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arxiv: 2502.06213 · v2 · submitted 2025-02-10 · 📊 stat.AP

Predicting Energy Demand with Tensor Factor Models

Mattia Banin , Matteo Barigozzi , Luca Trapin This is my paper

Pith reviewed 2026-05-23 04:26 UTC · model grok-4.3

classification 📊 stat.AP
keywords tensor factor modelselectricity demand forecastingseasonal patternshigh-dimensional time seriesfactor decompositionmulti-way arrays
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The pith

Restructuring hourly electricity demand into weekly tensors lets tensor factor models forecast more accurately than vector or functional benchmarks while isolating interpretable seasonal patterns.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Hourly electricity demand from multiple providers exhibits strong intra-day, intra-week, and cross-provider patterns. The paper converts the series into a sequence of weekly three-mode tensors and applies a factor decomposition along the hour, day, and provider dimensions. This separation produces forecasts that beat several standard methods at multiple horizons. The extracted factors also recover known features such as weekday-weekend differences and common provider dynamics. The framework supplies both improved predictions and a structured view of the processes driving demand.

Core claim

Representing the data as weekly tensors whose modes are hours of the day, days of the week, and providers permits a factor model whose loadings separately capture intra-day cycles, day-type effects, and shared provider movements; the resulting forecasts outperform vector factor models and functional time series methods at different horizons, and the factors align with established domain patterns.

What carries the argument

Three-mode tensor factor decomposition on fixed weekly blocks (hours × days × providers) that isolates seasonal structure along each mode.

If this is right

  • Forecasts at short and medium horizons become more accurate when multiple seasonalities are modeled simultaneously rather than sequentially.
  • Factor loadings along the provider mode identify groups of entities that share demand dynamics.
  • The same tensor construction supplies a general template for any high-dimensional series that contains intra-period, inter-period, and cross-sectional seasonalities.
  • Interpretability of the three sets of loadings supports diagnostic checks against known calendar effects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same weekly-tensor structure could be tested on non-calendar periodicities such as event-driven retail sales.
  • Varying the block length (bi-weekly or monthly tensors) would show whether weekly alignment is optimal or merely convenient.
  • If the provider-mode factors prove stable, they could serve as inputs for clustering or nowcasting exercises outside the original forecasting task.

Load-bearing premise

Turning the continuous hourly series into fixed weekly tensors captures every relevant dynamic without creating artifacts from the choice of weekly blocking.

What would settle it

Forecast accuracy comparison on the same data after the weekly blocks are randomly shifted or the day-of-week labels are permuted; if tensor-model gains disappear, the blocking assumption is the source of the reported improvement.

Figures

Figures reproduced from arXiv: 2502.06213 by Luca Trapin, Matteo Barigozzi, Mattia Banin.

