📡 Analytical Methods — A Primer

Why this topic? Wireless sensing systems don't just collect raw signals — they rely on mathematical and physical models to turn those signals into meaningful counts, locations, and behaviours. Understanding the analytical methods behind these systems is what separates a "black box" from a system you can actually reason about, extend, and trust.

Background

Imagine you drop a pebble into a still pond. The ripples spread outward in perfectly predictable rings — and if you know the physics of water waves, you can work backwards from those rings to figure out exactly where the pebble landed. Wireless sensing works on a similar principle: a WiFi router sends out radio waves, people or objects disturb those waves, and mathematical models let us decode what caused those disturbances.

This is harder than it sounds. Real rooms are not still ponds. Walls reflect signals, furniture scatters them, and a dozen people moving at once create overlapping disturbances that are nearly impossible to untangle without a principled analytical framework. The role of analytical methods is to provide that framework — equations and models that describe, from first principles, how a physical phenomenon (a person walking, a crowd gathering) affects a measurable signal.

Analytical approaches sit between two extremes. At one end are pure empirical or "black-box" machine learning methods that learn patterns from data without explaining why they work. At the other end are fully physics-based simulations that model every photon or radio wave but are computationally intractable. Analytical methods occupy the productive middle ground: they capture the dominant physics in tractable equations, make predictions that can be tested against real data, and generate insights that guide system design. In the wireless sensing domain, this means deriving closed-form expressions for how crowd size relates to CSI (Channel State Information — a fine-grained measure of how a WiFi signal changes as it travels from transmitter to receiver) variance, or writing differential equations that describe how pedestrian density flows through a corridor.

The papers in this vault span three research communities that each bring their own analytical traditions: the wireless sensor network community (signal propagation models, deployment geometry), the indoor spatial modelling community (graph-theoretic and topological representations of space), and the crowd dynamics community (continuum PDEs, agent-based force models). Understanding these analytical threads — and where they intersect — is essential for building sensing systems that are robust, interpretable, and deployable.

Key Methods

The analytical toolkit across these papers draws on several overlapping families of methods:

• Signal propagation and channel models — Fresnel zone theory, Doppler shift analysis, OFDM (Orthogonal Frequency Division Multiplexing, a technique that splits a WiFi signal across many parallel sub-channels called subcarriers) subcarrier modelling, and CSI ratio formulations
• Statistical and probabilistic models — Gaussian stochastic process models, KL-Divergence (a measure of how different two probability distributions are), variance and power spectral analysis, particle filters for state estimation
• Continuum and PDE-based crowd models — Eulerian flow equations, mean-field approximations, SEIR compartmental models for disease spread, Hamilton-Jacobi potential field equations
• Microscopic force-based models — Social Force Models, agent-based simulation, collision impulse mechanics
• Graph-theoretic and topological methods — Poincaré duality, Node-Relation Structures, cellular space decomposition (IndoorGML)
• Optimisation methods — Voronoi-based sensor deployment, parameter estimation as constrained optimisation, data assimilation via particle filters
• Physics-informed learning — hybrid models that embed physical equations as constraints inside neural networks

Active Research in the Vault

Analytical signal models for WiFi crowd counting. torun2026_72aa derives a closed-form relationship between the probability distribution of natural body fidgeting and the bandwidth of received WiFi signals, enabling calibration-free counting of seated crowds. Building on related signal theory, ling2024_4782 models each person's reflected CSI as an independent Gaussian stochastic process, then derives the Ratio of Dynamic power to Noise power (RDN) as the counting statistic — requiring only single-person prior data rather than large labelled datasets.

CSI propagation and sensing range models. wu2022_75d3 provides a systematic analytical account of how Fresnel zone geometry (concentric ellipsoidal shells around the transmitter-receiver pair) governs whether a target causes diffraction or reflection, directly determining which model class applies. zeng2021_1e4f derives the CSI ratio method analytically, showing it cancels hardware-induced phase noise and extends effective sensing range. li2022_4220 compares signal-level models (RSSI, CSI, radar) analytically to bound what each can theoretically resolve about human motion.

