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Calendar-Structured Sparse Principal Component Analysis for Interpretable Multi-Periodic Electricity Consumption Profiles

Carlos Quesada-Granja, Tony Castillo-Calzadilla, Carlos Rizo-Maestre

arXiv:2609.06060v2cs.LGcs.AI

TL;DR

Long-term electricity profiles contain simultaneous daily, weekly, and annual cycles, and representing these structures together is a challenge for interpretable dimensionality reduction. Calendar-SPCA embeds cyclic calendar geometry and sparse, locally coherent loadings in a joint factorization. Across two smart-meter datasets, its factors show interpretable calendar patterns while retaining substantial rank-matched PCA variance.

  • Problem

    Long-term electricity profiles exhibit daily, weekly, and annual seasonalities, which can be represented simultaneously when records span complete years.

    Source: §1 · p. 2

  • Method

    Calendar-SPCA combines sparse loading estimation and graph total variation in a low-rank factorization over cyclic calendar axes.

    Source: §1 · pp. 2–3, §3.2 · p. 6

  • Results

    Across GoiEner and Low Carbon London, Calendar-SPCA retained substantial rank-matched PCA variance and recovered readable daily, weekly, seasonal, and jointly localized patterns.

    Source: §7 · pp. 21–22

  • Takeaways & Limitations

    Calendar-SPCA aligns latent-factor structure with the calendar coordinates used to understand long-term electricity consumption.

    Source: §6 · pp. 20–21, §7 · p. 21

Abstract

from arXiv · show

Long-term electricity-consumption profiles exhibit several simultaneous periodic structures, including daily, weekly, and annual cycles. This work introduces Calendar-Structured Sparse Principal Component Analysis (Calendar-SPCA), a structured representation-learning method that incorporates this known multi-periodic geometry directly into a low-dimensional factorization. The method represents the feature domain as the Cartesian product of cyclic calendar axes and combines an L1 loading penalty with graph total variation, producing sparse, locally coherent, and directly interpretable latent factors. In this study, Calendar-SPCA is applied to the interpretable analysis of long-term electricity-consumption profiles and evaluated on two independent public smart-meter datasets, GoiEner and Low Carbon London, with different population sizes and temporal resolutions. A factorial experiment characterizes the effects of sparsity and calendar coherence and examines robustness across sample size, latent dimensionality, and repeated fits. At rank 15, Calendar-SPCA retains 96.92% and 82.90% of the explained variance of rank-matched principal component analysis (PCA) in GoiEner and Low Carbon London, respectively, with mean loading sparsities of 61.95% and 81.50%. Comparisons with classical sparse PCA and sparse PCA with total variation (SPCA-TV) show that Calendar-SPCA organizes latent factors into interpretable structures over the daily, weekly, and annual calendar axes while preserving substantial low-rank information.

1 Introduction

Long-term electricity profiles combine daily, weekly, and annual cycles, which can be represented explicitly as cyclic calendar axes. Calendar-SPCA uses that geometry in a sparse, coherent factorization whose components map back to temporal regions.

1 Introduction

  • The feature domain is organized as a 24 × 7 × 52 tensor over hour of day, day of week, and week of year.This makes the profile's daily, weekly, and annual coordinates explicit.Source: §1 · p. 2
  • Hours, days, and weeks connect cyclically across midnight, the weekly boundary, and the annual boundary, enabling calendar structure to inform dimensionality reduction.Source: §1 · p. 2
  • Calendar-SPCA combines low-rank reconstruction, an L1 loading penalty, and graph total variation over the calendar graph.The loading penalty selects calendar positions, while graph total variation promotes coherent values across neighboring positions.Source: §1 · pp. 2–3
  • Its loadings act as sparse, coherent masks over calendar regions, and the study evaluates the method on two datasets against PCA, sparse PCA, and SPCA-TV.The evaluation uses independent datasets with different populations and temporal resolutions.Source: §1 · p. 3

2 Related Work and Positioning

Calendar-SPCA builds on PCA, sparse loading estimation, structured regularization, and multiway representations. Its positioning combines these mechanisms on cyclic calendar axes and estimates factors jointly.

