arxiv: 2605.10562 · v1 · submitted 2026-05-11 · 🧮 math.NA · cs.NA

Recognition: no theorem link

Data-driven moving-window Bayesian inference for transient CO2-temperature network models of buildings

Clemens V. Verhoosel, Idoia Cortes Garcia, Stein K.F. Stoter, Zhijian Wang

Pith reviewed 2026-05-12 05:04 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Bayesian inferenceCO2-temperature modelRC networkbuilding thermal dynamicsmoving-window estimationsparse sensingairflow modelingoccupancy estimation

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The pith

Coupling CO2-informed airflow with thermal dynamics in a network model allows robust temperature prediction in buildings using sparse sensor data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a network model that connects multi-zone CO2 transport and thermal dynamics through shared latent variables for airflow and occupancy. Thermal parts use an RC network with added convective terms from airflow, while CO2 uses convective transport equations. A moving-window Bayesian procedure jointly infers the parameters, trajectories, and noise levels from limited measurements, providing forecasts with uncertainty bands. This setup is tested on synthetic data and a physical experiment, showing accurate reconstruction and quick adaptation after regime changes. The result supports practical building monitoring and energy assessment when full sensing is unavailable.

Core claim

The authors propose that linking CO2 and temperature dynamics via shared airflow and occupancy in a network model, calibrated by moving-window Bayesian inference, yields accurate posterior reconstructions and low-error forecasts while providing diagnostics for model mismatch at abrupt changes.

What carries the argument

The CO2-temperature network model with shared latent airflow and occupancy, calibrated via moving-window Bayesian inference that estimates thermal parameters, trajectories, and sensor noise.

Load-bearing premise

The RC network with airflow convective exchange and CO2 transport, linked by shared latent drivers, sufficiently captures the building's dynamics, and the moving-window inference tracks changes without major bias or delay.

What would settle it

Observing that forecast errors stay high or uncertainty bands do not adapt after a known regime shift in the physical experiment would indicate the claim fails.

Figures

Figures reproduced from arXiv: 2605.10562 by Clemens V. Verhoosel, Idoia Cortes Garcia, Stein K.F. Stoter, Zhijian Wang.

**Figure 1.** Figure 1: Schematic of the room layout. To formalize the model, we define the sets of nodal states as ϕ = {ϕi} and T = {Ti}. Denoting the “unknown” (to be inferred) parameters that determine the CO2 and temperatuer models as θ ϕ and θ T , and denoting the “known” model parameters as κ ϕ and κ T , the forward models are expressed as the residuals r ϕ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Schematic of local-to-global correspondence in the network. Here, ei denotes the ith oriented edge of the graph, where qi flows in the specified direction. The CO2 evolution is modeled as purely convective, and hence the network is described by an oriented graph whose nodes represent zones that store CO2 mass and whose edges represent inter-zonal convective exchange. For each (globally counted) node ni , t… view at source ↗

**Figure 3.** Figure 3: , we use data D = Dϕ ∪ DT contained in the current inference window to infer the parameter vector θ = θ ϕ ∪ θ T of the coupled CO2-temperature model. The inference step produces not only a single parameter estimate, but a range of plausible parameter values consistent with the noisy measurements. We use the resulting parameter samples to predict how CO2 and temperature behave beyond the inference window, a… view at source ↗

**Figure 4.** Figure 4: Schematic of Metropolis–Hastings sampling in one dimension. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Illustration of the benchmark problem room layout and network structure. 4.1.2. Parameter configuration and data generation Our benchmark problem considers two sets of parameters, parameters to be inferred (θ = θ ϕ ∪ θ T ), and constants or known parameters (κ = κ ϕ ∪ κ T ). The parameters to be inferred and the constants are listed as: θ = {nppl, q, R, C}, (12a) κ = {BCϕ , BCT , {Vi}, qexh, ϕexh, Qppl, cp… view at source ↗

**Figure 6.** Figure 6: Synthetic ground-truth benchmark results for rooms A, E, and F, with and without noise. 4.2.2. Moving-window prediction and parameter tracking [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

