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

Buildings are not static boxes; they are living systems that breathe, store heat, and react to the people inside them. Every time a door opens, a person walks in, or a window cracks, the air inside shifts, carrying both warmth and carbon dioxide. For decades, engineers have tried to predict how these changes affect indoor comfort, relying on models that treat heat and air quality as separate problems. They know that walls conduct heat and that people generate warmth, but they often struggle to track the invisible currents of air that move between rooms, especially when sensors are sparse or the building's behavior changes suddenly. The challenge has been to create a system that can learn from limited data in real time, distinguishing between a sensor glitch and a genuine shift in how the building is behaving, all while keeping the calculations fast enough to be useful for controlling heating and cooling systems.

A team of researchers at Eindhoven University of Technology has developed a new approach that treats a building's temperature and its carbon dioxide levels as two sides of the same coin. Instead of looking at heat and air quality in isolation, they created a model that links them together through the invisible drivers of airflow and human presence. In this framework, the movement of air is the common thread: the same air currents that carry carbon dioxide from one room to another also carry heat. By observing how both temperature and carbon dioxide change over time, the system can work backward to figure out where people are, how air is moving between zones, and how the building itself stores or releases energy. This method does not require a dense network of expensive sensors; it works with the sparse data typically available in office buildings, using a technique called moving-window Bayesian inference. This approach looks at a short, recent slice of time—like a forty-minute window—rather than trying to analyze the entire history of the building at once. This allows the model to adapt quickly when conditions change, such as when a meeting ends and people move from a small office to a large conference room.

To test their idea, the researchers first built a perfect, computer-generated version of a building with four small offices, two large meeting rooms, and a connecting hallway. In this simulated world, they knew the exact truth: they knew exactly how many people were in each room, exactly how fast the air was moving, and exactly how the walls conducted heat. They then introduced realistic sensor errors, mimicking the slight inaccuracies found in cheap, real-world sensors. When they ran their new model against this data, it successfully reconstructed the hidden patterns of airflow and occupancy. The model was able to track the movement of people from the small offices to the large meeting room, adjusting its internal estimates of air currents and heat storage to match the new reality. Crucially, when the system encountered a sudden change it hadn't seen before, it didn't just guess; it signaled its uncertainty. The range of possible answers widened, effectively telling the user, "I am not sure yet," until enough new data flowed in to confirm the new state. This ability to quantify doubt is a key feature, as it prevents the system from making confident but wrong predictions during chaotic transitions.

The researchers then compared their combined model against a simpler version that only looked at temperature. In the perfect, simulated environment, the temperature-only model performed reasonably well because the heat and carbon dioxide were so tightly linked by the physics of the simulation. However, the combined model proved superior in one specific way: it adapted much faster to the sudden change in occupancy. When the people moved, the model that used carbon dioxide data updated its understanding of the airflow and the number of people almost immediately, while the temperature-only model lagged behind. This suggested that even when temperature data is available, the extra layer of carbon dioxide information helps the system lock onto the correct explanation for what is happening inside the building more quickly.

To see if this held up in the real world, the team built a scaled-down physical model of the same building layout. They used small fans to drive air between rooms and a controlled source of carbon dioxide and heat to simulate a group of people gathering in one room. Unlike the perfect computer simulation, this physical setup had real-world imperfections: the air didn't mix perfectly, the sensors had their own quirks, and the building's behavior didn't always match the theoretical equations exactly. Despite these mismatches, the model performed impressively. It successfully tracked the rise and fall of carbon dioxide and temperature as the artificial "people" turned their source on and off. More importantly, the combined model provided a much clearer picture of what was happening than the temperature-only version. In the real experiment, the temperature data alone led the model to invent false sources of heat in rooms where no one was present, trying to explain the temperature changes with incorrect assumptions. The carbon dioxide data acted as a reality check, preventing these errors and narrowing the range of uncertainty around the airflow estimates. The model learned that the changes in temperature were driven by the known source in the main room, not by phantom heat sources elsewhere.

The study also explored how the size of the time window affected the results. They found that using a longer window of data generally improved accuracy, smoothing out the noise and allowing the model to better distinguish between random sensor errors and real trends. However, if the window was too long, the model became sluggish and slow to react to sudden changes. The sweet spot allowed the system to be both accurate and responsive. The researchers also discovered that the model could estimate the noise level of each individual sensor. If a sensor started acting up or if the building's behavior deviated significantly from the model's expectations, the system would automatically widen its uncertainty bands, signaling that the data was less reliable than usual.

While the results are promising, the researchers are careful to note the limitations. Their model assumes that the air in each room is perfectly mixed, which is rarely true in reality; in a real office, air near a vent might be cooler than air near a window. They also focused on a simplified scenario where the only heat source was the simulated people, ignoring other common factors like sunlight, computers, or lighting. In a real building, these additional heat sources could complicate the picture, making it harder to separate the effects of people from the effects of equipment. The researchers suggest that future work will need to incorporate these extra factors to make the model robust enough for full-scale deployment. They also acknowledge that the current computational method, while effective, is demanding and might need to be streamlined before it can run continuously on standard building control systems.

Ultimately, this work demonstrates a powerful new way to listen to a building. By treating temperature and carbon dioxide as a coupled system, the researchers have created a method that can infer the invisible dynamics of airflow and occupancy from limited data. It is a tool that doesn't just predict the future; it understands the present, adapting its confidence as the building changes. For the managers of office buildings, this could mean a future where heating and cooling systems respond not just to a thermostat, but to the actual flow of people and air, saving energy and improving comfort without needing to install a sensor in every corner. The path forward involves refining these models to handle the messier, more complex reality of full-scale buildings, but the core idea—that air and heat tell the same story—has been proven to work.