Figure 1
Figure 1. Figure 1: TIME SERIES OF HOURLY ENERGY DEMAND IN MEGA WATTS 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: WEAKLY AVERAGE (NORMALIZED) ELECTRICITY DEMAND OVER THE YEAR. SHADED AREAS AND SOLID LINES DISPLAY THE RANGE AND THE AVERAGE ACROSS COMPANIES, RESPECTIVELY. 1.4e−05 1.6e−05 1.8e−05 2.0e−05 Sat Sun Mon Tue Wed Thu Fri Day of Week Electricity Demand (normalized) Season Autumn Spring Summer Winter [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: AVERAGE (NORMALIZED) ELECTRICITY DEMAND FOR EVERY DAY OF THE WEEK IN EACH SEASON. SHADED AREAS AND SOLID LINES DISPLAY THE RANGE AND THE AVERAGE ACROSS COM￾PANIES, RESPECTIVELY. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: AVERAGE (NORMALIZED) ELECTRICITY DEMAND FOR EVERY HOUR OF THE DAY IN EACH SEASON. SHADED AREAS AND SOLID LINES DISPLAY THE RANGE AND THE AVERAGE ACROSS COM￾PANIES, RESPECTIVELY. Mean Median Standard Deviation Skewness Kurtosis AEP 14998.6 14749 2501.355 0.428 2.806 COMED 11383.48 11114 2278.45 1.131 5.038 DAYTON 2002.139 1973 378.478 0.518 3.142 DEOK 3104.468 3012 600.309 0.680 3.365 DOM 11049.34 10587 243… view at source ↗
Figure 5
Figure 5. Figure 5: TIME SERIES OF HOURLY ENERGY DEMAND IN MEGA WATTS (BLACK) AND FITTED VALUES FROM THE TWO-FACTOR MODEL (k = 1, k1 = 1, k2 = 2). 1.4e−05 1.6e−05 1.8e−05 2.0e−05 0 20 40 Week of Year Fitted values (normalized) [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: WEAKLY AVERAGE (NORMALIZED) FITTED VALUES FROM THE TWO-FACTOR MODEL (k = 1, k1 = 1, k2 = 2) OVER THE YEAR. SHADED AREAS AND SOLID LINES DISPLAY THE RANGE AND THE AVERAGE ACROSS COMPANIES, RESPECTIVELY. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: AVERAGE (NORMALIZED) FITTED VALUES FROM THE TWO-FACTOR MODEL (k = 1, k1 = 1, k2 = 2) FOR EVERY DAY OF THE WEEK IN EACH SEASON. SHADED AREAS AND SOLID LINES DISPLAY THE RANGE AND THE AVERAGE ACROSS COMPANIES, RESPECTIVELY. where w is the window of reference, n is the horizon of the forecast, W = T − n is the total number of rolling windows and Ye i,t+n|t is the n-step ahead forecast. Using our tensor factor… view at source ↗
Figure 8
Figure 8. Figure 8: AVERAGE (NORMALIZED) FITTED VALUES FROM THE TWO-FACTOR MODEL (k = 1, k1 = 1, k2 = 2) FOR EVERY HOUR OF THE DAY IN EACH SEASON. SHADED AREAS AND SOLID LINES DISPLAY THE RANGE AND THE AVERAGE ACROSS COMPANIES, RESPECTIVELY. series. The final forecasts for {Yi,t} are reconstructed by combining the predicted factors with the estimated loadings. • Vector factor model. Matrix and tensor time series factor models… view at source ↗
Figure 9
Figure 9. Figure 9: TIME SERIES OF ESTIMATED FACTORS FROM THE TWO-FACTOR MODEL (k = 1, k1 = 1, k2 = 2). (9) divided by the average standard deviation of the out-of-sample observations within each rolling window. This normalization accounts for variations in the scale of electricity demand across different time periods, allowing for a more meaningful comparison of forecasting accuracy. The results clearly indicate that our ten… view at source ↗
Figure 10
Figure 10. Figure 10: SAMPLE AUTOCORRELATION OF THE ESTIMATED FACTORS FROM THE TWO-FACTOR MODEL (k = 1, k1 = 1, k2 = 2). tensor-based approaches, as it does not fully exploit the multidimensional dependencies in the data. The functional time series model consistently exhibits the highest relative MSE across all horizons, indicating that it is less effective at capturing the intricate temporal structures required for accurate f… view at source ↗
read the original abstract

Hourly consumption from multiple providers displays pronounced intra-day, intra-week, and annual seasonalities, as well as strong cross-sectional correlations. We introduce a novel approach for forecasting high-dimensional U.S. electricity demand data by accounting for multiple seasonal patterns via tensor factor models. To this end, we restructure the hourly electricity demand data into a sequence of weekly tensors. Each weekly tensor is a three-mode array whose dimensions correspond to the hours of the day, the days of the week, and the number of providers. This multi-dimensional representation enables a factor decomposition that distinguishes among the various seasonal patterns along each mode: factor loadings over the hour dimension highlight intra-day cycles, factor loadings over the day dimension capture differences across weekdays and weekends, and factor loadings over the provider dimension reveal commonalities and shared dynamics among the different entities. We rigorously compare the predictive performance of our tensor factor model against several benchmarks, including traditional vector factor models and cutting-edge functional time series methods. The results consistently demonstrate that the tensor-based approach delivers superior forecasting accuracy at different horizons and provides interpretable factors that align with domain knowledge. Beyond its empirical advantages, our framework offers a systematic way to gain insight into the underlying processes that shape electricity demand patterns. In doing so, it paves the way for more nuanced, data-driven decision-making and can be adapted to address similar challenges in other high-dimensional time series applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces a tensor factor model for forecasting high-dimensional U.S. electricity demand by restructuring hourly consumption series into a sequence of weekly three-mode tensors (hours × days × providers). The decomposition is claimed to isolate intra-day cycles, weekday/weekend differences, and cross-provider commonalities. The authors assert that this yields superior out-of-sample forecast accuracy at multiple horizons relative to vector factor models and functional time series benchmarks, while also producing interpretable factors aligned with domain knowledge.

