TimeBundle/TimeMap: Temporal Data Structures

• TimeBundle/TimeMap are conceptual and technical constructs that bundle time-indexed data, enabling exploration and analysis of temporal phenomena across diverse domains.
• They integrate methodologies from neural self-organizing maps, fibre bundle mechanics, web archival systems, and tidy data practices to capture and quantify dynamic patterns.
• Applications span multivariate analytics, quantum mechanics, urban travel-time metrics, and hierarchical visualizations, offering actionable insights for research and practical deployment.

A TimeBundle or TimeMap is a conceptual and technical construct that supports the representation, exploration, and analysis of time-dependent data, covering a range of methodologies from neural abstractions and fibre bundles in mechanics to web archiving and time-annotated hierarchical visualizations. TimeBundles generally refer to structures that bundle data or states across time, while TimeMaps specifically encode versions, states, or patterns as a function of temporal progression. These constructs facilitate the paper and operational use of temporal phenomena in domains including multivariate temporal analytics, quantum mechanics, web archiving, travel-time metrics, tidy data management, and time-dependent hierarchical visualization.

1. Temporal Mapping Paradigms: Self-Organizing Time Maps

The Self-Organizing Time Map (SOTM) extends the traditional Kohonen Self-Organizing Map (SOM) to temporal multivariate contexts by linking SOM-type learning to ordered time units. For each time slice $t$, the cross-sectional data $Q(t)$ is mapped onto a one-dimensional array $A(t)$ using vector quantization and competitive learning, followed by batch updates of unit reference vectors:

$$m_i(t) = \frac{\sum_j h_{ic(j)}(t) x_j(t)}{\sum_j h_{ic(j)}(t)}$$

where $h_{ic(j)}(t)$ is a Gaussian neighborhood function. Time topology is preserved on the horizontal axis, and data topology on the vertical axis. The arrangement $[A(1), A(2), \ldots, A(T)]$ yields a two-dimensional grid encoding temporal and cross-sectional patterns.

SOTM leverages short-term memory by initializing $A(t)$ with parameters from $A(t-1)$, facilitating detection of gradual or abrupt temporal structural changes. Quantization errors, topographic errors, and the novel metric

$$\Delta(t) = \frac{1}{M} \sum_{i=1}^M \| m_i(t) - m_i(t-1) \|$$

quantify mapping quality and structural change. Visualization techniques include feature planes (color-coded spreads of variables over time), Sammon’s mapping (nonlinear MDS for explicit encoding of time and data relationships), and trajectory plots. SOTM is demonstrated on global development indicators, producing interpretable abstractions of temporal multivariate patterns and outperforming SOM for time-dependent structure analysis (Sarlin, 2012).

2. Mathematical Formulations: TimeBundle in Mechanics

The fibre bundle formulation recasts time-dependent classical and quantum mechanics by expressing configuration spaces as fibre bundles $Q \rightarrow \mathbb{R}$, locally modeled as $(t, q_i)$. The velocity space utilizes the jet manifold $J^1 Q$; phase space is $V^* Q$, a vertical cotangent bundle suited for arbitrary time-dependent transformations. Reference frames are encoded as connections:

$$D_I: J^1 Q \ni (t, q_i, \dot{q}_i) \mapsto (\dot{q}_i - T_i)$$

Quantum mechanics is geometric on Banach/Hilbert bundles $\mathcal{E} \to \mathbb{R}$. Connections of the form $V_t = d/dt + i H(t)$ induce Schrödinger dynamics for section evolution. Quantization is achieved via leafwise geometric quantization compatible with homogeneous cotangent bundles $T^* Q$ and Poisson vertical bundles $V^* Q$; Hamiltonian forms $H = p_k dq^k - H dt$ and corresponding operator representations are derived.

TimeBundle denotes the bundle structure over the temporal axis, while TimeMap refers to explicit time-parameterized transitions, supporting invariance under time-dependent transformations. Applications include the rigorous treatment of reference frame changes, instantwise quantum evolution, and integrable systems via action-angle formalism and Berry phase analysis (Sardanashvily, 2013).

3. Temporal Indexing and Caching: Web Archive TimeMaps

In the Memento Framework, a TimeMap is a machine-readable list aggregating mementos—archived, time-specific resource versions—with each tuple $(\text{URI-M}, \text{Memento-Datetime})$ capturing one version. TimeMaps provide consolidated timelines for web pages, supporting inter-archive querying.