**Figure 7.** Figure 7: Posterior predictive CO2 trajectories in rooms A, E and F for the synthetic-data experiment. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

**Figure 8.** Figure 8: Posterior predictive temperature trajectories in rooms A, E and F for the synthetic-data experiment. The corresponding evolution of the inferred parameters of the coupled CO2 and temperature model are shown in [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗

**Figure 8.** Figure 8: Tracking of model parameters under moving-window Bayesian inference, showing the posterior mean (solid lines) and 95% credible intervals (shaded bands). variability, which is expected because multiple resistance configurations can yield similar temperature trajectories over a finite window. Overall, it indicates that the moving-window Bayesian inference tracks RC parameters in a temporally coherent and phy… view at source ↗

**Figure 9.** Figure 9: Inferred CO2 noise levels in rooms A, E and F. Solid lines denote posterior means and shaded bands indicate 95% credible intervals as a function of the window start time. 4.2.4. Change window size and noise level To evaluate the sensitivity of the Bayesian inference to the moving-window size and measurement noise, we vary the inference window size from 10 to 60 data points under noise level of synthetic CO… view at source ↗

**Figure 10.** Figure 10: nRMSE of prediction and ground-truth data, evaluated under different inference window size 10–60 and paired measurement-noise settings (σCO2 , σT ). (a) shows CO2 and (b) shows temperature. 5. Experimental validation To assess the predictive performance of our modeling framework on real measurement data that correspond to time-varying operating conditions, we now implement the movingwindow Bayesian infer… view at source ↗

**Figure 11.** Figure 11: Experiment setup Atmospheric reference sensors are installed outside the enclosure to record background CO2 and temperature for boundary conditions. To simulate human-body CO2 and heat production in room F, a premixed CO2 gas cylinder and a heating module were used. The premixed gas had a nominal CO2 concentration of 5000 ppm, as supplied by Nippon Gases [31]. The dosing flow rate is regulated to 25 mL/m… view at source ↗

**Figure 12.** Figure 12: Posterior predictive CO2 concentration (left column) and temperature (right column) trajectories in rooms A, D, E, and F, for the scaled physical experiment. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗

**Figure 13.** Figure 13: Posterior mean (solid lines) and 95% credible intervals (shaded bands) of the inferred occupancies and fa¸cade airflow, for the scaled physical experiment. 5.2.3. Prediction evaluation To assess how well each posterior generalizes beyond the inference interval, we compute normalized root-mean-square error (nRMSE) between the model prediction and experimental measurements over a horizon of 80 samples aft… view at source ↗

**Figure 14.** Figure 14: Mean nRMSE over an 80 samples post-window prediction horizon, averaged over 8 rooms, for CO2 and temperature. room-wise inspection (not shown in the figure) indicates that the largest forecast errors concentrate in the directly actuated zone (room F), which is expected because it experiences the strongest source terms and the sharpest regime change. The remaining zones show smaller errors driven primaril… view at source ↗

read the original abstract

In this work, we proposes a CO2-temperature network model that links multi-zone mass transport and thermal dynamics through shared latent drivers, airflow and occupancy. The thermal component is formulated as a resistance-capacitance (RC) network augmented with airflow-driven convective exchange, while the CO2 component is governed by inter-zonal convective transport. To calibrate the model and track time-varying operating conditions based on sparse sensing, we introduce a moving-window Bayesian inference procedure that jointly estimates thermal parameters, airflow and occupancy trajectories. The estimation also provides room-specific sensor noise levels, yielding posterior predictive forecasts with credible intervals. The framework is assessed using a controlled synthetic benchmark, and a scaled physical validation experiment using CO2 and temperature sensing. In both settings, the posterior accurately reconstructs trajectories within windows and delivers low forecast errors. When inference windows overlap abrupt regime transitions, the widened uncertainty bands and increased inferred noise levels provide an interpretable diagnostic of model-data mismatch, followed by rapid recovery once the new regime is observed. Overall, coupling CO2-informed airflow with thermal dynamics yields a robust approach for conductive and advective temperature prediction, supporting practical monitoring and energy-performance assessment under limited sensing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The moving-window Bayesian inference for the coupled CO2-temperature model works in the reported tests, but joint estimation of parameters and latents risks confounding unless the thermal RC values are held fixed globally.

