Triangle‑Based Geometric Semantic Modeling for Time‑Series Analysis

Triangle‑Based Geometric Semantic Modeling for Time‑Series Analysis | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 10 January 2026 V1 Latest version Share on Triangle‑Based Geometric Semantic Modeling for Time‑Series Analysis Author : Kayode Adepoju 0000-0001-7906-3415 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176806278.86444285/v1 371 views 69 downloads Contents Abstract Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We formalize Triangle-based Geometric Semantic Modeling (TGSM), a novel approach to time-series analysis that encodes temporal transitions as geometric primitives. Consecutive observations are mapped to right-angled triangles, linking time span and change magnitude to a unified measure of movement strength. By organizing these primitives into directional classes and aggregating them across scales, TGSM provides a transparent bridge between raw signals and semantic structure. This framework offers a new lens for interpreting dynamic behavior and establishes an audit-ready foundation for semantic compression, model transparency, and robust feature design in explainable AI. The paper highlights the formulation, key derivations, and potential applications of this approach. Triangle‑Based Geometric Semantic Modeling for Time‑Series Analysis Kayode Adepoju Correspondence: Date: December 21, 2025 Abstract We formalize Triangle-based Geometric Semantic Modeling (TGSM), a novel approach to time-series analysis that encodes temporal transitions as geometric primitives. Consecutive observations are mapped to right-angled triangles, linking time span and change magnitude to a unified measure of movement strength. By organizing these primitives into directional classes and aggregating them across scales, TGSM provides a transparent bridge between raw signals and semantic structure. This framework offers a new lens for interpreting dynamic behavior and establishes an audit-ready foundation for semantic compression, model transparency, and robust feature design in explainable AI. The paper highlights the formulation, key derivations, and potential applications of this approach. Keywords: Geometric semantic modeling Time‑series geometry, Triangle primitives, Structural decomposition, Directional transitions, Semantic representation, Multiscale analysis 1.0 Introduction There is increasing interest in geometric and topological perspectives for analyzing temporal data. These approaches aim to capture structural characteristics that extend beyond raw numeric values, offering insights into shape, persistence, and multi-scale organization. Shapelet‑based classification emphasizes short, discriminative subsequences that capture characteristic forms in time‑series segments [Ye & Keogh, 2009]. Motif discovery extends that idea by identifying frequently recurring patterns, offering a compact way to describe repeated behaviors [Mueen et al., 2011]. Topological data analysis (TDA) contributes a global perspective, using algebraic summaries of shape to characterize persistence and connectivity in trajectories [Carlsson, 2009]. Parallel developments in multiscale analysis—such as wavelet transforms and fractal characterizations—highlight frequency content and complexity, revealing phase relationships and scale interactions that are not evident in pointwise summaries. Recent work continues to broaden these foundations. For instance, Majumdar et al. (2020) study clustering and classification of temporal signals using TDA‑derived features, illustrating how global shape summaries can guide grouping and labeling in complex series. El‑Yaagoubi et al. (2023) extend this line to multivariate time‑series, showing how topological invariants help expose joint structure across variables. Together, these contributions underscore the value of treating structure as information—not merely as a by‑product of statistical modeling—while motivating representations that remain interpretable at the step level and coherent across scales. While geometric and topological methods provide powerful tools for pattern recognition and feature extraction, many contemporary treatments depend on high‑dimensional embeddings or statistical abstractions that can blur the local geometric transitions driving interpretability. We propose an alternative lens: treat pairs of adjacent observations as geometric entities and explicitly construct a sequence of right‑angled triangles whose base reflects elapsed time and whose height reflects change magnitude. In this view, time is encoded as shape, and triangle area serves as a compact, additive proxy for transition strength. By reversing the usual resolution of a directional vector and materializing the underlying triangle for each step, we obtain geometric primitives that carry localized semantics—movement magnitude, orientation, temporal span—and can be aggregated consistently across windows and scales. Organized into direction‑only classes and summarized cumulatively, these primitives form a semantic field that enables semantic compression, phase-aware reasoning, and interpretable forecasting. 