Space-Time Accessibility Metrics

• Space-Time Accessibility (SPA) metrics are quantitative tools that combine spatial configurations and temporal constraints to assess access to opportunities.
• They integrate geometric formulations, dynamic network models, and equity-driven statistical approaches to capture real-world access disparities.
• SPA metrics inform urban planning and policy, enabling simulation of network changes and targeted interventions to improve transit and service delivery.

Space-Time Accessibility (SPA) Metric

Space-Time Accessibility (SPA) metrics quantify how regions, individuals, or opportunities are accessible when both spatial configuration and temporal constraints are considered. SPA formally integrates geometric structures, dynamic elements of time and demand, multi-modal networks, and socio-demographic disparities to advance the understanding of access across domains ranging from theoretical physics and pedestrian dynamics to urban transit, equity, and service delivery.

1. Theoretical Foundations in Space-Time Geometry

SPA metrics, in their geometric formulation, build upon the event manifold structure (Rau, 2010). An event manifold $M$ is a differentiable manifold endowed with a physical structure $\Phi$ characterized by:

• Volume element: Specifies the measure, $dV = \sqrt{|\det(g_{ab})|} \ d^n x$, for regions in space–time.
• Causal structure: Encoded by the light cone $g_{ab}dx^a dx^b=0$, which demarcates accessible regions through future and past separation (convex sets $j^+$, $j^-$).

Accessibility in this context becomes a function of whether one event is reachable from another along causal or timelike geodesics. The affine connection $\omega$ (defined via $d\theta + \omega \wedge \theta = 0$) prescribes geodesic curves $\gamma$ whose proper time $\tau$ is given by

$\tau = \int_\gamma \sqrt{ - g_{ab} dx^a dx^b }.$

This structure is dynamic—matter distributions modify $g_{ab}$, changing causal cones, thereby altering SPA. SPA thus quantifies accessibility as a function of geometric location, physical structure variability, and causal ordering.

2. Statistical and Equity-Driven SPA Models

Contemporary SPA metrics in service or urban contexts employ models sensitive to both spatial and temporal variation, equity, and demographic heterogeneity (1111.7120).

A typical SPA regression model is:

$$E[Y(t, s)|X] = \gamma_1(t, s)X_1(t, s) + \ldots + \gamma_R(t, s)X_R(t, s)$$

where regression coefficients $\gamma_r(t, s)$ vary over time $t$ and space $s$, decomposed into additive temporal, spatial, and interaction terms via radial basis kernels. Equity assessment links accessibility ($Y(U, t)$) to variables such as income or racial composition, with significant coefficients signaling inequity (e.g., higher travel cost for lower-income or racialized groups).

Inference leverages simultaneous confidence bands to identify nonlinear, linear, or constant effects over $t$ or $s$:

$$P\{\alpha(t) \in CB_\gamma \ \forall t \in T\} \geq 1-\gamma$$

and interaction effects are tested using mixed-model likelihood ratio tests.

These models dissect spatial-temporal patterns of inequity, allowing policy targeting for improved distribution of opportunities.

3. Algorithmic and Network-Based SPA Computation

SPA metrics operationalized for cities and transport systems involve scalable algorithms and network models for multi-modal transit, public transport, and shared mobility (Lang et al., 2020, Diepolder et al., 2023, Šfiligoj et al., 11 Jul 2025).

For example, accessibility from origin $p$ to destination $d$ is computed as:

• Road network: OSRM for travel times, filtered initially by geodesic $d_{geo}(p, d)$, and refined via K-NN selection:

$d^* = \min_{d \in D_K} d_{net}(p, d)$

• Public transport: Aggregates walking times to stops, transfer times, and scheduled bus/rail times, maintaining order constraints for feasible paths.