read the letter

The main takeaway is that this paper gives a concrete way to do joint Bayesian estimation of building thermal parameters and latent airflow/occupancy using CO2 and temperature measurements in a network model, with moving windows to handle changing conditions. They model the thermal part as an RC network with added convective terms from airflow, and the CO2 part as convective transport between zones. The shared latents tie them together. The inference procedure estimates everything together and gives posterior predictions with intervals. It does well on the validation side. Both the synthetic case and the physical scaled experiment show accurate reconstruction inside the windows and decent forecasts ahead. The uncertainty increases at abrupt changes, which serves as a useful flag, and then it recovers. The potential issue is in how the parameters are handled across windows. Since the procedure jointly estimates the thermal parameters with the trajectories, it is not clear if those parameters stay constant over the whole time series or get updated in each window. If they can adjust locally, then the good performance at transitions might come from the model refitting to the new data segment instead of the CO2 data properly informing the airflow and thus the temperature predictions. That would make the central claim about the coupling less convincing. The abstract mentions interpretable uncertainty at transitions but does not quantify how much is due to the linkage versus parameter flexibility. Overall the work is solid for what it sets out to do. It is aimed at people doing practical building monitoring and energy assessment who need to deal with sparse sensors and time-varying conditions. The combination of network modeling and Bayesian tracking is a reasonable extension of existing RC and data assimilation techniques. I would bring this to a reading group for the method details and the experiment setup. It should go to peer review. The validation is there and the concerns are fixable with more explanation on the estimation setup.

Referee Report

2 major / 2 minor

Summary. The paper proposes a CO2-temperature network model for multi-zone buildings that couples thermal RC dynamics (augmented with airflow-driven convective exchange) to inter-zonal CO2 convective transport through shared latent drivers (airflow and occupancy). A moving-window Bayesian inference procedure jointly estimates thermal parameters, airflow trajectories, occupancy, and sensor noise levels from sparse measurements, yielding posterior predictive forecasts with credible intervals. Validation on a controlled synthetic benchmark and a scaled physical experiment reports accurate trajectory reconstruction, low forecast errors, and interpretable widening of uncertainty bands at abrupt regime transitions followed by rapid recovery.

Significance. If the moving-window procedure can be shown to attribute predictive performance specifically to the CO2-informed airflow coupling rather than to local parameter refitting, the approach would provide a practical, data-driven framework for transient building monitoring and energy assessment under limited sensing. The explicit handling of regime transitions via uncertainty diagnostics is a methodological strength that could support real-world deployment.

major comments (2)

[Abstract; moving-window Bayesian inference procedure] Abstract and the description of the moving-window Bayesian inference procedure: the claim that coupling CO2-informed airflow with thermal dynamics 'yields a robust approach' for prediction depends on the joint estimation correctly resolving convective exchange and transport via the shared latent drivers. However, the procedure jointly estimates thermal parameters alongside airflow and occupancy trajectories within each window. If the RC parameters are permitted to vary (or be re-estimated) locally rather than held globally fixed, discrepancies in advective terms can be absorbed into parameter adjustments instead of being enforced by the CO2-temperature linkage. This directly affects the central claim of robustness at regime transitions, where only widened uncertainty and 'rapid recovery' are reported without quantifying whether recovery is due to correct prediction or refitting.
[Validation on synthetic benchmark and physical experiment] Validation sections (synthetic benchmark and physical experiment): the reported low forecast errors and accurate reconstruction within windows do not include an ablation or sensitivity test that fixes thermal parameters globally across windows while allowing only airflow/occupancy to vary. Without such a control, it remains unclear whether the CO2 coupling is load-bearing for the performance or whether the joint estimation masks model mismatch.

minor comments (2)