2.0 Methodology 2.1 Triangle Definition and Classification Every triangle constructed from consecutive time-series points is a right-angled triangle. Each triangle is defined by: A base representing the time interval between two observations, a height representing the change in observed value, and a hypotenuse representing the transition vector between the two points. Special Case: When adjacent values are equal, the height becomes zero, and the structure degenerates into a polygonal segment rather than a triangle. Such cases are rare but important for identifying periods of stasis in the series. Classification: Triangles are classified into Rising and Falling classes. A Rising triangle indicates that the second point is greater than the first, while a Falling triangle indicates the opposite. Importantly, interpretation does not depend on numeric sign because the class name itself encodes directionality. Thus, a Falling triangle does not imply negativity in an absolute sense; it simply denotes a downward structural transition within the series. 2.1.1 Deriving Semantic Meaning from GSM Statistics This geometric construction transforms the time series into a structured sequence of semantic units, each of which can be characterized by measurable properties such as height, area, angle, orientation, momentum, and rate of change.Each metric offers insight into the structural and directional behavior of the time series: Counts: The number of rising versus falling triangles indicates directional dominance. A higher count of rising triangles suggests an upward trend, while a predominance of falling triangles signals a downward trend. Ratios of these counts provide a quick measure of trend bias. Area: Triangle area represents structural magnitude and directional energy. Larger cumulative areas in rising triangles imply strong upward momentum, whereas larger areas in falling triangles indicate strong downward pressure. Histograms of area distributions reveal variability and concentration of energy within each class. Height: Height reflects the absolute value difference between consecutive points, serving as a proxy for volatility. A series with consistently high triangle heights exhibits greater variability, while smaller heights suggest stability. Mean-Median Differences: Comparing mean and median areas or heights within each class highlights skewness and asymmetry. Significant differences may indicate the presence of extreme movements or outliers affecting trend interpretation. Correlation: Correlation between height and area within each class provides insight into proportionality. Strong positive correlation suggests that larger directional changes are consistently associated with longer time spans, reinforcing trend strength. Slope: The slope of cumulative metrics over time reflects structural evolution. A steep slope in cumulative area for rising triangles indicates accelerating upward energy, while flattening slopes may signal trend exhaustion or reversal. Together, these interpretations enable analysts to assess whether a trend is dominant, stable, or declining. For example, a series with high rising counts, large cumulative areas, and strong positive slopes likely represents a robust upward trend. Conversely, mixed signals—such as rising counts but declining cumulative slopes—may indicate instability or an impending reversal. These foundational diagnostics form the basis for advanced confidence scoring and erosion analysis, which will be addressed later. 2.2. Axioms and Theorems 2.2.1 Geometric Primitives Let {(t_i, x_i)} be ordered observations with t_{i+1} > t_i. Define: Base (time gap): b_i := t_{i+1} − t_i > 0. Raw change: Δx_i := x_{i+1} − x_i. Height (magnitude): h_i := |Δx_i|. Triangle area (right triangle): A_i := (1/2) · b_i · h_i. Slope: m_i := h_i / b_i. Flat segments: h_i = 0 ⇒ A_i = 0 (tracked as needed). [Figure 1: Primitive construction—right-angled triangles from adjacent points] 2.2.2. Class Partitioning Class rule: s_i := sign(Δx_i) ∈ {−1, 0, +1}. Assignment: Rising if s_i = +1; Falling if s_i = −1; Flat if s_i = 0. Partition properties: complete and disjoint; direction-only semantics ensure applicability regardless of numeric sign (binary/categorical extensions deferred to future work). [Figure 2: Class partition visualization—Rising vs Falling; Flat] 2.2.3 Axioms and Theorems A1 (Existence): For each consecutive pair, the TGSM primitive (A_i, C_i) exists with A_i = (1/2) b_i h_i and C_i ∈ {R, F, Flat}. A2 (Class Partition): Indices decompose into R = {i : s_i=+1}, F = {i : s_i=−1}, Flat = {i : s_i=0}. A3 (Adjacency): Primitives use adjacent observations; temporal order is preserved. A4 (Additivity): For disjoint windows W1, W2, E(W1 ∪ W2) = E(W1) + E(W2), where E(W) = Σ_{i∈W} A_i. A5 (Monotone Invariance): For strictly monotone transforms g, class labels and within-partition area rankings are preserved up to bounded distortion. A6 (Update Invariance): Appending a new observation creates a single new primitive; existing primitives remain unchanged. A7 (Multiscale Structural Consistency): Under suitable conditions, class-wise ratios remain stable under resampling within tolerance ε. 2.2.4 Theorems T1 (Directional Balance): If R_H(W) ≈ 1 or R_A(W) ≈ 1, directional energy is balanced in window W. T2 (Crossover): If ∃ t_k with Cum_R(t_k) = Cum_F(t_k), directional dominance switches at t_k. T3 (Scale Consistency): Class ratios invariant within ε under resampling/aggregation indicate multiscale consistency. T4 (Estimator Consistency): Plug-in estimators for totals and ratios converge in probability under refinement and mild dependence/variation. 2.25. Primary Statistics Counts: N_R(W) = # {i∈W : s_i=+1}, N_F(W) = # {i∈W : s_i=−1}. Heights: H_R(W) = Σ_{R} h_i; H_F(W) = Σ_{F} h_i; with summary (mean, median, min, max) per class. Areas: A_R(W) = Σ_{R} A_i; A_F(W) = Σ_{F} A_i. Ratios: R_H(W) = H_R(W)/H_F(W); R_A(W) = A_R(W)/A_F(W). Mean/Median ratio (per class C): MM_C = mean({A_i | C_i=C}) / median({A_i | C_i=C}). Correlation (default Spearman for robustness; Pearson optional): ρ_A(W) = corr({A_i | R}, {A_j | F}). Cumulative curves (areas): Cum_R(t) = Σ_{i: t_i ≤ t, s_i=+1} A_i; Cum_F(t) = Σ_{i: t_i ≤ t, s_i=−1} A_i. Slopes: per-step m_i trajectories per class to summarize local pace of change. [Table 1: Primary statistics—counts, height summaries, area totals, ratios, mean/median ratios, correlations, slopes] [Figure 3a/b: Histograms—heights and areas by class] 2.26. Spikes & Crossovers (secondary statistics) Spikes: local extremes in per-step area or height series on a shared axis; robust detection via median absolute deviation thresholds. Crossovers: intersections in cumulative area curves (Cum_R vs Cum_F) or sustained intersections in per-step area plots; detect via sign change in Cum_R−Cum_F with minimal persistence. Role: confirm or refute transient signals, reduce false alarms, and isolate outliers in structural evolution. [Figure 4: Per-step area plot (both classes) with spike markers] [Figure 5: Cumulative area curves with crossover markers] [Figure 5: Cumulative area curves with crossover markers] 2.27. Directional Energy & Dominance (Giant Triangles) For window W = [t_min, t_max] with duration B(W) = t_max − t_min: Rising giant triangle height: H_R* := 2 A_R(W) / B(W); Falling giant triangle height: H_F* := 2 A_F(W) / B(W). [Figure 6: Giant triangles (Rising vs Falling) showing directional dominance] 2.28. Resampling & Aggregation (Multiscale TGSM) By anchoring each step in a single triangle, TGSM provides a localized, additive summary whose meaning persists under resampling and aggregation, enabling multiscale comparisons without re‑interpreting what a step means. For instance, select interval partitions (day/week/month) and aggregators (sum/mean/median/first/last) can be used in constructing primitives on the resampled series to compare dominance across scales. Hierarchical intervals can bridge irregular sampling; semantics are preserved while noise is reduced at coarser scales. 3.0 Discussion Volatile, irregular, regime‑shifting contexts—such as flood escalation windows, coastal storm‑surge episodes, and power‑system balancing periods—require summaries that distinguish brief excursions from sustained movement, reveal where movement concentrates within decision windows, and remain comparable across temporal resolutions. This study has shown that a geometry‑first, adjacent‑change representation provides practical leverage in unstable systems marked by volatility, irregular sampling, and regime shifts. The geometric representation developed in this work provides a step‑level account of directional movement that remains additive and interpretable across temporal resolutions. By expressing each adjacent transition as a geometric primitive with measurable area, the approach preserves local structure that is often diluted in smoothing‑based or frequency‑domain