c("$\\textbf{1}$","$\\text{AvgRelA}^{[1]}_{1,j}$","$\\vdots$",
"$\\text{AvgRelA}^{[1]}_{i,j}$","$\\vdots$","$\\text{AvgRelA}^{[1]}_{n,j}$"),
col6 = c("$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"),
col7 = c("$\\mathbf{all}$","$\\text{AvgRelA}_{1,j}$","$\\vdots$",
"$\\text{AvgRelA}_{i,j}$","$\\vdots$","$\\text{AvgRelA}_{n,j}$"))
rownames(df) <- c("","$\\textbf{1}$","$\\vdots$","$\\textbf{i}$","$\\vdots$","$\\textbf{n}$")
knitr::kable(df,align='ccccccc',escape = F, col.names = rep("",7))
- Avg$_$k (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1}$","$\\text{AvgRelA}^{[m],1}_{j}$",
"$\\text{AvgRelA}^{[m],1}_{a,j}$","$\\text{AvgRelA}^{[m],1}_{b,j}$"),
col2 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col3 = c("", "$\\textbf{1}$","$\\text{AvgRelA}^{[k],1}_{j}$",
"$\\text{AvgRelA}^{[k],1}_{a,j}$","$\\text{AvgRelA}^{[k],1}_{b,j}$"),
col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col5 = c("", "$\\mathbf{h_k}$","$\\text{AvgRelA}^{[k],h_k}_{j}$",
"$\\text{AvgRelA}^{[k],h_k}_{a,j}$","$\\text{AvgRelA}^{[k],h_k}_{b,j}$"),
col6 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col7 = c("", "$\\textbf{1}$","$\\text{AvgRelA}^{[1],1}_{j}$",
"$\\text{AvgRelA}^{[1],1}_{a,j}$","$\\text{AvgRelA}^{[1],1}_{b,j}$"),
col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col9 = c("", "$\\mathbf{m}$","$\\text{AvgRelA}^{[1],m}_{j}$",
"$\\text{AvgRelA}^{[1],m}_{a,j}$","$\\text{AvgRelA}^{[1],m}_{b,j}$"))
rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{all}$","$\\textbf{a}$","$\\textbf{b}$")
knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
- Avg$_$k$_$cum (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1:1}$","$\\text{AvgRelA}^{[m],1:1}_{j}$",
"$\\text{AvgRelA}^{[m],1:1}_{a,j}$","$\\text{AvgRelA}^{[m],1:1}_{b,j}$"),
col2 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col3 = c("", "$\\textbf{1:1}$","$\\text{AvgRelA}^{[k],1:1}_{j}$",
"$\\text{AvgRelA}^{[k],1:1}_{a,j}$","$\\text{AvgRelA}^{[k],1:1}_{b,j}$"),
col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col5 = c("", "$\\mathbf{1:h_k}$","$\\text{AvgRelA}^{[k],1:h_k}_{j}$",
"$\\text{AvgRelA}^{[k],1:h_k}_{a,j}$","$\\text{AvgRelA}^{[k],1:h_k}_{b,j}$"),
col6 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col7 = c("", "$\\textbf{1:1}$","$\\text{AvgRelA}^{[1],1:1}_{j}$",
"$\\text{AvgRelA}^{[1],1:1}_{a,j}$","$\\text{AvgRelA}^{[1],1:1}_{b,j}$"),
col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"),
col9 = c("", "$\\mathbf{1:m}$","$\\text{AvgRelA}^{[1],1:m}_{j}$",
"$\\text{AvgRelA}^{[1],1:m}_{a,j}$","$\\text{AvgRelA}^{[1],1:m}_{b,j}$"))
rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{all}$","$\\textbf{a}$","$\\textbf{b}$")
knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
Examples
library(FoReco)
data(FoReco_data)
# Cross-temporal framework
oct_recf <- octrec(FoReco_data$base, m = 12, C = FoReco_data$C,
comb = "bdshr", res = FoReco_data$res)$recf
oct_score <- score_index(recf = oct_recf,
base = FoReco_data$base,
test = FoReco_data$test, m = 12, nb = 5)
# Cross-sectional framework#'
# monthly base forecasts
mbase <- FoReco2matrix(FoReco_data$base, m = 12)$k1
# monthly test set
mtest <- FoReco2matrix(FoReco_data$test, m = 12)$k1
# monthly residuals
mres <- FoReco2matrix(FoReco_data$res, m = 12)$k1
# monthly reconciled forecasts
mrecf <- htsrec(mbase, C = FoReco_data$C, comb = "shr", res = mres)$recf
# score
hts_score <- score_index(recf = mrecf, base = mbase, test = mtest, nb = 5)
# Temporal framework
data(FoReco_data)
# top ts base forecasts ([lowest_freq' ... highest_freq']')
topbase <- FoReco_data$base[1, ]
# top ts residuals ([lowest_freq' ... highest_freq']')
topres <- FoReco_data$res[1, ]
# top ts test ([lowest_freq' ... highest_freq']')
toptest <- FoReco_data$test[1, ]
# top ts recf ([lowest_freq' ... highest_freq']')
toprecf <- thfrec(topbase, m = 12, comb = "acov", res = topres)$recf
# score
thf_score <- score_index(recf = toprecf, base = topbase, test = toptest, m = 12)
References
Di Fonzo, T., Girolimetto, D. (2021), Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives, International Journal of Forecasting, in press.