Doppler and Integrated Sensing and Communications (ISAC). yang2026_6c4f presents a geometric derivation of Doppler shift in sparse multipath channels, extending classical formulas to realistic indoor geometries. aljarrah2023_e060 provides an information-theoretic analytical framework unifying sensing and communication performance under a single KL-Divergence bound. lee2026_14e8 uses an iterative maximum likelihood framework to estimate direction-of-arrival from a single RF chain — a setting where standard array processing theory breaks down. zhou2022_09d5 surveys the analytical trade-offs between OFDM waveform parameters (subcarrier spacing, cyclic prefix length) and achievable sensing resolution.

Continuum models of pedestrian flow. hughes2002_57b4 is the foundational paper deriving a PDE (partial differential equation) system for crowd motion: pedestrians minimise a "cost" function related to both travel time and local density, yielding a Hamilton-Jacobi equation coupled to a conservation law. huang2009_292f extends this by testing multiple speed-density relationships (Greenshields, Underwood, Edie) and proposing a numerically efficient solver. ghattassi2025_a886 generalises the Hughes model using mean-field game theory to allow pedestrian groups to choose between navigating through or around high-density regions.

Microscopic force-based and agent-based models. helbing1995_149d introduced the Social Force Model (SFM), in which pedestrians are treated as particles subject to attractive forces (destination) and repulsive forces (other pedestrians, walls) — a framework that has become the most-cited microscopic crowd model. helbing2000_a943 applied the SFM analytically to show that physical pressures in jammed crowds can reach 4,450 N/m, consistent with reported structural failures in crowd disasters. Importantly, wang2023_9e5f refutes the panic simulation results of Helbing 2000 by introducing collision impulse mechanics, arguing that the original SFM overestimates risk in certain configurations. echeverrahuarte2023_fcc4 couples a mechanical repulsive-force model with a decisional process model to analytically separate physical collision avoidance from cognitive distraction effects.

Hybrid and data-driven analytical models. makinoshima2022_7e21 combines a force-based simulation with a particle filter to sequentially estimate latent behavioural parameters from aggregate density maps — bridging analytical simulation and real-time data assimilation. malleson2020_7b38 demonstrates the same data assimilation architecture (Agent-Based Model + particle filter) for crowd management in real time. di2023_285b surveys how physical conservation laws can be embedded as loss-function constraints in neural networks, improving sample efficiency while preserving analytical interpretability. porzycki2023_6cf3 derives a data-driven "Interplay Floor Field" by fitting cellular automata transition probabilities directly to trajectory data, fusing model structure with empirical calibration. wolinski2014_f409 frames simulation parameter estimation as a formal optimisation problem, enabling fair analytical comparison across different model families.

Spatial and topological analytical frameworks. lee2005_2ce1 introduces a graph-theoretic combinatorial data model (Node-Relation Structure) for representing 3D indoor topology, enabling shortest-path and spatial query analysis. kang2017_8400 formalises indoor space analytically via Poincaré duality: the primal cellular decomposition (room geometry) is paired with a dual navigation graph, enabling both geometric and topological reasoning. diakit2018_003e extends this by analysing how different space subdivision strategies affect navigation graph properties analytically. ojagh2020_321f applies IndoorGML's topological model analytically to reason about surface contact exposure risk in indoor environments.

Sensor deployment optimisation. afghantoloee2021_8628 derives a coverage optimisation framework using 3D Voronoi decomposition (partitioning space into regions closest to each sensor) to analytically minimise coverage gaps in complex geometries. zhen2022_bb0b formulates BLE (Bluetooth Low Energy) beacon placement as a combinatorial optimisation problem and derives adaptive placement heuristics analytically tuned to crowd density distributions.