methods. This emphasis on localized transitions is in line with shape‑based perspectives such as shapelets and motifs (Ye & Keogh, 2009; Mueen et al., 2011), while extending them beyond supervised discrimination toward an unsupervised, cumulative summary of directional energy. The cumulative dominance display, which aggregates step‑areas into class‑wise directional contributions, offers a compact means of identifying where movement concentrates within a decision window. This improves upon classical decomposition and smoothing approaches (Hyndman et al., 2010), which can delay or absorb short‑lived structural changes. The resulting representation aligns with reconciliation principles in hierarchical forecasting (Wickramasuriya et al., 2017; Athanasopoulos et al., 2024), as it maintains consistent interpretation across scales without requiring post‑hoc adjustments. The confirmation layer—combining spike filtering with cumulative crossovers—strengthens early‑warning reliability in volatile settings. This contrasts with sequential control‑chart methods such as CUSUM (Aue & Kirch, 2024), which may trigger nuisance alarms under irregular sampling or transient bursts. By requiring persistence in directional energy before signaling a shift, the approach improves upon these methods while remaining compatible with multivariate change‑point detection frameworks (Alanqary et al., 2021). Semantic coherence under resampling is another key property. Because directional class and area remain invariant under admissible transformations, interpretations established at fine cadence continue to hold at coarser horizons. This complements reconciliation‑based strategies in hierarchical forecasting (Hyndman et al., 2010), where cross‑level consistency is essential for operational decision‑making. The geometric formulation achieves this consistency intrinsically, without relying on model‑based adjustments. Interpretability follows directly from construction rather than post‑hoc explanation. The resulting diagnostics—counts, heights, areas, ratios, cumulative curves, spike frequency, and crossover timestamps—form a transparent evidence chain from raw adjacency to directional energy. This stands in contrast to explainability toolkits that attempt to rationalize model outputs after the fact (Theissler et al., 2022; García‑García et al., 2025). The geometric features also integrate naturally with selective‑forecasting strategies that modulate confidence based on sustained directional movement (Brusokas et al., 2025), supporting both predictive flexibility and auditability. Overall, the framework consolidates structural themes across shape‑based, topological, sequential, and reconciliation literatures into a coherent account of temporal behavior. It extends local‑pattern methods by making directional energy additive, complements global‑structure approaches by restoring locality, improves upon smoothing‑based pipelines by preserving step‑level transitions, and enhances sequential detection by enforcing persistence. These properties make the approach well suited to volatile, irregular, and regime‑shifting environments where interpretability, timeliness, and cross‑scale consistency are essential. 4.0 Conclusion TGSM offers a constructive extension to existing time-series analysis frameworks by combining structural rigor with interpretability. While numeric smoothing and frequency-based methods provide valuable insights, this model focuses on geometric primitives that capture directional energy and systemic transitions. This approach facilitates early identification of trend reversals and volatility patterns while reducing false alarms through structural confirmation. Metrics such as directional balance, mean–median divergence, and semantic crossovers enable practical decision support in domains where stability and adaptability are critical. The framework’s emphasis on time‑as‑shape, step‑level additivity, and direction‑only organization provides a transparent bridge from raw signals to semantic structure—well suited to monitoring, forecasting, and explanation across diverse contexts. This dual capability—interpretability and predictive potential—underscores it’s potential contribution to advancing multiscale time-series analytics. Limitations include sensitivity to sampling resolution, the need for calibrated thresholds, and reduced expressiveness in strongly multivariate settings unless paired withit’s potential methods. Future work will address irregular sampling, reference‑relative dominance indices, binary/categorical settings, coordinated multivariate confirmation and abstraction of risk indices. Table 1: Visual Comparison Table: TGSM vs