Occupancy Matching: Models & Applications

Updated 24 July 2025

Occupancy matching is a methodology that quantifies and predicts the presence and distribution of agents or states using probabilistic and optimization frameworks.
It integrates diverse approaches—from Markov chains and combinatorial models to reinforcement learning—to optimize control systems, network design, and autonomous perception.
Applications demonstrate practical benefits such as energy savings, enhanced 3D mapping accuracy, and improved policy performance in imitation learning.

Occupancy matching denotes a broad class of methods and frameworks focused on aligning, inferring, or predicting the presence, arrangement, or distribution of agents, objects, or states (“occupancy”) within a spatial, temporal, or abstract environment. Approaches to occupancy matching vary widely by application—ranging from building energy management and autonomous driving perception, to imitation learning in reinforcement learning (RL), network modeling, combinatorial optimization, and matching theory in economics and operations research. The following sections provide an encyclopedic synthesis of core principles, mathematical models, and representative applications of occupancy matching as established across several research domains.

1. Mathematical Models and Formulations of Occupancy Matching

Occupancy matching frameworks are distinguished by the formalization of occupancy as a quantifiable, often probabilistic variable—such as the binary presence of an agent in a space, fractional occupation probabilities, or joint occupancy distributions over system states or graph elements. Key models include:

Markov and Semi-Markov Chain Models: In building energy and control (Dobbs et al., 2014, Kundu et al., 2021), occupancy at time $k$ is modeled as a binary Markov process, with transition probabilities (e.g., $p_k = P(\gamma_{k+1} = 1 | \gamma_k = 1)$ ) that can be estimated via Bayesian updating or extended to capture variable holding times via semi-Markov chains. Such models may employ online learning and “forgetting” factors to adapt to changing patterns.
Occupancy Fractions and Polynomials: In combinatorial settings—such as graph theory and networked systems—the expected occupancy fraction is defined as a derivative of partition functions. For the hard-core model on a $d$ -regular graph $G$ , the independence polynomial $P_G(\lambda)$ yields the occupancy fraction:

$\alpha_G(\lambda) = \frac{1}{|V(G)|} \left( \frac{d}{d\lambda} \log P_G(\lambda) \right)$

similar constructions exist for matchings and dimer models (Davies et al., 2015).

State-Occupancy Matching in RL: Denoting the (discounted) visitation distribution $d^\pi(s)$ under policy $\pi$ , occupancy matching in reinforcement learning aims to match $d^\pi(s)$ to a target (expert) distribution $d^E(s)$ , commonly via divergence minimization (e.g., $\min_\pi D_{KL}(d^\pi \| d^E)$ ) (Ma et al., 2022, Yan et al., 2023).
Geostatistical and Data Fusion Models: For public transport networks, occupancy matching leverages a combination of disparate data sources (AFC and APC), applying spatial interpolation (kriging) to estimate occupancy at unmeasured points (Dib et al., 6 Feb 2024).
Matching Mechanisms with Capacity and Preferences: In economics and operational research, occupancy reflects discrete allocation, such as assigning agents with size $s(a)$ to hospital positions, leading to definitions of blocking pairs and stability that are “occupancy-based” (Balasundaram et al., 4 Jun 2025).

2. Occupancy Matching in Building Energy and Control Systems

One of the earliest and most prominent uses of occupancy matching is in optimal operation of building HVAC systems. Control algorithms employ occupancy prediction to match the conditioning of thermal zones with expected human presence (Dobbs et al., 2014). A typical implementation involves:

An online-trained stochastic Markov chain occupancy model that predicts future occupancy states with continual adaptation via Bayesian updates and a linear forgetting factor.
Integration with a Model Predictive Control (MPC) system in which predicted occupancy probabilities directly modulate a cost function balancing energy and occupant comfort:

$g(\tilde{x}, u, \Gamma) = \Gamma \beta (x_{zone} - \tau)^2 + r |u|$

where $\Gamma$ is the occupancy probability, $x_{zone}$ the zone air temperature, $\tau$ the setpoint, and $r$ the energy cost penalty.

Real-time occupancy matching ensures that pre-conditioning (heating/cooling) anticipates arrivals and avoids unnecessary operation during vacancy.

Empirical simulations demonstrate that occupancy-predicting MPC achieves up to 19% reduced energy consumption compared to scheduled controllers, while maintaining low occupant discomfort and adapting to changing usage patterns.

3. Occupancy Matching in Graphs, Networks, and Combinatorics

In combinatorial optimization and statistical physics, occupancy matching arises in maximizing the expected number of occupied elements (e.g., independent sets, matchings) within discrete structures.

Occupancy Fractions measure the population-average probability that a vertex or edge is occupied under a probabilistic model. Extremal results show that for all $d$ -regular graphs and fugacity parameter $\lambda$ , the complete bipartite graph $K_{d,d}$ achieves the highest possible occupancy fraction (Davies et al., 2015).
The methodology involves mapping the global occupancy maximization problem to a constrained optimization over distributions of local random variables (such as the number of uncovered neighbors) and leveraging dual linear programming:

$\alpha = \frac{\lambda}{1+\lambda} \mathbb{E}[(1+\lambda)^{-Y}]$

Subject to constraints such as $\mathbb{E}[Y] = d \cdot \mathbb{E}[(1+\lambda)^{-Y}]$ .