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FoReco documentation built on May 31, 2023, 5:17 p.m.
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
Let $$ \hat{e}{i,j,t}^{[k],h} = y{i,t+h}^{[k]} - \hat{y}_{i,j,t}^{[k],h}, \quad \begin{array}{l} i=1,\ldots,n, \ j=0,\ldots,J, \end{array} \; t=1,\ldots, q , \; \begin{array}{l} k \in {\cal K}, \ h=1,\ldots,h_k , \end{array} $$ be the forecast error, where $y$ and $\hat{y}$ are the actual and the forecasted values, respectively, suffix $i$ denotes the variable of interest, $j$ is the forecasting technique, where $j=0$ is the benchmark forecasting procedure, $t$ is the forecast origin, ${\cal K}$ is the set of the time frequencies at which the series is observed, and $h$ is the forecast horizon, whose lead time depends on the time frequency $k$.
Denote by A${i,j}^{[k],h}$ the forecasting accuracy of the technique $j$, computed across $q$ forecast origins, for the $h$-step-ahead forecasts of the variable $i$ at the temporal aggregation level $k$. For example, A${i,j}^{[k],h} = MSE_{i,j}^{[k],h}$ (type="mse" in score_index() function, default), otherwise we might have A${i,j}^{[k],h} = MAE{i,j}^{[k],h}$ (type="mae" in score_index() function) or A${i,j}^{[k],h} = RMSE{i,j}^{[k],h}$ (type="rmse" in score_index() function), where
$$
\begin{array}{rcl}
MSE_{i,j}^{[k],h} & = & \displaystyle\frac{1}{q}\displaystyle\sum_{t=1}^{q} \left(\hat{e}{i,j}^{[k],h}\right)^2 \
MAE{i,j}^{[k],h} & = & \displaystyle\frac{1}{q}\displaystyle\sum_{t=1}^{q} \left|\hat{e}{i,j}^{[k],h}\right| \
RMSE{i,j}^{[k],h} & = & \sqrt{\displaystyle\frac{1}{q}\displaystyle\sum_{t=1}^{q} \left(\hat{e}_{i,j}^{[k],h}\right)^2}
\end{array}
$$
\noindent In any case, we consider the relative version of the accuracy index $A_{i,j}^{[k],h}$, given by: $$ r_{i,j}^{[k],h} = \displaystyle\frac{A_{i,j}^{[k],h}}{A_{i,0}^{[k],h}} , \quad i=1,\ldots,n, \quad j=0,\ldots, J, \quad k \in {\cal K}, \quad h=1, \ldots, h_k, $$ and use it to compute the Average relative accuracy index of the forecasting procedure $j$, for given $k$ and $h$, through the geometric mean: $$ \text{AvgRelA}{j}^{[k],h} = \left(\displaystyle\prod{i=1}^{n} r_{i,j}^{[k],h} \right)^{\frac{1}{n}} , \quad j=0,\ldots,J . $$
\noindent We may consider the following average relative accuracy indices for selected groups of variables/time frequencies and forecast horizons:
Average relative accuracy indices for a single variable at a given time frequency, for multiple forecast horizons $$ \label{AvgRelAikh1h2} \text{AvgRelA}{i,j}^{[k],q_1:q_2} = \left(\prod{h=q_1}^{q_2} r_{i,j}^{[k],h}\right)^{\frac{1}{q_2 - q_1 + 1}}, \; \begin{array}{l} i=1,\ldots,n, \ j=0,\ldots,J, \end{array} \; \begin{array}{l} k \in {\cal K}, \ 1 \le q_1 \le q_2 \le h_k \end{array} . %i=1,\ldots,n , \; j=0,\ldots, J, \; k \in {\cal K}, \; 1 \le q_1 \le q_2 \le h_k. $$ Average relative accuracy indices for a group of variables (either all, or selected groups, e.g. a: uts, b: bts) at a given time frequency, either for a single forecast horizon or across them $$ \begin{array}{rcll} \text{AvgRelA}^{[k],h}j & = & \left(\displaystyle\prod{i=1}^{n} r_{i,j}^{[k],h}\right)^{\frac{1}{n}}, & j=0,\ldots,J , \; k \in {\cal K}, \; h=1,\ldots,h_k \[.5cm] \text{AvgRelA}^{[k],h}{a,j} & = & \left(\displaystyle\prod{i=1}^{n_a} r_{i,j}^{[k],h}\right)^{\frac{1}{n_a}}, & j=0,\ldots, J, \; k \in {\cal K} \[.15cm] \text{AvgRelA}^{[k],h}{b,j} & = & \left(\displaystyle\prod{i=n_a+1}^{n} r_{i,j}^{[k],h}\right)^{\frac{1}{n_b}}, & j=0,\ldots, J, \; k \in {\cal K} \[.15cm] \text{AvgRelA}^{[k]}j & = & \left(\displaystyle\prod{i=1}^{n} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n h_k}}, & \; j=0,\ldots,J, \; k \in {\cal K} \[.15cm] \text{AvgRelA}^{[k]}{a,j} & = &\left(\displaystyle\prod{i=1}^{n_a} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_a h_k}}, & j=0,\ldots, J, \; k \in {\cal K} \[.15cm] \text{AvgRelA}^{[k]}{b,j} & = &\left(\displaystyle\prod{i=n_a+1}^{n} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_b h_k}}, & j=0,\ldots, J, \; k \in {\cal K} \end{array} $$
Average relative accuracy indices for a single variable or for a group of variables (all, a: uts, b: bts), across all time frequencies and forecast horizons $$ \begin{array}{rcll} \text{AvgRelA}{i,j} & = & \left(\displaystyle\prod{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{k^+m}}, & \begin{array}{l} i=1,\ldots,n \ j=0,\ldots,J \end{array} \[.15cm] \text{AvgRelA}j & = & \left(\displaystyle\prod{i=1}^{n} \prod_{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n(k^+m)}} , & j=0,\ldots,J \[.15cm] \text{AvgRelA}{a,j} & = & \left(\displaystyle\prod{i=1}^{n_a} \prod_{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_a(k^+m)}} , & j=0,\ldots,J \[.15cm] \text{AvgRelA}{b,j} & = & \left(\displaystyle\prod{i=n_a+1}^{n} \prod_{k \in {\cal K}} \prod_{h=1}^{h_k} r_{i,j}^{[k],h}\right)^{\frac{1}{n_b(k^+m)}} , & j=0,\ldots,J \end{array} $$
The score_index() function outputs
The score_index() function returns a summary table called Avg$_$mat (if compact option is TRUE, default), otherwise it returns a list of four tables. Fixed method $j$, we have:
- Avg$_$mat (default output)
df <- cbind(all = c("$\\textbf{all}$","$\\text{AvgRelA}^{[m]}_j$","$\\vdots$","$\\text{AvgRelA}^{[k]}_j$","$\\vdots$", "$\\text{AvgRelA}^{[1]}_j$","$\\vdots$","$\\text{AvgRelA}_j$"), uts = c("$\\textbf{uts}$","$\\text{AvgRelA}^{[m]}_{a,j}$","$\\vdots$","$\\text{AvgRelA}^{[k]}_{a,j}$","$\\vdots$", "$\\text{AvgRelA}^{[1]}_{a,j}$","$\\vdots$","$\\text{AvgRelA}_{a,j}$"), bts = c("$\\textbf{bts}$","$\\text{AvgRelA}^{[m]}_{b,j}$","$\\vdots$","$\\text{AvgRelA}^{[k]}_{b,j}$","$\\vdots$", "$\\text{AvgRelA}^{[1]}_{b,j}$","$\\vdots$","$\\text{AvgRelA}_{b,j}$")) rownames(df) <- c("","$\\textbf{m}$","$\\vdots$","$\\textbf{k}$","$\\vdots$","$\\textbf{1}$","$\\vdots$","$\\textbf{all}$") knitr::kable(df,align='cccc',escape = F, col.names = rep("",3))
- Rel$_$mat (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1}$","$\\text{RelA}^{[m],1}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[m],1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[m],1}_{n,j}$"), col2 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col3 = c("", "$\\textbf{1}$","$\\text{RelA}^{[k],1}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[k],1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],1}_{n,j}$"), col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col5 = c("", "$\\mathbf{h_k}$","$\\text{RelA}^{[k],h_k}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[k],h_k}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],h_k}_{n,j}$"), col6 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col7 = c("", "$\\textbf{1}$","$\\text{RelA}^{[1],1}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[1],1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],1}_{n,j}$"), col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col9 = c("", "$\\mathbf{m}$","$\\text{RelA}^{[1],m}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[1],m}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],m}_{n,j}$")) rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{1}$","$\\vdots$","$\\textbf{i}$","$\\vdots$","$\\textbf{n}$") knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
- Rel$_$mat$_$cum (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1:1}$","$\\text{RelA}^{[m],1:1}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[m],1:1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[m],1:1}_{n,j}$"), col2 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col3 = c("", "$\\textbf{1:1}$","$\\text{RelA}^{[k],1:1}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[k],1:1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],1:1}_{n,j}$"), col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col5 = c("", "$\\mathbf{1:h_k}$","$\\text{RelA}^{[k],1:h_k}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[k],1:h_k}_{i,j}$","$\\vdots$","$\\text{RelA}^{[k],1:h_k}_{n,j}$"), col6 = c("$\\dots$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col7 = c("", "$\\textbf{1:1}$","$\\text{RelA}^{[1],1:1}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[1],1:1}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],1:1}_{n,j}$"), col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col9 = c("", "$\\mathbf{1:m}$","$\\text{RelA}^{[1],1:m}_{1,j}$","$\\vdots$", "$\\text{RelA}^{[1],1:m}_{i,j}$","$\\vdots$","$\\text{RelA}^{[1],1:m}_{n,j}$")) rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{1}$","$\\vdots$","$\\textbf{i}$","$\\vdots$","$\\textbf{n}$") knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
- AvgRelIND$_$ik (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\text{AvgRelA}^{[m]}_{1,j}$","$\\vdots$", "$\\text{AvgRelA}^{[m]}_{i,j}$","$\\vdots$","$\\text{AvgRelA}^{[m]}_{n,j}$"), col2 = c("$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col3 = c("$\\textbf{k}$", "$\\text{AvgRelA}^{[k]}_{1,j}$","$\\vdots$", "$\\text{AvgRelA}^{[k]}_{i,j}$","$\\vdots$","$\\text{AvgRelA}^{[k]}_{n,j}$"), col4 = c("$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col5 = c("$\\textbf{1}$","$\\text{AvgRelA}^{[1]}_{1,j}$","$\\vdots$", "$\\text{AvgRelA}^{[1]}_{i,j}$","$\\vdots$","$\\text{AvgRelA}^{[1]}_{n,j}$"), col6 = c("$\\dots$","$\\dots$","","$\\dots$","","$\\dots$"), col7 = c("$\\mathbf{all}$","$\\text{AvgRelA}_{1,j}$","$\\vdots$", "$\\text{AvgRelA}_{i,j}$","$\\vdots$","$\\text{AvgRelA}_{n,j}$")) rownames(df) <- c("","$\\textbf{1}$","$\\vdots$","$\\textbf{i}$","$\\vdots$","$\\textbf{n}$") knitr::kable(df,align='ccccccc',escape = F, col.names = rep("",7))