Empirical studies reveal only 80.2% of TimeMaps are monotonically increasing, with reductions caused by archival redactions, restructuring, network downtimes, or transient errors (Brunelle et al., 2013). To optimize user access and archive load, the paper presents a conditional caching algorithm:

• Replace cached TimeMap only if the live version has more mementos:

$$\text{if} (|\text{TM}_{\text{live}}| > |\text{TM}_{\text{cache}}|) \Rightarrow \text{update cache}$$

$$\text{MemDays} = \sum_t \max(|\text{TM}_{\text{live}}(t)| - |\text{TM}_{\text{cache}}(t)|, 0)$$

TTL of 15 days minimizes both missed mementos and archival queries. This policy maintains cache integrity and maximizes completeness in TimeBundle/TimeMap-powered archives.

4. Enriched Aggregation: Negotiation and Multi-Attribute TimeMaps

Advancements in private/public web archive aggregation amend TimeMap syntax/semantics for increased expressiveness. TimeMaps may now include attributes for content (status code, content-type), derived metrics (e.g., Memento Damage quantifying quality by missing resources), and access requirements (e.g., OAuth tokens for private mementos).

The meta-aggregator (MMA) aggregates from diverse sources, allowing precedence models (e.g., privateFirst, publicFirst) and dynamic inclusion via HTTP headers. StarGate augments temporal negotiation with axes for raw content, access, and quality metrics, leveraging HTTP Prefer/Vary semantics.

The system enables secure, privacy-preserving access to private and personal web archives, short-circuiting requests based on user configuration and authentication workflows. Richly attributed TimeMaps support context-aware replay and aggregation, reflecting both public and private views (Kelly et al., 2018).

5. Data Structures: Tidy Temporal Representations

Structural principles for temporal data promote explicit mapping of time semantics, visible index variables, and relational "key" columns ensuring uniqueness across time. The tsibble R package implements these principles by extending data frames with index and key attributes.

Tidy temporal data supports operation as fluent data pipelines, with specific verbs for gap detection and handling (has_gaps, scan_gaps), facilitating analysis and model-ready transformation. The explicit index and key, together with functional pipeline compatibility, enable scalable visualization, modeling, and forecasting for heterogeneous and irregular temporal datasets (e.g., health surveillance, smart-meter data, airline schedules) (1901.10257).

Example tsibble construction in R:

library(tsibble) tb_tsibble <- tb_data %>% as_tsibble(index = year, key = c(country, gender))

6. Visual Encodings: Time-Dependent Treemaps

Time-dependent treemaps visualize hierarchical data with weights $a_i(t)$ evolving over time. Algorithms are evaluated for visual quality (aspect ratio $\rho(R_i) = \min(w, h)/\max(w, h)$) and stability (corner-travel distance $\delta_{CT}(R_i, R_i')$). Data features classified include hierarchy levels, weight variance, speed of change, and insertion/deletion rates; these characteristics predict visual/stability trade-offs.

Matrix and line plots summarize algorithm rankings across 2400+ datasets, supporting informed selection for TimeBundle/TimeMap interactive systems. A baseline treemap $T^*$ provides a reference for minimum layout change attributable to data rather than algorithmic reordering.

This integrated evaluation methodology enables practitioners to balance clarity and temporal coherence in visualization, leveraging explicit quantitative metrics and classification schemes (Vernier et al., 2019).

7. Time as a Metric: Travel-Time Maps

The travel-time metric eschews physical proximity for max-min aggregation of route-dependent travel times:

$$T(a, b) = \max_{(a',b') \in \{(a,b),(b,a)\}} \left\{ \max_{t \in P} \left[ \min_{r \in R(a', b', t)} T(r(a', b', t)) \right] \right\}$$

This formulation guarantees metric properties and operationalizes worst-best travel scenarios. Applications include urban planning, navigation, GIS, and time-metric-based visualization (TimeMaps/TimeBundles). Challenges involve boundary regularization, ensuring data existence and route composability, and potential future incorporation of capacity and stability effects (Halpern, 2015).

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TimeBundle/TimeMap constructs unify temporal indexing, representation, and operationalization across diverse fields, enabling rigorous management, visualization, and analysis of time-dependent phenomena. Integrative methods—spanning neural abstractions, fibre bundle mechanics, web archiving, temporal data structures, hierarchical visualization, and metric-based mapping—support both exploratory research and practical deployment in complex temporal data environments.