Evacuation and safety models. sun2021_1423 derives a two-layer analytical model: an upper potential-field layer (treating pedestrians as charged particles to pre-plan paths) combined with a lower SFM layer for local dynamics. agnelli2023_ea3a analytically derives a spatial kinetic model coupling SEIR (Susceptible-Exposed-Infected-Recovered) disease compartments with crowd flow, demonstrating formally that mean-field compartmental models fail for small-to-medium populations in bounded spatial domains. haghani2023_5c35 surveys the analytical models underpinning the Swiss Cheese Model of Crowd Safety, identifying gaps in current frameworks.

Network flow and occupancy estimation. park2022_a4e0 derives a probabilistic occupancy estimator from WiFi association counts without requiring ground-truth labels — using synthetic population simulations as the analytical prior. bin2008_8bb7 applies Euclidean-distance and variance-similarity metrics analytically to network traffic flows for anomaly detection. negi2024_b639 uses Péclet number analysis (the ratio of advective to diffusive transport) to analytically characterise distance-control regimes in active particle systems — a framework applicable to cognitive agents like pedestrians. wartelle2026_8b5e bridges queueing theory (analytical) and machine learning (predictive) for emergency department flow, identifying the conditions under which each approach is analytically appropriate. sun2026_2f5e proposes a neural radiance field framework for wireless channel modelling, with an analytical foundation in Okumura-Hata propagation theory. wang2015_48cf provides an early analytical taxonomy of WiFi sensing methods, classifying them by the signal feature (Doppler shift, CSI amplitude, fingerprint) and the corresponding physical model. maury2018_d24a is a graduate-level mathematical text that rigorously analyses microscopic and macroscopic crowd models — well-posedness, stability, and characteristic phenomena — serving as a theoretical anchor for the entire crowd dynamics literature in this vault. alattas2020_0e4e combines Land Administration Domain Model topology with WiFi log analysis to reason analytically about user movement during a real evacuation drill. barrachinamunoz2019_06bc provides an analytical simulation platform for WLAN (Wireless Local Area Network) performance under dense deployment — a necessary tool for evaluating the sensing infrastructure assumptions made by other methods. chaudhari2026_85b1 frames CSI sensing analytically as an infrastructure layer whose value can be modelled within real-estate investment frameworks. david2025_866a provides a formal analytical privacy proof — introducing "Timed-sequence-indistinguishability" as a stronger privacy notion than existing definitions for BLE beacon randomisation schemes. peng2026_e950 applies Chain-of-Thought and Retrieval-Augmented Generation (RAG) analytically to knowledge graph construction — relevant as a method for organising the structured knowledge produced by other analytical models in the vault. taktak2026_5e30 analyses the formal conditions under which distributed WSN algorithms can be safely reconfigured during operation — a correctness problem requiring analytical verification. ren2006_378b uses spherical harmonic basis functions to analytically approximate low-frequency visibility integrals in real-time rendering — a mathematical technique with structural analogies to spatial channel modelling. makinoshima2022_7e21 additionally demonstrates that particle filter convergence can be analytically bounded as a function of agent count and observation noise.

Open Problems & Gaps

• Can a single unified analytical framework span both the signal layer (CSI propagation) and the crowd layer (pedestrian dynamics)? Current work treats these as separate models — deriving, for instance, how CSI variance scales with crowd size (as in ling2024_4782) independently of how crowd density evolves over time (as in hughes2002_57b4). A joint analytical model would enable sensing-aware crowd simulation and simulation-aware signal processing.

• How do analytical crowd models fail at small population sizes, and can this be formally quantified for sensor system design? agnelli2023_ea3a shows that mean-field approximations break down for small-to-medium populations, but there is no agreed analytical threshold or correction term that sensor system designers can use to know when a continuum model is trustworthy.

• Can analytical signal propagation models (Fresnel zones, Doppler geometry) be inverted efficiently enough for real-time, room-scale crowd localisation? wu2022_75d3 and [[Dop

History