Shared mobility services (SMS) introduce spatial-temporal field estimation using Ordinary Kriging, producing expected wait and travel times at all centroids $u$:

$$\hat{w}_{t_k}^s(x) = \sum_{i \in \mathcal{O}_{t_k}^s} \lambda_i w_i$$

These estimates are used to construct virtual trips that are injected into time-expanded access graphs, enabling calculation of classic indicators:

$$\text{acc}(u) = \sum_{u' \in \mathcal{C}(u, t)} O_{u'}$$

where $\mathcal{C}(u, t)$ is the set accessible within time threshold $\tau$.

The access graph (Šfiligoj et al., 11 Jul 2025) represents each stop as a node, connecting pairs if travel time $d_{ij}$ is below budget $t_b$; node degree quantifies number of reachable opportunities, while the Gini coefficient evaluates access equity across the graph.

4. Gravity-Based and Floating Catchment SPA Approaches

Gravity-based metrics are a cornerstone of SPA in job and service accessibility (Hu et al., 2020, Li et al., 2022, Verma et al., 7 Apr 2024). The SPA metric typically takes the form:

$A_{i, t} = \sum_{j=1}^n R_{j, t} f_t(d_{ij, t})$

with ratios $R_{j, t}$ capturing supply (jobs, POIs) per demand (workers, population), and $f_t(d)$ as a distance decay function. This generalizes in (Verma et al., 7 Apr 2024) to:

$$a_{k,m,\tau}(i) = \sum_{j \in \mathcal{R}} w_{ij, k, m, \tau} o_j$$

where $w_{ij, k, m, \tau} = f_{k, m}(c_{ij})$ for cost $c_{ij} \leq \tau$.

Two-step floating catchment area (2SFCA) approaches delineate catchment boundaries (isochrones) and aggregate service ratios, while machine learning models test associations with demographic variables, revealing access disparities.

5. Dimensional and Integrative SPA Frameworks

Recent research emphasizes multi-dimensional SPA (Bruno et al., 15 Sep 2025), decomposing accessibility into:

• Proximity: Short-range access, quantified as

$$A^{prox}_\lambda = \int_0^\infty dt \ \rho_\lambda^{foot}(t) \ f^{foot}(2t)$$

• Opportunity: Access to distant or significant POIs via transit,

$$A^{opp}_\lambda = \langle \int_0^\infty dt \ \rho_\lambda^{pub}(t_0, t) \ f^{pub}(2t) \rangle_{t_0}$$

• Value: Weighting of POIs by quality or significance, incorporated into opportunity and proximity sums.

A composite SPA metric may be written as:

$$SPA_\lambda = \omega_1 A^{prox}_\lambda + \omega_2 A^{opp}_\lambda + \omega_3 V_\lambda$$

Challenges include context-dependent weighting, temporal variability, and data integration.

6. SPA and Individual Capability Sets

Emerging SPA measures incorporate person-centered, capability-based views (Liao et al., 11 Oct 2025). SPA for individual $i$ quantifies the set of leisure opportunities $k$ accessible within a time budget $tb_i$, accounting for chained trips:

$$a^k_i = \begin{cases} 1 & \text{if } t_{hw} + t_{wk} + t_{kh} \leq tb_i \ 0 & \text{otherwise} \end{cases}$$

and aggregate SPA as $A^k_i = \sum_k a^k_i$.

Structural equation models reveal that SPA increases leisure diversity directly, but also reduces total travel time, which in turn may lower diversity. SPA is thus integral to understanding real freedoms and social participation, and discrepancies highlight mobility inequities.

7. Visualization and Policy Applications

SPA metrics underpin decision-support for service delivery, urban planning, and equity analysis. Four-dimensional visualizations (space-time cubes, volumetric rendering) expose dynamic patterns (Hu et al., 2020). Policy tools employing SPA can simulate network changes or resource reallocation, test scenario interventions, and uncover hidden inequalities even in regions with seemingly ample overall access (Li et al., 2022, Verma et al., 7 Apr 2024).

SPA thus comprises a rigorously structured, multi-modal, and dynamic approach to quantifying accessibility. Its application spans mathematical foundations of space–time, operational transport networks, and equity-driven analysis, positioning SPA metrics as essential instruments for research and decision-making across diverse technical domains.