2 Related Work and Positioning

  • PCA captures dominant variation with reduced latent directions, while sparse PCA links components more closely to original features using selected loading supports,,,.Source: §2 · pp. 3–4
  • Structured sparse methods encode relationships among variables through group or fusion penalties, including fused and generalized lasso formulations [7],, [9],,.Source: §2 · p. 4
  • SPCA-TV combines sparse estimation with total variation but extracts components sequentially, whereas Calendar-SPCA estimates all calendar factors jointly on a cyclic Cartesian-product graph [12].Source: §2 · p. 4
  • Multiway methods organize data along explicit modes, [14]; Calendar-SPCA combines sparse loadings, local coherence, prior structure, cyclic axes, and joint factor estimation.Source: §2 · p. 4

3 Calendar-SPCA

Calendar-SPCA represents long-term electricity profiles on a cyclic multi-periodic calendar domain and learns jointly estimated sparse, structured factors. Its alternating fitting procedure updates scores and loadings while maintaining active components and checking convergence.

3 Calendar-SPCA

  • Calendar-SPCA combines a cyclic calendar graph, structured sparse factorization, and alternating optimization to estimate latent factors over long-term electricity profiles.Source: §3 · p. 5

3.1 Calendar geometry

  • Hourly profiles use 24 × 7 × 52 coordinates and half-hourly profiles use 48 × 7 × 52, with cyclic adjacency linking neighboring positions across daily, weekly, and annual boundaries.Source: §3.1 · p. 5

3.2 Structured sparse factorization

  • The reconstruction term captures dominant variation, the ℓ1 penalty selects calendar positions, and graph total variation promotes coherent loading values across neighboring positions.Source: §3.2 · p. 6

3.3 Optimization

  • Fitting alternates between score refinement and a structured loading update, using primal–dual splitting and component recovery when a component becomes degenerate.Source: §3.3 · p. 7

4 Experimental Design

The experiments use two smart-meter datasets and a factorial design to assess regularization, robustness, and cross-method performance. Evaluation criteria cover reconstruction, loading structure, stability, and numerical reliability.

4.1 Datasets

  • GoiEner contributes 87,187 hourly profiles, while LCL contributes 5,359 half-hourly profiles on 24 × 7 × 52 and 48 × 7 × 52 domains, respectively.Source: §4.1 · pp. 8–9

4.2 Experimental protocol

  • The GoiEner factorial experiment varies sparsity and graph-total-variation penalties and examines robustness across sample size, rank, and repeated fits.Source: §4.2 · pp. 9–10

4.3 Model selection and baselines

  • Regularization strength is selected independently for each dataset along a one-dimensional path, using maximum discrete Menger curvature among eligible interior L-curve points.Source: §4.3 · p. 10
  • At a common selected rank, Calendar-SPCA is compared with rank-matched PCA, sparse PCA, and SPCA-TV using the same cyclic calendar neighborhood for SPCA-TV.Source: §4.3 · pp. 10–11

4.4 Evaluation criteria

  • Reconstruction quality is measured through explained variance, including its ratio to rank-matched PCA explained variance in the cross-method comparison.Source: §4.4 · p. 11
  • Loading structure is assessed with sparsity, calendar coherence, and effective-region counts, while repeat stability and effective rank track reproducibility and numerical reliability.Source: §4.4 · p. 12

5 Results

The results characterize how Calendar-SPCA’s regularizers shape its representations, assess robustness and operating-point selection, and compare the learned calendar structures with existing methods. Across both datasets, it retains substantial rank-matched PCA variance while producing sparse, locally coherent factors.

5.1 What do the two regularizers do?

  • Increasing λ1 raises loading sparsity, while increasing λTV reduces relative total variation; together they control support selectivity and coherence alongside reconstruction variance.Source: §5.1 · pp. 12–13, Figure 2 · p. 13

5.2 Is the representation robust?

  • At N = 20,000, explained variance rises from 33.44% at K = 5 to 40.55% at K = 15 and 43.98% at K = 30, with similar trajectories across four sample sizes.Source: Figure 3 · p. 14, §5.2 · p. 14
  • At K = 15, all five fits converged at every N, repeat stability increased with sample size, and mean sparsity ranged from 71.35% (N = 5,000) to 62.46% (N = 20,000).Source: §5.2 · p. 14
  • At K = 10, factors occupy relatively broad calendar regions; higher rank yields narrower, more localized components, and K = 15 is adopted for the final representations.Source: Figure 4 · p. 15, §5.2 · p. 16

5.3 How is the operating point selected?

  • Maximum curvature along λ1 = λTV = λ selects λ∗ = 2.0 for GoiEner and λ∗ = 1.5 for Low Carbon London, retaining Keff = 15.Source: Figure 5 · p. 16, §5.3 · p. 17

