Beyond point forecasts: Uncertainty quantification in tensor extrapolation for relational time series data

Figures & data

Table 1 Tensor notation summary.

Symbol	Description
p	scalar (lowercase letter)
a	vector (boldface lowercase letter)
ai	element i of vector a
A	matrix (boldface capital letter)
${\mathbf{a}}_r$	rth column of matrix A
air	element ir of matrix A
$\underset{¯}{\mathbf{X}}$	higher-order tensor (underlined boldface capital letter)
xijk	element ijk of tensor $\underset{¯}{\mathbf{X}}$

Table

Input $\underset{¯}{\mathbf{X}}$, R, λ
Output $\underset{¯}{\widehat{\mathbf{X}}},{\underset{¯}{\widehat{\mathbf{X}}}}^{\downarrow },{\underset{¯}{\widehat{\mathbf{X}}}}^{\uparrow }$
1:	Preprocessing / transformation: ${\underset{¯}{\mathbf{X}}}_{\mathit{\text{trans}}}$ $\leftarrow$ $\underset{¯}{\mathbf{X}}$
2:	CP decomposition: A, B, C $\leftarrow$ ${\underset{¯}{\mathbf{X}}}_{\mathit{\text{trans}}}$, R
3:	Separate column vectors: ${\mathbf{c}}_1,{\mathbf{c}}_2,\dots ,{\mathbf{c}}_R$ $\leftarrow$ C
4:	for each $r\in 1,\dots ,R$ do
5:	Automatic exponential smoothing: ${\mathbf{c}}_r^{\mathit{\text{fit}}},{\widehat{\mathbf{c}}}_r$ $\leftarrow$ ${\mathbf{c}}_r$
6:	end for
7:	Merge column vectors: ${\mathbf{C}}^{\mathit{\text{fit}}}$ $\leftarrow$ ${\mathbf{c}}_1^{\mathit{\text{fit}}},{\mathbf{c}}_2^{\mathit{\text{fit}}},\dots ,{\mathbf{c}}_R^{\mathit{\text{fit}}}$; $\widehat{\mathbf{C}}$ $\leftarrow$ ${\widehat{\mathbf{c}}}_1,{\widehat{\mathbf{c}}}_2,\dots ,{\widehat{\mathbf{c}}}_R$
8:	Calculate tensors: ${\underset{¯}{\mathbf{X}}}_{\mathit{\text{trans}}}^{\mathit{\text{fit}}}$ $\leftarrow$ A, B, ${\mathbf{C}}^{\mathit{\text{fit}}}$; ${\underset{¯}{\widehat{\mathbf{X}}}}_{\mathit{\text{trans}}}$ $\leftarrow$ A, B, $\widehat{\mathbf{C}}$
9:	Back-transformation: ${\underset{¯}{\mathbf{X}}}^{\mathit{\text{fit}}}$ $\leftarrow$ ${\underset{¯}{\mathbf{X}}}_{\mathit{\text{trans}}}^{\mathit{\text{fit}}}$; $\underset{¯}{\widehat{\mathbf{X}}}$ $\leftarrow$ ${\underset{¯}{\widehat{\mathbf{X}}}}_{\mathit{\text{trans}}}$
10:	Calculate prediction errors based on residuals: $\underset{¯}{\mathbf{E}}$ $\leftarrow$ $\underset{¯}{\mathbf{X}},{\underset{¯}{\mathbf{X}}}^{\mathit{\text{fit}}}$
11:	Calculate interval limits: ${\underset{¯}{\widehat{\mathbf{X}}}}^{\downarrow },{\underset{¯}{\widehat{\mathbf{X}}}}^{\uparrow }$ $\leftarrow$ $\underset{¯}{\widehat{\mathbf{X}}},\underset{¯}{\mathbf{E}}$, λ

Table 2 Forecasting performance based on empirical data in terms of MAPE, DM, MSIS, and MSIW.

		Tensor extrapolation
Measure	ETSModel	R = 4	R = 5	R = 6	R = 7	R = 8
MAPE	272.7493	266.4299	266.7067	264.6828	195.7938	193.7586
80% pred. int.
DM	0.0603	–0.0017	–0.0043	–0.0077	–0.0207	–0.0116
MSIS (p = 5)	12.0713	11.5867	11.5985	11.5315	7.3062	7.2776
MSIS (p = 10)	21.9902	20.7811	20.8060	20.6866	11.8007	11.7394
MSIS (p = 40)	81.5042	75.9479	76.0508	75.6170	38.7677	38.5098
MSIW	2.1523	2.3922	2.3911	2.3764	2.8117	2.8159
95% pred. int.
DM	–0.0003	–0.0488	–0.0499	–0.0499	–0.0552	–0.0475
MSIS (p = 5)	11.7629	11.3958	11.3988	11.3254	6.7231	6.7076
MSIS (p = 10)	20.2342	19.1285	19.1363	19.0119	9.1407	9.1033
MSIS (p = 40)	71.0619	65.5247	65.5610	65.1310	23.6463	23.4776
MSIW	3.2916	3.6631	3.6614	3.6389	4.3054	4.3118

The estimation sample includes the years 2012–2015, the hold-out sample covers the year 2016. The results obtained from tensor extrapolation are listed with respect to the selected hyperparameter, the number of components R. The constant p affects the penalty in case the observation misses the prediction interval.

Fig. 1 Generation of synthetic data.

Table 3 Forecasting performance based on synthetic data in terms of MAPE, DM, MSIS, and MSIW.

		Tensor extrapolation
Measure	ETSModel	R = 3	R = 4	R = 5
MAPE	229.2167	185.4018	179.8527	181.2276
80% pred. int.
DM	0.0848	0.0438	0.0469	0.0468
MSIS (p = 5)	7.8454	7.2656	7.0958	7.1177
MSIS (p = 10)	8.7375	8.2508	7.9383	7.9805
MSIS (p = 40)	14.0899	14.1620	12.9931	13.1573
MSIW	6.9534	6.2804	6.2533	6.2550
95% pred. int.
DM	0.0292	–0.0016	–0.0007	–0.0010
MSIS (p = 5)	10.7367	9.9298	9.8693	9.8826
MSIS (p = 10)	10.8392	10.2426	10.1631	10.1873
MSIS (p = 40)	11.4539	12.1198	11.9261	12.0154
MSIW	10.6343	9.6169	9.5754	9.5779

The estimation sample includes the first 80 observations, and the hold-out sample covers the next 20 observations. The results obtained from tensor extrapolation are listed with respect to the selected hyperparameter, the number of components R. The constant p affects the penalty in case the observation misses the prediction interval.