Significance. If the empirical superiority and interpretability claims hold after addressing the modeling assumptions, the framework could provide a systematic approach to multi-seasonal high-dimensional forecasting with potential extensions to other domains. The explicit separation of seasonal modes is a conceptual strength, but the absence of supporting details on data, estimation, and validation limits the assessed contribution at present.

major comments (2)
  1. [Abstract] Abstract: The assertion of 'rigorous' benchmark comparisons and 'superior forecasting accuracy' supplies no information on data period, number of providers, tensor rank selection procedure, loss function, or statistical significance tests for the reported gains. This absence is load-bearing because the central claim rests on these comparisons.
  2. [Abstract] Abstract: The weekly tensor construction (hours × days × providers) is presented as capturing pronounced annual seasonalities via the time-series evolution of factor scores across weeks, yet no mode exists for the annual cycle and no analysis is given of potential interactions (e.g., holiday or temperature effects that modulate intra-week loadings). If such interactions exist, the decomposition necessarily averages or misattributes them, which directly affects the validity of both the forecasting gains and the interpretability claims.
minor comments (1)
  1. [Abstract] The abstract would benefit from a brief statement of the forecasting horizons examined and the number of providers in the dataset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our paper. We address each of the major comments below and indicate the revisions we plan to make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The assertion of 'rigorous' benchmark comparisons and 'superior forecasting accuracy' supplies no information on data period, number of providers, tensor rank selection procedure, loss function, or statistical significance tests for the reported gains. This absence is load-bearing because the central claim rests on these comparisons.

    Authors: We agree that the abstract would benefit from additional details to better support the claims of rigorous comparisons and superior accuracy. In the revised manuscript, we will expand the abstract to specify the data period, the number of providers included, the procedure for selecting the tensor rank, the loss function employed for evaluation, and the statistical tests used to assess the significance of the forecasting gains. revision: yes

  2. Referee: [Abstract] Abstract: The weekly tensor construction (hours × days × providers) is presented as capturing pronounced annual seasonalities via the time-series evolution of factor scores across weeks, yet no mode exists for the annual cycle and no analysis is given of potential interactions (e.g., holiday or temperature effects that modulate intra-week loadings). If such interactions exist, the decomposition necessarily averages or misattributes them, which directly affects the validity of both the forecasting gains and the interpretability claims.

    Authors: The model captures annual seasonalities through the temporal evolution of the factor scores across the sequence of weekly tensors, which allows the factors to adapt to longer-term patterns including annual cycles without requiring an explicit annual mode. This design choice enables the separation of intra-day and intra-week patterns while letting the time series of scores handle slower variations. We acknowledge that interactions with external factors such as holidays or temperature could influence the loadings and are not explicitly modeled, potentially leading to some averaging of effects. To address this, we will add a discussion section in the revised paper on these potential limitations and their implications for the results, along with suggestions for future extensions that incorporate such covariates. revision: partial

Circularity Check

0 steps flagged

No significant circularity; forecasting claims rest on external benchmark comparisons.

full rationale

The paper restructures hourly demand into weekly tensors (hours × days × providers), applies a tensor factor decomposition to extract mode-specific loadings, and then generates forecasts by modeling the evolution of factor scores across the sequence of weeks. Predictive accuracy is assessed via direct comparison against vector factor models and functional time series benchmarks on held-out data, with no equations or procedures that define the target forecast error in terms of the fitted tensor parameters themselves. No self-citations are invoked as load-bearing uniqueness theorems, no fitted inputs are relabeled as predictions, and the annual seasonality noted in the abstract enters only through the time-series dynamics of the extracted factors rather than through any definitional closure. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the modeling assumption that a low-rank tensor decomposition along the three modes adequately captures the dominant seasonal and cross-sectional structure; no explicit free parameters, axioms, or invented entities are stated.

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Reference graph

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22 extracted references · 22 canonical work pages

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