Other Methods Interpretability High (geometric primitives) Low (numeric smoothing) Moderate (frequency domain) Low (global persistence) Multiscale Adaptability Native (resampling) Limited High Moderate Noise Robustness High (aggregation + confirmation) Low Moderate Moderate Early Trend Detection Yes (spikes + crossovers) Limited Moderate Limited Domain Agnostic Yes Yes Yes Yes Acknowledgments The author thanks collaborators and mentors who supported the development of TGSM. Funding No external funding was received for this study. Conflict of Interest The author declares no conflict of interest. Appendix A: Notation t_i: time at index i; x_i: value at time t_i; b_i: base (time gap); Δx_i: raw change; h_i: height = |Δx_i|; s_i: sign of change; A_i: area = (1/2)·b_i·h_i; m_i: slope = h_i/b_i; N_C: count in class C; H_C: total height in class C; A_C: total area in class C; Cum_C(t): cumulative area for class C up to time t; B(W): window duration. Appendix B: Axiom–Theorem Linkage Table Existence; Class Partition Directional Balance; Crossover Enables class-wise totals, ratios, and dominance switches. Adjacency Estimator Consistency Local construction supports stable plug-in estimators and windowed comparisons. Additivity Directional Balance Area totals are decomposable across disjoint windows; auditing is straightforward. Monotone Invariance Scale Consistency Unit/scale transforms preserve class labels and within-partition rankings. Update Invariance Crossover Streaming updates do not alter prior primitives; only new segments affect cumulative curves. Multiscale Structural Consistency Scale Consistency Ratios remain stable across resampling levels within tolerance. Appendix C: Validation Strategy To ensure robustness and interpretability, TGSM validation is grounded in its diagnostic classes and structural statistics. Validation includes mapping GSM metrics to observable statistics and applying statistical confidence tests. Full empirical results will appear in Part 2. Correlation Significance: Compute Pearson or Spearman correlation between Rising and Falling metrics; report p-values and confidence intervals. Slope Reliability: Fit linear models to area-based slope series; report R² and confidence intervals. Distribution Tests: Apply Kolmogorov–Smirnov or Shapiro–Wilk tests for normality and structural shifts. Trend Significance: Use Mann–Kendall tests to confirm slope trends. Bootstrap Resampling: Estimate confidence intervals for TGSM metrics across multiple windows. - Benchmarking: Compare TGSM outputs with ARIMA and Wavelet models. References Athanasopoulos, G.; Hyndman, R.J.; Kourentzes, N.; Panagiotelis, A. Forecast reconciliation: A review. Int. J. Forecast. 2024, 40, 430–456. Wickramasuriya, S.L.; Athanasopoulos, G.; Hyndman, R.J. Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization. Working Paper 2017. 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Deep learning for multivariate time series anomaly detection: An evaluation of reconstruction‑based methods. Artif. Intell. Rev. 2025, 58, 400. Brusokas, J.; Tirupathi, S.; Zhang, D.; Pedersen, T.B. The Time‑Energy Model: Selective time‑series forecasting using energy‑based models. Trans. Mach. Learn. Res. 2025. Ye, L., & Keogh, E. (2009). Time series shapelets: A new primitive for data mining. Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. Mueen, A., Keogh, E., & Young, N. (2011). Logical-shapelets: An expressive primitive for time series classification. Proceedings of the 17th ACM SIGKDD Conference. Carlsson, G. (2009). Topology and data. Bulletin of the American Mathematical Society, 46(2), 255-308. Majumdar, S., et al. (2020). Clustering and classification of time series using topological data analysis. Journal of Machine Learning Research. El-Yaagoubi, S., et al. (2023). Topological data analysis for multivariate time series data. Applied Network Science. Information & Authors Information Version history V1 Version 1 10 January 2026 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords directional transitions semantic representation structural decomposition time series geometry triangel primitives Authors Affiliations Kayode Adepoju 0000-0001-7906-3415 [email protected] Western Sydney University Hawkesbury Institute for the Environment View all articles by this author Metrics & Citations Metrics Article Usage 371 views 69 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Kayode Adepoju. Triangle‑Based Geometric Semantic Modeling for Time‑Series Analysis. Authorea . 10 January 2026. DOI: https://doi.org/10.22541/au.176806278.86444285/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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