[Abstract] Abstract: grammatical error in 'we proposes' should be corrected to 'we propose'.
[Abstract] The abstract mentions 'room-specific sensor noise levels' but does not clarify how these are parameterized or whether they are estimated per window or globally; this notation should be made explicit for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive feedback. We address the two major comments point-by-point below and commit to revisions that include the suggested ablation studies to better isolate the contribution of the CO2 coupling.

read point-by-point responses

Referee: Abstract and the description of the moving-window Bayesian inference procedure: the claim that coupling CO2-informed airflow with thermal dynamics 'yields a robust approach' for prediction depends on the joint estimation correctly resolving convective exchange and transport via the shared latent drivers. However, the procedure jointly estimates thermal parameters alongside airflow and occupancy trajectories within each window. If the RC parameters are permitted to vary (or be re-estimated) locally rather than held globally fixed, discrepancies in advective terms can be absorbed into parameter adjustments instead of being enforced by the CO2-temperature linkage. This directly affects the central claim of robustness at regime transitions, where only widened uncertainty and 'rapid recovery' are reported without quantifying whether recovery is due to correct prediction or refitting.

Authors: We appreciate this insightful comment highlighting a potential ambiguity in attributing performance to the model coupling versus local parameter estimation. The moving-window Bayesian inference is designed to adapt to changing conditions, but we recognize that without additional controls, the role of the CO2-temperature linkage in enforcing consistency may not be fully isolated. In the revised manuscript, we will add an explicit discussion of this aspect and perform a sensitivity analysis on the synthetic data to quantify the impact. This will support our claim of robustness by showing that the coupling aids in correct prediction rather than solely relying on refitting. revision: yes
Referee: Validation sections (synthetic benchmark and physical experiment): the reported low forecast errors and accurate reconstruction within windows do not include an ablation or sensitivity test that fixes thermal parameters globally across windows while allowing only airflow/occupancy to vary. Without such a control, it remains unclear whether the CO2 coupling is load-bearing for the performance or whether the joint estimation masks model mismatch.

Authors: We agree that the absence of such an ablation test leaves open the possibility that joint estimation is compensating for model discrepancies. To address this, we will implement and report an ablation study in the revised paper. Specifically, for the synthetic benchmark, we will re-run the inference with thermal RC parameters held fixed at their globally estimated values (from the full dataset or initial window) and compare the resulting forecast errors and reconstruction accuracy against the original results where parameters are allowed to vary per window. A similar analysis will be attempted for the physical experiment to the extent possible with available data and computation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; data-driven inference is self-contained

full rationale

The paper describes a moving-window Bayesian procedure that jointly estimates thermal RC parameters, airflow, and occupancy trajectories directly from sparse CO2 and temperature measurements. Validation occurs on independent synthetic benchmarks and physical experiments, with performance assessed via posterior predictive forecasts and credible intervals. No derivation step reduces a claimed prediction to a fitted input by construction, no self-citation is invoked as a uniqueness theorem or load-bearing premise, and the model equations (RC network augmented with convective terms) are calibrated rather than tautologically defined from the outputs they produce. The approach therefore remains externally falsifiable against held-out data segments and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The central claim rests on standard building physics modeling assumptions and Bayesian statistical procedures; free parameters are the quantities estimated from data rather than ad hoc choices.

axioms (3)

domain assumption Building thermal dynamics can be represented as an RC network augmented with airflow-driven convective exchange.
Core of the thermal component formulation in the abstract.
domain assumption CO2 dynamics are governed by inter-zonal convective transport linked to the same airflow.
Core of the CO2 component and coupling in the abstract.
domain assumption Moving-window Bayesian inference can jointly estimate time-varying parameters, trajectories, and noise from sparse sensing.
Core of the calibration procedure described.

invented entities (1)

shared latent drivers (airflow and occupancy) no independent evidence
purpose: Link multi-zone mass transport and thermal dynamics
Introduced as inferred variables connecting the CO2 and temperature components.

pith-pipeline@v0.9.0 · 5521 in / 1750 out tokens · 56481 ms · 2026-05-12T05:04:40.603231+00:00 · methodology

discussion (0)

Reference graph

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