5.4 What does Calendar-SPCA learn?

  • GoiEner factors often align with individual calendar axes, whereas Low Carbon London factors more often localize jointly across temporal coordinates; cyclic boundaries can connect apparently separated regions.Source: §5.4 · p. 17

5.5 How does Calendar-SPCA compare with existing methods?

  • Calendar-SPCA retains 96.92% of rank-matched PCA explained variance on GoiEner and 82.90% on Low Carbon London, with the highest mean sparsity and lowest RTV among the three structured representations on both datasets.Source: Table 3 · p. 19, §5.5 · p. 19
  • PCA and SPCA retain broad mixed-sign patterns, while SPCA-TV can yield fragmented patches; Calendar-SPCA organizes interpretable factors spanning the calendar domain.Source: §5.5 · pp. 19–20, Figure 7 · p. 20

6 Discussion

Calendar-SPCA aligns the geometry used in learning with the calendar coordinates used to interpret factors, while retaining data-driven flexibility. Its factors and scores offer complementary temporal and observation-level views, and the construction extends to other periodic domains.

6 Discussion

  • Calendar graph neighbourhoods define the learning geometry, while the optimization selects active regions and combines temporal coordinates within each factor.The resulting geometry guides factor organization while preserving data-driven flexibility.Source: §6 · pp. 20–21
  • Loading sparsity controls temporal selectivity, while graph total variation promotes local coherence across calendar positions.Joint estimation coordinates both properties; the score matrix provides observation-level embeddings complementary to the temporal meanings in the loading matrix.Source: §6 · p. 21
  • The construction extends to four or more periodic coordinates when an application supplies them, with examples of multi-periodic structure across other domains.Examples include telecommunications, industrial activity, traffic, building occupancy, and ecological observations.Source: §6 · p. 21

7 Conclusions

Calendar-SPCA represents long-term electricity profiles on cyclic daily, weekly, and annual coordinates, combining sparse loadings and graph total variation in a joint low-rank factorization. Experiments across two datasets report systematic behavior and readable calendar patterns while retaining substantial variance captured by rank-matched PCA.

7 Conclusions

  • Calendar-SPCA combines sparse loading estimation and graph total variation in a joint low-rank factorization over cyclic calendar axes.Its components are learned directly on daily, weekly, and annual coordinates and interpreted as coherent temporal patterns.Source: §7 · p. 21
  • Across GoiEner and Low Carbon London, experiments showed controllable sparsity and calendar coherence, systematic behavior across sample sizes and latent dimensionalities, and high repeat stability in the identified operating region.The experiments also report reproducible operating-point selection and substantial retention of variance captured by rank-matched PCA.Source: §7 · pp. 21–22
  • Recovered components revealed readable daily, weekly, seasonal, and jointly localized patterns across datasets with different population sizes and temporal resolutions.The conclusion presents Calendar-SPCA as an example of incorporating known feature geometry into representation learning.Source: §7 · p. 22

CRediT authorship contribution statement

The authors' contribution statement assigns conceptual, methodological, and supervisory responsibilities to Carlos Quesada-Granja, and writing responsibilities to all three listed authors.

CRediT authorship contribution statement

Funding

The supplied material states that the research received no external funding and outlines optimization, numerical safeguards, component recovery, and additional robustness and ablation analyses.

Funding

A Optimization details

  • The optimization appendix specifies loading updates, step-size selection, convergence checks, initialization, component recovery, and numerical settings.Source: A Optimization details · p. 25

A.1 Primal–dual loading update

  • For fixed scores, the loading update uses a primal–dual splitting scheme that applies sparsity to loadings and controls graph differences through the dual update.The regularizers are handled in the same convex loading solve.Source: §A.1 · p. 25

A.2 Step-size selection

  • Primal and dual step sizes are selected using the smooth loading objective and calendar incidence operator's spectral properties, with the prescribed condition checked during each update.The appendix gives settings for both calendar domains used in the experiments.Source: §A.2 · p. 26

A.3 Convergence and numerical safeguards

  • Convergence monitoring covers score refinement, the structured loading solve, and stabilization of the full factorization, with descent safeguards and reconstruction-change checks.The reconstruction difference can be evaluated from consecutive factorization Gram matrices without constructing the full reconstructed matrices.Source: §A.3 · pp. 26–28

A.4 Initialization and component recovery

  • The procedure starts from a randomized truncated SVD and can recover degenerate or redundant components using residual variation.A deterministic sign convention is applied after fitting for consistent stored solutions and visualizations.Source: §A.4 · pp. 28–29

A.5 Numerical settings

  • Numerical controls were fixed across experiments, with a larger outer-iteration budget for full-data regularization-selection fits.Table 4 collects settings for refinement, optimization, convergence, initialization, and recovery.Source: §A.5 · p. 29

B Additional robustness and ablation

  • The additional appendix analyses assess numerical reliability, isolate sparsity and calendar-coherence contributions through matched penalty ablation, and examine objective balance across sample sizes.Source: B Additional robustness and ablation · p. 29

B.1 Reliability of the factorial experiment

The factorial reliability checks indicate numerical stabilization across converged and budget-limited fits, while matched configurations distinguish the penalties’ roles in support size and calendar organization.