These approaches yield almost tight upper bounds for the number of independent sets or matchings of a given size and provide local interpretations for classical partition functions.

This general paradigm finds broader use in designing optimal network codes, spin systems, and analyzing system capacity under resource sharing.

4. Occupancy Matching for 3D Perception and Autonomous Driving

In autonomous driving and robotics, occupancy matching has become central for unified, geometry-aware representations of the environment and downstream decision making.

3D Occupancy Prediction involves inferring a voxel grid where each cell is labeled as occupied, free, or assigned a semantic class. Dense occupancy maps—quantized at resolutions down to 0.25–0.5 m—capture both fine-grained object shapes and background surfaces (Sima et al., 2023).
Frameworks such as AdaOcc adopt adaptive-resolution occupancy, combining coarse holistic scene occupancy with high-detail point cloud reconstruction in regions of interest, thus balancing computational efficiency and perceptual fidelity (Chen et al., 24 Aug 2024).
Novel loss functions and training regimes support the matching of predicted occupancy not only to ground truth LiDAR points (using sampling strategies and binary cross-entropy) but to semantic labels, by first learning geometry (binary occupancy) and subsequently mapping to semantics using either few annotated data points or massive binary datasets (Jury et al., 2023).
The introduction of flow matching modules, selective state space models, and robust mask training (for handling sensor loss) enables occupancy prediction even with sparse or occluded observations, as in the FMOcc architecture (Chen et al., 3 Jul 2025).

Empirical benchmarks on nuScenes, Occ3D, and OpenOcc reveal substantial improvements in Intersection over Union (IoU) and Hausdorff distance, as well as lower collision rates in motion planning tasks.

5. Occupancy Matching in Offline Imitation Learning and RL

A major recent theme connects occupancy matching to sample-efficient, robust imitation and reinforcement learning.

In offline imitation learning, the core target is to match the state occupancy measure $d^\pi(s)$ of the learner’s policy to that of an expert, even when expert actions are missing (imitation from observation). Approaches such as SMODICE and PW-DICE solve $\min_\pi D(d^\pi \| d^E)$ , where $D$ may be an $f$ -divergence (KL, chi-square) or the Wasserstein metric, the latter often using contrastively learned state embeddings for improved modeling (Ma et al., 2022, Yan et al., 2023).
The optimization is frequently formulated via duality, as in Fenchel-conjugate based objectives, or via linear programming in the primal Wasserstein formulation, capable of integrating BeLLMan flow constraints across large state spaces.
Pre-training large “flow occupancy” models via ODE-based generative modeling (flow matching) over temporally distant state trajectories, and conditioning on latent intentions (capturing multi-modal behaviors in heterogeneous data), further enhances adaptation in downstream RL tasks (Zheng et al., 10 Jun 2025).
Applications range from skill discovery, transfer learning, and robust policy distillation, to modeling foundation models for RL analogously to large-scale pre-trained models in natural language processing.

Reported results consistently show state-of-the-art performance in benchmark tasks, improved stability, and the ability to operate in domains where expert actions are not directly observed.

6. Data Fusion, Forecasting, and Resource Allocation via Occupancy Matching

Beyond perception and RL, occupancy matching appears as a central concept in several practical domains:

Public Transport Occupancy Estimation: Integrating automated fare collection (AFC) and automatic passenger counting (APC) data, adjusted by locally or spatially estimated fraud rates, yields “unified occupancy” across lines and schedules (Dib et al., 6 Feb 2024). Spatial interpolation (ordinary kriging) is used where direct APC data is absent, yielding robust estimations with weighted Mean Absolute Percentage Error (wMAPE) below 15% even with sparse APC deployment.
Matching Markets and Resource Allocation: In hospital-residents assignment with sized agents, occupancy-stable matchings guarantee that hospital capacity is utilized efficiently, refining classic blocking pair definitions to exclude capacity-wasting allocations (Balasundaram et al., 4 Jun 2025). Algorithmic results include both hardness results and efficient 3-approximation schemes for maximum-occupancy matchings.

In each case, the unifying thread is the careful alignment or prediction of occupancy—matching inferred, predicted, or assigned occupancy states with operational targets or ground-truth structures, often under strong constraints, cost or welfare objectives, and in the presence of partial or mismatched information.

7. Implications and Future Directions

The spread of occupancy matching methodologies has far-reaching implications:

In energy systems and building controls, adaptive occupancy matching directly enables substantial energy savings and comfort optimization at scale.
In autonomous driving and robotic perception, increasingly sophisticated occupancy representations drive advances in robust scene understanding, safety, and planning.
In RL and imitation learning, occupancy matching unifies divergence-based, Wasserstein, and flow-based generative modeling, enabling sample-efficient, intention-adaptive learning from both observations and multimodal datasets.
In transport and resource allocation, hybrid models and occupancy-based stability criteria foster fairer, more efficient, and automated allocation under real-world constraints.

Open challenges include scaling occupancy matching to high-dimensional spaces, integrating more complex sensor modalities, further improving robustness to distribution shifts and missing data, and unifying occupancy matching across physical, semantic, and abstract domains. Recent work suggests promising avenues in instance-level scene optimization, continual adaptation, and broader integrations with large-scale, pre-trained models for complex decision making.