- Avg$_$k (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1}$","$\\text{AvgRelA}^{[m],1}_{j}$", "$\\text{AvgRelA}^{[m],1}_{a,j}$","$\\text{AvgRelA}^{[m],1}_{b,j}$"), col2 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col3 = c("", "$\\textbf{1}$","$\\text{AvgRelA}^{[k],1}_{j}$", "$\\text{AvgRelA}^{[k],1}_{a,j}$","$\\text{AvgRelA}^{[k],1}_{b,j}$"), col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col5 = c("", "$\\mathbf{h_k}$","$\\text{AvgRelA}^{[k],h_k}_{j}$", "$\\text{AvgRelA}^{[k],h_k}_{a,j}$","$\\text{AvgRelA}^{[k],h_k}_{b,j}$"), col6 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col7 = c("", "$\\textbf{1}$","$\\text{AvgRelA}^{[1],1}_{j}$", "$\\text{AvgRelA}^{[1],1}_{a,j}$","$\\text{AvgRelA}^{[1],1}_{b,j}$"), col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col9 = c("", "$\\mathbf{m}$","$\\text{AvgRelA}^{[1],m}_{j}$", "$\\text{AvgRelA}^{[1],m}_{a,j}$","$\\text{AvgRelA}^{[1],m}_{b,j}$")) rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{all}$","$\\textbf{a}$","$\\textbf{b}$") knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
- Avg$_$k$_$cum (if compact = FALSE)
df <- cbind(col1 = c("$\\textbf{m}$","$\\textbf{1:1}$","$\\text{AvgRelA}^{[m],1:1}_{j}$", "$\\text{AvgRelA}^{[m],1:1}_{a,j}$","$\\text{AvgRelA}^{[m],1:1}_{b,j}$"), col2 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col3 = c("", "$\\textbf{1:1}$","$\\text{AvgRelA}^{[k],1:1}_{j}$", "$\\text{AvgRelA}^{[k],1:1}_{a,j}$","$\\text{AvgRelA}^{[k],1:1}_{b,j}$"), col4 = c("$\\textbf{k}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col5 = c("", "$\\mathbf{1:h_k}$","$\\text{AvgRelA}^{[k],1:h_k}_{j}$", "$\\text{AvgRelA}^{[k],1:h_k}_{a,j}$","$\\text{AvgRelA}^{[k],1:h_k}_{b,j}$"), col6 = c("$\\dots$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col7 = c("", "$\\textbf{1:1}$","$\\text{AvgRelA}^{[1],1:1}_{j}$", "$\\text{AvgRelA}^{[1],1:1}_{a,j}$","$\\text{AvgRelA}^{[1],1:1}_{b,j}$"), col8 = c("$\\textbf{1}$","$\\dots$","$\\dots$","$\\dots$","$\\dots$"), col9 = c("", "$\\mathbf{1:m}$","$\\text{AvgRelA}^{[1],1:m}_{j}$", "$\\text{AvgRelA}^{[1],1:m}_{a,j}$","$\\text{AvgRelA}^{[1],1:m}_{b,j}$")) rownames(df) <- c("${\\cal K}$","$\\textbf{h}$","$\\textbf{all}$","$\\textbf{a}$","$\\textbf{b}$") knitr::kable(df,align='ccccccccc',escape = F, col.names = rep("",9))
Examples
library(FoReco) data(FoReco_data) # Cross-temporal framework oct_recf <- octrec(FoReco_data$base, m = 12, C = FoReco_data$C, comb = "bdshr", res = FoReco_data$res)$recf oct_score <- score_index(recf = oct_recf, base = FoReco_data$base, test = FoReco_data$test, m = 12, nb = 5) # Cross-sectional framework#' # monthly base forecasts mbase <- FoReco2matrix(FoReco_data$base, m = 12)$k1 # monthly test set mtest <- FoReco2matrix(FoReco_data$test, m = 12)$k1 # monthly residuals mres <- FoReco2matrix(FoReco_data$res, m = 12)$k1 # monthly reconciled forecasts mrecf <- htsrec(mbase, C = FoReco_data$C, comb = "shr", res = mres)$recf # score hts_score <- score_index(recf = mrecf, base = mbase, test = mtest, nb = 5) # Temporal framework data(FoReco_data) # top ts base forecasts ([lowest_freq' ... highest_freq']') topbase <- FoReco_data$base[1, ] # top ts residuals ([lowest_freq' ... highest_freq']') topres <- FoReco_data$res[1, ] # top ts test ([lowest_freq' ... highest_freq']') toptest <- FoReco_data$test[1, ] # top ts recf ([lowest_freq' ... highest_freq']') toprecf <- thfrec(topbase, m = 12, comb = "acov", res = topres)$recf # score thf_score <- score_index(recf = toprecf, base = topbase, test = toptest, m = 12)
References
Di Fonzo, T., Girolimetto, D. (2021), Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives, International Journal of Forecasting, in press.
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