B.1 Reliability of the factorial experiment

  • Of 5,880 factorial fits, 3,499 (59.5%) satisfied the complete convergence criterion, while 2,381 completed the 500-iteration budget.Table 5 summarizes the final numerical diagnostics for both groups.Source: §B.1 · pp. 29–30, Table 5 · p. 30
  • Converged fits had very small final objective and reconstruction changes; budget-limited fits retained nearly all requested components and typically showed small final changes.Aggregated factorial values required at least four of five repetitions to satisfy the complete convergence criterion.Source: §B.1 · p. 30

B.2 Isolating sparsity and calendar coherence

  • With λTV = 0, raising λ1 from 0 to 3 increased mean loading sparsity from 0 to 0.691, identifying ℓ1 as the support-size and temporal-selectivity control.The comparison averaged nine reliable (N, K) scenarios common to all four configurations.Source: §B.2 · p. 31, Table 6 · p. 31
  • With λ1 = 0, raising λTV from 0 to 3 lowered mean relative total variation from 0.779 to 0.204 while supports remained dense.Graph-TV primarily organizes loading values across adjacent calendar positions.Source: §B.2 · p. 31
  • With both penalties active, mean loading sparsity was 0.657 and RTV was 0.386 versus 0.910 under sparsity alone, while effective regions fell from 7.35 to 3.64.Calendar coherence organized selected support into fewer dominant temporal structures.Source: §B.2 · p. 31
  • The matched comparison assigns ℓ1 to the amount of active calendar domain and graph total variation to how that support is connected and organized.Together, the penalties yield sparse, coherent calendar patterns.Source: §B.2 · p. 31

B.3 Sample-size dependence of the objective balance

At K = 15 and λ1 = λTV = 1, the relative contribution of both regularizers declined as sample size grew. The appendix documents the rank-matched reference methods and their regularization-selection procedures.

B.3 Sample-size dependence of the objective balance

  • The combined penalty-to-reconstruction ratio fell from 0.157 at N = 5,000 to 0.096 at N = 20,000, so fixed (λ1, λTV) implied a sample-size-dependent balance.The same progression appeared independently for the sparsity and calendar-TV terms.Source: §B.3 · pp. 31–32, Table 7 · p. 32

C.1 Reference methods

  • At K = 15, PCA provided the rank-matched reconstruction reference; SPCA used sparse alternating singular-vector updates, while SPCA-TV extracted rank-one components with ℓ1, quadratic, and total-variation regularization.SPCA-TV used the same cyclic calendar neighborhood as Calendar-SPCA, with its native isotropic total variation.Source: §C.1 · pp. 32–33

C.2 Regularization selection

  • SPCA and SPCA-TV parameters were selected before map inspection using method-specific one-dimensional L-curves; GoiEner paths used 5,000 profiles before transfer, unlike full-data London paths.The selected configurations retained all 15 components.Source: §C.2 · pp. 33–34

C.3 Final cross-method summary

The final rank-15 comparison describes distinct structural profiles across PCA, SPCA, SPCA-TV, and Calendar-SPCA. Calendar-SPCA combines loading selectivity and calendar coherence while retaining substantial explained variance on both datasets.

C.3 Final cross-method summary

  • Table 8 reports explained variance, PCA retention, loading sparsity, and relative total variation over the common cyclic calendar graph.PCA retention expresses explained variance relative to the corresponding PCA solution.Source: Table 8 · p. 34
  • At K = 15, Calendar-SPCA combined high loading selectivity and strong calendar coherence with substantial explained variance on both datasets.Table 8 reports the final comparison for all methods on the complete datasets.Source: §C.3 · p. 34, Table 8 · p. 34
  • PCA served as the rank-matched reconstruction reference, SPCA added substantial loading sparsity while preserving a high fraction of PCA variance, and SPCA-TV emphasized local calendar regularity.These are the distinct structural profiles described for the four representations.Source: §C.3 · p. 34
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